(2 years), he will be paid 4% interest compounded over four periods, each of

154 PART 2 Important Financial Concepts

TABLE 4.5 The Future Value from Investing

$100 at 8% Interest Compounded

Semiannually Over 24 Months

(2 Years)

Beginning Future value Future value at end

principal interest factor of period [(1) (2)]

Period (1) (2) (3)

6 months $100.00 1.04 $104.00

12 months 104.00 1.04 108.16

18 months 108.16 1.04 112.49

24 months 112.49 1.04 116.99

which is 6 months long. Table 4.5 uses interest factors to show that at the end of

12 months (1 year) with 8% semiannual compounding, Fred will have $108.16;

at the end of 24 months (2 years), he will have $116.99.

Quarterly Compounding

Quarterly compounding of interest involves four compounding periods within

quarterly compounding

Compounding of interest over the year. One-fourth of the stated interest rate is paid four times a year.

four periods within the year.

Fred Moreno has found an institution that will pay him 8% interest com-

EXAMPLE

pounded quarterly. If he leaves his money in this account for 24 months (2

years), he will be paid 2% interest compounded over eight periods, each of

which is 3 months long. Table 4.6 uses interest factors to show the amount Fred

TABLE 4.6 The Future Value from Investing

$100 at 8% Interest Compounded

Quarterly Over 24 Months

(2 Years)

Beginning Future value Future value at end

principal interest factor of period [(1) (2)]

Period (1) (2) (3)

3 months $100.00 1.02 $102.00

6 months 102.00 1.02 104.04

9 months 104.04 1.02 106.12

12 months 106.12 1.02 108.24

15 months 108.24 1.02 110.40

18 months 110.40 1.02 112.61

21 months 112.61 1.02 114.86

24 months 114.86 1.02 117.16

155

CHAPTER 4 Time Value of Money

TABLE 4.7 The Future Value at the End of

Years 1 and 2 from Investing $100

at 8% Interest, Given Various

Compounding Periods

Compounding period

End of year Annual Semiannual Quarterly

1 $108.00 $108.16 $108.24

2 116.64 116.99 117.16

will have at the end of each period. At the end of 12 months (1 year), with 8%

quarterly compounding, Fred will have $108.24; at the end of 24 months (2

years), he will have $117.16.

Table 4.7 compares values for Fred Moreno™s $100 at the end of years 1

and 2 given annual, semiannual, and quarterly compounding periods at the 8

percent rate. As shown, the more frequently interest is compounded, the greater

the amount of money accumulated. This is true for any interest rate for any

period of time.

A General Equation for

Compounding More Frequently Than Annually

The formula for annual compounding (Equation 4.4) can be rewritten for use

when compounding takes place more frequently. If m equals the number of times

per year interest is compounded, the formula for annual compounding can be

rewritten as

mn

i

FVn PV 1 (4.18)

m

If m 1, Equation 4.18 reduces to Equation 4.4. Thus, if interest is com-

pounded annually (once a year), Equation 4.18 will provide the same result as

Equation 4.4. The general use of Equation 4.18 can be illustrated with a simple

example.

The preceding examples calculated the amount that Fred Moreno would have at

EXAMPLE

the end of 2 years if he deposited $100 at 8% interest compounded semiannually

and compounded quarterly. For semiannual compounding, m would equal 2 in

Equation 4.18; for quarterly compounding, m would equal 4. Substituting the

appropriate values for semiannual and quarterly compounding into Equation

4.18, we find that

1. For semiannual compounding:

22

0.08

0.04)4

FV2 $100 1 $100 (1 $116.99

2

156 PART 2 Important Financial Concepts

2. For quarterly compounding:

42

0.08

0.02)8

FV2 $100 1 $100 (1 $117.16

4

These results agree with the values for FV2 in Tables 4.5 and 4.6.

If the interest were compounded monthly, weekly, or daily, m would equal 12,

52, or 365, respectively.

Using Computational Tools

for Compounding More Frequently Than Annually

We can use the future value interest factors for one dollar, given in Table A“1,

when interest is compounded m times each year. Instead of indexing the table for

i percent and n years, as we do when interest is compounded annually, we index

it for (i m) percent and (m n) periods. However, the table is less useful,

because it includes only selected rates for a limited number of periods. Instead, a

financial calculator or a computer and spreadsheet is typically required.

Fred Moreno wished to find the future value of $100 invested at 8% interest

EXAMPLE

compounded both semiannually and quarterly for 2 years. The number of com-

pounding periods, m, the interest rate, and the number of periods used in each

case, along with the future value interest factor, are as follows:

Compounding Interest rate Periods Future value interest factor

period m (i · m) (m n) from Table A“1

Input Function

Semiannual 2 8% 2 4% 2 2 4 1.170

100 PV

Quarterly 4 8% 4 2% 4 2 8 1.172

4 N

4 I

CPT

Table Use Multiplying each of the future value interest factors by the initial

FV

$100 deposit results in a value of $117.00 (1.170 $100) for semiannual com-

pounding and a value of $117.20 (1.172 $100) for quarterly compounding.

Solution

116.99

Calculator Use If the calculator were used for the semiannual compounding

calculation, the number of periods would be 4 and the interest rate would be

Input Function

4%. The future value of $116.99 will appear on the calculator display as

100 PV

shown in the first display at the left.

8 N

For the quarterly compounding case, the number of periods would be 8 and

2 I

the interest rate would be 2%. The future value of $117.17 will appear on the calcu-

CPT

lator display as shown in the second display at the left.

FV

Spreadsheet Use The future value of the single amount with semiannual and

Solution

quarterly compounding also can be calculated as shown on the following Excel

117.17

spreadsheet.

157

CHAPTER 4 Time Value of Money

Comparing the calculator, table, and spreadsheet values, we can see that the

calculator and spreadsheet values agree generally with the values in Table 4.7 but

are more precise because the table factors have been rounded.

