CONTEMPORARY FINANCIAL MANAGEMENTChapter 4:

Time Value of Money

INTRODUCTION This chapter introduces the concepts and skills

necessary to understand the time value of money and its applications.

2

PAYMENT OF INTEREST Interest is the cost of money

Interest may be calculated as: Simple interest Compound interest

3

SIMPLE INTEREST

Interest paid only on the initial principal

Example: $1,000 is invested to earn 6% per year, simple interest.

4

-$1,000 $60 $60 $60

0 1 2 3

COMPOUND INTEREST

Interest paid on both the initial principal and on interest that has been paid & reinvested.

Example: $1,000 invested to earn 6% per year, compounded annually.

5

-$1,000 $60.00 $63.60 $67.42

0 1 2 3

FUTURE VALUE

6

The value of an investment at a point in the future, given some rate of return.

nn 0FV = PV (1 + i)

FV = future valuePV = present valuei = interest rate n = number of periods

× ×n 0 0FV = PV +(PV i n)

FV = future valuePV = present valuei = interest rate n = number of periods

Simple Interest Compound Interest

FUTURE VALUE: SIMPLE INTEREST

7

Example: You invest $1,000 for three years at 6% simple interest per year.

( )× ×

+ × ×3 0 0FV = PV +(PV i n)

= $1,000 $1,000 0.06 3

= $1,180.00

-$1,000

0 1 2 36% 6% 6%

FUTURE VALUE: COMPOUND INTEREST

8

Example: You invest $1,000 for three years at 6%, compounded annually.

( )+

n3 0

3

FV = PV (1 + i)

= $1,000 1 0.06

= $1,191.02

-$1,000

0 1 2 36% 6% 6%

FUTURE VALUE: COMPOUND INTEREST

9

Future values can be calculated using a table method, whereby “future value interest factors” (FVIF) are provided.

See Table 4.1 (page 135)

n 0 i,nFV = PV (FVIF ), where: ( ) ni,nFVIF = 1+i

FV = future valuePV = present valueFVIF = future value interest factori = interest rate n = number of periods

FUTURE VALUE: COMPOUND INTEREST

10

Example: You invest $1,000 for three years at 6% compounded annually.

Table 4.1 Excerpt: FVIFs for $1

End of Period (n) 5% 6% 8%

2 1.102 1.124 1.166 3 1.158 1.191 1.260 4 1.216 1.262 1.360

3 0 6%,3FV = PV (FVIF )

=$1,000(1.191) =$1,191.00

PRESENT VALUE

What a future sum of money is worth today, given a particular interest (or discount) rate.

11

( )= n

0 n

FVPV

1+i

FV = future valuePV = present valuei = interest (or discount) rate n = number of periods

PRESENT VALUE

12

Example: You will receive $1,000 in three years. If the discount rate is 6%, what is the present value?

( ) ( )= = =

+3

0 n 3

FV $1,000PV $839.62

1+i 1 0.06

$1,000

0 1 2 36% 6% 6%

PRESENT VALUE

13

● Present values can be calculated using a table method, whereby “present value interest factors” (PVIF) are provided.

● See Table 4.2 (page 139)

0 n i,nPV = FV (PVIF ), where:( )i,n n

1PVIF =

1+iFV = future valuePV = present valuePVIF = present value interest factori = interest rate n = number of periods

PRESENT VALUE

14

Example: What is the present value of $1,000 to be received in three years, given a discount rate of 6%?

0 3 6%,3PV = FV (FVIF )

=$1,000(0.840) =$840.00

A NOTE OF CAUTION

Note that the algebraic solution to the present value problem gave an answer of 839.62

The table method gave an answer of $840.

15

Caution: Tables provide approximate answers only. If more accuracy is required, use algebra!

ANNUITIES

The payment or receipt of an equal cash flow per period, for a specified number of periods.

Examples: mortgages, car leases, retirement income.

16

ANNUITIES

Ordinary annuity: cash flows occur at the end of each period

Example: 3-year, $100 ordinary annuity

17

$100 $100 $100

0 1 2 3

ANNUITIES

Annuity Due: cash flows occur at the beginning of each period

Example: 3-year, $100 annuity due

18

$100 $100 $100

0 1 2 3

DIFFERENCE BETWEEN ANNUITY TYPES

19

0 1 2 3

$100 $100 $100

$100 $100 $100

0 1 2 3

$100$100

Ordinary Annuity

Annuity Due

ANNUITIES: FUTURE VALUE

20

● Future value of an annuity - sum of the future values of all individual cash flows.

