Contemporary Financial Management

© 2004 by Nelson, a division of Thomson Canada Limited

Contemporary Financial Management

Chapter 4:Time Value of Money

2© 2004 by Nelson, a division of Thomson Canada Limited

Introduction

● This chapter introduces the concepts and skills necessary to understand the time value of money and its applications.


Payment of Interest

● Interest is the cost of money

● Interest may be calculated as:●Simple interest●Compound interest


Simple Interest

● Interest paid only on the initial principal

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

-$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.

-$1,000 $60.00 $63.60 $67.42

0 1 2 3


Future Value

● 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

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

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 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


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


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

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.

n

0 n

FVPV

1+i

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


Present Value

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

● 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

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.

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.


Annuities

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

Example: 3-year, $100 ordinary annuity

$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

$100 $100 $100

0 1 2 3


Difference Between Annuity Types

0 1 2 3

$100 $100 $100

$100 $100 $100

0 1 2 3

$100$100

Ordinary Annuity

Annuity Due


Annuities: Future Value

● 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

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



● 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



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


Annuities: Future Value – Table

● 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

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

● 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

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

● 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


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



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



● Present value of an annuity due:

-n

AnnuityDue

1- 1+iPV = PMT 1+i

i

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



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


Annuities: Present Value – Table

● 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



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



● 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



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.

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.

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

$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).

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?

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

-$1,000 $60.00 $63.60

0 0.5 16% 6%

$67.42

1.56%

12%



● Increased interest payment frequency requires future and present value formulas to be adjusted to account for the number of compounding periods per year (m).

Future Value Present Value

n

nomn 0

mi

F PV 1m

V

n0 n

no

m

m

FVPV

i1+

m



Example: What is a $1,000 investment worth in five years if it earns 8% interest, compounded quarterly?

mn

nomn 0

(4)(5)

iFV PV 1

m

0.08$1,000 1

4

$1, 485.95



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?

n0 mn

nom

(12)(20)

FVPV

i1+

m

$10,000$1,364.62

0.101+

12


Impact of Compounding Frequency

$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.

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?

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

Contemporary Financial Management

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