Time Value of Money

Chapter 6

TIME VALUE OFMONEY

Alex Tajirian

Time Value of Money 6-2

© morevalue.com, 1997

1. OBJECTIVE

# Derive a valuation (pricing) equation based on cash flow (amount,timing, & risk).

# Time Value of Money analysis involves:

! What is $1 worth 10 years from today (Future Value)?

! What is $1 to be received in 10 years worth today (PresentValue)?

# Applications

! Loan amortization

! stated vs. effective interest charged

! rebate vs. low financing

! pricing of bonds (Chapter 7)

! pricing of stocks/firms (Chapter 7)

! What is the value of a particular division within a firm?

! How much value does a new project contribute to a firm?

# In this chapter we assume the following are given:

! cash flow: amount, timing, and risk as reflected in k

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2. TYPES OF VALUATION

L Based on investors' preferences and attitudes towards consumptionand risk.

! Demand & Supply analysis

L Based on "cash flow (CF)",

! CF: stream of promised future income

today = time of analysis

| | | | time

period

s

0 1 2 3

CF $100 200 -100 . . .

where periods can be hours, days, weeks, etc 7

Note. Positive CF means receiving $ (inflow), negative CFmeans paying $ (outflow)

Thus, given the CFs and how good the promise is, its risk, everyonewould agree on the value (price) of the income stream.

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FV1 ' Future Value of a CF 1 period from today' principal % interest payment' principal % (interest rate) × (principal)' $100 % (.1)($100)' $100 × (1 % .1) ' $110

(1)

FV2 ' [FV1](1% i) ' [100(1% i)](1% i) ' 100(1% i)2

' 100(1% i)2 ' 100(1.1)2 ' 100(1.21) ' $121

3. FUTURE VALUE (FV)

L Put $100 (CF) in a bank for one year at interest (i) = 10%

What is value of $100 one year from today; (FV1) ?

where, subscript 1 denotes # of periods in the future

Thus, the CF is compounded at rate "i".

L What is value of $100 two years from today; (FV2)?

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FVn ' CF (1% i)n ' CF [FVIFi, n] (2)

For i ' 10%, n ' 2, Y FVIF10, 2 ' 1.2100

In general for a single CF,

where,

i / re-investment rate, return on investment,cost of borrowing, opportunity cost,compounding rate, interest rate

n / number of periods in the future

(1+i)n / FVIFi, n / FV of interest Factor/ compounding factor

# How to calculate FVIF?

! Use calculator

! use table

3 Is (1+i)n >, = , or < 1 ?

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FV of $1

# of periods

$1 i = 0 %

i = 10%

i > 10%

Notes:

(a) If interest rate "i" = 0, then FV of a CF is constant irrespectiveof how far in the future you would be receiving it. sThehorizontal line above represents this.

(b) Given "i", the greater the "n", # of periods in the future, thegreater the FV. Thus, FV and "n" are positively related.

(c) Given "n", the higher the "i" the higher the FV. Thus, FV and"i" are positively related. i.e., they move in the samedirection.

Alex Tajirian



Y CF0 'FV1

(1 % i)(3)

Y PV 'CF1

1% i'

1001% .1

'1001.1

< 100

' 1&period discounted CF

4. PRESENT VALUE (PV)

L You are promised $100 one year from today

What is value of $100 today?; PV=?

! It better be < 100; time value of money

! from (2)

FV1 = CF0 (1 + i)

but CF0 = PV (value today) and FV1 = CF1 in future

substitute in (3),

where, i = discount rate

Alex Tajirian



FV2 ' CF0(1% i)2 ' PV(1% i)2

Y PV 'FV2

(1% i)2'

CF2

(1% i)2'

100

(1.1)2< 100

1.1< 100

PV 'CFn

(1 % i)n' CFn

1

(1 % i)n' CFn [ PVIFi,n ]

L You are promised $100 two years from today. PV = ?

L In general for a single CF,

where,

i / discount rate

PVIFi, n / PV of Interest Factor which depends on i, n.

// discount factor

? Is PVIF >,=, or < 1?

? How is risk reflected?7 Higher risk implies higher risk premium implies higher “i.”

! Given the CFs, the higher the i, the lower the value (PV).

Alex Tajirian



PV of $1

# of periods

$1 i = 0%

i = 10%

i > 10%

PV and "i" are inversely related ] They move in opposite direction.PV and "n" are inversely related ]They move in the opposite direction

Notes:

(a) If i = 0, then PV of a CF, say CF = $1, is constant at $1,irrespective of how far in the future it is received.