Continuous Compounding

In the extreme case, interest can be compounded continuously. Continuous

compounding involves compounding over every microsecond”the smallest time

continuous compounding

Compounding of interest an period imaginable. In this case, m in Equation 4.18 would approach infinity.

infinite number of times per year Through the use of calculus, we know that as m approaches infinity, the equation

at intervals of microseconds.

becomes

(ei n)

FVn (continuous compounding) PV (4.19)

where e is the exponential function10, which has a value of 2.7183. The future

value interest factor for continuous compounding is therefore

ei n

FVIFi,n (continuous compounding) (4.20)

To find the value at the end of 2 years (n 2) of Fred Moreno™s $100 deposit

EXAMPLE

(PV $100) in an account paying 8% annual interest (i 0.08) compounded

continuously, we can substitute into Equation 4.19:

e0.08 2

FV2 (continuous compounding) $100

2.71830.16

$100

$100 1.1735 $117.35

Calculator Use To find this value using the calculator, you need first to find the

value of e0.16 by punching in 0.16 and then pressing 2nd and then ex to get 1.1735.

10. Most calculators have the exponential function, typically noted by ex, built into them. The use of this key is espe-

cially helpful in calculating future value when interest is compounded continuously.

158 PART 2 Important Financial Concepts

Next multiply this value by $100 to get the future value of $117.35 as shown at the

Input Function

left. (Note: On some calculators, you may not have to press 2nd before pressing ex.)

2nd

0.16

ex Spreadsheet Use The future value of the single amount with continuous com-

pounding also can be calculated as shown on the following Excel spreadsheet.

1.1735

100

Solution

117.35

The future value with continuous compounding therefore equals $117.35. As

expected, the continuously compounded value is larger than the future value of

interest compounded semiannually ($116.99) or quarterly ($117.16). Continu-

ous compounding offers the largest amount that would result from compounding

interest more frequently than annually.

Nominal and Effective Annual Rates of Interest

Both businesses and investors need to make objective comparisons of loan costs

or investment returns over different compounding periods. In order to put inter-

est rates on a common basis, to allow comparison, we distinguish between nomi-

nal and effective annual rates. The nominal, or stated, annual rate is the contrac-

nominal (stated) annual rate

Contractual annual rate of tual annual rate of interest charged by a lender or promised by a borrower. The

interest charged by a lender or effective, or true, annual rate (EAR) is the annual rate of interest actually paid or

promised by a borrower.

earned. The effective annual rate reflects the impact of compounding frequency,

whereas the nominal annual rate does not.

effective (true) annual rate (EAR)

The annual rate of interest Using the notation introduced earlier, we can calculate the effective annual

actually paid or earned.

rate, EAR, by substituting values for the nominal annual rate, i, and the com-

pounding frequency, m, into Equation 4.21:

im

EAR 1 1 (4.21)

m

We can apply this equation using data from preceding examples.

Fred Moreno wishes to find the effective annual rate associated with an 8% nom-

EXAMPLE

inal annual rate (i 0.08) when interest is compounded (1) annually (m 1);

(2) semiannually (m 2); and (3) quarterly (m 4). Substituting these values into

Equation 4.21, we get

1. For annual compounding:

1

0.08

0.08)1

EAR 1 1 (1 1 1 0.08 1 0.08 8%

1

159

CHAPTER 4 Time Value of Money

2. For semiannual compounding:

2

0.08

0.04)2

EAR 1 1 (1 1 1.0816 1 0.0816 8.16%

2

3. For quarterly compounding:

0.08 4

0.02)4

EAR 1 1 (1 1 1.0824 1 0.0824 8.24%

4

These values demonstrate two important points: The first is that nominal and

effective annual rates are equivalent for annual compounding. The second is that

the effective annual rate increases with increasing compounding frequency, up to

a limit that occurs with continuous compounding.11

At the consumer level, “truth-in-lending laws” require disclosure on credit

card and loan agreements of the annual percentage rate (APR). The APR is the

annual percentage rate (APR)

The nominal annual rate of nominal annual rate found by multiplying the periodic rate by the number of

interest, found by multiplying the

periods in one year. For example, a bank credit card that charges 1 1/2 percent per

periodic rate by the number of

month (the periodic rate) would have an APR of 18% (1.5% per month 12

periods in 1 year, that must be

months per year).

disclosed to consumers on credit

“Truth-in-savings laws,” on the other hand, require banks to quote the

cards and loans as a result of

“truth-in-lending laws.” annual percentage yield (APY) on their savings products. The APY is the effective

annual rate a savings product pays. For example, a savings account that pays 0.5

annual percentage yield (APY)

percent per month would have an APY of 6.17 percent [(1.005)12 1].

The effective annual rate of

Quoting loan interest rates at their lower nominal annual rate (the APR) and

interest that must be disclosed to

consumers by banks on their savings interest rates at the higher effective annual rate (the APY) offers two

savings products as a result of

advantages: It tends to standardize disclosure to consumers, and it enables finan-

“truth-in-savings laws.”

cial institutions to quote the most attractive interest rates: low loan rates and high

savings rates.

Review Questions

4“12 What effect does compounding interest more frequently than annually

have on (a) future value and (b) the effective annual rate (EAR)? Why?

4“13 How does the future value of a deposit subject to continuous compound-

ing compare to the value obtained by annual compounding?

4“14 Differentiate between a nominal annual rate and an effective annual rate

(EAR). Define annual percentage rate (APR) and annual percentage yield

(APY).

11. The effective annual rate for this extreme case can be found by using the following equation:

ek

EAR (continuous compounding) 1 (4.21a)

For the 8% nominal annual rate (k 0.08), substitution into Equation 4.21a results in an effective

annual rate of

e0.08 1 1.0833 1 0.0833 8.33%

in the case of continuous compounding. This is the highest effective annual rate attainable with an

8% nominal rate.

160 PART 2 Important Financial Concepts

Special Applications of Time Value

LG6

Future value and present value techniques have a number of important applica-

tions in finance. We™ll study four of them in this section: (1) deposits needed to

accumulate a future sum, (2) loan amortization, (3) interest or growth rates, and

(4) finding an unknown number of periods.

