CHAPTER 5

Time Value of Money

SOLUTIONS TO PROBLEMS

4-3 LG 2: Future value calculation

FVn = PV × (1+i)n

Case Calculator solution Table value

A FVIF 12%,2 periods = (1 +0.12)2 = 1.2544 1.254
B FVIF 6%,3 periods = (1 +0.06)3 = 1.1910 1.191
C FVIF 9%,2 periods = (1 +0.09)2 = 1.1881 1.188
D FVIF 3%,4 periods = (1 + 0.03)4 = 1.1255 1.126

4-5 LG 2: Future values

FVn = PV × (1 + i)n or FVn = PV × (FVIFi%,n)

Case Case
A FV20 = PV × (1+i)n. B FV7 = PV × (1+i)n.
= $200 × (1+0.05)20 = $4,500 × (1+0.07)7
= $530.66 = $7,712.21

C FV10 = PV × (1+i)n D FV12 = PV × (1+i)n.

= $10,000 × (1.09)10 = $25,000 × (1.10)12
= $23,673.64 = $78,460.71

E FV5 = PV × (1+i)n. F FV9 = PV × (1+i)n.

= $37,000 × (1.11)5 = $40,000 × (1.12)9
= $62,347.15 =$110,923.15

4-6 LG 2: Future value

(a) FV=$3000X(1.12)3 =$4215

𝑖
(b) FV=PV(1+ )n*m
𝑚

0.12
6 mth, FV=$3000X(1+ )3x2 = $4257
2

0.12
4 mth, FV=$3000X(1+ )3x3 = $4269
3

0.12
3 mth, FV=$3000X(1+ )3x4 = $4278
4

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 0.12
2 mth, FV=$3000X(1+ )3x6 = $4284
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The higher the compounding frequency, the greater the future sum realised.
𝑃𝑀𝑇
(c) We know, FV of ordinary annuity= [(1 + 𝑖)𝑛 − 1]
𝑖

Here, PMT=$3,000; i=0.12, n=3

$3,000
FV of ordinary annuity= [(1 + 0.12)3 − 1]
0.12
=$10,122

4-7 LG 2: Future value

$2,000
FVA = [(1 + 0.12)20 − 1]= $144,104
0.12

4-10 LG 2: Time value

Deposit now: Deposit in 10 years:

FV40= $10,000 × (1.09)40 FV30 =PV10 × (1.09)30=$10,000× (1.09)30
=$314,094.20 =$132,676.78
You would be better off by $181,417.42 ($314,094.20 – $132,676.78) by investing
the $10,000 now instead of waiting for 10 years to make the investment.

4-19 LG 2: Time value and discount rates

PV = FVn × (1+i)-n or PV = FVn / (1+i)n (1+i)-n = 1/(1+i)n

a. (i) PV = $1,000,000 × (1.06)-10 (ii) PV = $1,000,000 × (1.09)-10

= $1,000,000 × (0.558) = $1,000,000 × (0.422)
= $558,394.78 = $422,410.81

(iii) PV = $1,000,000 × (1.12)-10

= $1,000,000 × (0.322)
= $321,973.24

b. (i) PV = $1,000,000 × (1.06)-15 (ii) PV = $1,000,000 × (1.09)-15

= $1,000,000 × (0.417) = $1,000,000 × (0.275)
= $417,265.06 = $274,538.04
(iii) PV = $1,000,000 × (1.12)-15
= $1,000,000 × (0.183)
= $182,696.26

c. As the discount rate increases, the present value becomes smaller. This decrease is
due to the higher opportunity cost associated with the higher rate. The future sum is
being discounted more heavily to find the present value. Also, the longer the time
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 until the lottery payment is collected, the less the present value due to the greater
time over which the opportunity cost applies. In other words, the larger the discount
rate and the longer the time until the money is received, the smaller will be the
present value of a future payment.

4-22 LG 3: Future value of an annuity

Future value of lump sum Future value of ordinary annuity
𝑃𝑀𝑇
FVn = PV × (1+i)n FVAn = 𝑖 [(1 + 𝑖)𝑛 − 1]

$1,00,000
FV25 = $1,300,000 × (1.05)25 FVA25 = 0.05 [(1 + 0.05)25 − 1]
= $1,300,000 × (3.386355) = 100,000 × (47.726)
= $4,402,261.42 = $4,772,709.88

Gabrielle would be better off taking the $100,000 25-year annuity.

