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Page 1 of 111
The Future for Southwest Airlines
The Unknown Story of Rising Costs
2012 FORTUNE 500 List Rank 167
Southwest has always prided itself for having the lowest cost
structures in the domestic airline industry. It consistently offers the
lowest fares and has one of the best overall customer service records.
Nonetheless, this detailed case study, analyzes the profits-revenues data
for all forty-one (41) years, from 1971-2011, using a new methodology,
based on a universal mathematical law relating profits and revenues that
has been shown to describe the financial performance of many leading
companies in the Fortune 500 list. It is shown that “costs” have been
rising continuously for Southwest, going back all the way to 1974. What
is this “costs”? Costs = (Revenues – Profits), with revenues and profits
being obtained from the annual reports. Addressing this “cost” issue, in
innovative ways, based on a greater scientific understanding of the
financial dynamics, will make Southwest Airlines even more profitable
in the coming years. There is no contradiction between “rising costs”
noted here and being the “lowest cost” domestic airliner.
Page 2 of 111
Table of Contents
§ No. Topic Page No. 1. Summary 11
2. Brief version of the story 13
3. Introduction 18
4. Profits and Revenues over time 19
5. The Profits-Revenues Linear Law 22
6. The Unknown Costs-Revenues Story of Southwest Airlines 30
7. Quarterly data and Total Operating Expenses 35
8. Maximum Point on Profits-Revenues Graph 39
9. Brief Discussion 43
10. Appendix 1: Further discussion of quarterly data 2007-2012 49
11. Appendix 2: Growth of profits and revenue in a single year 54
12. Appendix 3: Type II Behavior and Type I to Type II transition 65
13. Appendix 4: Profits and Passengers flown 71
14. Appendix 5: Profits-Revenues in Early years: Nonlinear law
The entire 41 years of profits-revenues data from 1971-2011
75
15. Now a word about Air Tran 89
16. Bibliography of related articles 105
http://upload.wikimedia.org/wikipedia/commons/5/53/Southwest_Airl
ines_Flight_1248_-1.jpg Found this interesting image of a plane landing
(doctored?) during a Google search “Southwest airlines images”.
Page 3 of 111
Let’s Make this Easy
Although this document has grown to an amazing 108 pages (it even
includes a small section on AirTran now!), there is really nothing to
it. So, here are some “tips” about how to STUDY this (assuming you
still want to study it!)
1. Ok, just go straight to page 100 of the Air Tran section! Swoosh!
2. Fun Predictions on page 10 gives the formula for revenues, R = kF.
This can be deduced from the Southwest Fact Sheet on page 9.
3. The key to the whole analysis is P = hR + c, the profits-revenues relation. It
is derived in § 5 and requires only a knowledge of the “breakeven” analysis
for profitability. (All of it in just one paragraph!) The P-R formula is then
tested and shown to hold true using actual financial data. (It has also been
tested with many other companies.) Both P and R can thus be predicted!
4. Everything else, starting with §6, is aimed at showing, through different
types of calculations, that “costs” have indeed been going up for Southwest
Airlines, actually ever since operations began.
5. Southwest today is among the major low-cost domestic airlines that has
delivered a profit consistently for 39 years in a row. Both the level of profits
and the profit margins can be increased if we understand how and why
“costs” have been increasing and, more importantly, develop INNOVATIVE
SOLUTIONS to control these “rising costs”.
6. More importantly, just look at the graphs. There is nothing mysterious about
this. Just read the captions with each graph. They tell the whole story. Study
the Southwest initial year graph, Figures 28 and 29, and the Air Tran initial
years graph in Figures 38 and 39.
With lots of LUV in the air!
Page 4 of 111
Happy Birthday
Southwest Airlines http://www.swamedia.com/channels/Our-History/pages/our-history-sort-by
Celebrating and
Living the
Southwest Way
Celebrating is an important part of the Southwest Airlines Culture. It’s
been that way from our very first flight in 1971, and it’s true today.
We’ve always subscribed to the mantra of “Work hard, play hard.”
Page 5 of 111
Quite coincidentally, this report was
completed on June 18, 2012,
the 41st anniversary of Southwest
Airlines first flight on June 18, 1971;
see History and Timeline.
Wishing even more LUV in the air!
What's LUV?
Southwest has been in LUV with our Customers from the very
beginning. Therefore, it's fitting that we began service to San Antonio
and Houston from Love Field in Dallas on June 18, 1971. As our
Company and Customers grew, our LUV grew too! With the prettiest Flight
Attendants serving "Love Bites" on our planes, and determined Employees issuing
tickets from our "Love Machines," we changed the face of the airline industry
throughout the 1970s. Then in 1977, our stock was listed on the New York Stock
Exchange under the ticker symbol "LUV." Over the ensuing years, our LUV has
spread from coast to coast and border to border thanks to our hardworking
Employees and their LUV for Customer Service.
Page 6 of 111
Our History
Select from a Category Select By Date
1966 to 1971
1967
March 15,
1967 Air Southwest Co. is incorporated.
November
27, 1967
With $500,000 in the bank, Herb files the application with the Texas
Aeronautics Commission (TAC) to serve DAL, IAH, and SAT.
Page 7 of 111
1968
January 15,
1968 Hearing before TAC begins.
February 20,
1968
TAC votes unanimously to grant Air Southwest a certificate of public
convenience and necessity.
February 21,
1968
Braniff, Trans Texas (later Texas International), and Continental Airlines obtain
a temporary restraining order from Travis County District Court prohibiting
TAC from delivering our Certificate.
August 06,
1968 Austin State District Court rules against Air Southwest.
August 06,
1968
Air Southwest files an appeal with the Third Court of Civil Appeals over the
State District Court's Aug. 6 decision.
1969
March 12,
1969
Herb files appeal with the Texas Supreme Court and offers to represent the
Company free of charge and pay all costs out of his own pocket.
March 12,
1969
State Court of Civil Appeals rules against Air Southwest, upholding the lower
court's decision.
1970
May 13, 1970 The Texas Supreme Court unanimously votes to overturn the lower courts'
findings and rules in favor of Air Southwest.
December 07,
1970
The United States Supreme Court denies appeal by Braniff and Texas
International (TI) of Texas Supreme Court decision.
1971
January 01,
1971 Lamar Muse joins Air Southwest as President.
March 10,
1971
Lamar Muse sells promissory notes for aircraft and startup costs, raising $1.25
million.
March 29,
1971 Air Southwest Co. changes its name to Southwest Airlines Co. (Southwest).
March 29,
1971
Boeing offers to sell Southwest three 737-200s with Boeing carrying 90% of the
financing.
March 29,
1971
Lamar Muse hires Dick Elliot, Jack Vidal, Donald Ogden, and Bill Franklin.
They become known as the "Over the Hill Gang."
June 08, 1971 Jun. 8, 1971 Initial Public Offering of 650,000 shares of Southwest stock at $11
Page 8 of 111
per share ($6.5 million). Thomson McKinnon Auchincloss, Inc. and Model,
Roland & Co., Inc. were the Principal Underwriters. The exchange was traded
over the counter, and we did not have a ticker symbol.
June 16, 1971
The Civil Aeronautics Board (CAB), refusing to interfere, throws out complaints
filed by Braniff and TI that Southwest's operation might violate its intrastate
exclusivity. Within hours, lawyers for the two win a restraining order from an
Austin judge barring Southwest from beginning service.
June 17, 1971
Herb pleads case to the Texas Supreme Court. Later that day, the Texas Supreme
Court overrules the State District Court's injunction preventing Southwest from
commencing service.
June 18, 1971 Dallas Provisioning base opens.
June 18, 1971
Southwest Airlines begins service to DAL, SAT, and IAH. Our flight
schedule starts with six roundtrips DAL-SAT and 12 roundtrips DAL-IAH
with $20 one-way fares.
June 18, 1971 First uniforms for hostesses and ticket agents introduced. The "love airline" is
born. Captain Emilio Salazar flies the inaugural flight.
September
29, 1971 Southwest receives fourth aircraft.
October 01,
1971
Southwest implements every-hour service DAL-IAH with 14 roundtrips and
every-other-hour service DAL-SAT with 7 roundtrips.
November
14, 1971 Begins service between HOU-SAT - closing triangle.
November
14, 1971
Southwest "revitalizes" Houston's Hobby airport (HOU) by providing air service
and transfers one-half of service from IAH to HOU.
November
21, 1971 Introduces $10 "night fare" between HOU-DAL.
November
22, 1971 Cancels Saturday service.
December 31,
1971
1971 Milestones Net Loss: $3,753,000 Revenue
passengers carried: 108,554 Trips flown: 6,051
Fleet: 4 aircrafts Employees: 195 at
year end. Cities opened: DAL, SAT, IAH,
HOU Advertising budget: $700,000
Page 9 of 111
Southwest Airlines Fact Sheet http://www.southwest.com/html/about-southwest/history/fact-sheet.html#fleet
Operates 558 Boeing 737 (as of March 30, 2012)
Fleet type Number Seats
737-300 158 137
737-500 25 122
737-700 372 137
(Beginning February 2, 2012 capacity is being increased to 143). Two 737-800s
began service April 11, 2012.
Southwest currently flies to 73 cities in 38 states.
More than 3200 flights per day.
Southwest aircrafts fly an average of six flights per day (6.18/day) or an
average of 11 hours and 12 minutes per day. (558 times 6 equals 3348.)
The average trip length is 679 miles and the average duration is 1 hour
and 58 minutes.
Southwest consumed about 1.8 billion gallons of jet fuel in 2011.
The average passenger fare is $141.72 one-way and average trip is
approx. 939 miles.
Other related studies: http://www.thomashauck.net/pdfs/1southwest.pdf
Southwest Airlines: Case Study
by Garrison & Keller, 5567 Beechmont Ave, Cinncinnati, OH 45276
http://www.dtic.mil/dtic/tr/fulltext/u2/a273125.pdf
An estimate of the MAXIMUM daily and annual revenues is readily arrived at using the
data compiled here.
Page 10 of 111
Fun Prediction Fun Formula for Revenues ( R = kF )
From Southwest Airlines Fact Sheet http://www.southwest.com/html/about-southwest/history/fact-sheet.html#fleet
Based on the information compiled in the Fact Sheet, the following formula for
total annual revenues, let’s call it R, can be easily deduced. R = kF where F is the
average fare per seat and k is a numerical constant which depends on the numbers
compiled in the Fact Sheet.
Annual Revenues ($, billions) R = kF
= 0.14683 × (Fare per seat in $)
Average fare per seat, F $ Annual Revenue R ($, billions)
$100 $14.683 B
$125 $18.358 B
$150 $22.025 B
It is assumed that 120 seats are sold per flight and that there are 3350 flights
per day, each day of the year.
The change in average seats per flight will affect the total revenues in exact
proportion.
Now, here’s the formula for predicting the profits P. It is given by P = hR + c
Here h and c are constants that can be deduced from the two line items that are
now being reported routinely in the annual and quarterly financial statements. This
can be appreciated by studying this document carefully.
HOMEWORK PROBLEM: Repeat above for Jet Blue (ok, Air Tran!) and
compare it with Southwest! Also, check out the 2011 revenues for Southwest.
Page 11 of 111
§ 1. Summary
Southwest Airlines is an amazing company which has been in service for 40 years
and has been able to report a profit year-after-year for 39 years. It is focused on
offering low fares with exemplary customer service in an industry that is extremely
competitive and notorious for its low profitability. A recent article by Seth
Stevenson, in the Slate magazine, which discusses the “keep it simple” philosophy
of this airline, prompted this analysis of the profits and revenues behavior. The
data for the twenty year period, 1992-2011 and first quarter 2012, is studied here.
The revenues have increased consistently since 1992 and revenues growth actually
seems to have accelerated since 2009. Unfortunately, the same cannot be said
about profits. Profits increased consistently from 1992 and reached a peak in 2000
after which profits have been varying wildly, showing large fluctuations. For the
period 1992-2001, a simple linear law y = hx + c = 0.123x – 0.141, where x is
revenues and y is profits, both in billions, can be shown to describe the data.
Profits increase at a fixed and steady rate with increasing revenues, once a cut-off
or “breakeven” revenue was exceeded (given by y = 0 and x = $1.15 billion).
This has been called the Type I behavior here and signifies a period of steadily
increasing profits with increasing revenues (h > 0 and c < 0). Thus, one could also
conceive of a Type II behavior (h > 0, c > 0), where profits increase at a lower rate
than in the Type I phase and also a Type III behavior (h < 0, c > 0), where profits
actually decrease with increasing revenues. Indeed, for the post-2000 period, a
careful analysis of the profits-revenues data reveals that Southwest Airlines is now
in the Type III mode: profits-revenues graph actually has a negative slope for the
period 2007-2011. Extrapolating from this recently established trend, it is also
conceivable that Southwest Airlines will soon report an annual loss, as revenues
increase further. The recently completed acquisition of Air Tran thus takes on
added significance and one would be tempted to blame this acquisition if there is
any historical first reporting of a loss.
This impending situation has been studied carefully to understand how costs have
been increasing with increasing revenues. It can be shown that costs are actually
increasing faster than revenues and this also explains the low “absolute” level of
Page 12 of 111
profits and the rather low profit margins. The total Operating Expenses, one of the
Items reported in the Consolidated Statement of Operation, and the Cost, computed
from the relation Costs = Revenues – Profits (where profits is the same as the item
called Net income), are both studied carefully to draw some conclusions that
should engage the immediate attention of Southwest Airlines management.
While the unprecedented reporting of profits, year-after-year, is unprecedented and
to be highly commended, focus must now be shifted to increasing the profit
margins and a return to the Type I behavior of the pre-2000 era. Many issues that
affect the cost structure need to be addressed in the coming years to sustain the
history of profitability. It is hoped that Southwest management will benefit from
these findings (especially those in the more detailed Appendices).
Perhaps, the most important finding here is that Southwest Airlines, like some
other companies (notably Ford Motors, Verizon Communications, Yahoo, and
Kroger) shows a maximum point on its profits-revenues (P-R) graph. Air Tran,
recently acquired by Southwest, also reveals a maximum point. The P-R graph is a
simple x-y graph of these two items, reported routinely in the consolidated
financial statements (quarterly and annual). It is truly amazing that the existence of
such a maximum point has escaped attention to date. (The present author began
these recent studies on May 18, 2012, following the disappointing Facebook IPO
launch and the general media discussion about its potential revenues growth.)
Why is there a maximum point on the profits-revenue graph?
Why would a company want to continue operations if profits actually decrease
with increasing revenues?
The appearance of a maximum point in the radiation spectrum for a blackbody
puzzled physicists in the closing years of the 19th century. Classical physics was
unable to explain the existence of such a maximum point. Now, we have, in the
humble opinion of the present author, a finding of far reaching significance that
should engage the attention of business leaders, and the finance and economics
community, both academic and day-to-day practitioners. From such an
understanding, there can be no doubt, will emerge a new, as yet unimagined, view
of how the financial world behaves. Perhaps, we can start building real “Profits
Engines”. Southwest Airlines can lead the way.
Page 13 of 111
§ 2. Brief Version of the Story
The purpose here is to make the good better and the better the best.
Southwest Airlines is already widely recognized as a highly successful low-cost
domestic airline. It has been in service for 40 years and has delivered profits, every
single year, for 39 consecutive years. Although some quarterly losses have been
reported, the company has always reported a profit for the year, taken as a whole.
A somewhat unconventional approach will therefore be taken here to show that
costs have actually been rising for Southwest Airlines, especially over the last
decade, with the emergence of what is described here as the Type III behavior.
Hence, we will first present four key findings, in the form of simple x-y graphs.
The figure captions are self-explanatory. The figure numbers used later in the text
are retained here. This is then followed by a more detailed presentation.
Figure 1: Revenues growth for Southwest Airlines for the last
twenty years (1992-2011). The rate of increase of revenues seems
to have accelerated since 2009, as seen by the increased slope.
0
2
4
6
8
10
12
14
16
18
1990 1995 2000 2005 2010 2015
Time, t [in years]
Re
ven
ues, x [
$,
billio
ns]
Page 14 of 111
Figure 2: Profits growth for Southwest airlines for the period
1992-2011. Profits increased steadily with increasing revenues
until 2000. Since then profits have been varying erratically with
the recent three years (2009-2011) yielding very low profits
compared to the historical values, with revenues having increased to record levels.
Now let us compare the profits and revenues data for the years 2000 and 2010,
obtained from the quarterly reports for each year. Specifically, we will
consider how profits “grow” during the year, with increasing revenues, when
we take a “snapshot” of the financial behavior of the company in 3 month, 6
month, 9 month, and 12 month intervals.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1990 1995 2000 2005 2010 2015
Time, t [in years]
Pro
fits
, y [
$,
billi
on
s]
Page 15 of 111
Figure 19: Composite plot comparing the evolution of profits, with
increasing revenues, during 2000 and 2010 (from the 3 months, 6
months, 9 months, and 12 month quarterly data for each year). A
linear profits-revenues equation, y = hx + c, is implied by the classical breakeven
analysis for profitability. If a is the fixed cost and b the unit variable cost, the total
cost C = a + bN where N is the number of units offered. If p is unit price, the
revenues generated from the sale of the N units is R = pN. Also, N = R/p. Hence,
the profits P = R – C = [1 – (b/p)]R – a which means the intercept c = - a and the
slope h = 1 – (b/p). The higher intercept made on the x-axis implies a higher fixed
cost. The lower slope means a lower rate of conversion of additional revenues
(beyond breakeven, or cut-off value) into profits. Since, h = 1 – (b/p), the lower
slope means a higher unit variable cost b or a lower unit price p (due to
competitive pressures). The inescapable conclusion from the above is that costs
have gone up for Southwest Airlines during the last decade although the company
is considered a major low-cost domestic airline. Is this a contradiction? NO!
