e136l7 Bond Pricing

8/13/2019 e136l7 Bond Pricing

1/21

Note: Read theappendage to this

document about thedifference between zero

rates and forward rates

e ore rev ew ng t e

lecture material about

forward rates This is

difficult and conceptually

complex material.

Chapter 4 from Hull

(c)2006-2013, Gary R. Evans. This material may not be used by others without permission of the author.

Exam resultsExam results

13

12

14

185 A 13

175 A 7

170 B+ 4

7

6

8

10A

B+

B

B

C+

C

165 B 4

160 B 4

150 C+ 4

140 C 2

120 C 2

Below ? 0

2 2

00

2

4

1

C

?


2/21

Where we are going in this lecture Review of interest rate formulas

Calculating "zero rates"

Calculating "forward rates"

Calculating "duration"

Justif in ield s reads and ield curves

Discussing yield arbitrage (intro)

What we are skipping from Hull chapter 4What we are skipping from Hull chapter 4

Section 4.7 Forward Rate Agreements

requires knowledge of the LIBOR/swap zero curve, which isn't

covered until chapter 7.

Section 4.9 Convexity

not very important and I don't want to throw too much at you all at

once.

On duration, I am using discrete examples compared to

' out there in the real world you are more likely to see discrete andthe two approaches are similar anyway.

also in chapter 5 I am using discrete rather than continuous upper

and lower limit calculations.


3/21

Types of Risk Embodied in the Yields of YBFAsTypes of Risk Embodied in the Yields of YBFAs

Default risk reflected in corporate and municipal YBFAs

Market (maturity) risk

due to capital gains and losses when interest rates fluctuate

the longer the maturity, the higher the risk

common to all YBFAs

Economic risk

ue to n at on or s m ar econom c or po t ca s oc s

the longer the maturity, the higher the risk(the more likely you

will be holding this asset during the "event"). Treasuries may experience "flight to safety" during political

shocks.

Yield SpreadsYield Spreads

YBFAs with long maturities have a higher probability of

gain or loss when interest rates fall or rise. They are also

subject to a higher level of economic risk.

Therefore, according to our notions of risk, longer term

YBFA maturities are seen as having higher risk, therefore

their yields have arisk premium.

Therefore, typically when we map the yields of a full

maturity range for a class of YBFA securities, like

Treasuries, from short-term to long-term, that mapping will

r se see next s e). These are calledyield spreads oryield curves and the

mapping is called theterm structure of interest rates.

Sometimes we see an atypicalflat or even invertedterm

structure of interest rates.


4/21

The yield curveThe yield curve today: March 2013today: March 2013

The configuration is normal.

but the s read between lon

and short is atypically narrow

because of QE2/QE3.

Note this configuration

compared to 2012 a carbon

copy, which shows the

stability of QE3.

Flat and Inverted YieldFlat and Inverted Yield Spreads: 1990Spreads: 199010.00

Jan 1990 yield curve flat

4.00

6.00

.

3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr

longposition

shortposition

This tends to happen during periods of high inflation or when

the FRS is aggressively raising short term rates to combat

inflation. Inverted (1980) at end of hyper-inflation period.

Presents an ideal spread arb situation.


5/21

Treasury YieldTreasury Yield Spreads: 2004Spreads: 2004--0505

6.00

This slide was from the 2005 lecture.

2.00

3.00

4.00

5.00

Mar-05

Jan-04

FRS raises

FFR targets

As can be seen, this Fed action

is influencing only short-term

rates.

0.00

1.00

1 mo 3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr

"Short on short term" means that you are betting that 3 mo prices will fall and yields will rise.

TheThe 20062006 yield spreadyield spread

6.00

March 28, 2006

4.774.754.744.724.724.734.50 4.59

2.00

3.00

4.00

5.00

A flat yield curve after the 2-year ... largely due to

overseas purchases of Treasury Securities (especially

China and OPEC favorin the much thinner lon end

loaded

to riseFRS raising

0.00

1.00

3 Mon 6 Mon 2 Yr 3 Yr 5 Yr 10 Yr 20 Yr 30 Yr

of the spectrum. This is tied very strongly to our

Merchandise Trade Deficit.

rates this end


6/21

Yield curve in March 2007Yield curve in March 2007

March 6, 2007: Yield curve is

inverted from 2 Year Note out.

