Time Value Of Money

Chapter 6

TIME VALUE OF MONEY

Alex Tajirian

Time Value of Money

6-2

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 (Present Value)?

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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Time Value of Money

6-3

2. TYPES OF VALUATION

L

Based on investors' preferences and attitudes towards consumption and risk. !

L

Demand & Supply analysis

Based on "cash flow (CF)", !

CF: stream of promised future income

today = time of analysis | period

|

|

0

1

$100

200

time

| 2

3

s CF

-100

...

where periods can be hours, days, weeks, etc Note.

7

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

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

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Time Value of Money

6-4

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) ?

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)

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)? 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

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Time Value of Money

6-5

In general for a single CF, FVn ' CF (1% i)n ' CF [FVIFi, n]

(2)

where,

#

i

/

n

/ number of periods in the future

(1+i)n

/ FVIFi, n / FV of interest Factor / compounding factor

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

How to calculate FVIF? !

Use calculator

!

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

3

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

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Time Value of Money

6-6

FV of $1

i > 10% i = 10% $1

i=0%

# of periods

Notes: (a)

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

(b)

Given "i", the greater the "n", # of periods in the future, the greater 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 same direction. © morevalue.com, 1997 Alex Tajirian

Time Value of Money

6-7

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) Y CF0 '

FV1

(3)

(1 % i)

but CF0 = PV (value today) and FV1 = CF1 in future substitute in (3),

Y PV '

CF1

'

1% i

100 100 ' < 100 1% .1 1.1

' 1& period discounted CF where, i = discount rate

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Time Value of Money

L

6-8

You are promised $100 two years from today. PV = ? FV2 ' CF0(1% i)2 ' PV(1% i)2 Y PV '

L

FV2

CF2

'

(1% i)2

100

'

(1% i)2

(1.1)2

<

100 < 100 1.1

In general for a single CF, PV '

CFn (1 % i) n

' CF n

1 (1 % i)n

' CF n [ PVIFi,n ]

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). © morevalue.com, 1997 Alex Tajirian

Time Value of Money

6-9

PV of $1 i = 0%

$1

i = 10%

i > 10%

# of periods

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.

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

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Time Value of Money

6-10

Example 1: Calculation of PV The IRS screwed up your tax return by $100. They offer you a choice between $100 today or $102 next year. If 1-year government guaranteed loans are being offered at 4.0%, which alternative would you choose? Choose $100 today, as PV ($102) < $100. PV '

102 < 100 (1% .04)

Example 2: Calculating PV The IRS makes you a new offer: $100 today or $105 next year. Which would you choose?

PV '

$105 $105 ' > $100 (1 % i) 1 % .04

ˆ choose $105.

L

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

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Time Value of Money

6-11

Example 3: Calculating PV ATT owes you $100, and makes you an offer of $100 today or $105 next year. Which would you choose? Assume that return on similar risky investments is 6%. PVATT '

ˆ L

$105 < $100 (1 % .06)

choose $100 as

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 were 4%, 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 that bankruptcy is "bad", thus you would require a higher risk premium to accept the ATT deal, which explains the difference between the two interest rates.

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Time Value of Money

6-12

5 ANNUITY Definition:

Equal CF over a # of equal length periods, paid at end of period.

periods

0

1

2

3

CFs

0

$100

$100

...

periods

0

1

2

3

CFs

0

CF

CF

CF

For FV,

value = ?

For PV, periods

0

1

2

3

CFs

0

CF

CF

...

value = ? Note.

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

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Time Value of Money

6-13

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

CFs

0

$100

$100

$100

FV = 100 100(1% .1) = 110

100(1% .1)2 = 121 FV Thus, 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

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331

Time Value of Money

6-14

In general for an annuity, 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 ]

where, FVIFA is FVIF of an annuity #

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

#

How to calculate [...] ! !

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

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

!

calculator or computer FVIFAi,n

(1% i)n & 1 ' i

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Time Value of Money

6-15

Example: Calculating FVIFA Using Formula Given: i=6%, n=2 FVIFA6%, 2 = ?

Substituting in above formula, we have

FVIFA6%,2

(1% .06)2 & 1 ' ' 2.06 .06

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Time Value of Money

6-16

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, PV '

100 (1% i)1

%

100 (1% i)2

' 100

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1 (1% i)1

%

1 (1% i)2

Time Value of Money

6-17

In general, for an annuity: PV '

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

' Sum of discounted CFs ' CF ×

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

' CF [PVIFAi,n ] #

Thus, if CFs are equal you do not need to discount each 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

PVIFAi,n '

1 1 & i i(1% i)n

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Time Value of Money

6-18

Example: Calculating PVIFA Using Formula For i = .5%, n =5, PVIFA = ?

PVIFA.5% , 5 '

1 1 & .005 .005(1 % .005)5

' 200 & 195.07 ' 4.93

Note:

You have to use this formula if interest rates is not an integer, as tables cannot accommodate for all possible value ranges.

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Time Value of Money

6-19

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 the loan 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;

ˆ

i = 14%.

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Time Value of Money

6-20

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 (1 % EAR) '

Y

1 %

EAR '

iNom

m

APR 1 % m

'

m 1 %

i Nom m

m

m

& 1

Intuitively:

Step 1:

Convert annual rates to period rates. Thus, divide annual 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 to power m: (. . . )m.

Note:

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

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Time Value of Money

6-21

Example: Calculating EAR Given:

EAR '

Bank charges 8% on loans, compounded quarterly. What is the EAR on the loan?