-0.20
0.00
0.20
0.40
0.60
0.80
0 2 4 6 8 10 12 14
2000
2010
2000 a
nd
2010
Pro
fits
, y [
$,
billi
on
s]
Cu
mu
lati
ve
valu
es d
uri
ng
th
e y
ear
2000 and 2010 Revenues, x [$, billions] Cumulative values during the year
Page 16 of 111
Figure 26: A very clear nonlinear growth of profits with increasing
revenues for Southwest Airlines for the period 1971-1992. Costs
have been rising for a long time now, as is obvious from this graph which is
prepared using two line items from the Annual Reports (revenues and net income
or profits). There is an unmistakable deceleration in the rate of growth of profits
with increasing revenues, i.e., the slope of the mathematical curve describing the
profits-revenues relation is decreasing. Amazingly, this has escaped attention to
date. The maximum point on the profits-revenue graph (see main text, this is
OUTSIDE the range of revenues covered in this graph) is another dramatic
example of this same trend. Southwest Airlines is now operating past its maximum
point, in the region where profits decrease even as revenues increase! Some other
examples of leading Fortune 500 companies that exhibit this maximum point are
Ford Motor Company, Verizon Communications, Yahoo, and Kroger.
-20
0
20
40
60
80
100
120
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Revenues, x [$, millions]
Pro
fits
, y [
$,
mil
lio
ns
]
Page 17 of 111
http://216.139.227.101/interactive/luv2009/198/page_003.jpg
Ready for Take-off !
The Southwest Secret How the airline manages to turn a profit, year
after year after year
By Seth Stevenson Posted Tuesday, June 12, 2012, at 11:45 AM ET
e-mail: [email protected]
http://www.slate.com/articles/business/operations/2012/06/southwest_airlines_prof
itability_how_the_company_uses_operations_theory_to_fuel_its_success_.html
http://216.139.227.101/interactive/luv2009/ Financial Data for 2000-2009
Page 18 of 111
§ 3. Introduction
A recent article in the Slate magazine by Seth Stevenson, on the enviable profits
http://www.slate.com/articles/business/operations/2012/06/southwest_airlines_prof
itability_how_the_company_uses_operations_theory_to_fuel_its_success_.html )
record of Southwest Airlines, caught my attention and is largely responsible for the
analysis being offered here. This airline, which is mostly focused on domestic
routes, has reported a profit, year-after-year, for 39 consecutive years. This point
has also been proudly highlighted by the company in its 2011 Annual Report (see
http://southwest.investorroom.com/ ) and also in all earlier year reports (35th year,
36th year, 37
th year, and so on.) This is no small achievement, especially in the
airline industry. The reason for this success, as discussed nicely by Stevenson, is in
the company’s basic philosophy of keeping things simple. For example,
1. The airline uses only one single type of aircraft, the Boeing 737. This
introduces all kinds of cost saving s and also offers operational flexibilities
(even in aircraft maintenance, crew training, etc.).
2. No seat numbers are assigned. Passengers can sit wherever they choose.
3. The “bags fly free” policy reduces checked bags at the gate and eliminates
delays and reduces wasted time.
4. There is no hub through which flights are routed, eliminating the resulting
congestions, snags, breakdowns, and hence delayed flights. A plane can be
readied and turned around in as little as 25 minutes after landing. After all,
an airline only makes money when its planes are flying.
All of this sounded too good to be true (never had a chance to fly with them). With
my ongoing interest in analyzing the financial performance of companies in the
2012 Fortune 500 list (a report on 13 companies in the 2012 list may be found at
http://www.scribd.com/doc/95906902/Simple-Mathematical-Laws-Govern-
Corporate-Financial-Behavior-A-Brief-Compilation-of-Profits-Revenues-Data ), I
decided to take a closer look at this airline.
Happy Customers! Consistent Profits! Here was a real “Profits Engine” that I have
been fantasizing about since circa 1998, when I first started studying financial data
of companies, big and small, in all sectors of the economy, in many parts of the
Page 19 of 111
world. The results of my study, as we will see shortly, are counterintuitive and
certainly NOT what I had expected either for Southwest Airlines.
§ 4. Profits and Revenues Over time
The profits and revenues data for the past twenty years (1992-2011) have been
compiled in Table 1. This profits and revenues data can also be used arrive at the
cost, using the fundamental equation Profits = Revenues – Costs. This “computed”
cost-revenue data may be found in Table 2, see also Appendix 1 where this point is
discussed clearly with reference made to the consolidated statement of operations
for first quarter 2012. As seen in Figure 1, revenues have been increasing steadily
year-after-year. Indeed, after a small dip between 2008 and 2009, the revenue
growth seems to have accelerated since 2009.
Figure 1: Revenues growth for Southwest Airlines for the last
twenty years (1992-2011). The rate of increase of revenues seems
to have accelerated since 2009, as seen by the increased slope.
0
2
4
6
8
10
12
14
16
18
1990 1995 2000 2005 2010 2015
Time, t [in years]
Re
ven
ues, x [
$,
billio
ns]
Page 20 of 111
Unfortunately, the same cannot be said about profits growth, although the airline
has reported a profit for 39 consecutive years. Profits increased steadily with
increasing revenues, from 1992 to 2000 but profits have been varying erratically
since then, see Figure 2. Profits declined sharply after 2000 and began to increase
again between 2002 and 2007, but in a much more erratic fashion. Since 2007,
profits have been decreasing although revenues have increased significantly.
These trends and the profits-revenues relationships can be better understood further
by the various x-y graphs presented in Figures 3 to 10.
Figure 2: Profits growth for Southwest airlines for the period
1992-2011. Profits increased steadily with increasing revenues
until 2000. Since then profits have been varying erratically with
the recent three years (2009-2011) yielding very low profits
compared to the historical values, with revenues having increased to record levels.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1990 1995 2000 2005 2010 2015
Time, t [in years]
Pro
fits
, y [
$,
billi
on
s]
Page 21 of 111
Table 1: Profits-Revenues data for Southwest Airlines
for the twenty year period (1992-2011)
Year Revenues,
x ($, bil)
Profits, y
($, bil)
Profit
Margin, y/x
% Profits
100(y/x)
Comments
2011 15.658 0.178 0.0114 1.14 Type III trend
2010 12.104 0.459 0.0379 3.79 is getting
2009 10.350 0.099 0.0096 0.96 established
2008 11.023 0.178 0.0161 1.61 Between
2007 9.861 0.645 0.0654 6.54 2007 to 2011
2006 9.086 0.499 0.0549 5.49 2005 7.584 0.548 0.0723 7.23 2004 6.530 0.313 0.0479 4.79 2003 5.937 0.442 0.0744 7.44 2002 5.522 0.241 0.0436 4.36 2001 5.555 0.511 0.0920 9.20 Highest profit
2000 5.650 0.625 0.1106 11.06 margins were
1999 4.736 0.474 0.1001 10.01 between 1998
1998 4.164 0.433 0.1040 10.40 to 2001
1997 3.817 0.318 0.0833 8.33 1996 3.406 0.207 0.0608 6.08 Type I behavior
1995 2.873 0.183 0.0637 6.37 1992-2001
1994 2.592 0.179 0.0691 6.91 All data from 1993 2.297 0.154 0.0670 6.70 Annual reports 1992 1.803 0.097 0.0538 5.38
Source: http://216.139.227.101/interactive/luv2009/ ten years 2000-2009 and
also http://southwest.investorroom.com/ 2011 Annual Report; 2007-2011
Page 22 of 111
2002 5.522 0.241
2004 6.530 0.313
2006 9.086 0.499
2007 9.861 0.645
§ 5. The Profits-Revenues Linear Law
Before we proceed with our analysis and discussion, let us consider the following
classical “breakeven” analysis for the profitability of a company making and
selling N units of a product (this could be airline seats in the case of Southwest). If
p is the unit price, the revenues generated R = pN. Let “a” denote the fixed cost
and “b” the unit variable cost. Then the total cost C = a + bN, the sum of the fixed
cost and the total variable cost. Hence, the profits P = R – C = (p – b)N – a, or
eliminating N using R = N/p, we get the relation P = [(p – b)/p] R – a . This
implies a linear law relating revenues, say x, and profits, let’s call it y. Thus,
y = hx + c …… Linear law for profits and revenues
Slope h = 1 – (b/p) .….. determined by unit price p and unit variable cost b.
Intercept c = - a .….. determined by the fixed costs of the operation.
The linear law y = hx + c, implied by this classical breakeven analysis, suggests
three different possibilities.
Type I: Positive slope, negative intercept (h > 0 and c < 0, positive intercept on
revenues-axis).
Type II: Positive slope and positive intercept (h > 0 and c > 0, positive intercept
on profits-axis).
Type III: Negative slope, positive intercept (h < 0, c > 0, positive intercepts on
both the profits and revenues axes).
Examples of companies obeying these three types of linear laws have been
discussed in another recent study (see Refs.[1,2] cited at the end of this article). We
are observing Type I behavior here with Southwest Airlines. In the first period
(before the peak profits in 2000), profits increase with increasing revenues once the
revenues exceed the “breakeven” value of $1.154 billion (intercept made on the x-
axis). Beyond this revenue level, profits increase at a fixed rate with 12.3% of the
additional revenues being converted into profits.
The profits-revenues figures for the four years listed
in the mini-table to the left again reveals a Type I
Page 23 of 111
relation (h > 0 and c > 0), y = 0.093x – 0.273 = 0.093 (x – 2.934), if we consider
the extreme (x, y) values for 2002 and 2006. The slope h has decreased slightly but
the cut-off, or “breakeven” revenue to report a profit, has increased to $2.934
billion, see the dashed line in Figure 4. This also means lower profits.
The transition from one Type I behavior to another Type I behavior, with a higher
intercept on the x-axis (which means higher “fixed costs”) and a lower slope
(which means additional revenues are converted at a lower rate into profits),
appears to have been the first noticeable effect of the fluctuations that started after
the first peak in profits in the year 2000.
Figure 3: The profits-revenues graph for the period 1992-2000.
Profits increase with increasing revenues. The equation of the
best-fit line through these points is y = hx + c = 0.123x – 0.141 =
0.123 (x – 1.154) with the linear regression coefficient having a
very high value of r2 = 0.935. Attention should also be called to the nearly
PERFECT profits-revenues graph for Apple Inc., with r2 = 0.99985, reported in
the articles cited in the references; see The Perfect Apple.
-0.2
0
0.2
0.4
0.6
0.8
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Revenues, x [$, billions]
Pro
fits
, y [
$,
billi
on
s]
Type I behavior y = 0.123x – 0.141 = 0.123 (x – 1.154)
r2 = 0.936
Page 24 of 111
Figure 4a: Profits-revenues graph with added data for the
period 2002-2011. This represents the profits data in Figure 2
where we see rapid fluctuations in the profits levels. A second
Type I straight line (dashed line) joining the (x, y) pairs for 2002 and 2006, with
the equation y = 0.093x – 0.273 = 0.093 (x – 2.934) captures this trend. This was
established after the first peak in profits we see in Figure 2.
All of the 20-year data is considered in Figures 4 and 5 to highlight this difference
between the two Type I behaviors and the increase in “breakeven” point, which
implies increase in the “fixed costs” for Southwest Airlines.
The fundamental significance of the intercept c in the linear law y = hx + c, or
positive intercept x = x0 = -c/h when y = 0 made on the x-axis and its relationship
to the idea of a “breakeven” revenue (x = x0 when y = 0 or revenues R = R0 when
Profits P = 0) can be appreciated if we consider the profits-revenues data for the
first few years of operation. This has been compiled from the annual reports from
1973, 1975 and 1978 and is discussed separately in Appendix 5. It is sufficient to
note here that Southwest did report a loss (on an annual basis) in 1972 before
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 12 14 16 18 20
Revenues, x [$, billions]
Pro
fits
, y [
$,
billi
on
s]
Change in the Type I behavior
Lower slope means higher “costs” due
to higher intercept on the x-axis.
Page 25 of 111
reporting its first year of profits in 1973. The cover page of the 1973 Annual
Report says “Southwest Airlines Turned the Corner in 1973”. An extract of
this early profits-revenues data may be found in the mini-table after the graph.
Figure 4b: Profits-revenues data for the first few years of
operation, obtained from 1973, 1975, and 1978 Annual Reports
reveals even more clearly the significance of the intercept c and its relation to the
“fixed” costs of operation. A loss was reported in both 1971 (partial year of
operations) and in 1972, the first full year of operations. The solid blue line
connects the 1971 and 1973 data (y = 0.5548x – 4.934). The dashed line connects
the 1974 and 1978 data (y = 0.2245x – 1.194). The loss means revenues did not
exceed the “breakeven” level. The first “breakeven” revenue calculated from the
equation of the line connecting the 1971 and 1973 data is $8.893 million (x = x0 =
- c/h = -4.934/0.555 = 8.893). The revenue for 1973 was $9.209 million. Both
slope h and intercept c of the graph are clearly changing as the revenues increase
and profits increase.
-10
-5
0
5
10
15
20
0 20 40 60 80 100
Revenues, x [$, billions]
Pro
fits
, y [
$,
billi
on
s]
Page 26 of 111
Page 27 of 111
The slope of the straight line joining the points (x1, y1) and (x2, y2) is:
h = (y2 – y1)/(x2 – x1). Knowing h we can determine the intercept.
c = (y2 – hx2) since the line passes through (x2, y2).
It is also given by
c = (y1 – hx1) since the line passes through (x1, y1).
Conversely, if h is known, the future value y2 for a future value x2 can
be predicted. y2 = y1 + h(x2- x1).
These simple algebraic relations are very useful for our analysis.
Year Revenues, x
$, millions
Profits, y
$, millions
Costs (x –y)
$, millions
Comments
1971 2.129 -3.753 5.882 Data from
1972 5.994 -1.591 7.585 1973, 1975
1973 9.209 0.175 9.034 and 1978
1974 14.852 2.14 12.712 Annual
1975 22.828 3.40 19.428 Reports
1976 30.92 4.939 25.981
1977 49.047 7.545 41.502
1978 81.065 17.004 64.061
Operations began on June 18, 1971. The first full year of operations was 1972.
This first year data indicates a loss with a small profit in 1973.
Next, a Type III behavior is very evident when we consider the most recent data
for the years 2007-2011; see also Figure 5 and the numbers in Table 1. Profits are
decreasing with increasing revenues. Now, extrapolating along this Type III
profits-revenues (P-R) line, if the current trend continues, we can conclude that
Southwest Airlines might actually report its first ever annual loss when its
revenues exceed about $18 billion or about $2.3 to $2.5 billion over the 2011 level.
(This revenue level could be reached in 2012 or 2013.)
This “precarious” profits situation with Southwest Airlines is also highlighted by
the rather low values, less than 2%, of the profit margins (see Table 1) reported in
recent years, with the exception of 2010. This should be compared to the profit
margins in the range of 8% to 11% reported in the earlier period and with much
significantly lower revenue levels. Notice that both the absolute level of profits
Page 28 of 111
($625 million) and the profit margin (11.06%) were higher in 2000 than in 2011
($178 million and 1.14%, respectively).
Essentially the above brief review of the profits-revenues situation (using the rarely
used but simple tool of x-y graph in financial data analysis, and aided by the
classical breakeven analysis) tells us that, notwithstanding the great many cost
efficiencies arising from the “keeping it simple” philosophy, the costs for this
airline are still too high, see Table 2. Although the company has reported a profit
for 39 consecutive years, the profit levels are actually quite low, especially as a
percent of revenues, see both Tables 1 and 2. Regardless of the fact that the
company is operating in the airline industry, where just reporting a profit has been
a major issue (see discussion of Delta Airlines in Ref. [1]), this cannot be an
excuse for the low levels of profits that are being reported.
Figure 5: The profits-revenues graph for the period 2000-2011 is
lacking the remarkable Type I trend revealed for the earlier
period. As seen in Figure 2, and also in Table 1, profits have been
fluctuating wildly and decreasing (after reaching a peak in 2007)
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 12 14 16 18 20
Revenues, x [$, billions]
Pro
fits
, y [
$,
billi
on
s]
Type III behavior
y = - 0.081x + 1.439
Joining 2007 and 2011
Page 29 of 111
even as revenues have increased. This is Type III behavior and is described by the
straight line with the negative slope. Using the (x, y) values for 2007 and 2011, the
equation of this Type III straight line is y = - 0.081x + 1.439.
Southwest management must therefore take a careful look at the reasons for the
wild fluctuations in the profits since 2000, and the cost issues implied and
address them earnestly to avoid reporting a loss in the near future.
It would be very easy, yes very, very easy, to blame such a uncharacteristic and
historically first loss on the recently completed acquisition of AirTran, if this
prediction (made on June 14, 2012) is borne out when the next annual report is
filed, or in the next couple of years at most. The profits-revenues graph for the
all forty-one years of operation, from 1971-2011, has also been prepared (see
Appendix 5) and reveals the same unmistakable trend.
Company Profile
With 40 years of service,
Southwest Airlines Co.
(Southwest), a low-fare major
domestic airline, continues to
differentiate itself from other
low-fare carriers, offering a
reliable product with
exemplary Customer Service.
Southwest was incorporated in
Texas and commenced
Customer Service on June 18,
1971 with three Boeing 737 aircraft serving three Texas cities - Dallas, Houston, and San
Antonio. Today, Southwest is the nation's largest carrier in terms of originating domestic
passengers boarded serving 73 cities in 38 states. On May 2, 2011, Southwest completed the
acquisition of AirTran Holdings, Inc., and now operates AirTran Airways as a wholly
owned subsidiary. Southwest has among the lowest cost structures in the domestic airline
industry, consistently offers the lowest and simplest fares, and has one of the best overall
Customer Service records. LUV is our stock exchange symbol, selected to represent our home at
Dallas Love Field, as well as the theme of our Employee and Customer relationships. Southwest
is one of the most honored airlines in the world known for its commitment to the triple bottom
line of Performance, People, and Planet. To read more about how Southwest is doing its part to
be a good citizen, click on the tab above to read the Southwest Airlines 2011 One Report™.
Page 30 of 111
§ 6. The Unknown Cost-Revenue Story of Southwest Airlines
The “cost” of producing the sales is highlighted in different ways in the financial
statements of a company. Here we will use the fundament equation of the financial
world, Profits = Revenues – Costs, or P = (R – C) with “profits” always being the
net income that is applied to determine the earnings per share (EPS). This also
ensures that all changes in tax laws are fully accounted for when we discuss the
profits-revenue or the costs-revenue relations. The “costs” that we will discuss now
therefore are the “overall” or the “effective” costs, after all obligations have been
met. Or, we can all it the “computed” cost to make it clear that it comes from this
simple computation.
