The Time Value of Money

Chapter 6

Learning Objectives

• Explain why a dollar today is worth more than a dollar in thefuture

• Define the terms future value

• Calculate the future value of an amount and an annuity

The Time Value of Money

• Consume Today or Tomorrow?

• TVM is based on the belief that people preferto consume goods today rather than wait toconsume the same goods tomorrow

•An apple we can have today is morevaluable to us than an apple we can havein one year.

•Money has a time value because buyingan apple today is more important thanbuying an apple in one year.

The Time Value of Money

• Consume Today or Tomorrow?

• A dollar someone has today can be spent forconsumption or loaned to earn interest

• A dollar loaned earns interest that increaseswealth and the ability to consume

• The rate of interest determines the trade-offbetween consumption today and saving(investing)

The Time Value of Money

• Future Value versus Present Value

• Cash-flows are evaluated based on future value or present value

• Future value measures what cash-flows are worth after a certain amount of time has passed

• Present value measures what future cash-flows are worth before a certain amount of time has passed

Comparison of Future Value and Present Value

The Time Value of Money

• Future Value versus Present Value

• Compounding is the process of increasing cash-flows to a future value

• Discounting is the process of reducing future cash-flows to a present value

Future Value

• What a dollar invested today will be worth in the future depends on

Length of the investment period

Method to calculate interest

Interest rate

2 types of methods to calculate interest

Simple method-calculated only on the original principal each year

Compound interest-calculated on both the original principal and on any accumulated interest earned up to that point. Future value implies the compound method

Interest rate

Simple interest is paid on the original principal amount only

Compound interest consists of both simple interest and interest-on-interest

Present Value of an Amount to be Received in the Future

• Taking future values back to the present is called discounting

Present Value=Future Value x Present Value Factor

PV = Present value (initial investment amount)

i = the interest rate

n= no. of time periods of the investment

Example 1

• Suppose we place $10000 in a savings account that pays 10%interest compounded annually. How will our savings grow?

Value at the end of year 1 = present value X (1+i)

i = interest rate

• = $10000 𝟏 + 𝟎. 𝟏 𝟒

• = $10000 (1.4641)

• = $14,641

Future Value: Example 2

• If we place $1,000 in a savings account paying 5% interestcompounded annually, how much will our account accrue to in 10years?

• Future value = present value X (𝟏 + 𝒓)𝒏

• 𝑭𝑽𝒏 = $1,000 (𝟏 + 𝟎. 𝟎𝟓)𝟏𝟎

• = $1,000 (1.62889)

• = $1,628.89

13

Refer Table B-1

Example 3: • What is the future value of $10,000 with an interest rate of 16

percent and one annual period of compounding? With anannual interest rate of 16 percent and two semiannual periodsof compounding? With an annual interest rate of 16 percentand four quarterly periods of compounding?

Annually:𝑭𝑽 = 𝑷𝑽 ∗ (𝟏 + 𝒊)𝒏

𝑭𝑽 = $𝟏𝟎, 𝟎𝟎𝟎 ∗ (𝟏+. 𝟏𝟔)𝟏

𝑭𝑽 = $𝟏𝟎, 𝟎𝟎𝟎 ∗ (𝟏. 𝟏𝟔)𝟏

𝑭𝑽 = $𝟏𝟏, 𝟔𝟎𝟎

Semi-annually:

𝑭𝑽 = 𝑷𝑽 𝑿 𝟏 +𝒊

𝒎

𝒏𝑿𝒎

• 𝑭𝑽 = $𝟏𝟎, 𝟎𝟎𝟎 𝑿 𝟏 +.𝟏𝟔

𝟐

𝟏𝑿𝟐

• 𝑭𝑽 = $𝟏𝟎, 𝟎𝟎𝟎 𝑿 𝟏. 𝟎𝟖 𝟐

• 𝑭𝑽 = $𝟏𝟎, 𝟎𝟎𝟎 𝑿 𝟏. 𝟏𝟔𝟔𝟒

• 𝑭𝑽 = $𝟏𝟏, 𝟔𝟔𝟒

• Quarterly:

• 𝑭𝑽 = 𝑷𝑽 𝑿 𝟏 +𝒊

𝒎

𝒏𝑿𝒎

• 𝑭𝑽 = $𝟏𝟎, 𝟎𝟎𝟎 𝑿 𝟏 +.𝟏𝟔

𝟒

𝟏𝑿𝟒

𝑭𝑽 = $𝟏𝟎, 𝟎𝟎𝟎 𝑿 𝟏. 𝟎𝟒 𝟒

𝑭𝑽 = $𝟏𝟎, 𝟎𝟎𝟎 𝑿 𝟏. 𝟏𝟔𝟗𝟗𝑭𝑽 = $𝟏𝟏, 𝟔𝟗𝟗

Present Value and Discounting

• Present Value concepts• A present value calculation takes end-of-the-period cash flows

and reverses the effect of compounding to determine theequivalent beginning-of-the-period cash flows

•Present Value Equation

•P𝒓𝒆𝒔𝒆𝒏𝒕 𝑽𝒂𝒍𝒖𝒆 = 𝑭𝑽𝒏 𝑿 𝑷𝑽𝑭𝒊,𝒏

•P𝒓𝒆𝒔𝒆𝒏𝒕 𝑽𝒂𝒍𝒖𝒆 = 𝑭𝑽𝒏 [𝟏

(𝟏+𝒓)𝒏]

•This is discounting and the interest rate i is called the discount rate.

•Present value (PV) is often referred to as the discounted value offuture cash-flows.

Present Value and Discounting

• Time and the discount rate affect present value

•The greater the amount of time before a cash flow is to occur,the smaller the present value of the cash-flow.

•The higher the discount rate, the smaller the present value of afuture cash-flow.

Example 4

What is the value of $500 to be received 10 years from today if ourdiscount rate is 6%.

P𝒓𝒆𝒔𝒆𝒏𝒕 𝑽𝒂𝒍𝒖𝒆 = 𝑭𝑽𝒏 [𝟏

(𝟏+𝒓)𝒏]

FV = $500, n = 10, r = 6% or 0.06

= $500 [𝟏

(𝟏+𝟎.𝟎𝟔)𝟏𝟎]

= $500 (0.558)

= $279.20

18

Refer Table B -3

Example 5

What is the present value of an investment that yields $1,000 to bereceived in 7 years and $1,000 to be received in 10 years if the discountrate is 6 percent?

𝒑𝒓𝒆𝒔𝒆𝒏𝒕 𝒗𝒂𝒍𝒖𝒆 = 𝑭𝑽𝒏 [𝟏

(𝟏+𝒓)𝒏] + 𝑭𝑽𝒏 [

𝟏

(𝟏+𝒓)𝒏]

PV= $𝟏, 𝟎𝟎𝟎 [𝟏

(𝟏+𝟎.𝟎𝟔)𝟕] + $1,000 [

𝟏

(𝟏+𝟎.𝟎𝟔)𝟏𝟎]

• = $1,000 (0.665) + $1,000 (0.558)

• = $665 + $558

• = $1223

Future and Present Values of Annuities

• Annuity- A series of equal payment made or received atregular time intervals

• Ordinary Annuity- A series of equal annuity payments madeor received at the end of each period

• Future value of an annuity-What an equal series of paymentswill be worth at some future date

• Future Value Factor of an Annuity (FVFA)-A factor that whenmultiplied by a stream of equal payments equals the futurevalue of that stream

• Future Value of an Annuity Table- Table of factors that showsthe future value of equal flows at the end of each period,given a particular interest rate

Example 6Carlos Menendez is planning to invest $3,500 every year for the next six years in

an investment paying 12 percent annually. What will be the amount he will have at

the end of the six years? (Round to the nearest dollar.)

A) $21,000

B) $28,403

C) $24,670

D) $26,124

Example 7You plan to save $1,250 at the end of each of the next three years to pay for a

vacation. If you can invest it at 7 percent, how much will you have at the end of

three years? (Round to the nearest dollar.)

A) $3,750

B) $3,918

C) $4,019

D) $4,589

Example 8: Maricela Sanchez needs to have $25,000 in five years. If she can earn 8 percent on

any investment, what is the amount that she will have to invest every year at the

end of each year for the next five years? (Round to the nearest dollar.)

A) $5,000

B) $4,261

C) $4,640

D) $4,445

Example 9Jane Ogden wants to save for a trip to Australia. She will need $12,000 at the end of

four years. She can invest a certain amount at the beginning of each of the next four

years in a bank account that will pay her 6.8 percent annually. How much will she

have to invest annually to reach her target? (Round to the nearest dollar.)

