Solution manual financial management 10e by keown chapter 05

Mathematically, the future value of an investment if compounded annually at a rate of i for n years will be where n = the number of years during which the compounding occurs i = the annu

CHAPTER 5 The Time Value of Money

CHAPTER ORIENTATION

In this chapter the concept of a time value of money is introduced, that is, a dollar today isworth more than a dollar received a year from now Thus if we are to logically compareprojects and financial strategies, we must either move all dollar flows back to the present orout to some common future date

CHAPTER OUTLINE

I Compound interest results when the interest paid on the investment during the first

period is added to the principal and during the second period the interest is earned onthe original principal plus the interest earned during the first period

A Mathematically, the future value of an investment if compounded annually at

a rate of i for n years will be

where n = the number of years during which the compounding

occurs

i = the annual interest (or discount) rate

PV = the present value or original amount invested at the

beginning of the first periodFVn = the future value of the investment at the end of n

years

1 The future value of an investment can be increased by either

increasing the number of years we let it compound or bycompounding it at a higher rate

2 If the compounded period is less than one year, the future value of an

investment can be determined as follows:

where m= the number of times compounding occurs during the

year

II Determining the present value, that is, the value in today's dollars of a sum of money

to be received in the future, involves nothing other than inverse compounding Thedifferences in these techniques come about merely from the investor's point of view

A Mathematically, the present value of a sum of money to be received in the

future can be determined with the following equation:

where: n = the number of years until payment will be

received,

i = the interest rate or discount rate

PV = the present value of the future sum of moneyFVn = the future value of the investment at the end of n

years

1 The present value of a future sum of money is inversely related to

both the number of years until the payment will be received and theinterest rate

III An annuity is a series of equal dollar payments for a specified number of years

Because annuities occur frequently in finance, for example, bond interest payments,

we treat them specially

A A compound annuity involves depositing or investing an equal sum of money

at the end of each year for a certain number of years and allowing it to grow

1 This can be done by using our compounding equation, and

compounding each one of the individual deposits to the future or byusing the following compound annuity equation:

t

i)(1

where: PMT = the annuity value deposited at the end of each

year

i = the annual interest (or discount) rate

n = the number of years for which the annuity will

lastFVn = the future value of the annuity at the end of the

nth year

B Pension funds, insurance obligations, and interest received from bonds all

involve annuities To compare these financial instruments we would like toknow the present value of each of these annuities

1 This can be done by using our present value equation and discounting

each one of the individual cash flows back to the present or by usingthe following present value of an annuity equation:

1

where: PMT = the annuity deposited or withdrawn at the end

of each year

i = the annual interest or discount rate

PV = the present value of the future annuity

n = the number of years for which the annuity will

last

C This procedure of solving for PMT, the annuity value when i, n, and PV are

known, is also the procedure used to determine what payments are associatedwith paying off a loan in equal installments Loans paid off in this way, inperiodic payments, are called amortized loans Here again we know three ofthe four values in the annuity equation and are solving for a value of PMT,the annual annuity

IV Annuities due are really just ordinary annuities where all the annuity payments have

been shifted forward by one year Compounding them and determining their presentvalue is actually quite simple Because an annuity, due merely shifts the paymentsfrom the end of the year to the beginning of the year, we now compound the cashflows for one additional year Therefore, the compound sum of an annuity due is

FVn(annuity due) = PMT (FVIFAi,n) (1 + i)

A Likewise, with the present value of an annuity due, we simply receive each

cash flow one year earlier – that is, we receive it at the beginning of eachyear rather than at the end of each year Thus the present value of an annuitydue is

PV(annuity due) = PMT (PVIFAi,n) (1 + i)

V A perpetuity is an annuity that continues forever, that is every year from now on this

investment pays the same dollar amount

A An example of a perpetuity is preferred stock which yields a constant dollar

dividend infinitely

B The following equation can be used to determine the present value of a

perpetuity:

where: PV = the present value of the perpetuity

pp = the constant dollar amount provided by the perpetuity

i = the annual interest or discount rate

VI To aid in the calculations of present and future values, tables are provided at the

back of Financial Management (FM)

A To aid in determining the value of FVn in the compounding formula

FVn = PV (1 + i)n = PV (FVIFi,n)tables have been compiled for values of FVIFi,n or (i + 1)n in Appendix B,

"Compound Sum of $1," in FM

B To aid in the computation of present values

tables have been compiled for values of

or PVIFi,nand appear in Appendix C in the back of FM

C Because of the time-consuming nature of compounding an annuity,

t

t

i)(1

t

t

i)(1

or FVIFAi,nfor various combinations of n and i

D To simplify the process of determining the present value of an annuity

V Spreadsheets and the Time Value of Money

A While there are several competing spreadsheets, the most popular one is

Microsoft Excel Just as with the keystroke calculations on a financial calculator, a spreadsheet can make easy work of most common financial calculations Listed below are some of the most common functions used with Excel when moving money through time:

Calculation: Formula:

Present Value = PV(rate, number of periods, payment, future value, type)Future Value = FV(rate, number of periods, payment, present value, type)Payment = PMT(rate, number of periods, present value, future value,

type)Number of Periods = NPER(rate, payment, present value, future value, type)Interest Rate = RATE(number of periods, payment, present value, future

value, type, guess)

where: rate = i, the interest rate or discount rate

number of periods = n, the number of years or periodspayment = PMT, the annuity payment deposited or received at the

end of each periodfuture value = FV, the future value of the investment at the end of n

periods or yearspresent value = PV, the present value of the future sum of moneytype = when the payment is made, (0 if omitted)

0 = at end of period

1 = at beginning of periodguess = a starting point when calculating the interest rate, if

omitted, the calculations begin with a value of 0.1 or 10%

ANSWERS TO END-OF-CHAPTER QUESTIONS

5-1 The concept of time value of money is recognition that a dollar received today is

worth more than a dollar received a year from now or at any future date It existsbecause there are investment opportunities on money, that is, we can place our dollarreceived today in a savings account and one year from now have more than a dollar.5-2 Compounding and discounting are inverse processes of each other In compounding,

money is moved forward in time, while in discounting money is moved back in time.This can be shown mathematically in the

compounding equation:

FVn = PV (1 + i)n

We can derive the discounting equation by multiplying each side of

this equation by and we get:

5-3 We know that

FVn = PV(1 + i)nThus, an increase in i will increase FVn and a decrease innwill

decrease FVn

5-4 Bank C which compounds daily pays the highest interest This occurs because,

while all banks pay the same interest, 5 percent, bank C compounds the 5 percentdaily Daily compounding allows interest to be earned more frequently than theother compounding periods

5-5 The values in the present value of an annuity table (Table 5-8) are actually derived

from the values in the present value table (Table 5-4) This can be seen, by

examining the values represented in each table The present value table gives values

for various values of i and n Thus the value in the present value of annuity table for

an n-year annuity for any discount rate i is merely the sum of the first n values in thepresent value table PVIFA 10%,10yrs = 6.145 

5-6 An annuity is a series of equal dollar payments for a specified number of years

Examples of annuities include mortgage payments, interest payments on bonds,fixed lease payments, and any fixed contractual payment A perpetuity is an annuitythat continues forever, that is, every year from now on this investment pays the samedollar amount The difference between an annuity and a perpetuity is that aperpetuity has no termination date whereas an annuity does

SOLUTIONS TO END-OF-CHAPTER PROBLEMS

Solutions to Problem Set A

5-1A (a) FVn = PV (1 + i)n

Thus n = 15 years (because the value of 2.079 occurs in the 15

year row of the 5 percent column of Appendix B)

Thus, i = 12% (because the Appendix B value of 3.896 occurs in

the 12 year row in the 12 percent column)

t

t

0.05) (1

n

0

t

i) (1

5

0

t

0.1) (1

n

0

t

i) (1

7

0

t

0.07) (1

n

0

t

i) (1

1 3

0

t

t (1 i)

1

1

t (1 0.03)

1

compounded for 1 year

(c) There is a positive relationship between both the interest rate used to

compound a present sum and the number of years for which the

compounding continues and the future value of that sum.

(e) An increase in the stated interest rate will increase the future value of a given

sum Likewise, an increase in the length of the holding period will increasethe future value of a given sum

PV = $8,500 (6.492)

Since the cost of this annuity is $50,000 and its present value is $55,182,given an 11 percent opportunity cost, this annuity has value and should beaccepted

1

PV = $7,000 (8.442)

Since the cost of this annuity is $60,000 and its present value is only

$59,094, given an 11 percent opportunity cost, this annuity should not beaccepted

PV = $8,000 (7.963)

Since the cost of this annuity is $70,000 and its present value is only

$63,704, given an 11 percent opportunity cost, this annuity should not beaccepted

5-11A Year 1:FVn = PV (1 + i)n

FV1 = 15,000(1 + 0.2)1

FV1 = 15,000(1.200)

FV1 = 18,000 booksYear 2:FVn = PV (1 + i)n

FV2 = 15,000(1 + 0.2)2

FV2 = 15,000(1.440)

FV2 = 21,600 booksYear 3: FVn = PV (1 + i)n

FV3 = 15,000(1.20)3

FV3 = 15,000(1.728)

FV3 = 25,920 books

Book sales 25,000 20,000 15,000

years

The sales trend graph is not linear because this is a compound growth trend Just as compound interest occurs when interest paid on the investment during the first period

is added to the principal of the second period, interest is earned on the new sum Book sales growth was

compounded; thus, the first year the growth was 20 percent

of 15,000 books for a total of 18,000 books, the second year

20 percent of 18,000 books for a total of 21,600, and the third year 20 percent of 21,600 books for a total of 25,920

n

0

t

i) (1

15

0

t

0.06) (1

Thus, PMT = $644.44

$1,079.50 = $500 (FVIF i%, 10 yr.)