Figure 6: The costs C = Revenues R – Profits P were deduced
from the financial data compiled in Table 1 from the annual
reports. Thus, Costs C = (x – y). The graph of costs C versus
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8
Revenues, x (or R), [$, billions]
Co
sts
, (x
– y
) (o
r C
), [
$,
billio
ns
]
C = 0.877 R + 0.1414 With r
2 = 0.9987
1992-2001 with 2000 excluded from
regression
Page 31 of 111
revenues R is being considered here for the two time periods, before and after the
peak in profits in 2000 that we see in Figure 2. Even with the wild fluctuations in
profit, the costs-revenues graph is remarkably linear and seemingly unaffected by
the fluctuations (this is due to low profits margins as well) and hence P as a
percent of R or C is very small).
Thus, the Computed Cost C = Revenues – Profits = (R – P) = (x – y) is given in
Table 2. The reason for using the twin notations C, R and P and also x and y will
become obvious in a moment.
Let us take a look at the data for 2011 and 2010. The revenues increased by the
amount ∆R = $3.554 billion but the profits decreased. Why?
Let us determine the C for each year using C = (R – P). After determining the Cs
for each year, we determine the increase in the cost ∆C = $3.835 billion. In other
words ∆C > ∆R. This surprising trend is confirmed if we consider the (C, R) pairs
for other years. Costs seem to be increasing faster than revenues and the slope of
the graph of costs versus revenues, ∆C/∆R, is therefore greater than unity (or 1).
This is the situation with Southwest Airlines for the most recent period 2001-2011,
see Figure 6. If the slope equals unity (1), then costs = revenues and there is zero
profits. If the slope is less than unity, ∆C < ∆R, then costs do not increase as fast as
revenues increase and the company will be more profitable. This was the situation
with Southwest Airlines in the earlier period, 1992-2000, see Figure 7.
The above calculations of the increase in costs ∆C and the revenues ∆R during the
post-2000 and pre-2000 periods also illustrates, perhaps, the reason for the wild
fluctuations in profits that we see in Figure 2. Although Southwest Airlines has
reported a profit, every year, for the past 39 years, the costs are actually increases
faster than revenues, especially since 2000 and this also explains the reversal in the
slope of the profits-revenues graphs – the change from Type I with a high slope to
Type I with a slightly lower slope and then to Type III.
Page 32 of 111
Table 2: Revenues-Profits-Costs data for Southwest Airlines
for the twenty year period (1992-2011)
Year Revenues,
x ($, bil)
Profits,
y
($, bil)
Costs,
C
(x –y)
∆C
Delta C
(cost)
∆R
Delta R
(revenue)
Comments
Changes are
relative 2011
2011 15.658 0.178 15.480
2010 12.104 0.459 11.645 3.835 3.554 ∆C > ∆R
2009 10.350 0.099 10.924
2008 11.023 0.178 10.172
2007 9.861 0.645 9.216 6.264 5.797 ∆C > ∆R
2006 9.086 0.499 8.587 2005 7.584 0.548 7.036 2004 6.530 0.313 6.217 9.263 9.128 ∆C > ∆R 2003 5.937 0.442 5.495 2002 5.522 0.241 4.897 10.583 10.136 ∆C > ∆R 2001 5.555 0.511 5.044 10.436 10.103 ∆C > ∆R
2000 5.650 0.625 5.025
1999 4.736 0.474 4.262 2.556 2.933 ∆C < ∆R
1998 4.164 0.433 3.731 Between 1999
1997 3.817 0.318 3.499 and 1992
1996 3.406 0.207 3.199
1995 2.873 0.183 2.69
1994 2.592 0.179 2.413 1993 2.297 0.154 2.143 1992 1.803 0.097 5.044
Since y = hx + c is the profits-revenue relation, the costs-revenue relation will
become C = (x – y) = x – hx – c = (1 – h)x – c or C = (1- h)R + a. In other words,
the slope of the C-R graph will be (1 – h) where h is the slope of the P-R graph and
the intercept of the C-R graph will be the negative of the intercept for the P-R
graph. This is exactly what we see here. The slope is 0.877 = (1 – 0.123) in
agreement with the slope of the graph (Type I behavior) in Figure 3. Now, we are
ready to consider the 2000-2011 period in the costs-revenue space.
Page 33 of 111
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 12 14 16 18 20
-5
0
5
10
15
20
25
0 2 4 6 8 10 12 14 16 18 20 22 24
Revenues, x (or R), [$, billions]
Revenues, x (or R), [$, billions]
Pro
fits
, y (
or
P),
[$, b
illi
on
s]
Co
sts
, (x
– y
) (o
r C
), [
$,
billio
ns
]
C = 1.032 R – 0.698 r2 = 0.9977
∆C/∆R = 1.032 > 1
A
B
Page 34 of 111
Figure 7: The costs-revenues graph for the period 2000-2011. The
exact same data is considered in the upper (profits-revenues) and
lower parts (costs-revenues) of the graph. The profits graph
reveals a large “scatter”(see A) but all the points line up nicely along an upward
sloping straight line, see B, in the costs graph. (This is observed repeatedly in the
analysis of such financial data for many companies.) The red dashed line is the
costs = revenue line. Since the data fall below this line, costs are lower than
revenues and the company is reporting a profit. However, and amazingly, the slope
of the best-fit line is greater than 1. C = 1.0321R – 0.698 with r2= 0.9977. The
unmistakable conclusion is that costs are increasing at a faster than the revenues.
The discussion here about the costs of Southwest Airlines also shows why
although the company is reporting a profit each year, the level of profits is also
going down year after year. The only trend that is obvious from profits-revenues
graph is the Type III straight line for 2007-2011. The other points appear to be
“outliers” to this striking Type III behavior. However, a statistically significant P-R
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 12 14 16 18 20
Revenues, x (or R), [$, billions]
Pro
fits
, y (
or
P),
[$, b
illi
on
s]
y = -0.032x + 0.698 Statistically significant
Deduced from C-R graph
y = 0.081x + 1.439
Joining 2007-2011
C
Page 35 of 111
relation can now be deduced from the extremely statistically significant C-R that
we have been able to deduce (since the linear regression coefficient is r2 = 0.9977).
From P = R – C, we get P = R – 1.0321R + 0.0698= - 0.0321R + 0.698. This
means that the statistically significant P-R graph for 2001-2011 is a straight line
with a negative slope h = - 0.0321 and a positive intercept of + 0.698. This is now
superimposed to the profits-revenues graph (see graph C in this figure).
§ 7. Quarterly Data and Total Operating Expenses
Let us now consider the quarterly data for years 2007-2011 (during which Type III
behavior seems to been established) to see if this intriguing finding of costs
increasing faster than revenues holds even if we use the quarterly time frame.
Firstly, it should be noted although Southwest Airlines has reported a profit for 39
consecutive years, the same cannot be said about the quarterly profits. Southwest
Airlines has reported a loss on several occasions, on a quarterly basis, but this was
compensated in the other quarters to yield an overall profit for the year.
In this context, we will consider both the “computed” cost C, as defined in the last
section, and also the line item referred to as Total Operating Expenses (OE) given
in the consolidated statement of operations for each quarter (and for each year). An
example of this consolidated statement of operations may be found in Appendix 1.
This illustrates the results reported for first quarter 2012. The OE is Item no. 2 and
profits (or Net Income), which was of interest thus far, is Item no. 7. The
“computed” cost (in Table 2) is the sum of Items 2, 4, 5, and 6 and accounts for
some additional expenses not included in the Total Operating Expenses (OE).
Revenues, Profits, the Total Operating Expenses and the “computed” Costs, for ten
consecutive quarters ending March 2012, are all listed in Table 3 below. Let us
first consider the overall change between the quarters ending December 2009 and
March 2012. The OE increased from $2.545 billion to $3.969 billion, an increase
of ∆(OE) = $1.424 billion. Revenues, on the other hand, increased only by $1.279
billion, from $2.7 12 billion to $3.991 billion. Hence, ∆(OE)/∆R > 1 and the rate of
increase of the total operating expenses is greater than the rate of increase of
revenues. The overall ratio ∆(OE)/∆R = 1.114 > 1 from Dec 2009 to Mar 2012.
Page 36 of 111
Table 3: Quarterly Data from Southwest Airline 10-Q SEC Filings
Quarter
ending
Revenues, x
$, billions
Profits, y
Net Income
$, billions
Total Operating
Expenses (OE)
$, billions
Computed
Costs
(x – y)
Mar 2012 3.991 0.098 3.969 3.893
Dec 2011 4.108 0.152 3.961 3.956
Sep 2011 4.311 -0.140 4.086 4.451
Jun 2011 4.136 0.161 3.929 3.975
Mar 2011 3.103 0.005 2.989 3.098
Dec 2010 3.114 0.131 2.898 2.983
Sep 2010 3.192 0.205 2.837 2.987
Jun 2010 3.168 0.112 2.805 3.056
Mar 2010 2.630 0.011 2.576 2.619
Dec 2009 2.712 0.116 2.545 2.596
An expanded version of this Table 3 is given in Appendix 1, with all of the data
from the quarters ending Sep 2006 to March 2012
We arrive at an exactly similar conclusion for ∆C/∆R. Between Dec 2009 and Mar
2012, ∆C = 1.297 and ∆R = 1.279 and ∆C/∆R = 1.297/1.279 = 1.014 > 1.
The graph of increasing Total Operating Expense (OE) with increasing revenues is
presented in Figure 8, along with the best-fit line deduced from linear regression.
The dashed red line is the y = x line, or R = OE line, when revenues equal the Total
Operating Expenses. If the data point falls below this line, the company should be
reporting a profit (some Other Expenses, Item 3, see Appendix 1, still have to be
accounted). If it falls above, it will most likely report a loss. However, the
interesting finding here is a slope greater than unity. Both the OE-revenues and
cost-revenues equations have a slope greater than unity, see Figures 8 and 9.
Total Operating Expenses (y) versus Revenues x yields, y = 1.0064x -0.209
Costs (x –y) versus Revenues x yields: C = 1.022 R – 0.162
The costs-revenue graph has a higher slope than the OE-revenues graph and both
slopes are greater t han unity if we consider the most recent period, Dec 2009 to
March 2012. This confirms the rather unpleasant trend revealed of costs rising
faster than revenues, from the analysis of the annual financial data.
Page 37 of 111
There is good news, however. If we consider the data for all quarters (Sep
2006 to Mar 2012, which span the years 2007-2011 over which Type III
behavior was established), we find a slope less than unity. This segment of the
present study is being presented in Appendix 1 for further and careful review.
Figure 8: The quarterly values of the Total Operating Expenses
(OE), our variable y, one of the line items in the consolidated
statement of operations for Southwest Airlines (and all other
companies as well) is plotted here as a function of the quarterly revenues, our
variable x. We consider the ten consecutive quarters from Dec 2009 to Mar 2012.
The best-fit line has the equation y = mx + c = 1.0064x – 0.209.The slope (m =
1.0064 > 1) is greater than unity suggesting OE rising faster than revenues. The
dashed red line is the graph of y = x with m = 1. Data points must fall below this
line for a profit. (It can be shown, in Excel computations, that the two straight lines
will indeed intersect at high x.)
-1
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
Quarterly Revenues, x [$, billions] To
tal O
pera
tin
g E
xp
en
se
s (
OE
) [$
, b
illio
ns]
Page 38 of 111
Figure 9: Since Profits = Revenues – Costs (P = R – C), we can
“compute” a cost C = R – P from the reported values of the
revenues (our variable x) and the net income or profits (variable
y in all earlier graphs). This cost C is now plotted versus the
quarterly revenues for the quarters Dec 2009 to Mar 2012. The dashed red line is
the graph of C = R. A profit is reported only when the data point falls below this
dashed line. This is a firm statement, and is always true, unlike the situation with
the Item called Total Operating Expenses where some Other Expenses still have to
be accounted for. Again, it is of interest to note that the slope of the best-fit line is
greater than unity. y = mx + c = 1.0223x – 0.162. It is actually significantly
greater than unity m = 1.0223 compared to the slope found for the graph of the
Total Operating Expenses.
-1
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
Quarterly Revenues, x [$, billions]
“C
om
pu
ted
” C
osts
(x –
y)
[$,
billio
ns
]
Page 39 of 111
§ 8. Maximum point on Profits-Revenue Graph
The Type I to Type III transition (with, perhaps, an intervening Type II which we
have not highlighted as being too significant) suggests that there must be a real and
continuous smooth curve relating profits and revenues, with both a negative slope
and a positive slope. In other words, the profits-revenues graph must have a
maximum point. This is illustrated in Figure 10 by the dashed curve.
Figure 10: The profits-revenues data for Southwest Airlines for
the 20-year period 1992-2011can be explained by invoking the
Type I and the Type III behaviors, as just discussed. The dashed
curve is not based on any mathematical calculations but
illustrates schematically a smooth nonlinear behavior with a maximum point.
Thus, the Type I, Type II, and Type III straight lines are short segments of this
more general curve. Indeed, one can postulate the existence of a family of such
curves, each with its own maximum point, all described by a simple mathematical
equation, y = mxn [ e
-ax/(1 + be
-ax)]+ c.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6 8 10 12 14 16 18
Revenues, x [$, billions]
Pro
fits
, y [
$,
billi
on
s]
Page 40 of 111
A parabola, described mathematically by the general equation, y = ax2 + bx + c, or
(x – h)2 = 2p(y – k), is a familiar example of a curve with a maximum point. The
trajectory of a golf ball, or a projectile, is believed to follow this “ideal”
mathematical curve.
http://www.trajectoware.com/Screenshot.gif
Golf ball trajectories: http://www.golf-simulators.com/images/t9_1.jpg
Page 41 of 111
http://en.wikipedia.org/wiki/Laffer_curve
http://www.tennessean.com/article/20120527/BUSINESS01/305270039/Economis
t-Arthur-Laffer-enjoys-renewed-popularity Laffer curve: t* represents the rate of
taxation at which maximum revenue is generated. This is the curve as drawn by
Prof. Arthur Laffer, but in reality the curve need not be single peaked nor
symmetrical at 50%.
There is another famous curve, called the Laffer curve, which is believed to have
led to the birth of so-called supply side economics in the late 1970s. This was
embraced by President Reagan as a governing philosophy after he got elected in
November 1980. This too has a maximum point, and is traditionally represented
using a parabola, a symmetric curve, although this would, no doubt, be an extreme
“idealization” for such a complex problem as the effect of tax rates on the
government revenues.
Then, there is the famous catenary curve. It looks like a parabola but it is not a
parabola. This describes the shape taken by a heavy cable when it is suspended
between two poles. Yeah, we see them hanging over our heads, every day!
A hanging chain forms a catenary. Freely-hanging transmission lines also
form catenaries.
http://en.wikipedia.org/wiki/Catenary
Page 42 of 111
There is one more curve with a maximum point, Planck’s blackbody radiation
curve, which also describes the relics of the Big Bang radiation (see below).
http://conferences.fnal.gov/lp2003/forthepublic/cosmology/cobe_wmap.jpg
http://www.learner.org/courses/physics/visual/img_lrg/CMB_spectrum.jpg
The blackbody radiation curve, or the curve for the COBE spectrum, along with
the Maxwell-Boltzmann velocity distribution curve (from the kinetic theory of
gases, for the distribution of molecular velocities in a gas) are the only known
examples of non-symmetric theoretical curves (familiar to this author) with a
maximum point. Both the Planck curve and the Maxwell-Boltzmann distribution
curve can be derived using straightforward statistical arguments.
Obviously, such a non-symmetric curve, with a maximum point (with some
fundamental statistics based justification), should be of great interest to us now to
arrive at a quantitative picture of the profits-revenue situation with Southwest
Airlines and, more generally, with any company, see Figure 10.
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§ 9. Brief Discussion
The simplest equation for a straight line is y = mx and for a curve with a changing
slope it is y = mxn where n = 1 gives the straight line and n > 1 yielding a curve
with increasing slope (acceleration of y) and n < 1 yield a curve with decreasing
slope (deceleration of y), see Figure 11. This is called the power law and “n” is
called the power law index. Frictional, or air, resistance (or the drag), experienced
by a moving object such as a golf ball, car, truck, aircraft, or a rocket is often
expressed using the power law.
Figure 11: Simple illustration of the power-law behavior, y = mxn for the three
cases, n = 1 (linear law), n >1 yielding accelerating growth of profits with
increasing revenues and n <1 yielding decelerating growth of profits with
increasing revenues. In all three cases, there is NO limit to the maximum revenues
or profits. This situation is certainly far more desirable than the situation where a
maximum point appears on the profits-revenue curve, as with the power-
exponential equation. The Type I, Type II, and Type III behavior may all be
thought of a small linear segment of the more general power-exponential law. The
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0.0 5.0 10.0 15.0 20.0 25.0
y = 0.1x
m = 0.1 and n = 1
y = 0.1x1.15
m = 0.1 and n = 1.15
y = 0.1x0.85
m = 0.1 and n = 0.85
Revenues, x
Pro
fits
, y
Page 44 of 111
appearance of a maximum point and the transition to Type III behavior, are
clearly not desirable.
The graph of speed versus time, in road tests routinely performed to assess the
performance of vehicles, can also be described by a power law. (I have confirmed
this with several vehicle road test data.) The acceleration a = ∆v/∆t is the slope of
the graph of speed (or more correctly velocity, hence the symbol v) versus time t.
Road tests show that the acceleration is not a constant but actually decreases with
time (or increasing speed). This is due to the air resistance (aerodynamic drag) just
mentioned and gives the nonlinear power law curve for v-t relation. Many other
examples of such nonlinear laws can be cited.