(b) For a given "n", the higher the "i", the lower is PV.

(c) For a given "i", the larger the "n", the smaller the PV.

ˆ̂

Alex Tajirian



PV '102

(1%.04)< 100

PV '$105

(1 % i)'

$1051 % .04

> $100

Example 1: Calculation of PV

The IRS screwed up your tax return by $100. They offer you achoice between $100 today or $102 next year. If 1-year governmentguaranteed loans are being offered at 4.0%, which alternative wouldyou choose?

Choose $100 today, as PV ($102) < $100.

Example 2: Calculating PV

The IRS makes you a new offer: $100 today or $105 next year.Which would you choose?

ˆ choose $105.

L Remember: you discount by a rate reflecting riskiness of CFs.Alternatively, an investment with similar risk yields 4% return.

Alex Tajirian



Example 3: Calculating PV

ATT owes you $100, and makes you an offer of $100 today or $105next year. Which would you choose? Assume that return on similarrisky investments is 6%.

ˆ choose $100 as

PVATT '$105

(1 % .06)< $100

L Note the discount rates in examples 2 and 3.

! The latter is higher reflecting default/bankruptcy risk.Obviously if interest on similar investment as the ATT were4%, then you would choose $105.

! How to calculate "i" will be discussed in the chapter on Risk &Return: Debt. The point I am trying to make here is thatbankruptcy is "bad", thus you would require a higher riskpremium to accept the ATT deal, which explains the differencebetween the two interest rates.

Alex Tajirian



5 ANNUITY

Definition: Equal CF over a # of equal length periods, paid at endof period.

periods 0 1 2 3

CFs 0 $100 $100 . . .

For FV,

periods 0 1 2 3

CFs 0 CF CF CF

value = ?

For PV,

periods 0 1 2 3

CFs 0

value = ?

CF CF . . .

Note. The book defines two different types of annuities: at thebeginning and at the end. I think it is just moreconfusing than it should be. My approach is easier.

Alex Tajirian



100(1% .1)

100(1% .1)2

FV3 ' 100 % 100(1% i)1 % 100(1% i)2

' 100[ 1 % (1% i)1 % (1% i)2]

' 100[FVIFA10%,3] ' 100[3.310]' $331

Future Value of an Annuity

Illustration 1: FV3 = ?

At end of each year, for 3 years, you put $100 in a bank (i=10%)

periods 0 1 2 3 FV

CFs 0 $100 $100 $100 = 100

= 110

= 121

FV 331

Thus,

Alex Tajirian



FVn ' Sum of Compounded Cash Flows

FVn ' CF % CF (1% i) % CF (1% i)2 % ... % CF (1% i)n&1

' CF × [ 1 % (1% i) % (1% i)2 % ... % (1% i)n&1 ]' CF × [ FVIFAi, n ]

for CF ' $100, i'6%, n'2;Y FV2 ' 100[ FVIFA6%,2 ] ' 100[ 2.0600 ] ' $206

FVIFAi,n '(1% i)n & 1

i

In general for an annuity,

where, FVIFA is FVIF of an annuity

# Thus, if CFs are equal, you do not need to compound each CFseparately as in Illustration 1.

# How to calculate [...]

! Each term separate! (As in Illustration 1: “long” method)! Tables for FVIFA

! calculator or computer

Alex Tajirian



FVIFA6%,2 '(1% .06)2 & 1

.06' 2.06

Example: Calculating FVIFA Using Formula

Given: i=6%, n=2

FVIFA6%, 2 = ?

Substituting in above formula, we have

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PV '100

(1% i)1%

100

(1% i)2' 100 1

(1% i)1%

1

(1% i)2

Present Value of Annuity

Illustration 2

i=6%

| | |

period 0 1 2

CF 0 100 100

Value ?

100x[1/(1+.06)]= 94.3

100x[1/(1+.06)2]= 89.0

PV $183.3

Thus,

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PV 'CF

(1 % i)%

CF

(1 % i)2% ... % CF

(1 % i)n

' Sum of discounted CFs

' CF × 1(1 % i)

%1

(1 % i)2% ...% 1

(1 % i)n

' CF [PVIFAi,n ]

PVIFAi,n '1i&

1

i(1% i)n

In general, for an annuity:

# Thus, if CFs are equal you do not need to discounteach CF separately as in Illustration 2.

# How to calculate [...]

! Each term separate ! (As in Illustration 2)! Tables for PVIFA (PVIF of an Annuity)! calculator or computer

Alex Tajirian



PVIFA.5% , 5 '1

.005&

1

.005(1 % .005)5

' 200 & 195.07 ' 4.93

Example: Calculating PVIFA Using Formula

For i = .5%, n =5,

PVIFA = ?