Deposits Needed to Accumulate a Future Sum

Suppose you want to buy a house 5 years from now, and you estimate that an ini-

tial down payment of $20,000 will be required at that time. To accumulate the

$20,000, you will wish to make equal annual end-of-year deposits into an account

paying annual interest of 6 percent. The solution to this problem is closely related

to the process of finding the future value of an annuity. You must determine what

size annuity will result in a single amount equal to $20,000 at the end of year 5.

Earlier in the chapter we found the future value of an n-year ordinary annu-

ity, FVAn, by multiplying the annual deposit, PMT, by the appropriate interest

factor, FVIFAi,n. The relationship of the three variables was defined by Equation

4.14, which is repeated here as Equation 4.22:

FVAn PMT (FVIFAi,n) (4.22)

We can find the annual deposit required to accumulate FVAn dollars by solv-

ing Equation 4.22 for PMT. Isolating PMT on the left side of the equation gives us

FVAn

PMT (4.23)

FVIFAi,n

Once this is done, we have only to substitute the known values of FVAn and

FVIFAi,n into the right side of the equation to find the annual deposit required.

As just stated, you want to determine the equal annual end-of-year deposits

EXAMPLE

required to accumulate $20,000 at the end of 5 years, given an interest rate

of 6%.

Table Use Table A“3 indicates that the future value interest factor for an

ordinary annuity at 6% for 5 years (FVIFA6%,5yrs) is 5.637. Substituting

Input Function

20,000 FV

FVA5 $20,000 and FVIFA6%,5yrs 5.637 into Equation 4.23 yields an annual

5 N required deposit, PMT, of $3,547.99. Thus if $3,547.99 is deposited at the end of

each year for 5 years at 6% interest, there will be $20,000 in the account at the

6 I

end of the 5 years.

CPT

PMT

Calculator Use Using the calculator inputs shown at the left, you will find the

Solution

annual deposit amount to be $3,547.93. Note that this value, except for a slight

3547.93

rounding difference, agrees with the value found by using Table A“3.

Spreadsheet Use The annual deposit needed to accumulate the future sum also

can be calculated as shown on the following Excel spreadsheet.

161

CHAPTER 4 Time Value of Money

Loan Amortization

The term loan amortization refers to the computation of equal periodic loan pay-

loan amortization

The determination of the equal ments. These payments provide a lender with a specified interest return and

periodic loan payments repay the loan principal over a specified period. The loan amortization process

necessary to provide a lender

involves finding the future payments, over the term of the loan, whose present

with a specified interest return

value at the loan interest rate equals the amount of initial principal borrowed.

and to repay the loan principal

Lenders use a loan amortization schedule to determine these payment amounts

over a specified period.

and the allocation of each payment to interest and principal. In the case of home

loan amortization schedule

mortgages, these tables are used to find the equal monthly payments necessary to

A schedule of equal payments to

amortize, or pay off, the mortgage at a specified interest rate over a 15- to 30-

repay a loan. It shows the alloca-

year period.

tion of each loan payment to

interest and principal. Amortizing a loan actually involves creating an annuity out of a present

amount. For example, say you borrow $6,000 at 10 percent and agree to make

equal annual end-of-year payments over 4 years. To find the size of the payments,

the lender determines the amount of a 4-year annuity discounted at 10 percent

that has a present value of $6,000. This process is actually the inverse of finding

the present value of an annuity.

Earlier in the chapter, we found the present value, PVAn, of an n-year annu-

ity by multiplying the annual amount, PMT, by the present value interest factor

for an annuity, PVIFAi,n. This relationship, which was originally expressed as

Equation 4.16, is repeated here as Equation 4.24:

PVAn PMT (PVIFAi,n) (4.24)

To find the equal annual payment required to pay off, or amortize, the loan,

PVAn, over a certain number of years at a specified interest rate, we need to solve

Equation 4.24 for PMT. Isolating PMT on the left side of the equation gives us

PVAn

PMT (4.25)

PVIFAi,n

Once this is done, we have only to substitute the known values into the righthand

side of the equation to find the annual payment required.

As just stated, you want to determine the equal annual end-of-year payments nec-

EXAMPLE

essary to amortize fully a $6,000, 10% loan over 4 years.

162 PART 2 Important Financial Concepts

TABLE 4.8 Loan Amortization Schedule ($6,000

Principal, 10% Interest, 4-Year Repayment

Period)

Payments

Beginning- End-of-year

End of-year Loan Interest Principal principal

of principal payment [0.10 (1)] [(2) (3)] [(1) (4)]

year (1) (2) (3) (4) (5)

1 $6,000.00 $1,892.74 $600.00 $1,292.74 $4,707.26

2 4,707.26 1,892.74 470.73 1,422.01 3,285.25

3 3,285.25 1,892.74 328.53 1,564.21 1,721.04

”a

4 1,721.04 1,892.74 172.10 1,720.64

aBecause of rounding, a slight difference ($0.40) exists between the beginning-of-year-4 principal

(in column 1) and the year-4 principal payment (in column 4).

Table Use Table A“4 indicates that the present value interest factor for an

annuity corresponding to 10% and 4 years (PVIFA10%,4yrs) is 3.170. Substituting

PVA4 $6,000 and PVIFA10%,4yrs 3.170 into Equation 4.25 and solving for

PMT yield an annual loan payment of $1,892.74. Thus to repay the interest and

principal on a $6,000, 10%, 4-year loan, equal annual end-of-year payments of

$1,892.74 are necessary.

Calculator Use Using the calculator inputs shown at the left, you will find the

annual payment amount to be $1,892.82. Except for a slight rounding difference,

Input Function

6000 PV

this value agrees with the table solution.

4 N The allocation of each loan payment to interest and principal can be seen in

columns 3 and 4 of the loan amortization schedule in Table 4.8. The portion of

10 I

each payment that represents interest (column 3) declines over the repayment

CPT

period, and the portion going to principal repayment (column 4) increases. This

PMT

pattern is typical of amortized loans; as the principal is reduced, the interest com-

Solution

ponent declines, leaving a larger portion of each subsequent loan payment to

1892.82

repay principal.

Spreadsheet Use The annual payment to repay the loan also can be calculated

as shown on the first Excel spreadsheet. The amortization schedule allocating

each loan payment to interest and principal also can be calculated precisely as

shown on the second spreadsheet.