Figure 12: Illustration of the three special cases of y = mxne
-ax, which is the
simplest form of the power-exponential law: the linear law y = 0.5x (n = 1, a = 0),
the power law y = 0.6x0.67
(a = 0) and the power-exponential, y = 3x0.67
e-0.125x
. The
maximum point occurs when x = n/a = 0.67/0.125 = 5.36. For n = 0, the law
0
1
2
3
4
5
6
7
8
0 5 10 15 20 25 30 35
Pro
fits
, y
Revenues, x [$, billions]
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becomes y = me-ax
and this is “pure” exponential behavior (example, radioactive
decay).
However, the linear and nonlinear laws do not reveal a maximum point. As x
increases y increases, indefinitely, with only the rate of increase ∆y/∆x, or dy/dx
according to the calculus notation, being dictated by the exact mathematical law.
One would like profits to increase as revenues increase. Even better, if profits
would increase at an accelerating rate. However, as we see here with Southwest
Airlines, which has reported a profit for every single year for 39 years, we do not
see any acceleration in the growth of profits with increasing revenues. At best we
see a linear law. This is also true if we study the data for many other companies
(see discussion of Apple in Refs. [1] and [2]). There is no company that I am
aware of, based on many such studies of hundreds of companies since 1998, that
has consistently shown an acceleration (n > 1), for a sustained period of time, in
the profits-revenue space.
Should we expect to see a maximum point on the profits-revenue graph? While it
is easy to speculate about this point, why would one expect to see a maximum
point? This is so counter-intuitive to whole idea of always trying to “maximize”
profits, which is just another way of saying (never mind the semantics) that we
want to see profits increasing indefinitely as revenues increase – all the way to
INFINITY!. An infinity of revenues with an infinity of profits! Wow!
Alas, over this past month (since the Facebook IPO), I find that many real world
companies do show a maximum point on their profits-revenues graph. Southwest
Airlines is the latest example of such a company that I have found. The others are
Ford Motor Company, Verizon, Yahoo, and Kroger. Each one of these companies
is seemingly profitable but each one is “struggling”.
Perhaps, there is some as yet undiscovered natural law (of behavioral
economics and finance) that is responsible for the maximum point. Also, think
about the indefinite increase in profits with increasing revenues. In mathematics,
we can say, and we often do, as x → ∞, y → ∞. This might be true as long as we
are dealing with purely conceptual mathematical entities. Now, apply it to finance
or money and we are dealing with an untenable “double” infinity of sorts, isn’t it?
How can infinite revenues yield infinite profits? Unless, of course, costs go to
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zero! May be now we have stumbled upon a simple (mathematical?) explanation
for the maximum point in the profits-revenues graph.
So, now, going a step further, the simplest equation for a curve with a maximum
point is y = mxne
-ax. This is called the power-exponential law. If a = 0, the
maximum point disappears and we get the power law. If a = 0 and n = 1, we get the
simple linear law or straight line. But, alas, straight lines are nice but they do not
always pass through the origin. Hence, we must add the nonzero constant c to our
equations. As seen here, the nonzero “c” in the profits-revenues graph is due to the
“fixed” costs. The general equation for a straight line is y = hx + c.
However, instead of the simpler y = mxne
-ax, it is more appealing (at least
intellectually speaking) to consider the slightly modified power-exponential
equation y = mxn [e
-ax /(1 + be
-ax) ] since this is nothing more than the generalized
statement of the famous equation from blackbody radiation, first derived by Max
Planck in December 1900. Planck’s original equation can be written as
u = (8πν2/c
3) U
where, U = ε [ e-ε/kT
/(1 – e-ε/kT
) ]
and, ε = hν
giving, u = (8πh/c3) ν
3 [ e
-ε/kT /(1 – e
-ε/kT) ] …………(1)
We do not have to understand everything about equation 1 (especially all the
subtleties of the physics that led to the discovery of this law) except to note that the
frequency ν is our variable x and the radiation energy density u is our variable y.
The frequency is raised to the power 3 which is to be replaced by the general
power law index n. The exponential factor within the square brackets is easily
recognized. In the original Planck equation a = h/kT (where T is the temperature
and k is a constant, called the Boltzmann constant). The constant m which precedes
all is given by (8πh/c3). In Planck’s theory c is the speed of light and “h” is now
called the Planck constant.
This modified form of the power-exponential equation is preferred since we can
readily derive Planck’s equation using the same statistical arguments that Planck
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used in 1900 and apply it instead to the problem of “profits” or “revenues” get
distributed among many different products that a company produces and sells.
Thus, the power-exponential equation y = mxn [e
-ax/(1 + be
-ax)] + c is the simplest
mathematical law relating x and y which also reveals a maximum point. The
significance of this law may be understood quite simply, as follows, by considering
various special cases.
Linear law: For n = 1, and a = b = 0, the power-exponential law reduces to the
linear law y = mx + c.
Power law: For a = 0 we get the simple power law, y = Mxn, where M = m/(1+b)
is a new constant that replaces m and absorbs the nonzero b in the denominator. In
this case, there is NO maximum point. The derivative dy/dx = n(y/x) is nonzero for
all values of x. Hence, the profits y will increase indefinitely without limit, with
increasing revenues x either at an accelerating rate (for n > 1) or at a decelerating
rate (for n < 1) with increasing revenues (see Figure 9, also Figures 26 and 27 in
Appendix 5). For n = 0, we get “pure” exponential behavior (e.g. radioactive
decay, charging of batteries, etc.)
Power-exponential law: A maximum point appears only when a > 0, however,
small. This can be appreciated by considering the simpler case of b = 0 and c = 0,
for which y = mxne
-ax. The derivative dy/dx = (n- ax)(y/x). Hence, there is a
maximum point on the graph at x = n/a. For x < n/a the derivative dy/dx > 0 and y
increases with increasing x. For x > n/a, the derivative dy/dx is negative and y
decreases with increasing x. This is the type of behavior that we are witnessing
with Southwest Airlines (see schematic graphs in Figures 10 and 12).
This simple nonlinear Planck equation, with a maximum point, may be used to
explain the appearance of the maximum point on the profits-revenues graph for
Southwest Airlines. Or, one might think in terms of the Maxwell-Boltzmann type
of distribution in the financial and economic world. Exactly, similar observations,
regarding the maximum point, have also been made (since I began this recent study
after the disappointing Facebook IPO on Friday May 18, 2012) with several
companies, most notably Ford Motor Company. The Ford data reveals a maximum
point, like we see here, again with a lot of scatter.
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In many ways, both Ford and Southwest Airlines are similar. They are both
currently profitable and appear to be doing quite well. However, they are not able
to report consistently high level of profits. There are too many wild fluctuations.
The reason, as we see here, lies in the fact that, for both companies, costs seem to
be increasing at a higher rate than revenues. In other words, the ratio ∆C/∆R > 1.
This trend must be reversed. As we see from the Southwest data, for the earlier
period (1992-2001), when ∆C/∆R < 1, profits were increasing with increasing
revenues and the profit margins were also higher.
Also, the negative (Type III) relation established since 2007 between profits and
revenues is a cause for some concern. This point has been discussed in more detail
in Ref. [1]. Many other companies, notably Ford, Verizon, Yahoo, and Kroger,
also reveal this “toxic” Type III behavior. This seems to be a precursor to company
reporting a continued losses even as revenues increase or, as with GM, the filing of
bankruptcy after a long period spent in the Type III mode, or, as with Air Tran,
becoming a target for merger/acquisition, see §15, page 89. (A discussion of the
“old GM” financial data, from 1991 to 2008, before the bankruptcy filing in June
2009 will be presented separately.)
This is what both Ford and Southwest can learn from the “old GM”. Type III
behavior is the recipe for eventual failure and should be reversed at the earliest.
http://1.bp.blogspot.com/_R2yHiPgsajA/SsFIHuETLaI/AAAAAAAAB5k/iaFISjQ
9pK8/s400/southwest+airlines+1981.bmp
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§ 10. Appendix 1: Southwest Airlines Consolidated Statement of Operations First Quarter 2012
Item
No.
Description
Three months
ending
March 2012
Three months
ending
March 2011
1 Total Operating Revenues 3,991 3103
2 OPERATING EXPENSES
2.1 Salary, Wages, and Benefits 1,141 954
2.2 Fuel and Oil 1.510 1038
2.3 Maintenance materials and repairs 272 199
2.4 Aircraft rentals 88 46
2.5 Landing fees and other rentals 254 201
2.6 Depreciation and amortization 201 155
2.7 Acquisition and integration 13 17
2.8 Other operating expenses 490 379
TOTAL Operating Expenses 3,969 2,989
3 Operating Income 22 114
4 OTHER EXPENSES (INCOME)
4.1 Interest Expense 40 43
4.2 Capitalized interest (5) (3)
4.3 Interest income (2) (3)
4.4 Other (gains) losses, net (170) 59
TOTAL Other (income) Expenses (137) 96
5 Income Before Income Taxes 159 18
6 Provision for Income Taxes 61 13
7 Net Income (Profits in this study) 98 5
All numerical values here are in millions of dollars.
This is an example of the consolidated statement of operations from the 10Q SEC
filings. Our interest in this study has been on Revenues, x, reported as Item no. 1
and the Net income reported as Item no. 7. It is this Net Income that has been
called the Profits, y. The Total Operating Expenses (Item no. 2) plus the items 4, 5
and 6 taken together would thus represent what has been referred to as Cost (or the
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“computed” Cost for clarity). Of these, Item no. 2, the Total Operating Expenses is
the major portion of the Cost.
Hence, let us now take a look at how Item 2, which will simply be referred to as
the Operating Expense (Op. Exp or OE) has varied over the last ten consecutive
quarters as revenues have either increased or decreased. The data that will be
analyzed is given below as Table 3.
Table 3: Quarterly Data from Southwest Airline 10-Q SEC Filings
Quarter
ending
Revenues, x
$, billions
Profits, y
Net Income
$, billions
Total Operating
Expenses (OE)
$, billions
Computed
Costs
(x – y)
Mar 2012 3.991 0.098 3.969 3.893
Dec 2011 4.108 0.152 3.961 3.956
Sep 2011 4.311 -0.140 4.086 4.451
Jun 2011 4.136 0.161 3.929 3.975
Mar 2011 3.103 0.005 2.989 3.098
Dec 2010 3.114 0.131 2.898 2.983
Sep 2010 3.192 0.205 2.837 2.987
Jun 2010 3.168 0.112 2.805 3.056
Mar 2010 2.630 0.011 2.576 2.619
Dec 2009 2.712 0.116 2.545 2.596
Sep 2009 2.666 -0.016 2.644 2.682
Jun 2009 2.616 0.054 2.493 2.562
Mar 2009 2.357 -0.091 2.407 2.448
Dec 2008 2.734 -0.056 2.664 2.790
Sep 2008 2.891 -0.120 2.805 3.011
Jun 2008 2.869 0.321 2.664 2.548
Mar 2008 2.530 0.034 2.442 2.496
Dec 2007 2.492 0.111 2.366 2.381
Sep 2007 2.588 0.162 2.337 2.426
Jun 2007 2.583 0.278 2.255 2.305
Mar 2007 2.198 0.093 2.114 2.105
Dec 2006 2.276 0.057 2.102 2.219
Sep 2006 2.342 0.048 2.081 2.294
Notice that quarterly losses (highlighted) have been reported by Southwest
Airlines but this was compensated by increased profits in the other quarters for
the same year. The “computed” cost C = (R – P) = (x – y) in the last column is
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always greater than The Total Operating Expenses (hereafter just OE Item no. 2
in the consolidated statements) since there are still some Other Expenses, Item
no. 4, which are not accounted for in the OE.
For the most recent quarters, from Dec 2009 to Mar 2012, the quarterly Operating
Expenses (OE) reveal the same trend that we saw with the annualized costs. The
OE is increasing faster than revenues since the linear regression reveals a slope
greater than unity, see Figure 8. Also, if we compute the cost again from Cost =
Revenues – Profits = (x – y), we again find that the slope of the cost-revenue graph
is also greater than unity. The quarterly costs are also increasing faster than the
quarterly revenues, see Figure 9.
However, if we consider all of the data in Table 3, from Sep 2006 to Mar 2012
(the period over which we saw the establishment of the Type III trend in the
annualized profits-revenues graph, Figures 3 to 7) we can derive some solace from
the fact that the best-fit lines for both these graphs have a slope of less than unity.
The best-fit equations are given below and the graphs have been included for
completeness as Figures 13 and 14.
Total Operating Expenses (OE) versus Revenues:
Mar 2012 and Dec 2009: y = mx + c = 1.1134x – 0.474 slope m > 1 for recent
quarters, which should be cause for concern and serious study by management.
Mar 2007 and Sep 2011 (highest and lowest revenues): y = 0.933x + 62.67
Best-fit equation for all quarters: y = 0.972x - 0.75 with r2 = 0.97
In all the above equations x is revenues and y is the Total Operating Expenses
(OE), Item No. 2 from the consolidated statements. For recent quarters the slope is
greater than unity, although when Type III has been established (with annualized
data), the quarterly data seems to reveal that OE is rising at a fixed rate and is
lower than the revenues (which means potential for profits, providing Other
Expenses, Item 4 and Taxes do not overwhelm and lead to a loss).
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Figure 13: Total Operating Expenses (hereafter just OE), Item no.
2 in the consolidated statements given earlier, increases at a
remarkably fixed rate as revenues (quarterly) increase. Data for
all quarters ending Sep 2006 to Mar 2012 has been plotted here.
This covers the period 2007-2011 during which the Type III behavior was
established in the annualize profits-revenues graph. The best-fit equation y = mx +
c = 0.972x – 0.075. The slope m = 0.972 < 1 which means that OE is increased at
a fixed rate with increasing revenues. It also means that less 3% of revenues is
available to cover “Other Expenses”, Item 4, taxes, Items 5 and 6, and still report
a profit, item no. 7. This trend also explains the very low profit margins that we
see being reported, although Southwest has consistently reported profits on an
annualized basis.
“Computed” Costs versus Revenues
Next, we consider the quarterly results and “compute” the costs C from the basic
equation Profits = Revenues – Costs, or P = R – C which means C = R – P. These
values are to be found in the last column of Table 3. The graph again reveals a
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7
Quarterly revenues, x [$, billions]
Tota
l Op
era
tin
g Ex
pe
nse
s (O
E) [
$, b
illio
ns]
y = 0.972x – 0.075, with r2 = 0.97
Quarters Sep 2006 to Mar 2012 Covers period for Type III pattern in annualized data
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remarkably consistent increase in costs (just as for the OE) with increasing
revenues. The best-fit line has the equation C = 0.996R – 0.64 with a very high
value for the correlation coefficient r2 = 0.9665 (perfect correlation r
2 = +1.0).
Figure 14: Since Profits = Revenues – Costs, P = R – C, we can
deduce costs C from the revenues (Item no. 1) and the profits, or
net income (Item no. 7) reported in the consolidated statement of
operations. The data for all the quarters from the quarters ending Sep 2006 to Mar
2012 is plotted here. This reveals a remarkably nice and consistent upward trend.
The best-fit line through the data has the equation C = 0.996R – 0.64 which means
the statistically significant relation for for quarterly profits is P = 0.004 R + 0.64.
The slope here is like the marginal tax rate. If revenues increase by $1 million,
profits only increase by $4000, or about 0.4% conversion rate for additional
revenues into profits. Hence, in addition to sustaining profitability, Southwest must
also focus into efforts on improving its profit margins.
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7
Quarterly revenues, x [$, billions]
“Co
mp
ute
d”
Co
sts
C =
(x
– y)
[$
, bill
ion
s]
Sep 2006 to Mar 2012 C = 0.996 R – 0.64
which means P = 0.004R + 0.64
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Since P = R – C, we can use the C-R equation to deduce the P-R equation and it is
therefore given by P = R – 0.996E + 0.64 = 0.004R + 0.64. This means that only a
very small amount of the additional revenues (beyond breakeven revenues) is
being converted into profits. The exact numerical value is 0.4%, less than one-half
percent.
This should be cause for concern for a company that is proud of its profits record,
of delivering profits consistently year-after-year for 39 years. Now, let the focus be
on improving this profit margin. As discussed in Refs. [1-3] there are many
companies with a slope h in the profits-revenues equation in excess of 0.20. The
goal should be to convert at least one-third of the additional revenues into profits.
In summary, the analysis here suggests that Southwest Airlines must take steps to
avoid what appears to be the inevitable reporting of an annualized loss for the first
time in its history. The maximum point revealed on the profit-revenues graph
should be taken very, very, seriously and the underlying reasons for this maximum
must be studied and understood.
§ 11. Appendix 2 Profits-Revenues-Costs-OE growth in a single year
Consolidated Statement of Operations First Quarter 2012
We usually look at financial data on a quarterly, or annual, basis. These are found
in what is known as the 10-K and 10-Q SEC filings of all companies. There are
many good reasons for doing this, one of them being the seasonal nature of
revenues due to what we all do (as consumers) during any given year. And,
depending on the nature of the business, there is a definitely a very marked
seasonality attached to both revenues and profits.
This also gives us a nice way to understand how exactly profits, revenues, costs
and the Total Operating Expenses (OE) “grow”, literally “grow” each year. We
will consider three-month, six-month, nine-month and the annual data for a single
year and study how profits and revenues evolve. We will use two approaches.
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1. Consider the data for each quarter (ending March, June, September and
December). We see significant variations between the four quarters in a
single year. For example, in 2009, Southwest Airlines reported losses for the
quarters ending March and September and profits for the quarters ending
June and December, yielding overall profit for the whole year. For 2011, the
loss in the third quarter was overcome by profits in the other three.
2. Consider the data for period ending three-months, six-months, nine-months,
and twelve-months. This second approach tells us about how “evolution” of
revenues and profits occur during a year, until the cycle is then repeated for
the next year.
The following min-table gives the data for 2011. Although the profits-revenues
graph yields a lot of scatter (as would be expected from a look at the numbers), the
costs-revenues graph reveals a nice upward sloping trend with hardly any scatter.
Revenues, x TOE Profits, y Costs ( x- y)
millions millions millions millions Mar 2011 3,103 2,989 5 3,098 Jun 2011 4,136 3,929 161 3,975 Sep 2011 4,311 4,086 -140 4,451 Dec 2011 4,108 3,961 152 3,956
The above table has data for three-months (one quarter) ending as indicated.