0.08 1% 4

4

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

Thus, the more frequent the compounding, the larger the difference.

6 APPLICATIONS Rebate vs. Low Financing Amortization Schedule

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Time Value of Money

6-22

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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Time Value of Money

6-23

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 at time 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).

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Time Value of Money

6-24

Rebate v. Low Financing (Continued)

PV = CF [ PVIFA ] Total loan = payment [ PVIFA ] Y payment '

total loan [ PVIFA ]

Alternative (a) low financing; payment '

$10,999 $10,999 ' $329.65 ' PVIFA 5% ,36 33.36 12

Alternative (b) rebate; payment '

$10,499 ' $338.77 PVIFA 10% ,36 12

Low Financing

Read fine print!

© morevalue.com, 1997 Alex Tajirian

Time Value of Money

6-25

1

Amortization Schedule for Fixed Payments

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

Beginning Balance

Total Payment(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

$13,189.87

$3,189.87

$10,000.00

Totals

loan 10,000 ' $2,637.97 ' PVIFA10% ,5 3.7908 (b) interest paid ' (Balance)(interest rate) ' (10,000)(.1) ' (c) principal ' total payment & interest (d) ending balance ' Beginning Balance & Principal paid (a) total payment '

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Alex Tajirian

Time Value of Money

6-26

1 T

SUMMARY

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

T 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 ]

T

i/

re-investment rate, discount rate, compounding rate, interest rate, return on investment, cost of borrowing, cost of financing, opportunity cost.

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Time Value of Money

T PV '

6-27

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

' Sum of discounted CFs ' CF ×

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

' CF [PVIFAi,n ]

T EAR '

1 %

i Nom m

m

& 1 '

APR 1 % m

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m

& 1

Time Value of Money

6-28

2

QUESTIONS

A. Agree/Disagree-Explain

1.

The more the frequency of compounding, the larger the difference between stated and effective interest rates.

2.

If you win a $4 m. State of California lottery, it would necessarily have the same value as winning $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 the lottery'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 for the aged on reaching the age of 80. Your aunt wants to decrease her savings by a constant amount each year for ten years, with a zero balance remaining. How much can she withdraw each year if she earns 6% annually on her savings? Her first withdrawal would be one year from today.

2.

Someone you know is about to retire. His firm has given him the option of retiring with a lump sum of $20,000 or an annuity of $2,500 for ten years. Which is worth more now, if an interest rate 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 years will 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 your investment, 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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Alex Tajirian

Time Value of Money

6-29

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 payment of $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 3 years. 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 payments of $2,636.97. Suppose the current interest rate on a similar loan is 12%. If the bank were to sell 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 annually in the bank. If the bank pays you 5% interest annually, how long would it be before you can take the trip?

14.

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

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Alex Tajirian

Time Value of Money

6-30

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. There is 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 the CFs by to obtain FV of PV.

4.

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

5.

Agree.

Solution 2:

B. Numerical 1. Step 1:

Step 2:

DCF; if expected inflation_ Y IP_ Y k _ Y PV of lottery CF ` Y you lose. inflation_ Y your purchasing power is less Y you lose.

Solution 1:

It is a PV problem. You are given value of a loan today, and asked to find the amount of payments. Use PV formulation to calculate payment.

PV ' CF [PVIFAk,n] Y CF ' Payment '

PV of loan 35,000 ' $4,755.37 ' PVIFA6%,10 7.3601

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Alex Tajirian

Time Value of Money 2.

6-31

Step 1:

It is a PV problem. You are given the value of a lump-sum today, and asked to compare it to another CF. If you calculate the PV of the CFs, you end with PVs that you need to compare. If you thought about it in terms of FV, then you would have realized that more information was required than provided by the question. 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.

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Alex Tajirian

Time Value of Money

6-32

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) Yt '

log(3) 1.098 ' ' 11.55years log(1.1) .0953

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Alex Tajirian

Time Value of Money 4.

6-33

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.

PV ' CF[PVIFAi,n] Y PVIFA '

PV ' CF

$50,000 ' 4.3436 $11,511.19

From Table for PVIFA we get i ' 16%

5.

CF '

PV $50,000 ' ' $11,511.19 PVIFA16%,8 4.3436

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Alex Tajirian

Time Value of Money 6.

6-34

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

PV ' CF[PVIFAi,n] Y PVIFAi,30 '

PV $85,000 ' ' 10.2737 CF $8,273.59

From Table for PVIFA we get i ' 9%

7.

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

Solution:

PV ' CF[PVIFAi,n] Y PVIFA9,n '

PV $12,000 ' ' 8.000 CF $1,500

From Table for PVIFA we get n ' 15 years

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Alex Tajirian

Time Value of Money 8.

6-35

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?

(a)from Chapter 2 realized return '

p1 & p0

'

p0

121 & 100 ' 21% 100

(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) ' Yk '

1.21

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 '

121 ' 1.21 100

From FVIF table, ?' 10%.

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Alex Tajirian

Time Value of Money

6-36

9.

See example p. 18.

10.

Similar to an EAR problem.

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

(1 % effective annual rate) ' (1 %

' (1 % .05)12 & 1

11.

Remember that FVIFA assumes 0 CFs at time 0 (see Section 4.0). Thus, you need to add the FV 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.

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Alex Tajirian

Time Value of Money

6-37

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

5,000 ' 4.3101 1,160.06

.

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

14. FV ' CF(FVIFA5%,2) Y CF '

FV $5,000 ' ' $2,439. FVIFA5%,2 2.05

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Alex Tajirian