Rates were slowly rising at this

time.

Yield Curve in March 2008Yield Curve in March 2008


7/21

The simple bond valuation formulaThe simple bond valuation formula

C Par n

r r

n

i 1 11where

MV = market valuepresently of the bond

n = number of years to maturity

C = cou on a ment ar times the cou on interest rater =present yield of this bond (market determined)

This formula assumes that there is only one interest paymentper year and that this bond was priced on the day after the

most recent interest payment was made.

The elementary bond formula (reduced form)The elementary bond formula (reduced form)

r n

11

1 1

r r n

1

whereMV: the present market value of the bondCR: cou on rate ori inal ield of the bond

Note: This formula cannot be used

to value an actual bond because it

assumes only one interest payment

r: current market rate on equivalent bondsn: number of remaining years to maturity

per year and that the bond is beingbought on the day of the coupon

payment. For actual bond pricing a

more complicated version of this is

shown at the end of this lecture.The original formula is a geometric series and

this is a reduced-form equation. To see its

justification and derivation, read the appendix in

the reading assigned for this lecture.


8/21

The formula when interest is paidThe formula when interest is paid

semisemi--annuallyannuallym

MV

r ri m

i

1 2 1 21

where

C = coupon interest payment (par X r/2)

r = present market yieldm = number of coupon payments remaining (years X 2)

TheThe final formula ask yield (final formula ask yield (ytmytm))

aCParC

MVapn

n

ai 1

1

This term represents an adjustment for accrued interest.

MV= market value, the quoted askprice of the bond,

C= the annual coupon payment, equal to the coupon rate times par,

r

m

r m1

ar=r = the prevailing annual market yield, expressed asask yieldoryield to maturity,

m = the number of coupon payments per year,

n = the number of remaining coupon payments,

p = the number of days in this coupon period (between 181-184, use 182 if unknown),

a = the number of days between the last coupon payment and the settlement day.


9/21

an examplean example

Purchase date: February 8, 2009 (146 days since last coupon payment, 36 to next),Next coupon date: March 15, 2009 (the first of 42 coupons remaining),

Redemption date: September 15, 2029 (for par and last coupon),

You are buying a 30-yr bond that was issued on September 15, 1999:

Coupon rate/amount: 8% yielding $4per coupon payment,Present market rate (askyield): 10%. 8022.0

182

146

p

a

8022.02

8

10.1

100

2

10.1

1

2

8099.05.20

42

18022.0

ii

MV

31.8321.3

10.1

100

05.1

1460.20

42

18022.0

i

iMV

ReducedReduced--form Version of the Complex Formulaform Version of the Complex Formula

aCr

Cn

11

11

Solving the same problem from the last slide (rounding error explains the difference):

pmr

ar

mr

mrm

apm

n

pa

3651

/

1

1

31.838022.0410.1

1100

05.1

05.0

05.111

4099.05.20

8022.0

42

MV


10/21

The continuous TThe continuous T--bond yield formulabond yield formula

P Ce eby i

i

n y n

21

2100

This formula above is the generalized continuous bond yield

formula shown in Hull (he only shows an example, below). This

formula is solved fory using an iterative technique. P is the price

of the bond, C is the semi-annual coupon payment (equal to the

coupon rate times 100 divided by 2, and n is the number of

3 3 3 103 98 390 5 1 0 1 5 2e e e ey y y y . . . .

coupon payments to e ma e ou e t e num er o years o

maturity). Compare this to the discrete bond formula.

The Market Value of a CMO or CDOThe Market Value of a CMO or CDO

MV PI ri

, 1 12360

A Collateralized Mortgage Obligation (called a CDO if made up of other

classes of debt) is a pass-through debt obligation where the principle and

interest payments (PI) made by mortgage holders is passed through to the

owner of the CMO.

A CMO differs from bonds in three respects: (1) The numerator is not a

known value and must be treated as a random variable, 2 the a ment is

monthly, and (3) there is no principal value at the end.

Because mortgages get paid off or refinanced, the nominal front-end value

of PIis considerably higher than at the tail.