Cumulative Revenues, x TOE Profits, y Costs ( x- y)
values millions millions millions millions Mar 2011 3,103 2,989 5 3,098 Jun 2011 7,239 6,918 166 7,073 Sep 2011 11,550 11,004 26 11,524 Dec 2011 15,658 14,965 178 15,480
The above table has data for cumulative values for 3, 6, 9, and 12 months. The
profits at the end of June 2011 were $166 million, the sum of the two quarters.
The cumulative profits dropped to $26 million because of the loss in third
quarter, and so on.
The costs-revenues equation is C = 0.986R + 37.757. The OE-revenues graph also
has a similar pattern with the equation y = mx + c = 0.954x +29.101 where x is
revenues and y is the OE. All values are in millions. These equations were
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obtained by simply joining the first and fourth quarter (x, y) pairs, after confirming
the linearity of the graphs.
Both slopes are less than unity. Converting the cost equation to the profits
equation, we get P = R – C = hR + c = 0.014R – 37.757. The negative intercept c =
-37.757 means that once revenues exceed the breakeven value (R0 = - c/h) $2740
million, about 1.4% of the additional revenues are converted into profits.
Figure 15: The evolution of profits and revenues during a single
year (2010). The straight line joining the two extreme points
reveals the overall trend nicely. One could determine the best-fit
equation but such a computational accuracy is not really
0
0.1
0.2
0.3
0.4
0.5
0.6
0 2 4 6 8 10 12 14 16
2010 Revenues, x [$, billions] Cumulative values during the year
2010 P
rofi
ts,
y [
$, b
illio
ns]
Cu
mu
lati
ve
valu
es d
uri
ng
ye
ar
Page 57 of 111
necessary to understand these basics. The profits-revenues equation, y = hx + c or
P = 0.0473 R – 0.113. The cut-off or breakeven revenue is obtained by setting P =
0 = -c/h = 0.113/0.0473 = $2.397 billion (see intercept where profits go to zero).
The data for 2010 can also be viewed in the same fashion. Only the “cumulative”
values are given below. The costs-revenues equation is C = 0.953R +113.37 or P =
0.0473 R – 113.37, see Figure 15. The “break even” revenue R0 = 113.37/0.0127 =
$2397 million is slightly lower than for 2011 but the slope h = 0.0473 is much
higher than for 2011.
In other words, costs increased between 2010 and 2011, C = 0.986R + 37.757 for
2011 versus C = 0.953R +113.37 for 2010. Both the slope and the intercept
increased in 2011 compared to 2010. What is the reason for these variations from
year-to-year?
Cumulative Revenues, x TOE Profits, y Costs ( x- y)
values billions billions billions billions
Mar 2010 2.630 2.576 0.011 2.619 Jun 2010 5.798 5.381 0.123 5.686 Sep 2010 8.990 8.218 0.328 8.785 Dec 2010 12.104 11.116 0.459 11.973
The above table has cumulative data through the month ending as indicated.
Finally, let us consider now the data for the year 2000, the year of peak profits
before the wild fluctuations started, see Table below and Figure 16.
Cumulative Revenues, x Profits, y Costs ( x- y)
values billions billions billions Mar 2000 1.428 0.121 1.307 Jun 2000 2.982 0.297 2.685 Sep 2000 4.461 0.481 3.980 Dec 2000 5.928 0.636 5.292
The above table has cumulative data through the month ending as indicated
The profits-revenues equation y = hx + c = 0.114x – 0.0424, determined by
joining the start and end values. The slope h = (0.636 – 0.121)/(5.928 – 1.428) =
0.114 and the intercept c is then fixed from the (x, y) values at either the start
or the end (since h is known, and x and y are known, c is calculated easily).
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Figure 16: The evolution of profits and revenues during a single
year (2000). The straight line joining the two extreme points again reveals the
overall trend nicely. Comparing this with Figure 15 for 2010, the most noticeable
difference is the healthy profit for the first quarter and the smaller intercept made
on the x-axis. Perhaps, this explains why the first quarter profits were higher in
2000 compared to 2010. The higher “fixed costs” seem to be a factor in this
difference. Obviously, the finding being reported here needs more careful study,
with data for many other companies and perhaps also for Southwest Airlines.
With modern computers one could analyze large volumes of data in a matter of
days, or hours, if not minutes. The present author took only a week (while engaged
in other important personal activities) to perform all these calculations, prepare
all the graphs and finish writing this entire report.
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 1 2 3 4 5 6 7
2000 Revenues, x [$, billions] Cumulative values during the year
20
00 P
rofi
ts,
y [
$, b
illio
ns]
Cu
mu
lati
ve
valu
es d
uri
ng
ye
ar
Page 59 of 111
Figure 17: The red dashed line P = 0.0851 R – 0.042, connecting the data for Mar
and Sep has a higher slope compared to the blue solid line, P = 0.0695R – 0.042,
connecting the Mar and Dec data. The intercepts made on the revenues-axis are
thus different. The data for Jun lies practically on the line joining the Mar-Dec
data. Why did the slope (or the intercept) change during the year, between the
various quarters? The best-fit line has an intermediate slope.
The quarterly data for 2007 (when Type III seems to be initiated) also reveal the
similar pattern, see table below. The slope is lower and intercept higher than 2000.
Cumulative Revenues, x Profits, y Costs ( x- y)
values billions billions billions Mar 2007 2.198 0.111 2.087 Jun 2007 4.781 0.273 4.508 Sep 2007 7.369 0.551 6.818 Dec 2007 9.861 0.644 9.217
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12
2007 Revenues, x [$, billions] Cumulative values during the year
20
07
Pro
fits
, y [
$, b
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Page 60 of 111
The profits-revenue graph reveals a nice linear behavior. The straight line
connecting the Mar and Dec (R, P) data yields P = 0.0695 R – 0.042 while the
Mar and Sep (R, P) data yields a higher slope, P = 0.0851R – 0.076. The best-
fit line through these four points has an intermediate slope, P = 0.0735R – 0.05
= 0.0735 (R – 0.68). The slope is lower than for 2000 and the cut-off or
breakeven revenue has also increased.
Figure 18: Best-fit line through the 2007 “intrayear” data has the
slope that falls between the high and low values of the slope
indicated in Figure 17. These detailed calculations are presented
to illustrate that “costs” are increasing although Southwest prides
itself as a low-cost airline. Now we have to understand what operational factors
contribute to these subtleties in “cost” increases.
These simple calculations based on “intrayear” data show that there is significant
change in both the slope of profits-revenue graph over the past decade and also the
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 12
2007 Revenues, x [$, billions] Cumulative values during the year
20
07
Pro
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, y [
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Page 61 of 111
intercept, suggesting changes in both the “fixed” costs and the “variable” costs.
Recall that, according to the classical breakeven analysis for profitability, the slope
of the profits-revenues equation y = hx + c, is given by h = 1 – (b/p) is related to
the unit price p and the unit variable cost b. The intercept c = - a related to the
fixed cost. The results are summarized below for convenience.
P = 0.014R – 37.757 = 0.014 (R – 2.740) for 2011
P = 0.0473 R – 113.37 = 0.0473 (R – 2.397) for 2010
P = 0.0734R – 0.05 = 0.0734 (R – 0.68) for 2007
P = 0.114R – 0.0424 = 0.114 (R – 0.371) for 2000
The decreasing value of the slope h and the increasing values of R = R0 the
minimum revenue to report a profit are unmistakable and quite obvious here.
The visual appearance of the first data point in the 2010 graph (just above the cut
off revenue level) versus the 2000 graph, the considerably lower scatter in the
graph for 2000 versus 2010, and, above all, the scale of the two graphs - revenues
have more than doubled between 2000 and 2010 - should all engage our attention,
see also the composite plot that follows.
Yet, profits in 2010 were lower than the profits in 2000, significantly lower - $459
million in 2010 and $636 million in 2000!
In summary, the simple calculations presented in this Appendix reveal the
“dynamic” nature of the changes in the operations of a company (especially the
cost structure which is of interest to us), occurring from quarter-to-quarter and also
year-to-year. Such intra-year variations, and inter-year variations, must be
carefully studied to more fully understand the basic “laws” governing financial
performance of various companies.
Page 62 of 111
Figure 19: Composite plot comparing the evolution of profits during 2000 and
2010 (from the 3 months, 6 months, 9 months, and 12 month quarterly data for
each year). A linear profits-revenues equation is implied by the classical
breakeven analysis for profitability. If a is the fixed cost and b the unit variable
cost, the total cost C = a + bN where N is the number of units offered. If p is unit
price, the revenues generated from the sale of the N units is R = pN. Also, N = R/p.
Hence, the profits P = R – C = [1 – (b/p)]R – a which means the intercept c = - a
and the slope h = 1 – (b/p). The higher intercept made on the x-axis implies a
higher fixed cost. The lower slope means a lower rate of conversion of additional
revenues (beyond breakeven) into profits. Since, h = 1 – (b/p), the lower slope
means a higher unit variable cost b or a lower unit price p (due to competitive
pressures). The inescapable conclusion from the above is that costs have gone up
for Southwest Airlines during the last decade although the company prides itself on
being a major low-cost domestic airline. Is this a contradiction? NO, if one
understands the meanings of “costs” being discussed here. The good can still
become better and the better can become the best!
-0.20
0.00
0.20
0.40
0.60
0.80
0 2 4 6 8 10 12 14
2000
2010
2000 a
nd
2010
Pro
fits
, y [
$,
billi
on
s]
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e y
ea
r
2000 and 2010 Revenues, x [$, billions] Cumulative values during the year
Page 63 of 111
Summary listing of Profits-Revenues Equations
The following table summarizes the profits-revenues equations deduced, as
discussed here, from the quarterly data for a single year, by considering the growth
in 3 month, 6 month, 9 month, and 12 month intervals.
Year Profits-revenues equation
Units of x and y are indicated
Slope h Breakeven
R0 = - c/h
$, millions
Comment
1974 y = 0.133x +157.6 in 000s 0.133 Type II
1975 y = 0.151x -39.52 in 000s 0.151 0.26 M Type I
1978 y = 0.22x – 826.73 in 000s 0.22 3.76 M Type I
1979 y = 0.121x + 230.12 in 000s 0.121 Type II
1980 y = 0.139x – 1302.5 in 000s 0.139 9.33 M Type I
1981 y = 0.130x - 1.088 millions 0.130 8.35 M Type I
1982 y = 0.115x - 4.222 millions 0.115 36.58 M Type I
1983 y = 0.101x - 4.477 millions 0.101 44.26 M Type I
1984 y = 0.096x – 1.80 millions 0.096 18.74 M Type I
1985 y = 0.077x – 5.14 millions 0.077 66.65 M Type I
1986 y = 0.079x -11.036 millions 0.079 138.93 M Type I
1987 y = 0.049x – 18.75 millions 0.049 375.12 M Type I
1988 y = 0.085x -14.965 millions 0.085 176.59 M Type I
1989 y = 0.066x + 4.487 millions 0.066 Type II
1990 y = 0.0453x -6.72 millions 0.045 148.22 Gulf War!
Made a Profit
1991 y = 0.034x – 17.85 millions 0.034 523.83 Type I
1992 y = 0.059x – 8.62 millions 0.059 145.84 M Type I
1992
y = 0.058x -6.86 millions
0.058
118.6 M
1993 & 1992
reports have
diff. values
1993 y = 0.019x – 17.97 millions 0.019 919.3 M Type I
1994 y = 0.0713x + 0.0046 billions 0.0713 Intercept c ≈ 0
1995 y = 0.075x + 0.071 billions 0.075 Type II
2000 y = 0.114x – 0.0424 billions 0.114 371 M Type I
2007 y = 0.0734 x – 0.05 billions 0.073 680 M Type I
2010 y = 0.0473 x – 113.37 billions 0.047 239.7 M Type I
2011 y = 0.014 x – 37.57 billions 0.014 274 M Type I
Page 64 of 111
First quarter 1987, first quarterly loss since first quarter of 1973.
http://www.chron.com/CDA/archives/archive.mpl/1987_459421/southwest-in-the-
red-for-first-time-since-73.html
There are frequent back-and-forth transitions from Type I to Type II
behavior between years. (This is like an engine running erratically and
misfiring.) We also see a general trend of decreasing slope h (less of the
revenues are converted into profits) and a rising value of the “breakeven”
revenues, i.e., intercept c, along implying higher costs and lower profits.
Southwest Airlines route network maps from key focus cities.
Goegraphy Lesson: Don’t be misled by this map and try to find the Pacific Ocean
near Tucson, Arizona. San Diego is southernmost city here in the continental USA,
on the Pacific coast. California’s southern border with Mexico (the nearly
horizontal stretch, east of San Diego) then begins. The Baja peninsula, part of
Mexico, sticks out of southern California into the Pacific Ocean. All the other states
to the east, all the way to Texas, share a border with Mexico.
Hahaha for fooling me today!
Page 65 of 111
§ 12. Appendix 3 Type II behavior and Type I to Type II transition
The following data, obtained from the Southwest Annual Report for 1995 is of
interest since it reveals an interesting example of Type II behavior when we
consider the “intrayear” data for 1995, i.e., the growth of revenues and profits
during the year, reported in quarterly reports and summarized in the annual report.
The annual report for 1995 also provides the data for the years 1991-1995 which
also, reveals, as we will see a Type II behavior.
Revenues, x Profits, y Costs ( x- y)
Billions billions billions Mar 1995 0.621 0.118 0.503 Jun 1995 0.738 0.060 0.678 Sep 1995 0.765 0.067 0.698 Dec 1995 0.749 0.043 0.706
The above table has data for three-months (one quarter) ending as indicated.
Cumulative Revenues, x Profits, y Costs ( x- y)
Values Billions billions billions Mar 1995 0.621 0.118 0.503 Jun 1995 1.359 0.178 1.181 Sep 1995 2.124 0.245 1.879 Dec 1995 2.873 0.288 2.585
The above table has data for cumulative values for 3, 6, 9, and 12 months. The
profits at the end of June 1995 were $0.118 + $0.060 = $0.178 billion, the sum
of the individual values for the two quarters, and so on.
Revenues, x Profits, y Costs ( x- y)
Billions billions billions 1995 2.873 0.183 2.690 1994 2.592 0.179 2.413 1993 2.297 0.169 2.128 1992 1.803 0.110 1.693 1991 1.379 0.033 1.346
The above table has annualized data for each year as indicated.
Page 66 of 111
Figure 20: The cumulative profits and costs (see table) are increasing during the
year. In 1995 every quarter was profitable. The profits and revenues vary from one
quarter to the next. Nonetheless, the profits-revenues graph, as we see above, and
also the costs-revenues reveal a nice linear behavior. The straight line, y = hx + c
= 0.0756x + 0.071, or equivalently, P = 0.0756R + 0.071, joining the Mar and
Dec data points makes a small positive intercept on the profits axis. This describes
the data quite well. This is Type II behavior (h > 0, c > 0), as discussed in §5 of
main text before analysis of the profits-revenues data was presented. (The best-fit
line will have a slightly higher slope and slightly smaller intercept, since one data
point, the nine-months ending data, is above this Type II line.) The “intrayear”
1994 data seems to reveal the “verge” of Type I to Type II transition. On a similar
profits-revenues graph, the best-fit line through the four data points makes a very
small positive intercept (h = 0.071 and c = 0.00458 > 0).
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1995 Revenues, x [$, billions] Cumulative values during the year
19
95
Pro
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$, b
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Page 67 of 111
Figure 21: The intrayear profits-revenues data for 1994 and 1995. The
“diamonds” are the values for 1995 and the “squares” the values for 1994. Profits
were higher in 1995 compared to 1994 for similar revenue levels. For 1994, we
get y = hx + c = 0.0713x + 0.00458 and for 1995 y = 0.0756x + 0.071. The nearly
zero value of the constant c implies that the company is going through the Type I to
Type II transition with c < 0 (Type I), c = 0, and c > 0 (Type II).
The data for the yeas 1991-1995 also shows and interesting pattern. As we see
from the table, between 1991 and 1992 there is a significant increase in profits but
this rate seems to have slowed down in subsequent years. One could describe this
with a Type I behavior going into a Type II behavior. However, it is preferable to
consider the alternative and more conservative viewpoint of an overall Type I
behavior with a smaller slope. As we know (after data is already in place, with the
benefit of hindsight, as they say) from the analysis of the entire 1991-2000 period,
the conclusion of a Type I behavior is a much better conclusion.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
1994 and 1995 Revenues, x [$, billions] Cumulative values during the year
1994 a
nd
1995
Pro
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1994
1995
Page 68 of 111
Figure 22: The profits-revenues data for the period 1991-1995,
obtained from the 1995 Annual Report. Profits increased rapidly,
as we see here, between 1991 and 1992. This was followed by a
period with a slower rate of increase of profits. This can be described
mathematically as a Type I to Type II transition (or by using a smooth nonlinear
curve, such as the power law y = mxn + c, with the index n < 1). However, one
should seek nonlinearity (or postulate linear transitions such suggested above)
only after careful consideration of the alternative viewpoint of a single linear trend
to describe the same data, see Figure 23.
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Annual Revenues, x [$, billions]
An
nu
al
Pro
fits
, y [
$, b
illi
on
s]
Page 69 of 111
Figure 23: The alternative viewpoint of a single Type I linear
trend is to be preferred and must be considered before accepting
the transition from Type I to Type II, in other words, a permanent
transition to a lower profitability mode.
Nonetheless, one must look for “clues” about such a transition and take appropriate
actions to avoid a “permanent” lapse into a less desirable mode.
The Type I to Type III transition, that seems to have occurred between 2007-2011,
and discussed more fully in this article, is a case in point. Southwest management
must address this issue immediately to avoid reporting a historical first annual loss
in the near future and also carefully consider and review the historical patterns
pointed out here. It should be noted that the 1995 Southwest Annual Report seems
to be unique in providing data for all the four quarters for year 1995 (and also for
1994). This is usually not readily available in annual reports, as seen below.