163

CHAPTER 4 Time Value of Money

Interest or Growth Rates

It is often necessary to calculate the compound annual interest or growth rate

(that is, the annual rate of change in values) of a series of cash flows. Examples

include finding the interest rate on a loan, the rate of growth in sales, and the rate

of growth in earnings. In doing this, we can use either future value or present

value interest factors. The use of present value interest factors is described in this

section. The simplest situation is one in which a person wishes to find the rate of

interest or growth in a series of cash flows.12

Ray Noble wishes to find the rate of interest or growth reflected in the stream of

EXAMPLE

cash flows he received from a real estate investment over the period 1999 through

2003. The following table lists those cash flows:

Year Cash flow

2003 $1,520

}4

2002 1,440

}3

2001 1,370

}2

2000 1,300

}1

1999 1,250

By using the first year (1999) as a base year, we see that interest has been earned

(or growth experienced) for 4 years.

12. Because the calculations required for finding interest rates and growth rates, given the series of cash flows, are

the same, this section refers to the calculations as those required to find interest or growth rates.

164 PART 2 Important Financial Concepts

In Practice

FOCUS ON PRACTICE Time Is on Your Side

For many years, the 30-year fixed-

rate mortgage was the traditional

Monthly principal Total interest paid

choice of home buyers. In recent

Term Rate and interest over the term of the loan

years, however, more homeown-

ers are choosing fixed-rate mort- 15 years 6.50% $1,742 $113,625

gages with a 15-year term when

30 years 6.85% $1,311 $271,390

they buy a new home or refinance

their current residence. They are

often pleasantly surprised to dis-

cover that they can pay off the Why isn™t everyone rushing to Yet another option is to make

loan in half the time with a monthly take out a shorter mortgage? Many additional principal payments

payment that is only about 25 per- homeowners either can™t afford whenever possible. This shortens

cent higher. Not only will they own the higher monthly payment or the life of the loan without commit-

the home free and clear sooner, would rather have the extra spend- ting you to the higher payments. By

but they pay considerably less in- ing money now. Others hope to do paying just $100 more each month,

terest over the life of the loan. even better by investing the differ- you can shorten the life of a 30-

year mortgage to 24 1/4 years, with

For example, assume you ence themselves. Suppose you in-

need a $200,000 mortgage and can vested $431 each month in a mu- attendant interest savings.

borrow at fixed rates. The shorter tual fund with an average annual

loan would carry a lower rate (be- return of 7 percent. At the end of

Sources: Daniela Deane, “Adding Up Pros,

cause it presents less risk for the 15 years, your $77,580 investment Cons of 15-Year Loans,” Washington Post

lender). The accompanying table would have grown to $136,611, or (October 13, 2001), p. H7; Henry Savage, “Is

15-Year Loan Right for You?” Washington

shows how the two mortgages $59,031 more than you contributed! Times (June 22, 2001), p. F22; Carlos Tejada,

compare: The extra $431 a month, However, many people lack the “Sweet Fifteen: Shorter Mortgages Are Gain-

ing Support,” Wall Street Journal (Septem-

or a total of $77,580, saves $157,765 self-discipline to save rather than

ber 17, 1998), p. C1; Ann Tergesen, “It™s Time

in interest payments over the life spend that money. For them, the to Refinance . . . Again,” Business Week

of the loan, for net savings of 15-year mortgage represents (November 2, 1998), pp. 134“135.

$80,185! forced savings.

Table Use The first step in finding the interest or growth rate is to divide the

amount received in the earliest year (PV) by the amount received in the latest year

(FVn). Looking back at Equation 4.12, we see that this results in the present value

interest factor for a single amount for 4 years, PVIFi,4yrs, which is 0.822

($1,250 $1,520). The interest rate in Table A“2 associated with the factor clos-

est to 0.822 for 4 years is the interest or growth rate of Ray™s cash flows. In the

row for year 4 in Table A“2, the factor for 5 percent is 0.823”almost exactly the

0.822 value. Therefore, the interest or growth rate of the given cash flows is

approximately (to the nearest whole percent) 5%.13

Calculator Use Using the calculator, we treat the earliest value as a present

value, PV, and the latest value as a future value, FVn. (Note: Most calculators

13. To obtain more precise estimates of interest or growth rates, interpolation”a mathematical technique for esti-

WW mating unknown intermediate values”can be applied. For information on how to interpolate a more precise answer

W

in this example, see the book™s home page at www.aw.com/gitman.

165

CHAPTER 4 Time Value of Money

require either the PV or the FV value to be input as a negative number to cal-

Input Function

culate an unknown interest or growth rate. That approach is used here.) Using

1250 PV

the inputs shown at the left, you will find the interest or growth rate to be

1520 FV

5.01%, which is consistent with, but more precise than, the value found using

4 N

Table A“2.

CPT

Spreadsheet Use The interest or growth rate for the series of cash flows also can

I

be calculated as shown on the following Excel spreadsheet.

Solution

5.01

Another type of interest-rate problem involves finding the interest rate asso-

ciated with an annuity, or equal-payment loan.

Jan Jacobs can borrow $2,000 to be repaid in equal annual end-of-year amounts

EXAMPLE

of $514.14 for the next 5 years. She wants to find the interest rate on this loan.

Table Use Substituting PVA5 $2,000 and PMT $514.14 into Equation 4.24

and rearranging the equation to solve for PVIFAi,5yrs, we get

PVA5 $2,000

PVIFAi,5yrs 3.890 (4.26)

PMT $514.14

The interest rate for 5 years associated with the annuity factor closest to 3.890 in

Input Function

514.14 PMT

Table A“4 is 9%. Therefore, the interest rate on the loan is approximately (to the

2000 PV nearest whole percent) 9%.

5 N

Calculator Use (Note: Most calculators require either the PMT or the PV value

CPT

to be input as a negative number in order to calculate an unknown interest rate

I

on an equal-payment loan. That approach is used here.) Using the inputs shown

Solution

at the left, you will find the interest rate to be 9.00%, which is consistent with the

9.00

value found using Table A“4.