Pieces of this can be sold off as Cash Pass-Thru Certificates and therefore

must be priced.


11/21

A very important question ...A very important question ...

Regarding the conventional bond formula or any similar formula ...

nnn rPar

r

C

r

C

r

C

r

C

r

CMV

111111

132

... is this piece, the third payment, considered by itself, being priced

properly, given that you are discounting it at a 20 year rate when the

yield on a 3-year asset is not the same as a 20-year asset.

Why is this relevant? Suppose you want to sell off this piece (atranche).

This issue is at the root of the importance of the difficult material

ahead, the derivation of thezero rate andforward rates.

Hull's "zero rate"Hull's "zero rate"Hull's "zero rate," more typically called "zero coupon rate," is the continuous

version of the traditional formula used to value a zero-coupon bond (like a US

Treasur STRIP . This is a lon -term bond that makes no interest a ment a s

no "coupon") but simply returns the par value at maturity.

This is the formula, where r is the current market

rate, and t is the time in years to maturity.

P

r t

100

1For example, a 30-year zero-coupon bond with22 years and three months remaining to

maturity with current yields of equivalents at 100

5.25% is worth: 1 0 0525 22 25. . .

Hull's continuous

rate equivalent is: P e ert 100 100 31090 0525 22 25. . .


12/21

Determining "Treasury Zero Rates"Determining "Treasury Zero Rates"

"Treasury Zero Rates," which are difficult to calculate (conceptually ...

canned software exists to do it for you), are used to give a correct value to

each individual payment of the cashflow stream. It consists of continuous

discounting to the present each payment from the known cashflow of

Treasury securities. Remember that a Treasury security is almost never

purchased at par and that yields on these securities are expressed as simple

interest compounded quarterly or semi-annually. Therefore to make the

conversion to continuous, the we go back to a conversion formula that we

saw in an earlier lecture:

r m r

mcd

ln 1

Note: This zero-rate conversion can be used for other types of securities as well.

Where we are going in the next fewWhere we are going in the next fewslidesslides

We are trying to define and calculate the rates for this zero rate

table. Compare this to Tables 4.3 and 4.4 in Hull. This step isnecessary for the calculation of forward rates, which are shown in

Table 4.5 in Hull.


13/21

First row in the table: 3-month TBill

The 3-mo TBill pays no coupon and is always sold at a discount

(97.50 in this example). Therefore the interest earned in one

quarter is 100-price (2.50 in this example). First we calculate the

simple quarterly interest:

r

P

Pq

4 100 4 2 50

97 5 10 256%

.

. .

e t en convert t s rate to cont nuous us ng t e convers on

formula:

r mr

c

q

ln ln

..1

4 4 1

010256

4 10127%

2nd row in the table: 62nd row in the table: 6--month TBillmonth TBill

This TBill also pays no coupon. Here we assume you paid

94.90 and because it has a maturity of 6 months it is paying

semi-annual interest:

r

P

Psa

2 100 2 510

94 90 10 748%

.

. .

which is converted to the continuous annual rate:

r mr

c

sa

ln ln

..1

2 2 1

010748

2 10 469%


14/21

3rd row in table: 13rd row in table: 1--year TBillyear TBill

This is easy. Here the conversion is direct from annual to

continuous. Assuming the TBill is worth for 90.00:

ra

100 90

90 1111%.

Once we get to TBonds, however, we have to take into

10536.1111.1ln1ln ac rr

-

payment equal to Par (100) times the initial coupon rate,

then pay a final payment equal to the Par value of thebond. For each six months we have to use an iterative

process where we take the semi-annual and annual rates

that we have already calculated and ...

4th row in the table: TBond with 1.5 years4th row in the table: TBond with 1.5 years

... discount the first coupon payment at the 6-month continuous rate, discount

the second coupon payment at the 1-year continuous rate, then solve for the

. .

In this example, we assume a coupon payment of $4 (8% annually) and the

current secondary market value of a 1.5 year bondof 96 (which implies that

the current yield is higher than 8% because this bond is at discount):

This is the value of the

remaining piece.

4 4 104 960 10469 0 5 0 10536 1 5e e e R

. . . .