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
An
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Pro
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, y [
$, b
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Annual Revenues, x [$, billions]
Page 70 of 111
1995 ANNUAL REPORT
ITEM 6. SELECTED FINANCIAL DATA (Pages 9 and 10)
YEARS ENDED DECEMBER 31,
-----------------------------------------------------------------------------
1995 1994 1993 1992 1991
---- ---- ---- ---- ----
FINANCIAL DATA:
(in thousands except per share amounts)
Operating revenues
$2,872,751 $2,591,933 $2,296,673 $1,802,979 $1,379,286
Operating expenses
2,559,220 2,275,224 2,004,700 1,609,175 1,306,675
------- ---------- ---------- ---------- ----------
Operating income . . . . . . . . . . . . . . 313,531
316,709 291,973 193,804 72,611
------ ---------- ---------- ---------- ----------
Net income (1)
$182,626 $179,331 $ 169,543 $ 109,923 $ 33,148
========== ========== ========== ========== ==========
ITEM 8 : FINANCIAL STATEMENTS AND SUPPLEMENTARY DATA (Page 11)
QUARTERLY FINANCIAL DATA (UNAUDITED)
(IN THOUSANDS EXCEPT PER SHARE AMOUNTS)
----------------------------------------------------
1995 MARCH 31 JUNE 30 SEPT. 30 DEC. 31
-------- ------- -------- -------
Operating revenues $620,999 $738,205 $764,975 $748,572
Operating income 23,409 103,425 114,098 72,599
Income before income taxes 20,034 100,801 114,215 70,090
Net income 11,826 59,724 67,717 43,359
Net income per common and .08 .41 .45 .29
common equivalent share
Page 71 of 111
§ 13. Appendix 4
Profits and Passengers Flown
One would expect to find a steady increase in profits as the number of passengers
flown by an airline increases. Surprisingly, the data in this regard, for Southwest
Airlines, shows a good bit of scatter, see http://www.swamedia.com/channels/By-
Category/pages/yearend-summary . This is summarized in Table 4.
Table 4: Profits and Passengers flown for Southwest Airlines
Year
Revenue
Passengers
(millions)
Net Income
($, millions)
Year
Revenue
Passengers
(in 000s)
Net Income
($, millions)
1971 108.554 -3.753
2010 88.191 459 1972 308.999 -1.591
2009 86.301 99 1973 543.407 0.175
2008 88.529 178 1974 759.721 2.141
2007 88.713 645 1975 1,136.32 3.4
2006 96.3 499 1976 1,539.11 4.939
2005 77.694 548 1977 2,339.52 7.545
2004 70.9 313 1978 3,528.11 17.004
2003 65.674 240.969 1979 5,000.09 16.652
2002 63.046 240.969 1980 5,976.62 28.447
2001 64.447 511.147 1981 6,792.93 34.165
2000 63.678 625.224 1982 7,965.55 34.004
1983 9,511.00 40.867
1984 10,697.54 49.724
1985 12,651.24 47.278
1986 13,637.52 50.035
While the number of (revenue producing) passengers did not vary significantly in
recent years (from 2007 to 2010, a low of 86.3 million to a high of 88.7 million),
notice the erratic variation in profits (net income), with a low of $99 million in
2009 and a high of $645 million in 2007. Compare also the profits for 2010 with
Page 72 of 111
the profits for 2008 with nearly the same number of passengers. Not surprisingly
the x-y graph in Figure 24 thus reveals a lot of scatter.
Figure 24: Net income versus number of passengers flown for the
period 2000-2010. Although there is a lot of scatter, a linear
trend, in agreement with data for earlier period (see 1971-1986 in Figure 25) is
suggested if we consider the data selectively. The straight line connecting the data
for 2002 and 2006 is shown here and has the equation y = 7.759 (x – 31.99). In
other words, a certain minimum number of revenue producing passengers, about
32 million, must be flown before a profit can be reported. After this “breakeven”
level, profits increase at a fixed rate of about $7.76 per passenger. This can now
be compared to the earlier period.
Nonetheless, a nice linear trend is revealed, if we overlook the scatter and consider
the data selectively. The same linear trend is also suggested by the data for the
early years (1971-1986) which can also be obtained at the same website.
0
100
200
300
400
500
600
700
800
0 20 40 60 80 100 120 140
Number of passengers, x [millions]
Ne
t In
co
me
(P
rofi
ts),
y [
$,
mil
lio
ns]
Recent years (2000-2010) y = 7.759x – 248.23 = 7.759 (x – 31.99)
Page 73 of 111
An estimate of the (additional) profits produced per (additional) passenger can be
obtained from the slope of such a graph. As we see from the Table 4, in 1971 and
1972, Southwest reported a loss. It reported its first profitable year in 1973. The
number of passengers increased steadily during this period. Thus, it is clear that a
certain minimum number of passengers is required (to cover the fixed costs, or the
breakeven revenue discussed earlier) to report a profit. Once this minimum is
exceeded, profits increase at a fixed rate per passenger.
For 2000-2010 period, y = 7.756x – 248.23 = 7.756 (x – 31.99) where x and y are
both in millions. Thus, profits increase at a fixed rate of about $7.76 per passenger
once the “breakeven” level, about 32 million passengers is reached. This can be
compared to the earlier period, 1971-1986, see Figure 25.
Figure 25: Net income versus number of passengers flown for the
first full sixteen years of operation (1971-1986), with 1971 being
only a partial year of operation.
-10
0
10
20
30
40
50
60
70
0 4,000 8,000 12,000 16,000
Number of passengers, x [thousands]
Early years 1971-1986
y = 0.00415 x – 0.475
= 0.00415 (x – 114.54)
Net
Inco
me (
Pro
fits
), y
[$,
milli
on
s]
Page 74 of 111
A nice upward trend is observed for this earlier period. The best-fit line through
the data points yields a slope of $4.15 per passenger, or about one-half the
current value of nearly $8 per passenger.
A number of line segments, with varying slopes and intercepts, can also be
conceived for this earlier period. For these “local” segments, an estimate as high as
$9.80 per passenger (example for the 1979 and 1981 data points, dashed line) is
possible. The results here thus yield a somewhat conflicting picture and should be
investigated more completely. While the per-passenger profit seems to have
improved in recent years, the absolute level of profits and also the profit margins
have also greatly decreased.
Improved statistical models must be developed to predict profitability per
passenger and also profits per unit of revenue. The latter now seems to be the more
predictable of the two.
******************************************************************
New Fashion Police: Southwest Airlines
A woman was booted off a Southwest
Airlines flight in Reno for wearing a
T-shirt with the pictures of President
Bush and Vice President Dick Cheney
and the F-word. Don't they have
anything better to do, like look for
potential hijackers? This was poor
judgment and the airline should apologize
for sticking their nose where it doesn't
belong. Have you ever seen the
ridiculous uniforms worn by most pilots and flight attendants? They are the last
people on earth who should be telling people what to wear.
Above found at http://www.waynebesen.com/blog/2005_10_02_archive.html
Page 75 of 111
The slope of the straight line joining the points (x1, y1) and (x2, y2) is:
h = (y2 – y1)/(x2 – x1). Knowing h we can determine the intercept.
c = (y2 – hx2) since the line passes through (x2, y2).
It is also given by
c = (y1 – hx1) since the line passes through (x1, y1).
Conversely, if h is known, the future value y2 for a future value x2 can
be predicted. y2 = y1 + h(x2- x1).
These simple algebraic relations are very useful for our analysis.
The straight line connecting 1971 and 1973 data therefore has the
equation y = 0.5548x – 0.00493 = 0.555(x – 0.008894) where x and y
are in billions. The “breakeven revenue was x0 = -c/h = $8.89 million
when Southwest only had 3 aircrafts and operated between 3 cities
in Texas: Dallas, San Antonio, and Houston. The first slope h = 0.555
is very high and 55.5% of additional revenues, beyond breakeven,
were being converted into profits.
§ 14. Appendix 5
Profits-Revenues for Early years The nonlinear Power law model
Year Revenues, x
$, millions
Profits, y
$, millions
Costs (x –y)
$, millions
Comments
1971 2.129 -3.753 5.882 Data from
1972 5.994 -1.591 7.585 1973, 1975
1973 9.209 0.175 9.034 and 1978
1974 14.852 2.14 12.712 Annual
1975 22.828 3.40 19.428 Reports
1976 30.92 4.939 25.981
1977 49.047 7.545 41.502
1978 81.065 17.004 64.061
Operations began on June 18, 1971. The first full year of operations was 1972.
This first year data indicates a loss with a small profit in 1973.
Page 76 of 111
Pictorial depiction of the distribution of expenses for Southwest
Airlines from their 1982 Annual Report. Fuel & Oil (36.3%) and
Employee Salaries and Benefits (27.7%) are the top two items.
Insurance and Taxes (the focus of much political debate) only
accounts for 2.1% of total expenses.
In this section we will first summarize certain operational features and
how Southwest Airlines has grown in the early years and then consider
the profits-revenues data.
For 1973, the first full year of operations with a small profits, the
following data was obtained from the Annual Report. There were 10,619
flights. Still there are a number of interesting issues that affect the
number of flights as noted in the 197s report. The company operated
three aircrafts in its first quarter of operation (in 1971) and four aircrafts
in its second quarter (in 1971). The fourth aircraft was disposed off in
May 1972. Saturday operations were discontinued until November 1972.
According to the 1980 annual report, the fleet size (made up of Boeing
737-200’s) increased from 6 in 1976, to 10 in 1977, to 13 in 1978 to 18
Page 77 of 111
in 1979 and to 23 in 1980 and aircraft utilization increased from 9 hours
and 22 mins per day in 1976 to a maximum on 11 hours 37 mins in
1979. According to the 1984 annual report, the Southwest route system
covered 24 cities and 25 airports in 11 states. The trips operated in 1984
was 200,124. Aircraft utilization varied from 11 hours and 16 mins per
day to 11 hours and 11 hours and 46 mins between 1980 and 1984. They
had 48 owned aircraft at year end, with three new and one used aircraft
being added to the fleet during the year. According to the 1986 Annual
Report, the airline was operating a fleet of 46 Boeing 737-200’s and 17
Boeing 737-300’s. Nine more Boeing 737-300’s were to be delivered in
1987. The trips operated were 262,082 in 1982 and 230,227 in 1981.
This means about 4000 trips per aircraft per year or about 10 to 11 trips
per aircraft per day – an amazingly high number of trips for one aircraft
in a single day!
From the 1984 Annual Report
Now you have to believe they offer Exemplary Customer Service!
Page 78 of 111
A careful examination of the overall growth in both profits and revenues, since
Southwest airlines began its operations 41 years ago, almost to the date, on June
18, 1971, reveals a clear non1inear trend, especially if we consider the data for the
period 1971-1992. This is indicated by the graph prepared in Figure 26 below.
Figure 26: A very clear nonlinear growth of profits with increasing
revenues for Southwest Airlines for the period 1971-1992.
Profits increase with increasing revenues but at a decreasing rate as revenues
increase. In other words, the slope of the profits-revenues graph is decreasing
continuously. If revenues increase by an amount ∆x (say by $10 million), profits
increase by an amount ∆y. What is the relation between ∆x and ∆y? Can we
predict the increase in profits? This is the whole point of discovering the
fundamental mathematical laws governing the profits-revenues growth. If the ratio
∆y/∆x = h, a constant, profits will always grow by exactly the same fixed amount if
revenues increase by a fixed amount. The law relating profits and revenues is y =
hx + c and the slope h = dy/dx = ∆y/∆x is a constant. However, this linear behavior
is clearly not being observed when we consider the data for the entire 1971-1992
-20
0
20
40
60
80
100
120
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Revenues, x [$, millions]
Pro
fits
, y [
$,
mil
lio
ns
]
Page 79 of 111
period. Hence, we must consider the possibility of fitting the data with a nonlinear
upward rising curve.
Figure 27: The profits-revenues data for the period 1971-1992
can be described using the power-law model y = mxn + c. First
we fix the power law index n = 0.66 ≈ 2/3. With this choice of n, it is easily shown
that the curves for m = 0.7 (lower curve, blue) and m = 0.87 (upper curve, red)
bracket the entire data set. The nonzero value of c = - 5 provides the best match
for the reported loss in 1971 and 1972. The value of n ≈ 2/3 is picked here since
such a value of n is commonly observed in many, well understood, physical process
where we observe deviations from the linear law; see text. The four year with low
profits are obviously exceptional cases.
The simplest such law is the power law, y = mxn + c, which was discussed earlier.
The nonzero constant c is included here since at very low revenue levels,
Southwest did report a loss, for both the partial year of operation in 1971 and the
first full year of operation in 1972. A profit was reported only in 1973 (even first
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quarter of 1973 was a loss). Choosing n = 0.66 ≈ 2/3 as the power law index, it is
readily shown that the entire data set is bracketed between the curves with m = 0.7
and 0.87. The nonzero constant c = - 5 and this value is the best choice to match
with the reported loss in 1972.
Why n ≈ 2/3? This value of n is commonly observed in many physical processes
where we see deviations from a linear law. The most common example of a liner
law is Ohm’s law for an electrical conductor, V = RI, or Newton’s law of viscosity.
If the voltage V applied to a conductor increases, the current I flowing through the
circuit increases proportionally. The constant of proportionality is the electrical
resistance R. This is mathematically the same as y = hx + c with c = 0. However,
not all electrical circuits are linear. Some also show nonlinear, or non-Ohmic,
behavior. The power law (or even the power exponential law) is a simple model for
such nonlinear electrical circuit.
When we stir a fluid like motor oil, or water, the viscous resistance we feel does
not increase as the stirring rate increases. Such a fluid is called a Newtonian fluid
and its behavior is described by the law y = μx where x is the shearing (stirring)
rate and y the shear stress (proportional to viscous resistance). The constant of
proportionality is called the viscosity μ. It is this law that is used to fix the
viscosity values of most common liquids like motor oil, machine oil, etc.
But there are more complex fluids, like paint, honey, molasses, etc. which show a
varying viscous resistance as the rate of shearing is increased. Fluids with small
dispersed particles (what is called two-phase mixtures, or dough or pancake mix
with added ingredients) are other examples. Fluids whose viscosity varies with the
shearing rate lead to what is known as shear thinning or shear thickening behavior.
Stirring a paint makes it seem less viscous and this actually allows us to paint a
wall. If we stop the brushing (i.e., shearing of the paint), the paint stops flowing
and stays on the wall! Some other fluids show the opposite behavior and become
very stiff if we stir them. The more vigorously it is stirred, the stiffer it gets.
Other examples are processes such as “coarsening” of particles in materials
science. If a system is made of very small microscopic particles, it also has a lot of
“energy” associated with the surface of the particles. Hence, given the opportunity,
the system will try to become “coarse” (to minimize its total energy and thus
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become more stable), with the bigger particles actually trying to absorb the smaller
particles. Coarsening rates often follows a fractional nonlinear law, with n ≈ 1/3.
A company like Southwest and its operations may be thought of in a similar way.
Although the company has a “keep it simple” philosophy, as fleet sizes grow, as
the number of routes increase, and as the number of cities covered increase, the
operations become more complex. For example, fuel costs (one of the two top cost
items) vary nonlinearly depending on the distances flown by the aircraft. Short
haul flights are different from longer hauls. Other processes similar to those just
described interact in complex ways. Various subunits of the organization (like the
many microscopically small particles in a physical system) interact with each other
in unpredictable ways. This increases the operational “resistance”, similar to that
seen in a shear thickening or shear thinning fluid, or as in the coarsening model.
These analogies from the physical and engineering sciences can be quite useful and
it is obvious that the choice of n ≈ 2/3 is eminently justified. Social systems, with a
complex organizational structure and hierarchy can be thought of in a similar way
– like many “particles” that interact with each other. How this interaction occurs
determines the value of the index n. Simple fractions such as ½, ⅓, ⅔, 3/2, or
whole numbers like 2 (quadratic or parabolic law) are often encountered.
Of course, one could also use n = 0.5, or 0.8 or 0.85 and try “to fit” the data. It will
soon become obvious that not all values of “n” are appropriate to describe the
overall trend. If we consider the data for a very different company (say Apple,
Google, or Facebook, see Refs. [1-3] cited in bibliography), a different value of “n
“might apply, because of its unique “culture” and “management philosophy”.
Costs have been going up for a long time now, ever since operations began in 1971
and this is reason for the changing slopes [mostly decreasing implying increasing
variable costs, mathematically, h = 1 – (b/p) where b is unit variable cost and p the
unit price in the simple breakeven analysis] and increasing values of the intercept
made on the revenues-axis (implying higher fixed costs). The nonlinearity we saw
for the period 1971-1992 is further evidence of the “rising costs”. Composite plots
for the entire 40+ years of operation are presented next to call attention to this
same message.
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Figure 28: The profits-revenues graph for Southwest Airlines for
all 41 years, 1971-2011, for which data is available. 1971 was a
partial year of operation. Every straight line on this graph, with a decreased slope,
implies lower rate of conversion of revenues into profits. The dashed red line with
the steepest slope highlights the initial rate of conversion of revenues to profits (y
= hx + c = 0.555x – 0.00493, i.e., 55% of revenues beyond breakeven was
converted into profits in this period). This portion of the graph is being presented
using an expanded scale, separately as Figure 29. The airline’s operations then
“settled” to a lower rate of profits generation (y = 0.142x – 0.16), indicated by the
reduced slope of the blue line (only about 14% of the revenues beyond breakeven
are now being converted into profits). The dashed blue line, joining the 2007 (x, y)
pair to the origin, has an even lower slope h = 0.065 and only about 6.5% of the
revenues appeared as profits. The current trend was then established (negative
slope h < 0, c > 0) with profits decreasing even as revenues have increased, i.e., a
Type III behavior, with the attendant appearance of a maximum point. Southwest
Airlines is now actually operating past this maximum point. (A smooth curve, with
a maximum point, can be envisioned on this graph, see Figure 10).