166 PART 2 Important Financial Concepts

Spreadsheet Use The interest or growth rate for the annuity also can be calcu-

lated as shown on the following Excel spreadsheet.

Finding an Unknown Number of Periods

Sometimes it is necessary to calculate the number of time periods needed to gener-

ate a given amount of cash flow from an initial amount. Here we briefly consider

this calculation for both single amounts and annuities. This simplest case is when

a person wishes to determine the number of periods, n, it will take for an initial de-

posit, PV, to grow to a specified future amount, FVn, given a stated interest rate, i.

Ann Bates wishes to determine the number of years it will take for her initial

EXAMPLE

$1,000 deposit, earning 8% annual interest, to grow to equal $2,500. Simply

stated, at an 8% annual rate of interest, how many years, n, will it take for Ann™s

$1,000, PV, to grow to $2,500, FVn?

Table Use In a manner similar to our approach above to finding an unknown

interest or growth rate in a series of cash flows, we begin by dividing the amount

deposited in the earliest year by the amount received in the latest year. This

results in the present value interest factor for 8% and n years, PVIF8%,n, which is

0.400 ($1,000 $2,500). The number of years (periods) in Table A“2 associated

with the factor closest to 0.400 for an 8% interest rate is the number of years

required for $1,000 to grow into $2,500 at 8%. In the 8% column of Table A“2,

the factor for 12 years is 0.397”almost exactly the 0.400 value. Therefore, the

number of years necessary for the $1,000 to grow to a future value of $2,500 at

8% is approximately (to the nearest year) 12.

Calculator Use Using the calculator, we treat the initial value as the present

Input Function

value, PV, and the latest value as the future value, FVn. (Note: Most calculators

1000 PV

require either the PV or the FV value to be input as a negative number to calcu-

2500 FV

late an unknown number of periods. That approach is used here.) Using the

8 I

inputs shown at the left, we find the number of periods to be 11.91 years,

CPT

which is consistent with, but more precise than, the value found above using

N Table A“2.

Solution

Spreadsheet Use The number of years for the present value to grow to a

11.91

specified future value also can be calculated as shown on the following Excel

spreadsheet.

167

CHAPTER 4 Time Value of Money

Another type of number-of-periods problem involves finding the number of

periods associated with an annuity. Occasionally we wish to find the unknown

life, n, of an annuity, PMT, that is intended to achieve a specific objective, such as

repaying a loan of a given amount, PVAn, with a stated interest rate, i.

Bill Smart can borrow $25,000 at an 11% annual interest rate; equal, annual

EXAMPLE

end-of-year payments of $4,800 are required. He wishes to determine how long it

will take to fully repay the loan. In other words, he wishes to determine how

many years, n, it will take to repay the $25,000, 11% loan, PVAn, if the pay-

ments of $4,800, PMT, are made at the end of each year.

Table Use Substituting PVAn $25,000 and PMT $4,800 into Equation 4.24

and rearranging the equation to solve PVIFA11%,n yrs, we get

PVAn $25,000

PVIFA11%,n yrs 5.208 (4.27)

PMT $4,800

The number of periods for an 11% interest rate associated with the annuity

Input Function

factor closest to 5.208 in Table A“4 is 8 years. Therefore, the number of peri-

4800 PMT

ods necessary to repay the loan fully is approximately (to the nearest year)

25000 PV

8 years.

11 I

Calculator Use (Note: Most calculators require either the PV or the PMT value

CPT

to be input as a negative number in order to calculate an unknown number of

N

periods. That approach is used here.) Using the inputs shown at the left, you will

Solution

find the number of periods to be 8.15, which is consistent with the value found

8.15

using Table A“4.

Spreadsheet Use The number of years to pay off the loan also can be calculated

as shown on the following Excel spreadsheet.

168 PART 2 Important Financial Concepts

Review Questions

4“15 How can you determine the size of the equal annual end-of-period deposits

necessary to accumulate a certain future sum at the end of a specified future

period at a given annual interest rate?

4“16 Describe the procedure used to amortize a loan into a series of equal peri-

odic payments.

4“17 Which present value interest factors would be used to find (a) the growth

rate associated with a series of cash flows and (b) the interest rate associ-

ated with an equal-payment loan?

4“18 How can you determine the unknown number of periods when you know

the present and future values”single amount or annuity”and the applic-

able rate of interest?

SUMMARY

FOCUS ON VALUE

Time value of money is an important tool that financial managers and other market partici-

pants use to assess the impact of proposed actions. Because firms have long lives and their

important decisions affect their long-term cash flows, the effective application of time-

value-of-money techniques is extremely important. Time value techniques enable financial

managers to evaluate cash flows occurring at different times in order to combine, compare,

and evaluate them and link them to the firm™s overall goal of share price maximization. It

will become clear in Chapters 6 and 7 that the application of time value techniques is a key

part of the value determination process. Using them, we can measure the firm™s value and

evaluate the impact that various events and decisions might have on it. Clearly, an under-

standing of time-value-of-money techniques and an ability to apply them are needed in

order to make intelligent value-creating decisions.

REVIEW OF LEARNING GOALS

agers rely primarily on present value techniques.

Discuss the role of time value in finance, the use

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Financial tables, financial calculators, and comput-

of computational tools, and the basic patterns

ers and spreadsheets can streamline the application

of cash flow. Financial managers and investors use

of time value techniques. The cash flow of a firm

time-value-of-money techniques when assessing the

can be described by its pattern”single amount, an-

value of the expected cash flow streams associated

nuity, or mixed stream.

with investment alternatives. Alternatives can be as-

sessed by either compounding to find future value or

Understand the concepts of future and present

discounting to find present value. Because they are

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value, their calculation for single amounts, and

at time zero when making decisions, financial man-

169

CHAPTER 4 Time Value of Money

the relationship of present value to future value. future value of a mixed stream of cash flows is the

Future value relies on compound interest to mea- sum of the future values of each individual cash

sure future amounts: The initial principal or deposit flow. Similarly, the present value of a mixed stream

in one period, along with the interest earned on it, of cash flows is the sum of the present values of the

becomes the beginning principal of the following individual cash flows.