0 949 0 90 26 241 5. . . e R

R1 5085196

15 10 681%.

ln( .

. .


15/21

5th row in the table:5th row in the table: TBondTBond with 2.0 yearswith 2.0 years

... discount the first coupon payment at the 6-month continuous rate, discountthe second coupon payment at the 1-year continuous rate, discount the third

coupon payment at the 1.5-year continuous rate, then solve for the 2 year

- - .

In this example, we assume a coupon payment of $6 (12% annually) and the

current secondary market value of a 2- year bondof 101.60 (which implies

that the current yield is lower than 12% because this bond is at premium):

60.101106666 25.110681.010536.05.010469.0 Reeee

9333.16667.178520.0900.0949.02

R

e

10808.0

2

80561.0ln2

R

... and later maturities... and later maturities

This same iterative process, which is called the "bootstrap

method," continues out (typically quarterly or semi-annually)

un e en o e ma ur y spec rum say years .

Clearly this would be rather easy to program in C++ or

equivalent.

For data, you simply use the coupon rates and current prices

for Treasuries of these various maturities quoted in The Wall

.

Note how this is different from the traditional bond formula ...

here each payment is discounted at the rate that is relevant

for its maturity, rather than at a single "market" discount rate.


16/21

... giving us the zero... giving us the zero--rate tablerate table

-... .

The implicit formula that we haveThe implicit formula that we havecreated ...created ...

i n

P Ce eb ii n

2

1

2100

Now each cashflow component of this bond is correctly valued, which is essential

if you plan to sell off pieces of this bond as cashflow certificates or even to

calculate the true effective ield that ou will be earnin if ou sell this bond

before maturity.

Note: You should realize that the Pb this bond should not and will not be different

than the equivalent in the original discrete formula ... it can't be, given the

technique we used. Each piece is priced differently than the original formula.


17/21

Review: What did we just do?Review: What did we just do?

In effect, we pretended that a ten-year bond is not a ten-year

on . nstea t s a co ect on o zero-coupon notes one or

each coupon payment and one much larger one for par) that are

each continuously discounted back to the present at the

appropriate interest rate for that maturity.

For example, the coupon payment that will be paid in two years is

discounted back to the present at the rate that is appropriate for 2-

year notes.

We determine that rate by taking the implied continuous marketrate for a 2-year note discounting the coupon payment back to the

present

A simple application of the zeroA simple application of the zero--couponcouponconcept (to motivate its acceptance)concept (to motivate its acceptance)

Many exotic but commonly used financial instruments, especially debt

obligations, like Collateralized Debt Obligations (CDOs ... often

mortgage pools) or Carry Trade paper (borrowing from one large lender

at a low rate to re-loan to many different smaller borrowers at a higher

rate), are often chopped up and sold off in pieces. These pieces are

commonly called "tranches."

Suppose you have a debt contract promising to pay you $100 per year

or t e next t ree years. uppose you ec e to se t e t r payment asa tranche to a third party? How would you price it?

If the 3-year zero rate is 5%, the tranche would be sold at $86.07:

P e 100 86 070 05 3. .


18/21

Step 2: Determining Forward RatesStep 2: Determining Forward Rates(warning: conceptually very difficult to grasp)(warning: conceptually very difficult to grasp)

As Hull points out in section 4.6 and 4.7, forward rates are used to price forward

rate agreements. To calculate forward rates, we have to have access to (or to

- - .

Here is an example: Suppose you have a debt instrument that pays you interest

for five years according to a zero coupon schedule (with zero rates) that goes out

for five years. [Note important to understand: This may be a traditional debt

instrument for which you have made a zero rate schedule as we did in the

previous section]. If you ask the question, "What is the interest yield that I will

earn in the 5th year alone?" you are asking "What is the 5-year forward rate?"

How does this differ from the five-year zero rate? There we are asking the

question, "What is my present interest yield on the coupon payment that will

made to me in the 5th year."

Stop and read the document entitled Justification of the technique used to

calculate forward rates from zero rates by example.