4-23 LG 3: Future value of an annuity

a. Future value of an ordinary annuity vs annuity due
(i) Ordinary annuity (ii) Annuity due
𝑃𝑀𝑇 𝑃𝑀𝑇
FVAn = [(1 + 𝑖)𝑛 − 1] FVAn = [(1 + 𝑖)𝑛 − 1](1+i)
𝑖 𝑖

$2,500
A) Ordinary annuity, FVA10 = [(1 + 0.08)10 − 1] = $36,216.41
0.08

$2,500
Annuity Due, FVA10 = [(1 + 0.08)10 − 1](1.08) = $39,113.72
0.08

b. The annuity due results in a greater future value in each case. By depositing the
payment at the beginning rather than at the end of the year, every payment has one
additional year of compounding.

4-24 LG 3: Present value of an annuity

a. Present value of an ordinary annuity vs annuity due

(i) Ordinary annuity (ii) Annuity due

𝑃𝑀𝑇 𝑃𝑀𝑇
PVAn = 𝑖 [1 − (1 + 𝑖)−𝑛 ] PVAn = 𝑖 [1 − (1 + 𝑖)−𝑛 ](1+i)

$12,000
A Ordinary Annuity, PVA7%,3= [1 − (1 + 0.07)−3] = $31,491.79
0.07

$12,000
Annuity Due, PVA7%,3= [1 − (1 + 0.07)−3] (1.07) = $33,696.22
0.07

b. The annuity due results in a greater present value in each case. By depositing the
payment at the beginning rather than at the end of the year, there is one less year
over which each payment is discounted to find the present value. Hence, the
payments are not discounted as much and the present value is greater.

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4-26 LG 3: Future value of a retirement annuity
$2,000 $2,000
a. FVA40 = 0.10 [(1 + 0.10)40 − 1] b. FVA30 = 0.10 [(1 + 0.10)30 − 1]
= $885,185.11 = $328,988.05

c. By delaying the deposits by 10 years the total opportunity cost is $556,197.06. This
difference is due to both the lost deposits of $20,000 ($2,000 × 10 years) and the lost
compounding of interest on all of the money that would otherwise have been deposited over
the first 10 years.

d. Annuity due:
$2,000
FVA40 = 0.10 [(1 + 0.10)40 − 1](1.10) = $973,703.62

Annuity due:
$2,000
FVA30 = 0.10 [(1 + 0.10)30 − 1](1.10) = $361,886.85

Both deposits increased due to the extra year of compounding from the beginning-of-year
deposits instead of the end-of-year deposits. However, the incremental change in the 40-
year annuity is much larger than the incremental change in the 30-year annuity ($88,518.51
vs $32,898.80) due to the larger sum on which the last year of compounding occurs.

4-28 LG 2, 3: Funding your retirement

$20,000
a. PVA = [1 − (1 + 0.11)−30] b. PV = FV × (1+i)-n
0.11
= $173,875.85 = $173,875.85× (1.08)-20
= $31,024.82

c. Both values would be lower. In other words, a smaller sum would be needed in 20
years for the annuity and a smaller amount would have to be put away today to
accumulate the needed future sum.

4-30 LG 3: Perpetuities
PV of Perpetuity = PMT  i

a. Case PV Factor b. PV of Perpetuity = PMT i

A 1  0.08 = 12.50 $20,000  0.08 = $ 250,000
B 1  0.10 = 10.00 $100,000  0.10 = $1,000,000
C 1  0.06 = 16.67 $3,000  0.06 = $ 50,000
D 1  0.05 = 20.00 $60,000  0.05 = $1,200,000

4-33 LG 4: Value of a mixed stream

a. Cash flow Number of years
stream Year to compound FV = CF × (1+i)n Future Value
A 1 2 $900 × (1.12)2 = $1,128.60
2 1 1,000 × 1.120 = 1,120.00

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 3 0 1,200 = 1,200.00
$3,448.60

B 1 4 $30,000 × (1.12)4 = $47,220.00

2 3 25,000 × (1.12)3 = 35,125.00
3 2 20,000 × (1.12)2 = 25,080.00
4 1 10,000 × 1.12 = 11,200.00'
5 0 5,000 = 5,000.00
$123,625.00
C 1 3 $1,200 × (1.12)3 = $1,686.00
2 2 1,200 × (1.12)2 = 1,504.80
3 1 1,000 × 1.12 = 1,120.00
4 0 1,900 = 1,900.00
$6,210.80

b. If payments are made at the beginning of each period, the present value of each of
the end-of-period cash flow streams will be multiplied by (1 + i) to get the present
value of the beginning-of-period cash flows.