period. The present value of a future amount is the

amount of money today that is equivalent to the Understand the effect that compounding inter-

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given future amount, considering the return that est more frequently than annually has on future

can be earned on the current money. Present value value and on the effective annual rate of interest.

is the inverse future value. The interest factor for- Interest can be compounded at intervals ranging

mulas and basic equations for both the future value from annually to daily, and even continuously. The

and the present value of a single amount are given more often interest is compounded, the larger the

in Table 4.9. future amount that will be accumulated, and the

higher the effective, or true, annual rate (EAR). The

Find the future value and the present value of annual percentage rate (APR)”a nominal annual

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an ordinary annuity and find the present value rate”is quoted on credit cards and loans. The an-

of a perpetuity. An annuity is a pattern of equal nual percentage yield (APY)”an effective annual

periodic cash flows. For an ordinary annuity, the rate”is quoted on savings products. The interest

cash flows occur at the end of the period. For an factor formulas for compounding more frequently

annuity due, cash flows occur at the beginning of than annually are given in Table 4.9.

the period. Only ordinary annuities are considered

in this book. The future value of an ordinary annu- Describe the procedures involved in (1) deter-

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ity can be found by using the future value interest mining deposits to accumulate a future sum,

factor for an annuity; the present value of an ordi- (2) loan amortization, (3) finding interest or growth

nary annuity can be found by using the present rates, and (4) finding an unknown number of peri-

value interest factor for an annuity. The present ods. The periodic deposit to accumulate a given fu-

value of a perpetuity”an infinite-lived annuity”is ture sum can be found by solving the equation for

found using 1 divided by the discount rate to rep- the future value of an annuity for the annual pay-

resent the present value interest factor. The interest ment. A loan can be amortized into equal periodic

factor formulas and basic equations for the future payments by solving the equation for the present

value and the present value of an ordinary annuity value of an annuity for the periodic payment. Inter-

and the present value of a perpetuity, are given in est or growth rates can be estimated by finding the

Table 4.9. unknown interest rate in the equation for the pre-

sent value of a single amount or an annuity. Simi-

Calculate both the future value and the present larly, an unknown number of periods can be esti-

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value of a mixed stream of cash flows. A mixed mated by finding the unknown number of periods

stream of cash flows is a stream of unequal periodic in the equation for the present value of a single

cash flows that reflect no particular pattern. The amount or an annuity.

SELF-TEST PROBLEMS (Solutions in Appendix B)

ST 4“1 Future values for various compounding frequencies Delia Martin has $10,000

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that she can deposit in any of three savings accounts for a 3-year period. Bank A

compounds interest on an annual basis, bank B compounds interest twice each

year, and bank C compounds interest each quarter. All three banks have a stated

annual interest rate of 4%.

170 PART 2 Important Financial Concepts

TABLE 4.9 Summary of Key Definitions, Formulas, and

Equations for Time Value of Money

Definitions of variables

e exponential function 2.7183

EAR effective annual rate

FVn future value or amount at the end of period n

FVAn future value of an n-year annuity

i annual rate of interest

m number of times per year interest is compounded

n number of periods”typically years”over which money earns a return

PMT amount deposited or received annually at the end of each year

PV initial principal or present value

PVAn present value of an n-year annuity

t period number index

Interest factor formulas

Future value of a single amount with annual compounding:

i)n

FVIFi,n (1 [Eq. 4.5; factors in Table A“1]

Present value of a single amount:

1

PVIFi,n [Eq. 4.11; factors in Table A“2]

(1 i)n

Future value of an ordinary annuity:

n

i)t 1

FVIFAi,n (1 [Eq. 4.13; factors in Table A“3]

t=1

Present value of an ordinary annuity:

n

1

PVIFAi,n [Eq. 4.15; factors in Table A“4]

t

t=1 (1 i)

Present value of a perpetuity:

1

PVIFAi,∞ [Eq. 4.17]

i

Future value with compounding more frequently than annually:

i mn

FVIFi,n 1 [Eq. 4.18]

m

∞:

for continuous compounding, m

ei n

FVIFi,n (continuous compounding) [Eq. 4.20]

to find the effective annual rate:

im

EAR 1 1 [Eq. 4.21]

m

Basic equations

Future value (single amount): FVn PV (FVIFi,n) [Eq. 4.6]

Present value (single amount): PV FVn (PVIFi,n) [Eq. 4.12]

Future value (annuity): FVAn PMT (FVIFAi,n) [Eq. 4.14]

Present value (annuity): PVAn PMT (PVIFAi,n) [Eq. 4.16]

171

CHAPTER 4 Time Value of Money

a. What amount would Ms. Martin have at the end of the third year, leaving all

interest paid on deposit, in each bank?

b. What effective annual rate (EAR) would she earn in each of the banks?

c. On the basis of your findings in parts a and b, which bank should Ms.

Martin deal with? Why?

d. If a fourth bank (bank D), also with a 4% stated interest rate, compounds

interest continuously, how much would Ms. Martin have at the end of the

third year? Does this alternative change your recommendation in part c?

Explain why or why not.

ST 4“2 Future values of annuities Ramesh Abdul wishes to choose the better of two

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equally costly cash flow streams: annuity X and annuity Y. X provides a cash

inflow of $9,000 at the end of each of the next 6 years. Y provides a cash inflow

of $10,000 at the end of each of the next 6 years. Assume that Ramesh can earn

15% on annuity X and 11% on annuity Y.

a. On a purely subjective basis, which annuity do you think is more attractive?

Why?

b. Find the future value at the end of year 6, FVA6, for both annuity X and

annuity Y.

c. Use your finding in part b to indicate which annuity is more attractive. Com-

pare your finding to your subjective response in part a.

ST 4“3 Present values of single amounts and streams You have a choice of accepting

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either of two 5-year cash flow streams or single amounts. One cash flow stream

is an ordinary annuity, and the other is a mixed stream. You may accept alterna-

tive A or B”either as a cash flow stream or as a single amount. Given the cash

flow stream and single amounts associated with each (see the accompanying

table), and assuming a 9% opportunity cost, which alternative (A or B) and in

which form (cash flow stream or single amount) would you prefer?