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Figure 29: The profits-revenues graph for the first four years,
1971, 1972, 1973, and 1974 using a “zoomed in” scale. Notice
how the first three data points line almost perfectly on the straight
line y = 0.555x – 4.934 = 0.555 (x – 8.894). As revenues increase profits increase
(or what is the same, losses decrease). Hence, a profit is reported only when the
revenues exceeded the minimum of $8.894 million. Hence, no profits were reported
in 1972 when revenue was only $6 million. The revenues for 1973 increased to
$9.2 million and hence Southwest was able to report a small profit. The slope h =
0.555 means that 55.5% of the additional revenues (beyond breakeven) were being
converted into profits. However, this high rate of conversion of revenues to profits
could not be sustained, even in 1974. Although both revenues and profits increased
in 1974, the profits for 1974 were lower than the prediction based upon the
extrapolation of the 1971-1973 P-R line.
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Figure 30: The data plotted here for 1971-1978 were obtained
from the 1975 and 1978 Annual Reports. The solid blue line has
the equation y = 0.555x – 4.934 = 0.555 (x – 8.893) and joins the
1971 and 1973 data. The profit y = 0 when revenues x = x0 = -c/h = 4.934/0.555 =
$8.893 million. The slope h = 0.555 indicates that 55% of the revenues (beyond
this breakeven level) were being converted into profits between 1971 and 1973.
However, this period of high revenues to profits conversion did not last very long
and the slope of the profits-revenues graph has been decreasing ever since. The
highest (x, y) data here is for 1978. The dashed blue line joining the (x, y) pairs for
1974 and 1978 has the equation y = 0.224x – 1.194 = 0.224 (x – 5.32). This means
that only 22.4% of the revenues were being converted into profits in this period.
The slope h deduced here is greater than the slope of h = 0.142 (solid blue line) in
Figure 28 which indicates further decrease in the rate of revenues-profits
conversion. Since then a Type III behavior has been established as discussed in
detail. Costs have thus been increasing but have largely escaped attention because
financial data are seldom analyzed using methods such as those being described
here – very common in the physical and engineering sciences.
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Figure 31: The data re-plotted here is for the period 1971-1984
and was obtained from the various Annual Reports. The red
dashed line with the highest slope is for the initial period 1971-1973. The solid
blue line connecting 1974 and 1980 data has the equation y = 0.133x + 0.00017
and essentially passes through the origin. Thus, the period 1974 to 1980 represents
a much lower rate of conversion of revenues to profits. The slope h = 0.133
deduced here is consistent with h = 0.123 deduced earlier for 1992-2001[see
Figure 3, best-fit equation y = 0.123x – 0.141 = 0.123 (x – 1.154)]. However, the
data for 1980 to 1984 begins to deviate from the line with slope h = 0.133, as
indicated by the blue dotted line connecting the 1980 and 1984 data. (Hence, a
continuous power law curve, with n < 1, can also be used to describe the data
instead of these line segments of various slopes. However, care must be exercised
when making predictions using nonlinear laws, which can lead to either overly
“bearish” or “bullish” predictions. The linear law is more reliable when making
short term extrapolations when a “fixed” rate seems to be observed. Nonlinear law
should be used only after a thorough understanding of the system’s behavior.)
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The forty-one year profits-revenue
history for Southwest Airlines
The first twenty years (1971-1990)
Year Revenues, x
$, billions
Profits, y
$, billions
Comments
1971 0.0021 -0.0037
1972 0.006 -0.0016
1973 0.0092 0.0002 First profit
1974 0.0149 0.0021
1975 0.0228 0.0034
1976 0.0309 0.0049
1977 0.0490 0.0075
1978 0.0811 0.0170
1979 0.1361 0.0167
1980 0.2130 0.0284
1981 0.2704 0.0342
1982 0.3312 0.0340
1983 0.4482 0.0409
1984 0.5359 0.0497
1985 0.6797 0.0473
1986 0.7688 0.0500
1987 0.7783 0.0202
1988 0.8604 0.0580
1989 0.9736 0.0745
1990 1.1868 0.0471 $1 billion Revenues
1991 1.3793 0.0370
1991 1.3136 0.0269
The data for 1992-2011 (and first quarter 2012) have been presented in Table
1 of the main text. Most of the data was obtained from the annual reports.
Sometimes discrepancies were noticed, especially with revenues between
various reports. Both values are included in such instances.
The following values were obtained from the ten-year summary in the 1993
Annual Report and show slight discrepancies with earlier reports.
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From 1993 Annual Report: Ten-year Summary
Year
Revenues, x
$, millions
Profits, y
$, millions
Comments
1984 519.106 49.724
1985 656.689 47.278
1986 742.287 50.035
1987 751.649 20.155
1988 828.343 57.952
1989 973.568 74.505
1990 1144.421 50.605 $1 billion Revenues
1991 1267.897 33.148
1991 519.106 49.724
From the 1974 Annual Report
SWA began its operations between just three Texas cities.
Service to Rio Grande Valley was added on February 11, 1975.
Now Southwest flies to 73 cities in 38 states.
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What a difference a decade makes (see page 77)
From the 1974 Annual Report Those were the days!!!
Somewhere in the skies over Texas With lots of LUV in the air!
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§ 15. Now, a word about AirTran!
Since the AirTran acquisition will play a major role in the future of Southwest
Airlines, let’s take a quick look at the AirTran 10-year profits-revenues data. The
results are presented here without much discussion, with the x-y graphs telling the
story for us. First, consider the following data from the 2010 Annual Report.
Year Revenues, x
$, millions
Profits, y
$, millions
Costs (x – y)
$, millions
2010 2619.712 38.543 2581.169
2009 2341.442 134.662 2206.78
2008 2552.478 -266.334 2818.812
2007 2309.983 52.683 2257.3
Figure 32: AirTran profits-revenues graph for 2007-2010 showing an interesting
criss-crossing Type III behavior (or a back and forth Air Tran “swooshing”). The
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horizontal red line is the x-axis. One expects profits to increase with increasing
revenues. This is the normal Type I behavior (positive intercept on revenues-axis)
or Type II behavior (positive intercept on the profits-axis) But, we also see the
opposite behavior with several companies, at least in some situations. If profits
decrease with increasing revenues, or increase with decreasing revenues (yes, it is
possible!), we have Type III behavior. This is what we see here with Air Tran.
We see a very interesting pattern in the time-evolution of profits and revenues.
This can be described by three straight line segments all with a negative slope.
For 2007 and 2008: y = -1.315x + 3092
For 2008 and 2009: y = -1.315x + 3092
For 2009 and 2010: y = -0.345x + 943.43
The revenues increased between 2007 and 2008 but the profits decreased – so
much that AirTran actually reported a huge loss! This is what was classified as
Type III behavior and is indicated by the line with the negative slope (h < 0, c > 0).
Using the formulas given earlier, for the equation of a straight line connecting any
two points, we get y = -1.315x + 3092. The arrow shows the direction of time.
Next, revenues decreased between 2008 and 2009 but now AirTran reported a nice
profit. The dashed line, with the negative slope, indicates this and takes us to the
point above the red line. This is an interesting variant of the Type III behavior,
with profits increasing with decreasing revenues. (This is observed with other
companies as well, see Yahoo discussed in Refs. [1-3].) The equation for the new
Type III line is y = -1.315x + 3092.
Next, revenues increased between 2009 and 2010 but profits decreased once again
but a loss was avoided. This too is Type III behavior, similar to that observed
between 2007 and 2008. The equation of this line is y = -0.345x + 943.43.
If we consider the data for prior years, the ten-year period 2001-2010, and prepare
a x-y graph, no real pattern is detected. The data is given in the table below.
A successful company must report a profit. AirTran has reported a profit for 8 out
of 10 years, with a small loss in 2001 and a much bigger loss in 2008. But look at
the profits-revenues graph. It is totally chaotic. The behavior is like that of an
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engine that works erratically each time you turn it on. It does not run smoothly and
misfires and so the ride is jerky! This seems to be the way some companies are
behaving today. In other words, the “Profits Engine” is not running smoothly.
Year Revenues, x
$, millions
Profits, y
$, millions
Costs (x – y)
$, millions
2006 1892.083 14.714 1877.369
2005 1450.544 1.722 1448.822
2004 1041.422 12.255 1029.167
2004 918.04 100.517 817.523
2002 733.37 10.745 722.625
2001 665.164 -2.757 667.921
Figure 33: Erratic profits-revenue graph for AirTran for the period 2001-2010.
The situation is like that of the old steam engines before James Watt started
studying the reasons for their erratic behavior. During his investigations, on the old
Newcomen engine (which was made available to the young James Watt for study,
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by the University of Glasgow; Watt had just finished his education and was looking
things to do to launch his professional career and earn a living), Watt learned
about a remarkable property of steam called its latent heat (Professor Black, from
what we would now call their physics department, had also been studying the
properties of steam). Armed with this knowledge, Watt built a steam condenser
and essentially started recycling the exhaust steam back which dramatically
reduced the consumption of coal. He also reduced heat losses by improving the
insulation used. These “scientific” studies on the steam engine led to dramatic
improvements in efficiency (it was more than doubled) and led to what we now
call the Industrial Revolution.
Alas, many companies today seem to be behaving like the old Newcomen engine
before Watt. The AirTran profits-revenues graph seems like a perfect example.
Figure 34: The costs-revenues graph for AirTran prepared using the profits and
revenues data from the Annual Reports. The straight line connecting the 2003 and
2010 data has the equation C = 1.0364 R – 133.952. The descriptive symbols are
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used here for clarity instead of x and C = (x – y). But, they are retained in the axis
labels to show the relationship with the profits-revenues graph.
Figure 35: The AirTran costs-revenues graph with the best-fit
equation C = 1.10292 – 57.98. The linear regression coefficient r2 = 0.9834 is very
high, indicating a high degree of confidence in the important conclusion here that
the slope of the graph dC/dR > 1. In other words, costs are increasing faster than
revenues and so the company’s profits vary erratically.
The situation, however, changes dramatically if we consider the “costs” in the last
column, determined using the simple equation which is universally applicable to
all companies: Profits = Revenues - Costs or C = (x – y). We will also use the
descriptive notation R, P and C in this context to minimize confusion. The C-R
graph shows a remarkably linear relationship. As revenues increase, costs also
increase, even if we find that profits are varying erratically and there is no pattern.
(A nice linear P-R relation, as we saw in the early years with Southwest, Figure 3,
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is the ideal situation; see also P-R graphs
for Apple discussed in the articles cited.
Even if P-R graph is a scatter, the C-R
relation is often linear, as we see here.)
Notice that the slope of the straight line
joining the (C, R) data for 2003 and 2010
is greater than 1. The revenues increased
by ∆R = (2619.712 – 918.04) = 1701.672
but ∆C = 1763.646. Thus, ∆C > ∆R, as
we also saw earlier with the Southwest
data in the post 2007 period. This means
that for AirTran, costs have been
increasing faster than revenues.
To test if this is a “fluke”, arising from
the specific choice of the data points, a
linear regression analysis was performed
to determine the equation of the best-fit
line through these 10 points. This is given
below and is now included in the new
graph prepared in Figure 35.
C = 1.0292 – 57.98 with r2 = 0.9834
With the high value for the linear
regression coefficient, it is clear that the
statistically significant value of the slope
of the graph is also greater than 1. The
best-fit line does not go exactly through
the points we picked but is very close to
them. The slope dC/dR = 1.0292 > 1, as
we learned in our elementary calculus.
The following alternative view is
possible if we ignore the loss in 2008.
To use a sports analogy, this
is like keeping golf scores.
A player is allowed a certain
number of strokes, say y, to play
a hole. This is called par for the
hole. A good player may take
fewer strokes (a birdie, - 1, or
eagle - 2, or rarely a double
eagle -3). Or, the player can
take extra strokes (bogey, +1,
double bogey +2, triple bogey
+3, quadrupule bogey, +4).
Birdies and bogeys arrive
erratically and the scores kept
in this way go from positive
(too many bogeys) to negative
(lots of birdies).
But, the total cumulative
strokes y will always increases
as the total holes played x
increases. This law cannot be
violated. The law is y = hx + c.
We can test this law with
several world class golfers and
arrive at the values of h and c
for them. (Or use “hypothetical”
scores.) We are doing the same
here. Profits and losses are like
keeping scores above and below
par. But, costs will always go up
as revenues increase, regardless
of profits or losses. This too is
an inviolable law.
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Figure 36: An alternative view of the AirTran profits-revenues
graph, if we completely ignore the 2008 data.
Now the profits data can be described by two straight lines. The red line, with the
equation y = hx + c = 0.024x + 78.5 simply joins the 2003 and 2009 data. This is
Type II behavior (h > 0, c > 0). Profits increase with increasing revenues but
usually at a lower rate than a Type I behavior (h > 0, c < 0).
The blue line is the best-fit line through the remaining data points, with the
equation y = 0.0214x – 14.48 = 0.0214 (x – 676.2). This is clearly a Type I
behavior. The line makes a positive intercept of x = x0 = -c/h = 14.48/0.0214 =
$676.2 million on the revenues-axis. The data for 2001 confirms this cut off since
AirTran had a small loss with revenues of $665.164 million. Beyond this cut-off,
or breakeven, revenues, AirTran was able to report a small profits (much less than
with the Type II situation, the difference being the large positive intercept).
The situation we see here with AirTran is thus unusual. There is clear evidence of
costs increasing faster than revenues, if we include all the profits and losses data. If
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we analyze it more selectively, AirTran seems to exhibit both Type I and Type II
behavior in overlapping periods. The slope h is roughly the same for both --- the
Type I and Type II lines seem to be roughly parallel in our graph. But, the situation
including the huge loss, with dC/dR > 1, suggests a Type III behavior (see below).
Figure 37: The profits-revenues graph for AirTran for the sixteen year (16) period
1995-2010. Air Tran reported a profit for only 2 out of the 6 additional years (in
2000 and 1995). All data are taken from the Annual Reports. The graph of costs
versus revenues, where Costs = Revenues – Profits = (x – y), again shows a nice
linearity with very little scatter. However, the slope of the best-fit line is now
slightly less than one. The C-R equation is C = 0.997R+ 7.53 = kR + A, with a
linear regression coefficient r2 = 0.9885. A very small positive intercept (A = 7.53)
is made on the cost-axis (when R = 0, C = 7.53) and so the best-fit C-R line
essentially passes through the origin. Costs increase with increasing revenues.
Since the linear regression coefficient is so high, this is a statistically significant
result. Using the C-R equation, we can deduce the profits-revenues equation from
P = R – C which yields P = 0.003R – 7.53 = hR + c, with a very small positive
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slope h = 0.003. In other words, only 0.3% of the revenues can be converted into
profits. Or, 99.7% of revenues are being absorbed as “costs” and very little
appears as profits. The analogy with Einstein’s work function from the
photoelectric law (discussed in the references cited) is very telling in this case.
Also, intriguing is the possibility of a maximum point, as suggested by the dashed
curves (NOT derived from any mathematical calculations). The four points along
the curve past the maximum are for 2007-2010 which led to the criss-crossing
Type III behavior discussed earlier.
Figure 38: Possible appearance of a maximum point on the profits-revenues graph
for AirTran. Perhaps, this was precursor to the “acquisition” of Air Tran. A
company cannot continue to operate for long in the Type III mode with profits
decreasing as revenues increase, or vice versa. (In the case of General Motors,
where a similar Type III mode was observed over several years, it eventually led to
its historic bankruptcy filing in June 2009.) Notice how different conclusions are
permitted by the 10-year data and the 16-year data. The falling part of the curve
can be envisioned with the 10-year data but the rising part is only revealed if we
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consider the 16-year data. Type III behavior is always preceded by Type I and/or
Type II. A rise precedes every fall. Hence, the Type III behavior usually suggests
the existence of a maximum point on the profits-revenues graph.
In summary, the profits-revenues and costs-revenues pattern for AirTran, in the
ten-year period immediately preceding its acquisition by Southwest, reveals
interesting challenges ahead for the new company.
The following is from the 2011 Southwest Annual Report. The issuance of the
SOC means there is only ONE company now, legally speaking, even if the rest of
the integration is not complete.
Since the AirTran acquisition on May 2, 2011, we have made tremendous progress on integrating it into Southwest Airlines. Our efforts to begin optimizing the combined network have resulted in significant changes to AirTran’s route network. In 2012, AirTran is closing 15 cities that proved unsustainable with today’s dramatically higher fuel prices. We will serve 97 cities total between our separate networks based on our joint schedules currently published through November. On March 1, 2012, we received approval from the Federal Aviation Administration (FAA) for a Single Operating Certificate (SOC), marking a key milestone in the integration of the two airlines. AirTran can now begin transferring aircraft to be converted to the Southwest livery, and we can begin transitioning AirTran airport facilities to Southwest, beginning with Seattle in August 2012.
http://www.airtran.com/common/pdf/SpreadingLowFares_FactSheet.pdf
Page 99 of 111
http://www.getfilings.com/o0000931763-98-000779.html
Airline Southwest Air Tran
Stock Symbol LUV AAI
Founded June 18,1971 Oct 26, 1993
Headquarters Dallas, Texas Orlando, Florida
Employees 34,636 8,083
Fleet (Active, as of Sep 27, 2010) 547 138
For completeness, the following is a summary of the profits-revenues data for
AirTran since its operations began in 1993.
Year Revenues, x
$, millions
Profits, y
$, millions
Costs (x – y)
$, millions
2000 624.094 47.436 576.658
1999 523.468 -99.394 622.862
1998 439.307 -40.738 480.045
1997 211.456 -96.663 308.119
1996 219.636 -41.469 261.105
1995 367.757 67.763 299.994
1994 133.901 20.732 113.169
1993 5.811 -0.894 6.705
The data for 2001-2010 has been presented earlier. The merger with
Southwest was approved overwhelmingly by shareholders on March 23, 2011.
http://travel.usatoday.com/flights/post/2011/03/airtran-shareholders-ok-southwest-
merger/148989/1
On May 11, 1996, the (predecessor) company (ValuJet Airlines) tragically lost
Flight 592, from Miami to Atlanta. The plane crashed shortly after take-off after a
cabin fire. http://en.wikipedia.org/wiki/ValuJet_Flight_592 . There were no
survivors. The ensuing adverse media coverage (about low cost airlines), and the
intense FAA scrutiny that followed, led to a total shutdown of all operations on
June 17, 1996. FAA returned the company’s operating certificate on Aug 29. The
DOT issued a “show cause” order about the fitness to be an air carrier and gave its
Page 100 of 111
It can be shown that the sum of all the deviations from the profits predicted by the initial
Type I line equals $3.624 billion (see tables given here for 1993-2010). The sum of all the
revenues equals $19.05 billion. Multiply $19 B by 20%, for a quick calculation. We get
$3.8 B. Or, use the exact slope and convert 18.9% of $19.05 B. We get $3.61 B. Add that
small intercept. We get $3.624 billion – the sum of all the profits NOT made. Case
CLOSED for this type of an analysis!
final approval on Sep 26, 1996. Operations then resumed on Sep 30 with flights
between Atlanta and four other cities. These tragic events explain the “sudden”
loss of revenues in 1996 and also the losses. But, this seems to have lingered
(through 1999???). Profitability was finally achieved only in 2000.