$100 $100 $100

0 1 2 3

FVFVFV

FV of Annuity

ANNUITIES: FUTURE VALUE – ALGEBRA

Future value of an ordinary annuity

( ) ÷ ÷

n

OrdinaryAnnuity

1+i -1FV = PMT

i

21

FV = future value of the annuityPMT = equal periodic cash flowi = the (annually compounded) interest raten = number of years

ANNUITIES: FUTURE VALUE

Example: What is the future value of a three year ordinary annuity with a cash flow of $100 per year, earning 6%?

( )

( ).

.

$ .

÷ ÷

−= ÷ ÷ =

n

OrdinaryAnnuity

3

1+i -1FV = PMT

i

1 06 1100

06

318 36

22

23

Annuities: Future Value – Algebra

● Future value of an annuity due:

( ) ( ) ÷ ÷

n

AnnuityDue

1+i -1FV = PMT 1 + i

i

FV = future value of the annuityPMT = equal periodic cash flowi = the (annually compounded) interest raten = number of years

ANNUITIES: FUTURE VALUE – ALGEBRA

Example: What is the future value of a three year annuity due with a cash flow of $100 per year, earning 6%?

( ) ( )

( ) ( )..

.

$ .

÷ ÷

−= ÷ ÷ =

n

AnnuityDue

3

1+i -1FV = PMT 1+i

i

1 06 1100 1 06

06

337 46

24

ANNUITIES: FUTURE VALUE – TABLE

25

● The future value of an ordinary annuity can be calculated using Table 4.3 (p. 145), where “future value of an ordinary annuity interest factors” (FVIFA) are provided.

n i,nFVAN = PMT(FVIFA ), where:

( )+ −n

i,n

1 i 1FVIFA =

iPMT = equal periodic cash flowi = the (annually compounded) interest raten = number of periods FVAN = future value (ordinary annuity)FVIFA = future value interest factor

ORDINARY ANNUITY: FUTURE VALUE

26

Example: What is the future value of a 3-year $100 ordinary annuity if the cash flows are invested at 6%, compounded annually?

( ) =n i,nFVAN = PMT(FVIFA )

=$100 3.184 $318.40

ANNUITY DUE: FUTURE VALUE

27

● Calculated using Table 4.3 (p. 145), where FVIFAs are found. Ordinary annuity formula is adjusted to reflect one extra period of interest.

( ) +n i,nFVAND = PMT FV 1 iIFA , where:

( )+ −n

i,n

1 i 1FVIFA =

iPMT = equal periodic cash flowi = the (annually compounded) interest raten = number of periods FVAND = future value (annuity due)FVIFA = future value interest factor

ANNUITY DUE: FUTURE VALUE

28

Example: What is the future value of a 3-year $100 annuity due if the cash flows are invested at 6% compounded annually?

( ) + = =

n i,nFVAND = PMT FVIFA 1 i

$100 3.184(1.06) $337.50

ANNUITIES: PRESENT VALUE

29

● The present value of an annuity is the sum of the present values of all individual cash flows.

$100 $100 $100

0 1 2 3

PVPVPV

PV of Annuity

30

Annuities: Present Value – Algebra

● Present value of an ordinary annuity

( ) ÷ ÷

-n

OrdinaryAnnuity

1- 1+iPV = PMT

i

PV = present value of the annuityPMT = equal periodic cash flowi = the (annually compounded) interest or discount raten = number of years

ANNUITIES: PRESENT VALUE – ALGEBRA

Example: What is the present value of a three year, $100 ordinary annuity, given a discount rate of 6%?

( )

( ) -- .

.

$ .

÷ ÷

= ÷ ÷ =

-n

OrdinaryAnnuity

3

1- 1+iPV =PMT

i

1 1 06100

06

267 30

31

ANNUITIES: PRESENT VALUE – ALGEBRA

( ) ( ) ÷ ÷

-n

AnnuityDue

1- 1+iPV = PMT 1+i

i

32

● Present value of an annuity due:

PV = present value of the annuityPMT = equal periodic cash flowi = the (annually compounded) interest or discount raten = number of years

ANNUITIES: PRESENT VALUE – ALGEBRA

Example: What is the present value of a three year, $100 annuity due, given a discount rate of 6%?

( ) ( )

( ) ( )..

.

$ .

−

÷ ÷

−= ÷ ÷ =

-n

AnnuityDue

3

1- 1+iPV = PMT 1+i

i

1 1 06100 1 06

06

283 34

33

ANNUITIES: PRESENT VALUE – TABLE

34

● The present value of an ordinary annuity can be calculated using Table 4.4 (p. 149), where “present value of an ordinary annuity interest factors” (PVIFA) are found.

0 i,nPVAN = PMT(PVIFA ), where:

( )

-n

i,n

1- 1+iPVIFA =

iPMT = cash flowi = the (annually compounded) interest or discount rate n = number of periods PVAN = present value (ordinary annuity)PVIFA = present value interest factor

ANNUITIES: PRESENT VALUE – TABLE

35

Example: What is the present value of a 3-year $100 ordinary annuity if current interest rates are 6% compounded annually?

