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0 Effective interest rate =
(proceeds, % X principal)  interest
18% X $80,000  $14,400  25%


(WO/,$80,000)  $14,400 $57,600
X
11. The cost of financing with a factor is:
Purchased receivable (0.05 X $400,000) $20,000
Lending fee (0.02 X $400,000 X 0.90) 7,200
Total cost $27,200
The cost of financing with a bank is:
Interest (0.025 X $400,000 X 0.90) $ 9,000
Processing charge (0.04 X $400,000 X 0.90) 14,400
Additional cost of not using factor
Credit costs 3,600
Bad debts (0.035 X $400,000) 14,000
Total cost $41,000
It is less expensive to use the factor.
Chapter 6
Time Value of Money
6.1 INTRODUCTION
Time value of money is a critical consideration in financial and investment decisions. For example,
compound interest calculations are needed to determine future sums of money resulting from an
investment. Discounting, or the calculation of present value, which is inversely related to compounding,
is used to evaluate future cash flow associated with capital budgeting projects. There are plenty of
applications of time value of money in finance. The chapter discusses the concepts, calculations, and
applications of future values and present values.
6.2 FUTURE VALUESCOMPOUNDING
A dollar in hand today is worth more than a dollar to be received tomorrow because of the interest
it could earn from putting it in a savings account or placing it in an investment account. Compounding
interest means that interest earns interest. For the discussion of the concepts of compounding and time
value, let us define:
F,, = future value = the amount of money at the end of year n
P = principal
i = annual interest rate
n = number of years
Then,
PI = the amount of money at the end of year 1
= principal and interest = P + i = P ( l + i )
P
F2 = the amount of money at the end of year 2
= Fl(I + i ) = P ( I + i ) ( l + i ) = P ( 1 + i ) 2
The future value of an investment compounded annually at rate i for n years is
+ i)" = PFVIFi,,,
F,, = P ( l
where FVIFj,, is the future value interest factor for $1and can be found in Appendix A.
EXAMPLE 6.1 George Jackson placed $1,0oO in a savings account earning 8 percent interest compounded
annually. How much money will he have in the account at the end of 4 years?
F" = P(1 + i"
)
F = $1,000(1 + 0.08)4 = $1,000 * FVIF*,4
4
From Appendix A, the FVIF for 4 years at 8 percent is 1.3605. Therefore,
F4= $1,000(1.3605) = $1,360.50
EXAMPLE 6.2 Rachael Kahn invested a large sum of money in the stock of TLC Corporation. The company paid
a $3 dividend per share. The dividend is expected to increase by 20 percent per year for the next 3 years. She wishes
to project the dividends for years 1 through 3.
+ i)"
F" = P(l
= $3(1 + 0.2)' = $3(1.2000) = $3.60
Fl
F2 = $3(1 + 0.2)2= $3(1.4400) = $4.32
F3 = $3(1 + 0.2)3= $3(1.7280) = $5.18
158
159
TIME VALUE OF MONEY
CHAP. 61
Intrayear Compounding
Interest is often compounded more frequently than once a year. Banks, for example, compound
interest quarterly, daily, and even continuously. If interest is compounded m times a year, then the
general formula for solving for the future value becomes
at a smaller interest rate per period (ilm).For
The formula reflects more frequent compounding (n em)
example, in the case of semiannual compounding (m = Z), the above formula becomes
i) n.2
F n = P (1 + = P * mPFi/2,n.2
As m approaches infinity, the term (1 + i/m)n.mapproaches e"", where e is approximately 2.71828,
and F, becomes
Fn= p 2'"
The future value increases as m increases. Thus, continuous compounding results in the maximum
possible future value at the end of n periods for a given rate of interest.
EXAMPLE 6 3 Assume that P = $100, i = 12% and n = 3 years. Then for
.
Annual compounding (rn 1): F3 = $100(1+ 0.12)3 = $100(1.404)3= $140.49
=
( + 0.06)6=2
Oi2Y
Semiannual compounding (rn = 2): F3 = $100 1+ 
= $100(1 $100(1.4185) = $141.85
(1+0i2)3'4
Quarterly compounding (rn = 4): F˜ = $100 $100(1+ 0.03)12
=
= $100(1.4257) = $142.57
F:, = $100 1+ 
(
Monthly compounding (m = 12):
0312
= $100(1+ 0.01)36 $100(1.4307) = $143.07
=
Continuous compounding (e""): F3 = $100.e(0.12'3) $100(2.71828)0.N
=
= $100(1.4333) = $143.33
Future Value of an Annuity
An annuity is defined as a series of payments (or receipts) of a fixed amount for a specified number
of periods. Each payment is assumed to occur at the end of the period. The future value of an annuity
is a compound annuity which 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.
Let S n = the future value of an nyear annuity
A = the amount of an annuity
Then we can write
+ A ( l + i)n* +  .+ A ( l + i)'
S, = A ( l + i)nl 8
+ (1 + i)n2+  .+ (1+]')i
= A[(i+ Z)nl
n1
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