Figure 39: Air Tran data reveals a Type I behavior between 1993 and 1995, as
indicated by the solid blue line. The profits-revenue equation y = 0.1897x – 1.996
= 0.1897 (x – 10.52). The reader (assuming we have one!!!) is strongly urged to
deduce this equation using the data presented in the tables here. Only then will its
impact be truly felt! Ideally, every point we see here should fall on this Type I line.
-400
-300
-200
-100
0
100
200
300
400
500
600
0 500 1000 1500 2000 2500 3000
Revenues, x [$, millions]
Pro
fits
, y [
$,
mil
lio
ns
]
Type I behavior 1993-1995
y = 0.189x – 1.996 = 0.1897 (x – 10.52)
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Nonetheless, the availability of this historical
data, since Air Tran began its operations in
1993, permits us to draw an interesting, but
cautionary, conclusion about the growth and
evolution of companies. As with Southwest
Airlines, between 1971 and 1974, we see
revenues increasing between 1993 and 1995.
Unlike Southwest, which reported a loss in
1972, its first full year of operation (it only
“turned the corner” in 1973 and reported a full
year of profits), Air Tran “turned the corner”
in its first full year of operations in 1994 and
reported a profit. Both profits and revenues
increased in 1995. This was followed by four
continuous years of losses with profits
returning only in 2000. But the profits were
lower: only $47.44 million versus $116.4
million obtained by extrapolation!
If Air Tran had continued to follow this initial profits-revenues line, in 2003, when
it reported a profit, it would have reported $172 million instead of $100 million.
This shows that something went seriously wrong (was it the accident in May
1996?) with the initial very successful plans and “costs” started increasing after
1995 and went totally out of control.
And, instead of reporting a historically high profit of about $436 million in 2007,
Air Tran reported a profit of only $52.7 million. And, instead of a profit of $482
million in 2008, it reported its highest loss of about $266 million.
The potential for high profits, even in the airlines business, stares us in the
face here! This can also be appreciated if we consider how profits evolve with
increasing revenues during a single year. The following tables summarize the data
(from the various annual reports) for 3 month, 6 month, 9 month, and 12 month
periods for1995, 2003, 2009 and 2010. Comparing the cumulative profits and
revenues for 1995 and 2003 and also 1995 and 2010 it is quite obvious that
From 1993-2010
Sum Total of
Revenues
$19,049.67 million
$19.05 billion
Profits
($6.477) millions
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1. The fixed costs have been rising from 1995 to 2003 to 2010, as indicated by
the increasingly more positive values of the “breakeven” revenue, given by
the intercept made on the revenues-axis.
2. The rate at which additional revenues (beyond the breakeven) are being
converted into profits has also decreased. This is indicated by the decreasing
values of the slope h of the graphs. It was 19% in 1995 (h = 0.1911),
decreased to 13.8% in 2003 (h = 0.1387) and had reduced to just 2.5% in
2010 (h = 0.0251).
Now all of this is water down the bridge!
Yes, the potential for high profits, even in the airlines business, stares us in the
right in the face here! Just imagine converting about 18% or
20%, okay let’s do just 10%, of additional revenues
beyond breakeven into profits! Air Tran did it in 1994 and 1995. It can be done
again in 2014 and 2015. Wow! Let’s do it with a Swoosh!
Air Tran Quarterly data: Profits-revenues growth during a single year
Quarter Revenues,
quarterly
$, millions
Profits,
quarterly
$, millions
Cum. Rev,
x
$, millions
Cum Profit,
y
$, millions
P-R equation
For growth
in the year
1Q2010 605.141 -12.025 605.141 -12.025 h = 0.0251
2Q2010 700.557 12.380 1305.698 0.355 c = -27.22
3Q2010 667.934 36.263 1973.632 36.618 y = hx + c
4Q2010 645.540 1.925 2619.172 38.543 x0 = - c/h
= 1084.1 All slopes h obtained from the lowest and highest (x, y) pairs.
1Q2009 541.955 28.707 541.955 28.707 h = 0.0589
2Q2009 603.653 78.438 1145.608 107.145 c = -3.204
3Q2009 597.402 10.426 1743.01 117.571 y = hx + c
4Q2009 598.432 17.091 2341.442 134.662 x0 = - c/h
= 54.41 Graphs were prepared in all cases, even if not included here.
1Q2003 208.002 2.036 208.002 2.036 h = 0.1387
2Q2003 233.901 57.191 441.903 59.227 c = -26.81
3Q2003 237.311 19.613 679.214 78.84 y = hx + c
4Q2003 238.826 21.677 918.04 100.517 x0 = - c/h
= 193.32
Page 103 of 111
Air Tran Quarterly data: Profits-revenues growth during a single year
Quarter Revenues,
quarterly
$, millions
Profits,
quarterly
$, millions
Cum. Rev,
x
$, millions
Cum Profit,
y
$, millions
P-R equation
For growth
in the year
1Q1995 60.747 9.071 60.747 9.071 h = 0.1912
2Q1995 86.913 16.860 147.66 25.931 c = -2.542
3Q1995 109.296 22.661 256.956 48.592 y = hx + c
4Q1995 110.801 19.171 367.757 67.763 x0 = - c/h
= 13.298 Slope h from the lowest and highest (x, y) pairs
Figure 40: Comparison of the evolution of profits and revenues in a single year at
three-month, six-month, nine-month, and 12-month intervals, using the quarterly
data. The revenues for just the first quarter of 2010 was almost double the revenue
for the entire year in 1995. Yet, profits were lower in 2010. Firstly, notice the large
intercept made on the revenues axis in 2010 compared to 1995. This means (using
-100
0
100
200
300
400
500
600
700
0 500 1000 1500 2000 2500 3000 3500
Cumulative Revenues, x [$, millions]
Growth during a single year
Cu
mu
lati
ve
Pro
fits
, y [
$, m
illi
on
s]
Gro
wth
du
rin
g a
sin
gle
ye
ar y = 0.1912x – 2.54 = 0.1912 (x – 13.3)
x0 = 13.3 in 1995 and h = 0.19
y = 0.025x – 27.2 = 0.025 (x – 1084) x0 = 1084 in 2010 (higher fixed cost) and h = 0.025 (lower slope means
higher unit variable cost)
Page 104 of 111
the breakeven analysis for profitability of a company making and selling N units of
a single product) that the fixed costs have gone up between 1995 and 2010.
Secondly, the high slope for 1995 is higher. Nearly 20% of the revenues (beyond
breakeven value) were being converted into profits in 1995 but only about 2.5%
was being converted into profits in 2010. According to the breakeven model the
slope h = 1 – (b/p), see §5 in main text (page 22). Hence, the unit variable cost b
has increased between 1995 and 2010, or more correctly, the ratio b/p, where p is
the unit price, has increased. It is to be hoped that Air Tran and Southwest can still
benefit from these findings! 1995 extrapolation equals $498 M profits in 2010.
Figure 41: A very unique Air Tran Type III profits-revenues graph, which lies
entirely in the fourth quadrant, with losses reported for every single quarter during
the year, in 2008. The cumulative quarterly loss was -$273.83 million (slight
discrepancy with the annual value given, -$266.33 million). The solid line joins the
-350
-300
-250
-200
-150
-100
-50
0
50
0 500 1000 1500 2000 2500 3000
Cumulative Revenues, x [$, millions]
Growth during a single year
Cu
mu
lati
ve
Pro
fits
, y [
$, m
illi
on
s]
Gro
wth
du
rin
g a
sin
gle
year
y =-0.122x + 38.07 Solid line y = -0.178x + 182.01 Dashed line
2008 Evolution of revenues-losses
Page 105 of 111
cumulative values for first and fourth quarters but the Type III dashed line begs
attention. This joins the second quarter to the fourth quarter. The general trend is
what is important here, not the exact numerical values. Type III behavior we see
here seems to be the precursor for either a merger (if someone is interested) or a
bankruptcy filing (when no suitors are available, as with General Motors. When
GM publicly announced its willingness to sell off some of its money losing
foundries, back in 1990s, which produce many critical automotive castings there
were no buyers! The latter is based on recollection of published reports during that
time.)This should be compared to the graph for 1995 when Air Tran was able to
report profits with very small revenues. The annual revenue for 1995 was only
$367.757 million, between 50% to 60% of the revenue for any one quarter in 2008.
Still, Air Tran could not report a profit in 2008.
Figure 42: The profitable year 1995, with Type I behavior revealed by the solid
upward sloping line, is compared with the Type III behavior observed in 1996,
-400
-300
-200
-100
0
100
200
0 500 1000 1500 2000 2500 3000
1995
2008
1997
1996
Cu
mu
lati
ve
Pro
fits
, y [
$, m
illi
on
s]
Gro
wth
du
rin
g a
sin
gle
year
Cumulative Revenues, x [$, millions]
Growth during a single year
Page 106 of 111
1997, and 2008 with annual losses. Even with greatly increased revenues, as
revealed by the fact that the data for 1995-1997 are crowded together near the
origin, Air Tran was unable to report a profit in 2008. (The Type III line for 1997
is an interesting “financial example” of a line with a negative slope which makes a
negative intercept on BOTH the profits and the revenues axes. Most Type III lines
make a positive intercept on both the axes. ) Instead of profits indicated by the
extrapolation of the Type I line for 1995, it reported its highest losses, revealed by
the two Type III lines for 2008. In other words, the quadrant four graph is to be
wholly avoided when we consider the cumulative data (from quarterly reports) in a
single year. It may be the precursor to either bankruptcy filing (if no one want to
take over the company) or a merger with a willing suitor.
Air Tran Quarterly data: Profits-revenues growth during a single year
Quarter Revenues,
quarterly
$, millions
Profits,
quarterly
$, millions
Cum. Rev,
x
$, millions
Cum Profit,
y
$, millions
P-R equation
For growth
in the year
1Q2008 589.115 -34.813 596.391 -34.813 h = -0.122
2Q2008 589.115 -13.538 1289.771 -48.351 c = 38.97
3Q2008 589.115 -107.087 1963.063 -155.438 y = hx + c
4Q2008 589.115 -118.391 2552.178 -273.829 x0 = - c/h
= 311.5 Slope h from the lowest and highest (x, y) pairs
In this case the 2Q and 4Q data beg attention, h = -0.179
How could I say all this without that a x-y graph?
If anyone is reading this, please go back now and enjoy the story
told by each of these graphs!
Page 107 of 111
If anyone is reading this, please go back now and enjoy the story
told by each of these graphs!
Illustration of DC-9 ValuJet, Flight 592. The plane was observed crashing.
It crashed on a lovely Saturday afternoon, the day before Mother’s day.
The improper placement and loading of canisters with chemicals (that are used to produce Oxygen
for the Emergency System), in the cargo compartment below the passenger cabin,
http://en.wikipedia.org/wiki/ValuJet_Flight_592
seems to have contributed to the spark that led to the fire and the crash.
Sadly, this is, perhaps, the clearest example of incompetent and/or ignorant employees,
working without proper training or supervision, doing what they think is best! They just
did not know how stupid it was to do what they did! The loss of the space shuttle
Challenger, on January 28 1986, seconds after launch, is another example of a similar
tragedy that could have been avoided.
It is really the story of what we all do each day, as employees, to
make the company we work for profitable, to enrich our own lives,
and to enrich the communities we live in.
Lurking behind each of our actions is that demon called the “costs”.
And before we know it h > 0 turns into h < 0
And the day of reckoning arrives!
It happened between 1993 and 2011!
http://psiresearcher.files.wordpress.com/2011/09/flighteverglades-plane-crash-
2_05320299.jpg?w=640
http://www.theatlantic.com/magazine/archive/1998/03/the-lessons-of-valujet-
592/6534/ Time to get over that fateful 1996 crash!
Page 108 of 111
§ 16. Bibliography
Related Internet articles posted at this website
Since the Facebook IPO on May 18, 2012
1. http://www.scribd.com/doc/95906902/Simple-Mathematical-Laws-Govern-
Corporate-Financial-Behavior-A-Brief-Compilation-of-Profits-Revenues-
Data Current article with all others above cited for completeness, Published
June 4, 2012 with several revisions incorporating more examples.
2. http://www.scribd.com/doc/94647467/Three-Types-of-Companies-From-
Quantum-Physics-to-Economics Basic discussion of three types of
companies, Published May 24, 2012. Examples of Google, Facebook,
ExxonMobil, Best Buy, Ford, Universal Insurance Holdings
3. http://www.scribd.com/doc/96228131/The-Perfect-Apple-How-it-can-be-
destroyed Detailed discussion of Apple Inc. data. Published June 7, 2012.
4. http://www.scribd.com/doc/95140101/Ford-Motor-Company-Data-Reveals-
Mount-Profit Ford Motor Company graph illustrating pronounced
maximum point, Published May 29, 2012.
5. http://www.scribd.com/doc/95329905/Planck-s-Blackbody-Radiation-Law-
Rederived-for-more-General-Case Generalization of Planck’s law, Published
May 30, 2012.
6. http://www.scribd.com/doc/94325593/The-Future-of-Facebook-I Facebook
and Google data are compared here. Published May 21, 2012.
7. http://www.scribd.com/doc/94103265/The-FaceBook-Future Published
May 19, 2012 (the day after IPO launch on Friday May 18, 2012).
8. http://www.scribd.com/doc/95728457/What-is-Entropy Discussion of the
meaning of entropy (using example given by Boltzmann in 1877, later also
used by Planck to develop quantum physics in 1900). The example here
shows the concepts of entropy S and energy U (and the derivative T =
dU/dS) can be extended beyond physics with energy = money, or any
property of interest. Published June 3, 2012.
9. The Future of Southwest Airlines, Completed June 14, 2012 (to be
published).
Page 109 of 111
About the author
V. Laxmanan, Sc. D.
The author obtained his Bachelor’s degree (B. E.) in Mechanical Engineering from
the University of Poona and his Master’s degree (M. E.), also in Mechanical
Engineering, from the Indian Institute of Science, Bangalore, followed by a
Master’s (S. M.) and Doctoral (Sc. D.) degrees in Materials Engineering from the
Massachusetts Institute of Technology, Cambridge, MA, USA. He then spent his
entire professional career at leading US research institutions (MIT, Allied
Chemical Corporate R & D, now part of Honeywell, NASA, Case Western Reserve
University (CWRU), and General Motors Research and Development Center in
Warren, MI). He holds four patents in materials processing, has co-authored two
books and published several scientific papers in leading peer-reviewed
international journals. His expertise includes developing simple mathematical
models to explain the behavior of complex systems.
While at NASA and CWRU, he was responsible for developing material processing
experiments to be performed aboard the space shuttle and developed a simple
mathematical model to explain the growth Christmas-tree, or snowflake, like
structures (called dendrites) widely observed in many types of liquid-to-solid phase
transformations (e.g., freezing of all commercial metals and alloys, freezing of
water, and, yes, production of snowflakes!). This led to a simple model to explain
the growth of dendritic structures in both the ground-based experiments and in the
space shuttle experiments.
More recently, he has been interested in the analysis of the large volumes of data
from financial and economic systems and has developed what may be called the
Quantum Business Model (QBM). This extends (to financial and economic
systems) the mathematical arguments used by Max Planck to develop quantum
physics using the analogy Energy = Money, i.e., energy in physics is like money in
economics. Einstein applied Planck’s ideas to describe the photoelectric effect (by
treating light as being composed of particles called photons, each with the fixed
quantum of energy conceived by Planck). The mathematical law deduced by
Planck, referred to here as the generalized power-exponential law, might actually
Page 110 of 111
have many applications far beyond blackbody radiation studies where it was first
conceived.
Einstein’s photoelectric law is a simple linear law, as we see here, and was
deduced from Planck’s non-linear law for describing blackbody radiation. It
appears that financial and economic systems can be modeled using a similar
approach. Finance, business, economics and management sciences now essentially
seem to operate like astronomy and physics before the advent of Kepler and
Newton.
Cover page of AirTran 2000 Annual Report
Acknowledgements
With sincere thanks to the many Internet sources that have been
used to compile this document – as evident by all the corporate logos
and various photographs used here to make the presentation more
interesting. All of them have cited and are liberally and profusely
acknowledged.
Page 111 of 111
What are the Odds?
“Southwest Airlines is America’s largest domestic airline, as measured by
originating domestic passengers boarded (based on third quarter 2011 data from
the U.S. Department of Transportation (DOT)). We remain one of the lowest cost
producers among major airlines with one of the world’s largest mainline fleets.”
From the 2011 Annual Report
Gary C Kelly
Chairman of the Board, President
Chief Executive Officer
In Annual Report after annual report, Southwest Airlines has
proudly emphasized its status as the low-cost leader in the industry.
What then are the chances of any of these
ideas and the discussion here about rising
costs getting accepted?
Only time will tell.
If the predictions here have any validity, we can expect Southwest
Airlines to go through a period of reporting losses - with its acquisition
of AirTran and rapidly changing nature of its complex
operations being held at fault. But there is a different reason
and it lies buried in the story here. Costs have been rising
for a long long time, as evident from the nonlinear curve
for 1971-1992. Even if losses are avoided, Southwest can greatly
improve its profitability by studying some of the ideas outlined here.
With lots of LUV in the air!