Cash flow stream

End of year Alternative A Alternative B

1 $700 $1,100

2 700 900

3 700 700

4 700 500

5 700 300

Single amount

At time zero $2,825 $2,800

ST 4“4 Deposits needed to accumulate a future sum Judi Janson wishes to accumulate

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$8,000 by the end of 5 years by making equal annual end-of-year deposits over

the next 5 years. If Judi can earn 7% on her investments, how much must she

deposit at the end of each year to meet this goal?

172 PART 2 Important Financial Concepts

PROBLEMS

4“1 Using a time line The financial manager at Starbuck Industries is considering

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an investment that requires an initial outlay of $25,000 and is expected to result

in cash inflows of $3,000 at the end of year 1, $6,000 at the end of years 2 and

3, $10,000 at the end of year 4, $8,000 at the end of year 5, and $7,000 at the

end of year 6.

a. Draw and label a time line depicting the cash flows associated with Starbuck

Industries™ proposed investment.

b. Use arrows to demonstrate, on the time line in part a, how compounding to

find future value can be used to measure all cash flows at the end of year 6.

c. Use arrows to demonstrate, on the time line in part b, how discounting to

find present value can be used to measure all cash flows at time zero.

d. Which of the approaches”future value or present value”do financial man-

agers rely on most often for decision making? Why?

4“2 Future value calculation Without referring to tables or to the preprogrammed

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function on your financial calculator, use the basic formula for future value

along with the given interest rate, i, and the number of periods, n, to calculate

the future value interest factor in each of the cases shown in the following table.

Compare the calculated value to the value in Appendix Table A“1.

Case Interest rate, i Number of periods, n

A 12% 2

B 6 3

C 9 2

D 3 4

4“3 Future value tables Use the future value interest factors in Appendix Table A“1

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in each of the cases shown in the following table to estimate, to the nearest year,

how long it would take an initial deposit, assuming no withdrawals,

a. To double.

b. To quadruple.

Case Interest rate

A 7%

B 40

C 20

D 10

4“4 Future values For each of the cases shown in the following table, calculate the

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future value of the single cash flow deposited today that will be available at the

end of the deposit period if the interest is compounded annually at the rate speci-

fied over the given period.

173

CHAPTER 4 Time Value of Money

Case Single cash flow Interest rate Deposit period (years)

A $ 200 5% 20

B 4,500 8 7

C 10,000 9 10

D 25,000 10 12

E 37,000 11 5

F 40,000 12 9

4“5 Future value You have $1,500 to invest today at 7% interest compounded

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annually.

a. Find how much you will have accumulated in the account at the end of

(1) 3 years, (2) 6 years, and (3) 9 years.

b. Use your findings in part a to calculate the amount of interest earned in

(1) the first 3 years (years 1 to 3), (2) the second 3 years (years 4 to 6), and

(3) the third 3 years (years 7 to 9).

c. Compare and contrast your findings in part b. Explain why the amount of

interest earned increases in each succeeding 3-year period.

4“6 Inflation and future value As part of your financial planning, you wish to pur-

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chase a new car exactly 5 years from today. The car you wish to purchase costs

$14,000 today, and your research indicates that its price will increase by 2% to

4% per year over the next 5 years.

a. Estimate the price of the car at the end of 5 years if inflation is (1) 2% per

year, and (2) 4% per year.

b. How much more expensive will the car be if the rate of inflation is 4% rather

than 2%?

4“7 Future value and time You can deposit $10,000 into an account paying 9%

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annual interest either today or exactly 10 years from today. How much better

off will you be at the end of 40 years if you decide to make the initial deposit

today rather than 10 years from today?

4“8 Single-payment loan repayment A person borrows $200 to be repaid in 8 years

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with 14% annually compounded interest. The loan may be repaid at the end of

any earlier year with no prepayment penalty.

a. What amount will be due if the loan is repaid at the end of year 1?

b. What is the repayment at the end of year 4?

c. What amount is due at the end of the eighth year?

4“9 Present value calculation Without referring to tables or to the preprogrammed

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function on your financial calculator, use the basic formula for present value,

along with the given opportunity cost, i, and the number of periods, n, to calcu-

late the present value interest factor in each of the cases shown in the accompa-

nying table. Compare the calculated value to the table value.

174 PART 2 Important Financial Concepts

Opportunity Number of

Case cost, i periods, n

A 2% 4

B 10 2

C 5 3

D 13 2

4“10 Present values For each of the cases shown in the following table, calculate the

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present value of the cash flow, discounting at the rate given and assuming that

the cash flow is received at the end of the period noted.

Single cash End of

Case flow Discount rate period (years)

A $ 7,000 12% 4

B 28,000 8 20

C 10,000 14 12

D 150,000 11 6

E 45,000 20 8

4“11 Present value concept Answer each of the following questions.

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a. What single investment made today, earning 12% annual interest, will be

worth $6,000 at the end of 6 years?

b. What is the present value of $6,000 to be received at the end of 6 years if the

discount rate is 12%?

c. What is the most you would pay today for a promise to repay you $6,000 at

the end of 6 years if your opportunity cost is 12%?

d. Compare, contrast, and discuss your findings in parts a through c.

4“12 Present value Jim Nance has been offered a future payment of $500 three years

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from today. If his opportunity cost is 7% compounded annually, what value

should he place on this opportunity today? What is the most he should pay to

purchase this payment today?

4“13 Present value An Iowa state savings bond can be converted to $100 at maturity

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6 years from purchase. If the state bonds are to be competitive with U.S. Savings

Bonds, which pay 8% annual interest (compounded annually), at what price

must the state sell its bonds? Assume no cash payments on savings bonds prior

to redemption.

4“14 Present value and discount rates You just won a lottery that promises to pay

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you $1,000,000 exactly 10 years from today. Because the $1,000,000 payment

is guaranteed by the state in which you live, opportunities exist to sell the claim

today for an immediate single cash payment.

a. What is the least you will sell your claim for if you can earn the following

rates of return on similar-risk investments during the 10-year period?