Note: You have to use this formula if interest rates is not aninteger, as tables cannot accommodate for all possiblevalue ranges.

Alex Tajirian



Example: Determining Interest Rate

Given a loan with:

Amount of loan equal to $35,000; Payment = $4,998.1 per year

; n =30 years

What is the interest rate on the loan?

Solution:

Step 1: This is a PV problem. You know the value of theloan today.

Step 2: Use PV formulation

PV = CF [ PVIFAi,30 ]

35,000 = $4,998.1 [ PVIFAi,30 ]

PVIFAi,30 = $35000/$4,998.1 = 7.0027

From Table: looking for row for 30 periods, PVIFAi,30;

ˆ̂ i = 14%.

Alex Tajirian



(1 % EAR) ' 1 %iNom

m

m

' 1 %APR

m

m

Y EAR ' 1 %iNom

m

m

& 1

5 QUOTED vs. EFFECTIVE RATE

iNom = ( periodic rate ) x m = APR

m = # of periods in a year

if quarters, m=4; monthly, m=12

APR / Annual % Rate / Quoted Rate

EAR / Effective Annual Rate

Intuitively:

Step 1: Convert annual rates to period rates. Thus, divideannual rate by number of periods "m" in a year.

Step 2: Now for each year, you have "m" more periods.Thus, you have to compound "m" times, i.e. raise topower m: (. . . )m.

Note: (1 + y)(1 + y)(1 + y) = (1 + y)3 = compounding 3-times

Alex Tajirian



EAR ' 1%0.08

4

4

& 1 ' (1.02)4 & 1 ' .0824 ' 8.24%

Example: Calculating EAR

Given: Bank charges 8% on loans, compounded quarterly. Whatis the EAR on the loan?

Thus, the more frequent the compounding, the larger the

difference.

6 APPLICATIONS

Rebate vs. Low Financing

Amortization Schedule

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Rebate v. Low Financing

SALE! SALE!

5%* FINANCING OR $500 REBATE

FULLY LOADED CONVERTIBLE

only $10,999

5% APR on 36 month loan

SALE! SALE!

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6.1 Rebate v. Low Financing

L Banks are making 10%, 36 month, car loans

Solution:

Step 1: This is a PV problem, as it deals with value of loans attime of decision making (today) not in the future

.

Step 2: Alternatives

(a) low financing: $10,999 loan at 5%, n = 36

(b) Rebate: (10,999 - 500) = 10,499 bank loan at 10%, n=36

Thus, choose alternative with lowest monthly payment.

Step 3: It is an annuity (equal CFs).

Step 4: Use PV of an annuity setting to calculate the unknown CF(payment).

Alex Tajirian



Y payment 'total loan[ PVIFA ]

payment '$10,999

PVIFA 5%12

,36

'$10,999

33.36' $329.65

payment '$10,499

PVIFA 10%12

,36

' $338.77

Rebate v. Low Financing (Continued)

PV = CF [ PVIFA ]

Total loan = payment [ PVIFA ]

Alternative (a) low financing;

Alternative (b) rebate;

Low Financing

Read fine print!

Alex Tajirian



1 Amortization Schedule for

Fixed Payments

$10,000 loan, 10%, 5 years, annual payments

Year BeginningBalance

TotalPayment(a)

Interest Paid(b) Principal Paid(c) Ending

Balance(d)

1 $10,000.00 $2,637.97 $1,000.00 $1,637.97 $8,362.02

2 8,362.03 2,637.97 836.20 1,801.77 6,560.25

3 6,560.25 2,637.97 656.03 1,981.95 4,578.30

4 4,578.30 2,637.97 457.83 2,180.14 2,398.16

5 2,398.16 2,637.97 239.82 2,398.16 0.00

Totals$13,189.87 $3,189.87 $10,000.00

(a) total payment 'loan

PVIFA10% ,5

'10,0003.7908

' $2,637.97

(b) interest paid ' (Balance)(interest rate) ' (10,000)(.1) '

(c) principal ' total payment & interest(d) ending balance ' Beginning Balance & Principal paid

Alex Tajirian



FVn ' Sum of Compounded Cash Flows

FVn ' CF % CF (1% i) % CF (1% i)2 % ... % CF (1% i)n&1

' CF × [ 1 % (1% i) % (1% i)2 % ... % (1% i)n&1 ]' CF × [ FVIFAi, n ]

1 SUMMARY

T Value depends on! Amount of CF! Timing of CF! Risk of CF

T

T i / re-investment rate, discount rate, compounding rate, interest rate, return oninvestment, cost of borrowing, cost of financing, opportunity cost.