A) $3,000

B) $2,980

C) $2,538

D) $2,711

Annuities Continued

• Present Value of an Annuity- What the series of payments inthe future is worth today

• Present Value Factor of an Annuity (PVFA)- A factor thatwhen multiplied by a stream of equal payments equals thepresent value of that stream

• Present Value of an Annuity Table- Table of factors thatshows the value today of equal flows at the end of eachfuture period, given a particular interest rate

How to calculate present value of an annuity

30)1.6()1(

11

1

0

i

iCF

i

PVFACF

PVFACFPVA

n

o 𝑷𝑽𝒐𝒇 𝒂𝒏 𝒂𝒏𝒏𝒖𝒊𝒕𝒚 = 𝑷𝑴𝑻𝟏−𝒑𝒓𝒆𝒔𝒆𝒏𝒕 𝒗𝒂𝒍𝒖𝒆 𝒇𝒂𝒄𝒕𝒐𝒓

𝒓

o 𝑷𝑽𝒐𝒇 𝒂𝒏 𝒂𝒏𝒏𝒖𝒊𝒕𝒚 = 𝑷𝑴𝑻𝟏−(𝟏+𝒓)−𝒏

𝒓

19.154,5$0.08

0.08)1/(1 (1$2000PVA

3

3

Level Cash Flows: Annuities and Perpetuities

• Present Value of an Annuity Example

• A contract will pay $2,000 at the end of each year for threeyears and the appropriate discount rate is 8%. What is a fairprice for the contract?

31Refer Table B-4

Example 1

What is the value of 10-year $1,000 annuity discounted back tothe present at 5 percent?

𝑷𝑽𝒐𝒇 𝒂𝒏 𝒂𝒏𝒏𝒖𝒊𝒕𝒚 = 𝑷𝑴𝑻𝟏 − (𝟏 + 𝒓)−𝒏

𝒓

𝑷𝑽 = $𝟏, 𝟎𝟎𝟎𝟏 − (𝟏 + 𝟎. 𝟎𝟓)−𝟏𝟎

𝟎. 𝟎𝟓• = $1,000 (7.722)

• = $7,722

Example 2• Transit Insurance Company has made an investment in another company that will

guarantee it a cash flow of $37,250 each year for the next five years. If the company uses adiscount rate of 15 percent on its investments, what is the present value of thisinvestment? (Round to the nearest dollar.)

A) $101,766

B) $124,868

C) $251,154

D) $186,250

Annual payment = PMT = $37,250

No. of payments = n = 5

Required rate of return = 15%

Present value of investment = PVA5

5

11

(1 )

11

(1.15)$37,250 $37,250 3.3522

0.15

n

n

iPVA PMT

i

$124,867.78

Example 3Herm Mueller has invested in a fund that will provide him a cash flow of $11,700

for the next 20 years. If his opportunity cost is 8.5 percent, what is the present value

of this cash flow stream? (Round to the nearest dollar.)

A) $234,000

B) $132,455

C) $110,721

D) $167,884

20

11

(1 )

11

(1.085)$11,700 $11,700 9.4633

0.085

n

n

iPVA PMT

i

$110,721.04

Example 4Myers, Inc., will be making lease payments of $3,895.50 for a 10-year period,

starting at the end of this year. If the firm uses a 9 percent discount rate, what is

the present value of this annuity? (Round to the nearest dollar.)

A) $23,250

B) $29,000

C) $25,000

D) $20,000

10

11

(1 )

11

(1.09)$3,895.50 $3,895.50 6.4177

0.09

n

n

iPVA PMT

i

$24,999.99

Ordinary Annuity versus Annuity Due

• Present Value of Annuity Due

•Cash flows are discounted for one period less than in an ordinaryannuity.

• Future Value of Annuity Due

•Cash flows are earn compound interest for one period more thanin an ordinary annuity.

40

1

Due

1

Due

)(1FVA FVA

(6.4) )(1PVA PVA

i

i

Ordinary Annuity versus Annuity Due

41

Perpetuities

• A stream of equal cash flows that goes on forever

• Preferred stock and some bonds are perpetuities

• Equation for the present value of a perpetuity can be derivedfrom the present value of an annuity equation

42

).(i

CF

i

)(CF

i

i)(CF

ityor an annue factor fesent valuCFPVP

36

011

11

Pr0

Valuing Perpetuity Example 1

• Suppose you decide to endow a chair in finance. The goal of theendowment is to provide $100,000 of financial support per yearforever. If the endowment earns a rate of 8%, how much moneywill you have to donate to provide the desired level of support?