(1) 6% (2) 9% (3) 12%

175

CHAPTER 4 Time Value of Money

b. Rework part a under the assumption that the $1,000,000 payment will be

received in 15 rather than 10 years.

c. On the basis of your findings in parts a and b, discuss the effect of both the

size of the rate of return and the time until receipt of payment on the present

value of a future sum.

4“15 Present value comparisons of single amounts In exchange for a $20,000 pay-

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ment today, a well-known company will allow you to choose one of the alterna-

tives shown in the following table. Your opportunity cost is 11%.

Alternative Single amount

A $28,500 at end of 3 years

B $54,000 at end of 9 years

C $160,000 at end of 20 years

a. Find the value today of each alternative.

b. Are all the alternatives acceptable, i.e., worth $20,000 today?

c. Which alternative, if any, will you take?

4“16 Cash flow investment decision Tom Alexander has an opportunity to purchase

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any of the investments shown in the following table. The purchase price, the

amount of the single cash inflow, and its year of receipt are given for each invest-

ment. Which purchase recommendations would you make, assuming that Tom

can earn 10% on his investments?

Investment Price Single cash inflow Year of receipt

A $18,000 $30,000 5

B 600 3,000 20

C 3,500 10,000 10

D 1,000 15,000 40

4“17 Future value of an annuity For each of the cases shown in the following

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table, calculate the future value of the annuity at the end of the deposit

period, assuming that the annuity cash flows occur at the end of each

year.

Amount of Interest Deposit period

Case annuity rate (years)

A $ 2,500 8% 10

B 500 12 6

C 30,000 20 5

D 11,500 9 8

E 6,000 14 30

176 PART 2 Important Financial Concepts

4“18 Present value of an annuity For each of the cases shown in the table below, cal-

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culate the present value of the annuity, assuming that the annuity cash flows

occur at the end of each year.

Case Amount of annuity Interest rate Period (years)

A $ 12,000 7% 3

B 55,000 12 15

C 700 20 9

D 140,000 5 7

E 22,500 10 5

4“19 Future value of a retirement annuity Hal Thomas, a 25-year-old college

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graduate, wishes to retire at age 65. To supplement other sources of

retirement income, he can deposit $2,000 each year into a tax-deferred in-

dividual retirement arrangement (IRA). The IRA will be invested to earn an

annual return of 10%, which is assumed to be attainable over the next

40 years.

a. If Hal makes annual end-of-year $2,000 deposits into the IRA, how much

will he have accumulated by the end of his 65th year?

b. If Hal decides to wait until age 35 to begin making annual end-of-year

$2,000 deposits into the IRA, how much will he have accumulated by the end

of his 65th year?

c. Using your findings in parts a and b, discuss the impact of delaying making

deposits into the IRA for 10 years (age 25 to age 35) on the amount accumu-

lated by the end of Hal™s 65th year.

4“20 Present value of a retirement annuity An insurance agent is trying to sell you

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an immediate-retirement annuity, which for a single amount paid today will pro-

vide you with $12,000 at the end of each year for the next 25 years. You cur-

rently earn 9% on low-risk investments comparable to the retirement annuity.

Ignoring taxes, what is the most you would pay for this annuity?

4“21 Funding your retirement You plan to retire in exactly 20 years. Your goal is to

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create a fund that will allow you to receive $20,000 at the end of each year for

the 30 years between retirement and death (a psychic told you would die after

30 years). You know that you will be able to earn 11% per year during the 30-

year retirement period.

a. How large a fund will you need when you retire in 20 years to provide the

30-year, $20,000 retirement annuity?

b. How much will you need today as a single amount to provide the fund calcu-

lated in part a if you earn only 9% per year during the 20 years preceding

retirement?

177

CHAPTER 4 Time Value of Money

c. What effect would an increase in the rate you can earn both during

and prior to retirement have on the values found in parts a and b?

Explain.

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LG2 4“22 Present value of an annuity versus a single amount Assume that you just won

the state lottery. Your prize can be taken either in the form of $40,000 at the

end of each of the next 25 years (i.e., $1,000,000 over 25 years) or as a single

amount of $500,000 paid immediately.

a. If you expect to be able to earn 5% annually on your investments over the

next 25 years, ignoring taxes and other considerations, which alternative

should you take? Why?

b. Would your decision in part a change if you could earn 7% rather than 5%

on your investments over the next 25 years? Why?

c. On a strictly economic basis, at approximately what earnings rate would you

be indifferent between the two plans?

4“23 Perpetuities Consider the data in the following table.

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Perpetuity Annual amount Discount rate

A $ 20,000 8%

B 100,000 10

C 3,000 6

D 60,000 5

Determine, for each of the perpetuities:

a. The appropriate present value interest factor.

b. The present value.

4“24 Creating an endowment Upon completion of her introductory finance

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course, Marla Lee was so pleased with the amount of useful and interesting

knowledge she gained that she convinced her parents, who were wealthy

alums of the university she was attending, to create an endowment. The endow-

ment is to allow three needy students to take the introductory finance course

each year in perpetuity. The guaranteed annual cost of tuition and books for the

course is $600 per student. The endowment will be created by making a single

payment to the university. The university expects to earn exactly 6% per year

on these funds.

a. How large an initial single payment must Marla™s parents make to the univer-

sity to fund the endowment?

b. What amount would be needed to fund the endowment if the university

could earn 9% rather than 6% per year on the funds?

178 PART 2 Important Financial Concepts

4“25 Future value of a mixed stream For each of the mixed streams of cash flows

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shown in the following table, determine the future value at the end of the final

year if deposits are made at the beginning of each year into an account paying

annual interest of 12%, assuming that no withdrawals are made during the

period.

Cash flow stream

Year A B C

1 $ 900 $30,000 $1,200

2 1,000 25,000 1,200

3 1,200 20,000 1,000

4 10,000 1,900

5 5,000

4“26 Future value of a single amount versus a mixed stream Gina Vitale has just

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contracted to sell a small parcel of land that she inherited a few years ago. The

buyer is willing to pay $24,000 at the closing of the transaction or will pay the