Chapter 4 Solutions
2.
To find the FV of a lump sum, we use:
FV = PV(1 +
r
)
t
a.
FV = $1,000(1.06)
10
= $1,790.85
b.
FV = $1,000(1.09)
10
= $2,367.36
c.
FV = $1,000(1.06)
20
= $3,207.14
d.
Because interest compounds on the interest already earned, the interest earned in part
c
is more than twice the interest earned in part
a
. With compound interest, future values
grow exponentially.
3.
To find the PV of a lump sum, we use:
PV = FV / (1 +
r)
t
PV = $15,451 / (1.07)
6
= $10,295.65
PV = $51,557 / (1.15)
9
= $14,655.72
PV = $886,073 / (1.11)
18
= $135,411.60
PV = $550,164 / (1.18)
23
= $12,223.79
4.
To answer this question, we can use either the FV or the PV formula. Both will give the
same answer since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 +
r
)
t
Solving for
r
, we get:
r
= (FV / PV)
1 /
t
– 1
FV = $307 = $242(1 +
r
)
2
;
r
= ($307 / $242)
1/2
– 1
= 12.63%
FV = $896 = $410(1 +
r
)
9
;
r
= ($896 / $410)
1/9
– 1
= 9.07%
FV = $162,181 = $51,700(1 +
r
)
15
;
r
= ($162,181 / $51,700)
1/15
– 1
= 7.92%
FV = $483,500 = $18,750(1 +
r
)
30
;
r
= ($483,500 / $18,750)
1/30
– 1
= 11.44%
5.
To answer this question, we can use either the FV or the PV formula. Both will give the
same answer since they are the inverse of each other. We will use the FV formula, that is:
FV = PV(1 +
r
)
t
Solving for
t
, we get:
t
= ln(FV / PV) / ln(1 +
r
)
FV = $1,284 = $625(1.06)
t
;
t
= ln($1,284/ $625) / ln 1.06
= 12.36 years
FV = $4,341 = $810(1.13)
t
;
t
= ln($4,341/ $810) / ln 1.13
= 13.74 years
FV = $402,662 = $18,400(1.32)
t
;
t
= ln($402,662 / $18,400) / ln 1.32 = 11.11 years
FV = $173,439 = $21,500(1.16)
t
;
t
= ln($173,439 / $21,500) / ln 1.16 = 14.07 years
9.
A consol is a perpetuity.
To find the PV of a perpetuity, we use the equation:
PV =
C
/
r
PV = $120 / .057
PV = $2,105.26
11.
To solve this problem, we must find the PV of each cash flow and add them. To find the PV
of a lump sum, we use:
PV = FV / (1 +
r)
t
[email protected]% = $1,200 / 1.10 + $730 / 1.10
2
+ $965 / 1.10
3
+ $1,590 / 1.10
4
= $3,505.23
[email protected]% = $1,200 / 1.18 + $730 / 1.18
2
+ $965 / 1.18
3
+ $1,590 / 1.18
4
= $2,948.66
[email protected]% = $1,200 / 1.24 + $730 / 1.24
2
+ $965 / 1.24
3
+ $1,590 / 1.24
4
= $2,621.17
13.
To find the PVA, we use the equation:
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PVA =
C
({1 – [1/(1 +
r)
]
t
} /
r
)
[email protected] yrs:
PVA = $4,300{[1 – (1/1.09)
15
] / .09} = $34,660.96
[email protected] yrs:
PVA = $4,300{[1 – (1/1.09)
40
] / .09} = $46,256.65
[email protected] yrs:
PVA = $4,300{[1 – (1/1.09)
75
] / .09} = $47,703.26
To find the PV of a perpetuity, we use the equation:
PV =
C
/
r
PV = $4,300 / .09
PV = $47,777.78
Notice that as the length of the annuity payments increases, the present value of the annuity
approaches the present value of the perpetuity. The present value of the 75-year annuity and
the present value of the perpetuity imply that the value today of all perpetuity payments
beyond 75 years is only $74.51.
14.
This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:
PV =
C
/
r
PV = $20,000 / .065 = $307,692.31
To find the interest rate that equates the perpetuity cash flows with the PV of the cash flows.
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