Alex Tajirian



TT PV 'CF

(1% i)%

CF

(1% i)2% ... %

CF

(1% i)n

' Sum of discounted CFs

' CF × 1(1% i)

%1

(1% i)2% ...% 1

(1% i)n

' CF [PVIFAi,n ]

TT EAR ' 1 %iNom

m

m

& 1 ' 1 %APR

m

m

& 1

Alex Tajirian



2 QUESTIONS

A. Agree/Disagree-Explain

1. The more the frequency of compounding, the larger the difference between stated and effectiveinterest rates.

2. If you win a $4 m. State of California lottery, it would necessarily have the same value aswinning $4 m. NY State lottery, assuming that the payments are identical.

3. "i" is referred to as the discount factor.

4. There is no advantage in distinguishing between annuities and non-annuity CFs.

5. "Congratulations! You have already won the California lottery." If inflation increases, then thelottery's payoff would be worth less.

B. Numerical

1. Your 69-year old aunt has savings of $35,000. She has made arrangements to enter a home forthe aged on reaching the age of 80. Your aunt wants to decrease her savings by a constantamount each year for ten years, with a zero balance remaining. How much can she withdraweach year if she earns 6% annually on her savings? Her first withdrawal would be one year fromtoday.

2. Someone you know is about to retire. His firm has given him the option of retiring with a lumpsum of $20,000 or an annuity of $2,500 for ten years. Which is worth more now, if an interestrate of 7% is utilized for the annuity? Do not consider taxes.

3. A firm's earnings are $5,000 and are growing at 10% a year. Approximately how many yearswill it take for earnings to triple?

4. You are considering the purchase of a $50,000 machine, which is expected to generate$11,511.19 annually for 8 years. What is the expected return on the investment?

5. A machine costs $50,000 and is expected to yield a 16% annual rate of return on yourinvestment, for 8 years. What is the annual income from the machine?

6. Your banker tells you that a $85,000 loan, for 30 years, has an annual payment of $8,273.59.

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What must be the interest rate on the loan?

7. The current balance on your loan is $12,000. It has an interest of 9%, and an annual paymentof $1,500. How long would it lake you to payoff the entire loan?

8. After two years, your $100 investment is now worth $121.(a) What is the total realized return on your investment?(b) What is the annual return on your investment?

9. What is the PVIFA for i = .5% and n = 3?

10. If the average monthly return on Widget Inc. is 5%, what is its effective annual rate?

11. You put $100 in a bank today and expect to contribute an additional $100 after 1, 2, and 3years. What is the FV of your investment after 3 years if the interest rate is 3%?

12. A bank had issued a $10,000 loan a year ago at 10% interest for 5 years with annual paymentsof $2,636.97. Suppose the current interest rate on a similar loan is 12%. If the bank were tosell this loan to another financial institution, how much would it be worth?

13. You want to take a $5,000 vacation to Europe. You can only afford to put $1,160.06 annuallyin the bank. If the bank pays you 5% interest annually, how long would it be before you cantake the trip?

14. You plan to take a $5,000 trip to Europe in 2 years. If banks pay 5% interest compoundedannually, and you want to make equal monthly contributions, how much should you put in thebank annually?

Alex Tajirian



10. ANSWERS TO QUESTIONS

A. Agree/Disagree-Explain

1. Agree. See EAR formula p. 20.

2. Disagree. You need to discount CFs by the appropriate k reflecting the risk of the CFs. Thereis no reason to believe that the State of California has the same default risk as the Sate of NY.Thus, the discount rates might be different.

3. Disagree. It is the discount rate not the discount factor. A factor is a number you multiply theCFs by to obtain FV of PV.

4. Disagree. The advantage of an annuity formulation is that you do not need to go through thecumbersome process of discounting each CF separately.

5. Agree. Solution 1: DCF; if expected inflation_ Y IP_ Y k _ Y PV of lottery CF `Y you lose.

Solution 2: inflation_ Y your purchasing power is less Y you lose.

B. Numerical1. Step 1: It is a PV problem. You are given value of a loan today, and asked to find the

amount of payments.

Step 2: Use PV formulation to calculate payment.