43

000,250,1$08.0

000,100$0

i

CFPVP

Example 2

• What is the value of a $500 perpetuity discounted back tothe present at 8 percent?

𝑷𝑽 =𝑪𝑭

𝒊

𝑷𝑽 =$𝟓𝟎𝟎

𝟎.𝟎𝟖= $6250

Example 3Your father is 60 years old and wants to set up a cash flow stream that would be

forever. He would like to receive $20,000 every year, beginning at the end of this

year. If he could invest in account earning 9 percent, how much would he have to

invest today to receive his perpetual cash flow? (Round to the nearest dollar.)

A) $222,222

B) $200,000

C) $189,000

D) $235,200

Annual payment needed = PMT = $20,000

Investment rate of return = i = 9%

Term of payment = Perpetuity

Present value of investment needed = PV

PMT $20,000PV of Perpetuity

0.09i

$222,222.22

Example 4A lottery winner was given a perpetual payment of $11, 444. She could invest the

cash flows at 7 percent. What is the present value of this perpetuity? (Round to

the nearest dollar.)

A) $112,344

B) $163,486

C) $191,708

D) $201,356

Annual payment needed = PMT =

$11,444

Investment rate of return = i = 7%

Term of payment = Perpetuity

Present value of investment needed = PV

PMT $11,444PV of Perpetuity

0.07i

$163,485.71

Cash Flows That Grow at a Constant Rate

• Growing Annuity

• equally-spaced cash flows that increase in size at a constant ratefor a finite number of periods

• Multiyear product or service contract with periodic cash flowsthat increase at a constant rate for a finite number of years

• Growing Perpetuity

• equally-spaced cash flows that increase in size at a constant rateforever

• Common stock whose dividend is expected to increase at aconstant rate forever

Cash Flows That Grow at a Constant Rate

• Growing Annuity

• Calculate the present value of growing annuity (only) when thegrowth rate is less than the discount rate.

51

(6.5)

i1

g11

g-i

CFPVA

n

1

n

Growing Annuity: Example• A coffee shop will operate for fifty more years. Cash flow was

$300,000 last year and increases by 2.5% each year. The discountrate for similar firms is 15%. Estimate the value of the firm.

128,452,2$

9968.0000,460,2$

15.1

025.11

025.015.0

500,307$

500,307$)025.01(000,300$1

50

0

PVA

CF

Cash Flows That Grow at a Constant Rate

• Growing Perpetuity

• Use Equation 6.6 to calculate the present value of growingperpetuity (only) when the growth rate is less than discountrate.

• It is derived from equation 6.5 when the number of periodsapproaches infinity

53

(6.6)

g-i

CFPVP 1

0

Cash Flows That Grow at a Constant Rate

• Growing perpetuity example

• A firm’s cash flow was $450,000 last year. You expect the cashflow to increase by 5% per year forever. If you use a discountrate of 18%, what is the value of the firm?

54

615,634,3$

13.0

500,472$

05.018.0

500,472$

500,307$)05.01(000,450$1

0

PVP

CF

The Effective Annual Interest Rate

• Describing interest rates

• The most common way to quote interest rates is in terms ofannual percentage rate (APR). It does not incorporate theeffects of compounding.

• The most appropriate way to quote interest rates is in terms ofeffective annual rate (EAR). It incorporates the effects ofcompounding.

55

The Effective Annual Interest Rate

• Calculate Annual Percentage Rate (APR)

APR = (periodic rate) x m

m is the # of periods in a year

• APR does not account for the number of compounding periodsor adjust the annualized interest rate for the time value ofmoney

• APR is not a precise measure of the rates involved in borrowingand investing

56

Future and Present Value Calculations and Excel Functions for Special Situations

• The more frequent the compounding for any given interestlevel and time period, the higher the future value.

• In the Excel RATE and NPER functions, the Payment box orloan payment box must be a negative value to represent cashoutflows

Summary

• Future values determine the value of dollar payments in thefuture

• Present value indicates the current value of future dollars

• Formulas are used to calculate both future and presentvalues

• All calculations can be made using tables or spreadsheets