Calculating the Forward Rate fromCalculating the Forward Rate fromZero RatesZero Rates

Time Zero Rate Forward Rate

1 2.000

2 2.500 3.000

3 3.000 4.000

4 3.500 5.000

5 4.000 6.000

Forward Rates for 5 Year Investment

.

to calculate forward rates in this

table is easy:

RF RZ t RZ tt t t 1 1

For example:

RF4 350 4 3 00 3 5% . ( . )

Why so simple?

See also Hull Table

4.5


19/21

Why so simple??Why so simple??

The value of the investment over all four years, equal to

principal value compounded at the four year zero rate, will

equal the principal value compounded at the three year zero

rate compounded again at the 4th year forward rate (because

that is the rate earned for only that year):

100 1000 030 3 0 035 44e e e

FR. .

Taking the natural logs and solving for FR4 leaves us with

the equation on the previous page. Normally thedenominator is 1 as in this example, but we do allow for

multi-period forward rates.

Macaulay DurationMacaulay Duration"Duration is a weighted average of the times that interest payments and the final

return of principal are received. The weights are the amounts of the payments

discounted by the yield-to-maturity of the bond."* Duration is a common measure

t at a ows ana ysts to compare on s w t erent matur t es an coupons. e

discrete form of the equation assuming annual interest payments is:

D

t c y

P

t c y

c y

i i

t

i

n

i i

t

i

n

i

tn

i i

i

1 1

1

1 1

whereD is duration,y is

current yield, c is the cash

payment (coupon or coupon

plus redemption when t=n),

P is the price of the bond,

*Source: Subject: Bonds - Duration Measure,by Rich Carreiro, 1998, at ht tp://invest-

faq.com/articles/bonds-duration.html. This article provides a good introduction to duration. The

numerical example above and the example on the next page is from that source.

i 1

Note: Not in Hull

and n is the number ofyears maturity.


20/21

Calculating durationCalculating duration

Like most of these summation formulas, duration is easy to

calculate in an Excel workbook or even easier et in a C++ or

V Basic program. For example, consider a 5-year annual note

with a 7% coupon and a current yield of 5%. Duration would

be equal to:

D years

6 677 2 6 349 3 6 047 4 5 759 5 83838

6 6 77 6 3 49 6 0 47 5 759 83 838 4 41

. ( . ) ( . ) ( . ) ( . )

. . . . . .

Memo Note: When payments arecompounded semi-annually or quarterly,there is a simple conversion formula that

you can apply (see Hull p. 91):

DD

y ma 1

Calculating Duration with ExcelCalculating Duration with Excel

Maturity value: 100

Calculating Duration fo r a 5-year Annual Note The weight of each

a ment relativeoupon rate: .

Current yield: 0.05

Years to maturity: 5

Year Cash payment Present Value Numerator Weight

1 7 6.667 6.667 0.061

2 7 6.349 12.698 0.058

3 7 6.047 18.141 0.056

4 7 5.759 23.036 0.053

5 107 83.837 419.186 0 .772

Totals Note value (denom): 108.659 479.728 1.000

total note value. It

must sum to 1.0

This number has

no inherent

meaning ... it is to.

Refer to this slide when doing your homework (after exam?).

Also compare it to Table 4.6 in Hull (where he is using a

continuous rather than discrete formula).

be compared toanother bond.


21/21

How is duration used?How is duration used?When comparing the duration of two bonds, a bond with a higher duration

carries more riskand is likely to have a higher price volatility.

Why do we need to calculate duration when we know that a long-term bond

as more r s an vo at ty t an a s orter-term note on t we nee to

know only the time to maturity?

No, because the yield relative to

the coupon also affects risk and

volatility. For example, a 10-year

bond with a low coupon rate and a

low yield will have a higher

volatility and risk to cashflow than

a 10-year bond with a high coupon

and high yield because the latterreceives payment proportions at a

faster rate.

Image source: investopedia.com, section on advanced bonds, duration.

Hull's Duration FormulaHull's Duration Formula

Hull's duration formula (4.12), which is continuous

is clearly a similar formula to our own, which is more commonly

used:

D

t c e

P

i i

yt

i

n

i

1

D

t c y

P

i it

i

i

11

All resulting theory is the same, of course.

e136l7 Bond Pricing

Documents

Transcript of e136l7 Bond Pricing