PV ' CF [PVIFAk,n]

Y CF ' Payment 'PV of loanPVIFA6%,10

'35,0007.3601

' $4,755.37

Alex Tajirian



2. Step 1: It is a PV problem. You are given the value of a lump-sum today, and asked tocompare it to another CF. If you calculate the PV of the CFs, you end with PVsthat you need to compare. If you thought about it in terms of FV, then youwould have realized that more information was required than provided by thequestion. Thus, it must be a PV problem.

Step 2: Calculate PV of CFs.

PV ' $2,500[PVIFA7%,10] ' 2,500×7.0236 ' $17,559

but $17,559 < $20,000

ˆ accept LUMP SUM

3. It is a FV problem. You want to know how long it takes to reach 3 times the current value.

Triple Y compounding factor ' (1% k)n ' FVIFi,n ' 3

From FV Table, for k'10%, find n' ?

In table you get 2.8531 for 11 years and 3.1384 for 12.

ˆ approximately 11.5 years.

Alex Tajirian



For those who desire Swiss precision:

FV ' CF(1% k)t

Y 15,000 ' 5,000(1% .1)t

Y 3 ' (1.1)t

Y log 3 ' t log(1.1)

Y t 'log(3)

log(1.1)'

1.098.0953

' 11.55years

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PV ' CF[PVIFAi,n]

Y PVIFA 'PVCF

'$50,000

$11,511.19' 4.3436

From Table for PVIFA we get i ' 16%

CF 'PV

PVIFA16%,8

'$50,0004.3436

' $11,511.19

4. Given: PV of new machine = $50,000,Expected to generate annuity CF = $11,511.19 for 8 years.What is expected rate of return on investment? i = ?

Solution:

Step 1: PV problem, you are given the value of a machine today.

Step 2: Realize that i= interest rate = return on investment

Step 3: Calculate i.

5.

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PV ' CF[PVIFAi,n]

Y PVIFAi,30 'PVCF

'$85,000

$8,273.59' 10.2737

From Table for PVIFA we get i ' 9%

PV ' CF[PVIFAi,n]

Y PVIFA9,n 'PVCF

'$12,000$1,500

' 8.000

From Table for PVIFA we get n ' 15 years

6. Given: $85,000 loan, 30 years, annual payments = $8,273.59What is the interest rate on the loan? i =?

Solution:

7. Given: $12,000 loan, i = 9%, annual payment = $1,500What is length of loan? n =?

Solution:

Alex Tajirian



(a)from Chapter 2

realized return 'p1 & p0

p0

'121 & 100

100' 21%

(b) Solution 1: based on DCF

(1% total return) ' (1%k)(1%k)' (1%k)2

Y (1% .21) ' (1%k)2

Y (1%k) ' 1.21

Y k ' 1.21 & 1 ' 1.1 & 1 ' .1 ' 10%

Solution 2: based on definition of FV

FV ' CF(1%k)2 ' CF(FVIF?,2)

Y FVIF?,2 '121100

' 1.21

From FVIF table, ?'10%.

8. Your $100 investment is now worth $121, after two years.(a) What is the total realized return on your investment?(b) What is the annual return on your investment?

Alex Tajirian



(1 % effective annual rate) ' (1 %annual rate

12)12

' (1 % monthly rate)12

Y effective annual rate ' (1 % monthly rate)12 & 1

' (1 % .05)12 & 1

9. See example p. 18.

10. Similar to an EAR problem.

11. Remember that FVIFA assumes 0 CFs at time 0 (see Section 4.0). Thus, you need to add theFV of CF0. Thus,

FV3 = 100(FVIF3%,3) + 100(FVIFA3%,3)

12. Note 1. (a) Distinguish between market value of loan and book value.(b) A bank might be interested in obtaining cash immediately, and thus

wants to sell the loan (i.e. the promised CFs) for immediate cash.

Step 1: It is a PV problem (i.e. market value of loan)

Step 2: Calculate PV of loan

PVloan ' $2,636.97 × (PVIFA12%,4)

Note 2.(a) PV is not calculated based on PV of ending balance but the Sum of DCF.

Alex Tajirian



FV ' CF(FVIFA)5,000 ' 1,160.06(FVIFA5,?) Y

FVIFA '5,000

1,160.06' 4.3101

Y need to use FVIFA table ,which gives us n ' 4 years

.

FV ' CF(FVIFA5%,2

) Y

CF 'FV

FVIFA5%,2

'$5,0002.05

' $2,439.

13.

14.

Alex Tajirian

Time Value of Money

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