стр. 78 
P, PVIFAj,,
*
10, and i = 10%. Therefore,
Here A = $500, n =
Pi0 = $SOO(PVIFA1,10) = $500(6.1446) = $3,072.30
Present Value. Calculate the present value of the following future cash inflows discounted at 10
6.4
percent: ( a ) $1,000 a year for years 1through 10; (6) $1,000 a year for years 5 through 10; and
( c ) $1,000 a year for years 1through 3, nothing in years 4 through 5, then $2,000 a year for years
6 through 10.
SOLUTION
P, PVIFAj,,
=A*
(4
$1,000, i = 10%, and n
Here A 10 years. Therefore,
= =
Pi0 = $l,OOO(PVIFAlo,lo)= $1,000(6.1446) = $6,144.60
P = $1,OOO(PVIFAio,io) $1,000(PVIFAio.4)
= $1,000(6.1446)  $1,000(3.1699) = $6,144.60  $3,169.90 = $2,974.70
This type of annuity is called a deferred annuity.
= $1 ,00O(PVIFAio,3) + [$2,000(PVIFAlo.1") $˜,OOO(PVIFA˜O,S)]

P
= $1,000(2.4869) + [$2,000(6.1446) $2,000(3.7908)]
+ $4,707.60 = $7,194.5
= $2,486.90 + [$12,289.2 $7,581.601= $2,466.90
Present Value. Your favorite uncle has offered you the choice of the following options. He will
6.5
give you either $2,000 1year from now or $3,000 years from now. Which would you choose if
4
the discount rate is ( a ) 10 percent? ( b )20 percent?
SOLUTION
P F, * PVIFj.,
=
( a ) Option 1:$2,000 one year from now. In this case i = 10%, n = 1 , Fl = $2,000, and PVIFlo,l = 0.9091.
Therefore,
P $2,000(0.9091) = $1,818.20
=
167
CHAP. 61 TIME VALUE OF MONEY
Option 2: $3,000 four years from now. In this case i = 10%,n = 4, F4 = $3,000,and PVIFlo,4= 0.6830.
Therefore,
P = $3,000(0.6830) = $2,049
At 10 percent, the best choice is $3,000 four years from now.
( 6 ) Option 1:$2,000 one year from now. In this case i = 20%, n = 1,F1= $2,000, and PVIFm,l = 0.8333.
Therefore,
P = $2,000(0.8333) = $1,666.60
Option 2: $3,000 four years from now. In this case i = 20%, n $3,000, and PVIFm,4= 0.4823.
= 4, F4 =
Therefore,
P = $3,000(0.4823) = $1,446.9
At 20 percent, the best choice is $2,000 one year from now.
Present value. A 55yearold executive will retire at age 65 and expects to live to age 75. Assuming
6.6
a 10 percent rate of return, calculate the amount he must have available at age 65 in order to
receive $lO,OOO annually from retirement until death. (CFA, adapted.)
SOLUTION
This problem involves finding the present value of an annuity. The executive must have $61,446
available at age 65, calculated as follows:
P, = A PVIFA,,,
*
= $10,000, n = 10, i = 10%, and PVIFAlo,lo= 6.1446. Therefore,
Here A
= $10,000(6.1446) = $61,446
Pi0
Present Value. Your father is about to retire. His firm has given him the option of retiring with
6.7
a lump sum of $20,000or an annuity of $2,500 for 10 years. Which is worth more now, if an interest
rate of 6 percent is used for the annuity?
SOLUTION
P, = A PVIFj.,
*
= $2,500, i = 6%, n
Here A = 10, and PVIFA6,10 7.3601. Therefore,
=
= $2,500(PVIFA6,10) = $2,500(7.3601) = $18,400.25
Pi0
The lump sum of $20,000 is worth more now.
6.8 Perpetuities. What is the present value of a perpetuity of $80 per year if the discount rate is 11
percent.
SOLUTION
The present value of a perpetuity = A = = $727.27
$80
7
11%
1
Deposits Required. If you need $6,000 5 years from now, how much of a deposit must you make
6.9
in your savings account each year, assuming an 8 percent annual interest rate?
TIME VALUE O F MONEY [CHAR 6
168
SOLUTION
Solving S n = A * FVIFAi,, for A yields:
In this problem S5 = $6,000,i = 8%, n = 5, and FVIFA8,5= 5.8666. Therefore,
A =  $6вЂ™000 $1,022.74
=
5.8666
61
.0 Sinking Fund. A $1million bond issue is outstanding. Assume deposits earn 8 percent per annum.
Calculate the amount to be deposited to a sinking fund each year in order to accumulate enough
money to retire the entire $1million issue at the end of 20 years. (CFA, adapted.)
SOLUTION
стр. 78 