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value of an annuity.
Sn = A FVIFAi,,
Solving for A , we obtain:
Sn
Sinking fund amount = A =
FVIFAi,,
EXAMPLE 6.9 Mary Czech wishes to determine the equal annual endofyear deposits required to accumulate
$5,O00 at the end of 5 years when her son enters college. The interest rate is 10 percent. The annual deposit is:
TIME VALUE OF MONEY [CHAP. 6
162
ss = $5,000
(from Appendix B)
FVIFAlo,S= 6.1051
A =  $5'000  $818.99 = $819

6.1051
In other words, if she deposits $819 at the end of each year for 5 years at 10 percent interest, she will have
accumulated $5,000 at the end of the fifth year.
Amortized Loans
If a loan is to be repaid in equal periodic amounts, it is said to be an amortized loan. Examples
include auto loans, mortgage loans, and most commercial loans. The periodic payment can easily be
computed as follows:
P n = A * PV1FAi.n
Pn
Amount of loan = A =
Solving for A , we obtain:
PVIFAi.0
EXAMPLE 6.10 Jeff Balthness has a 40month auto loan of $5,000 at a 12 percent annual interest rate. He wants
to find out the monthly loan payment amount.
i = 12% + 12 months = 1%
Pa = $5,000
(from Appendix D)
PVIFA1,&= 32.8347
$152.28
A==$5000
Therefore,
32.8347
So, to repay the principal and interest on a $5,000, 12 percent, 40month loan, Jeff Balthness has to pay $152.28
a month for the next 40 months.
Assume that a firm borrows $2,000 to be repaid in three equal installments at the end of each
EXAMPLE 6.11
of the next 3 years. The bank wants 12 percent interest. Compute the amount of each payment.
P3 = $2,000
PVIFA12.3 = 2.4018
A =  $2'ooo $832.71
=
Therefore,
2,4018
Each loan payment consists partly of interest and partly of principal. The breakdown is often
displayed in a loan amortization schedule. The interest component is largest in the first period and
subsequently declines, whereas the principal portion is smallest in the first period and increases
thereafter, as shown in the following example.
EXAMPLE 6.12 Using the same data as in Example 6.11, we set up the following amortization schedule:
Repayment Remaining
Year Payment Interest of Principal Balance
$832.71 $240,00"
1 $592.71 $1,407.29
2 $832.71 68.88 $663.83 $ 743.46
$1
3 $832.68* $ 89.22 $743.46'
Interest is computed by multiplying the loan balance at the beginning
a
of the year by the interest rate. Therefore, interest in year 1 is
$2,000(0.12) = $240; in year 2 interest is $1,407.29(0.12) = $168.88; and in
year 3 interest is $743.46(0.12) = $89.22. All figures are rounded.
Last payment is adjusted downward.
Not exact because of accumulated rounding errors
163
TIME VALUE OF MONEY
CHAP. 61
Annual Percentage Rate (APR)
Different types of investments use different compounding periods. For example, most bonds pay
interest semiannually; banks generally pay interest quarterly. If an investor wishes to compare
investments with different compounding periods, he or she needs to put them on a common basis. The
annual percentage rate (APR), or effective annual rate, is used for this purpose and is computed as
follows:
A P R = (l+kTl.O
where r = the stated, nominal or quoted rate and m = the number of compounding periods per year.
EXAMPLE 6.13 If the nominal rate is 6 percent, compounded quarterly, the APR is
( 1 + â€˜406)â€˜
(1 +  1.0 = (l.01S)4 1.0 = 1.0614  1.0 = 0.0614 = 6.14%
1.O=
APR= J)nl
This means that if one bank offered 6 percent with quarterly compounding, while another offered 6.14 percent with
annual compounding, they would both be paying the same effective rate of interest.
Rates of Growth
In finance, it is necessary to calculate the compound annual interest rate, or rate of growth, associated
with a stream of earnings.
EXAMPLE 6.14 Assume that the Geico Company has earnings per share of $2.50 in 19x1,and 10years later the
earnings per share has increased to $3.70. The compound annual rate of growth of the earnings per share can be
computed as follows:
Fn= P *FVIFj,,
Solving this for FVIF, we obtain:
FVIFj,lo= = 1.48
$3â€™70
$2.50
From Appendix A an FVIF of 1.48 at 10 years is at i = 4%. The compound annual rate of growth is therefore 4
percent.
Bond Values
Bonds call for the payment of a specific amount of interest for a stated number of years and the
repayment of the face value at the bondâ€™s maturity. Thus, a bond represents an annuity plus a lump sum.
Its value is found as the present value of this payment stream. The interest is usually paid
semiannually.
M
V=C I
â€œ
+(1+ r)â€œ
(1 + r)l
t=l
= I(PVIFA,,,) + M(PVIF,,)
where I = interest payment per period
M = par value, or maturity value, usually $1,000
r = investorâ€™s required rate of return
n = number of periods
This topic is covered in more detail in Chapter 7, â€œRisk, Return, and Valuation.â€
TIME VALUE OF MONEY [CHAR 6
164
EXAMPLE 6.15 Assume there is a 10year bond with a 10 percent coupon, paying interest semiannually and
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