10

0

t

.10) (1

n

0

t

i) (1

10

0

t

.09) (1

$10,000,000 = PMT(15.193)

Thus, PMT = $658,197.855-18A One dollar at 12.0% compounded monthly for one year

= $1(1.127)

= $1.127One dollar at 13.0% compounded annually for one year

= $10,000 + $50,000 + $10,000

PV = FVn (PVIFi,n)

PV = $2,000(PVIF10%, year 1) + $3,000(PVIF10%, year 2) +

$4,000(PVIF10%, year 3) - $5,000(PVIF10%, year 4) +

$5,000(PVIF10%, year 5)

= $2,000(.909) + $3,000(.826) + $4,000(.751) - $5,000(.683) + $5,000(.621)

= $1,818 + $2,478 + $3,004 - $3,415 + $3,105

= $6,990

Investment B:

PV = $2,000(PVIF10%, year 1) + $2,000(PVIF10%, year 2) +

$2,000(PVIF10%, year 3) + $2,000(PVIF10%, year 4) +

$5,000(PVIF10%, year 5)

= $2,000(.909) + $2,000(.826) + $2,000(.751) + $2,000(.683) + $5,000(.621)

= $1,818 + $1,652 + $1,502 + $1,366 + $3,105

= $9,443

Investment C:

PV = $5,000(PVIF10%, year 1) + $5,000(PVIF10%, year 2) -

$5,000(PVIF10%, year 3) - $5,000(PVIF10%, year 4) +

1

t t 15

1

1 .06)

(1

1

1

1

$30,000 = $10,000 (PVIFAi%, 5 yr.)

(1 1

= $25,000 (.074)

= $1,850Thus take the $10,000 in 12 years

n

0

t

i) (1

5

0

t

.12) (1

n

0

t

i) (1

15

0

t

.07) (1

n

0

t

i) (1

$30,330 = PMT (FVIFA7%, 15 yr.)

5-33A (a) This problem can be subdivided into (1) the compound value of the $100,000

in the savings account (2) the compound value of the $300,000 in stocks, (3)the additional savings due to depositing $10,000 per year in the savingsaccount for 10 years, and (4) the additional savings due to depositing

$10,000 per year in the savings account at the end of years 6-10 (Note the

$20,000 deposited in years 6-10 is covered in parts (3) and (4).)(1) Future value of $100,000

FV10 = $100,000 (1 + 07)10

(2) Future value of $300,000

n 0

10 0

t

.07)(1

= $10,000 (13.816)

= $138,160(4) Compound annuity of $10,000 (years 6 - 10)

5

0

t

.07) (1

t (1 10)

1

5-40A FVn = PV (FVIFi%, n yr.)

$6,500 = 12(FVIFi%, 37 yr.)solving using a financial calculator:

= $50,000 (8.365-3.170) + $250,000 (.149) + $50,000 (0.112 + 102) + $100,000 (.092)

= $259,750 + $37,250 + $10,700 + $9,200

= $316,900(b) If you live longer than expected you could end up with no money later on in

life

5-42A rate (i) = 8%

number of periods (n) = 7payment (PMT) = $0present value (PV) = $900type (0 = at end of period) = 0

Future value = $1,542.44Excel formula: =FV(rate,number of periods,payment,present value,type)

Notice that present value ($900) took on a negative value

5-43A In 20 years you’d like to have $250,000 to buy a home, but you only have $30,000

At what rate must your $30,000 be compounded annually for it to grow to $250,000 in

20 years?

number of periods (n) = 20payment (PMT) = $0present value (PV) = $30,000future value (FV) = $250,000type (0 = at end of period) = 0

guess =

i = 11.18%

Excel formula: =RATE(number of periods,payment,present value,future

value,type,guess)

Notice that present value ($30,000) took on a negative value

5-44A To buy a new house you take out a 25 year mortgage for $300,000 What will your

monthly interest rate payments be if the interest rate on your mortgage is 8 percent?

Two things to keep in mind when you're working this problem: first, you'll have toconvert the annual rate of 8 percent into a monthly rate by dividing it by 12, andsecond, you'll have to convert the number of periods into months by multiplying 25times 12 for a total of 300 months

Excel formula: =PMT(rate,number of periods,present value,future value,type)

type (0 = at end of period) = 0

monthly mortgage payment = ($2,315.45)

Notice that monthly payments take on a negative value because you pay them