( ) =0 i,nPVAN = PMT(PVIFA )

=$100 2.673 $267.30

ANNUITIES: PRESENT VALUE – TABLE

36

● Calculated using Table 4.4 (p. 149), where PVIFAs are found. Present value of ordinary annuity formula is modified to account for one less period of interest.

+0 i,nPVAND = PMT PVI (1 i)FA

( ) −

n

i,n

1- 1+iPVIFA =

iPMT = cash flowi = the (annually compounded) interest or discount rate n = number of periods PVAND = present value (annuity due)PVIFA = present value interest factor

ANNUITIES: PRESENT VALUE – TABLE

37

Example: What is the present value of a 3-year $100 annuity due if current interest rates are 6% compounded annually?

( ) + = =

0 i,nPVAND = PMT PVIFA (1 i)

$100 2.673 1.06 $283.34

OTHER USES OF ANNUITY FORMULAS

Sinking Fund Problems: calculating the annuity payment that must be received or invested each year to produce a future value.

38

Ordinary Annuity Annuity Due

n

i,n

FVANPMT=

FVIFA ( )+n

i,n

FVANPMT=

FVIFA 1 i

OTHER USES OF ANNUITY FORMULAS

Loan Amortization and Capital Recovery Problems: calculating the payments necessary to pay off, or amortize, a loan.

39

0

i,n

PVANPMT=

PVIFA

PERPETUITIES Financial instrument that pays an equal cash flow per

period into the indefinite future (i.e. to infinity).

Example: dividend stream on common and preferred stock

40

$60 $60 $60

0 1 2 3

$60

4 ∞

PERPETUITIES Present value of a perpetuity equals the sum of the

present values of each cash flow.

Equal to a simple function of the cash flow (PMT) and interest rate (i).

41

=0

PMTPVPER

i

∞

=

= ∑0 n1

PMTPVPER

(1+i)t

PERPETUITIES Example: What is the present value of a $100

perpetuity, given a discount rate of 8% compounded annually?

42

= = =0

PMT $100PVPER $1,250.00

i 0.08

MORE FREQUENT COMPOUNDING

Nominal Interest Rate: the annual percentage interest rate, often referred to as the Annual Percentage Rate (APR).

Example: 12% compounded semi-annually

43

-$1,000 $60.00 $63.60

0 0.5 16% 6%

$67.42

1.56%

12%

MORE FREQUENT COMPOUNDING Increased interest payment frequency requires

future and present value formulas to be adjusted to account for the number of compounding periods per year (m).

44

Future Value Present Value

= + ÷

n

nomn 0

mi

F PV 1m

V=

÷

n0 n

no

m

m

FVPV

i1+

m

MORE FREQUENT COMPOUNDING Example: What is a $1,000 investment worth in five

years if it earns 8% interest, compounded quarterly?

45

= + ÷

= + ÷ =

mn

nomn 0

(4)(5)

iFV PV 1

m

0.08$1,000 1

4

$1,485.95

MORE FREQUENT COMPOUNDING Example: How much do you have to invest today in

order to have $10,000 in 20 years, if you can earn 10% interest, compounded monthly?

46

= ÷

= = ÷

n0 mn

nom

(12)(20)

FVPV

i1+

m

$10,000$1,364.62

0.101+

12

IMPACT OF COMPOUNDING FREQUENCY

47

$1.097

$1.098

$1.099

$1.100

$1.101

$1.102

$1.103

$1.104

$1.105

$1.106

Annual Semi-Annual

Quarterly Monthly Daily

$1,000 Invested at Different 10% Nominal Rates for One Year

EFFECTIVE ANNUAL RATE (EAR)

The annually compounded interest rate that is identical to some nominal rate, compounded “m” times per year.

48

= − ÷

m

nomeff

ii 1+ 1

m

==

eff

nom

i effective annual rate

i nominal interest rate

m = compounding frequency per year

EFFECTIVE ANNUAL RATE (EAR)

EAR provides a common basis for comparing investment alternatives.

Example: Would you prefer an investment offering 6.12%, compounded quarterly or one offering 6.10%, compounded monthly?

49

= − ÷

= − ÷ =

m

nomeff

4

ii 1+ 1

m

0.06121+ 1

4

6.262%

= − ÷

= − ÷ =

m

nomeff

12

ii 1+ 1

m

0.0611+ 1

12

6.273%

MAJOR POINTS The time value of money underlies the valuation of

almost all real & financial assets

Present value – what something is worth today

Future value – the dollar value of something in the future

Investors should be indifferent between: Receiving a present value today Receiving a future value tomorrow A lump sum today or in the future An annuity

50