A. $3,448.60 (1 + 0.12) = $3,862.43

B. $123,325.00 (1 + 0.12) = $138,460.00
C. $6,210.80 (1 + 0.12) = $6,956.10

4-36 LG 4: Present value – mixed streams

a. Cash Flow
Stream Year CF × (1+i)-n = Present Value
-1
A 1 $50,000 × (1.15) = $43,500.00
2 40,000 × (1.15)-2 = 30,240.00
3 30,000 × (1.15)-3 = 19,740.00
4 20,000 × (1.15)-4 = 11,440.00
5 10,000 × (1.15)-5 = 4,970.00
$109,890.00

Cash Flow
Stream Year CF × PVIF15%,n = Present Value
B 1 $10,000 × (1.15)-1 = $8,700.00
2 20,000 × (1.15)-2 = 15,120.00
3 30,000 × (1.15)-3 = 19,740.00
4 40,000 × (1.15)-4 = 22,880.00
5 50,000 × (1.15)-5 = 24,850.00
$91,290.00

b. Cash flow stream A, with a present value of $109,890, is higher than cash flow
stream B’s present value of $91,290 because the larger cash inflows occur in A in
the early years when their present value is greater, while the smaller cash flows are
received further in the future.

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4-40 LG 5: Changing compounding frequency

𝑖
(i) Compounding frequency: FVn = PV × (1+ )n*m
𝑚

Initial deposit: $5,000

Annual Semi-annual Quarterly
a. Periodic interest rate: 0.12 0.12  2 =0.06 0.12  4 = 0.03
Number of periods: 5 5 × 2 = 10 5 × 4 = 20
FV5 $5000(1.12)5 $5000(1.06)10 $5000(1.03)20
FV5 $8,811.71 $8,954.24 $9,030.56

b. Periodic interest rate: 0.16 0.16  2 = 0.08 0.16  4 = 0.04

Number of periods: 6 6 × 2 = 12 6 × 4 = 24
FV5 $5000(1.16)6 $5000(1.08)12 $5000(1.04)24
FV5 $12,181.98 $12,590.85 $12,816.52

c. Periodic interest rate: 0.20 0.20  2 = 0.10 0.20  4 =0.05

Number of periods: 10 10 × 2 = 20 10 × 4 = 40
FV5 $5000(1.20)10 $5000(1.10)20 $5000(1.05)40
FV5 $30,958.68 $33,637.50 $35,199.94

(ii) Effective interest rate: ieff = (1 + i/m)m – 1

Annual Semi-annual Quarterly

a. ieff = (1 + 0.12/1)1 – 1 = (1 + 0.12/2)2 – 1 = (1 + 0.12/4)4 – 1
= (1.12)1 – 1 = (1.06)2 – 1 = (1.03)4 – 1
= (1.12) – 1 = (1.124) – 1 = (1.126) – 1
= 0.12 = 12% = 0.124 = 12.4% = 0.126 = 12.6%

b. ieff = (1 + 0.16/1)1 – 1 = (1 + 0.16/2)2 – 1 = (1 + 0.16/4)4 – 1

= (1.16)1 – 1 = (1.08)2 – 1 = (1.04)4 – 1
= (1.16) – 1 = (1.166) – 1 = (1.170) – 1
= 0.16 = 16% = 0.166 = 16.6% = 0.170 = 17.0%

c. ieff = (1 + 0.20/1)1 – 1 = (1 + 0.20/2)2 – 1 = (1 + 0.20/4)4 – 1

= (1.20)1 – 1 = (1.10)2 – 1 = (1.05)4 – 1
= (1.20) – 1 = (1.210) – 1 = (1.216) – 1
= 0.20 = 20% = 0.210 = 21.0% = 0.216 = 21.6%

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4-41 LG 5: Changing compounding frequency

Pars’ Savings Bank:

mn
 i 
FV1 = PV  1 + 
 m
= $12,000  (1 + 0.03 2 )
2

= $12,000  1.027689 = $12,362.70

Sel’s Bank:
FV1 = PV  (ein )
= $12,000  (2.7183) = $12,334.58
0.0275

Joseph should choose the 3% rate with semi-annual compounding.

4-46 LG 5: Compounding frequency and future value

a. (i) FV10 = $2,000 × (1.08)10 (ii) FV10 = $2,000 × (1+0.08/2)10*2

= $4,317.85 = $4,382.25

(iii) FV10 = $2,000 × (1+0.08/365)10*365 =$4,450.69

(iv) FV10 = PV× (eixn) =$2,000 × (e0.08x10) = $2,000 × (2.71830.8)= $4,451.08

b. (1) ieff = (1 + 0.08/1)1 – 1 (2) ieff = (1 + 0.08/2)2 – 1

= (1 + 0.08)1 – 1 = (1 + 0.04)2 – 1
= (1.08) – 1 = (1.082) – 1
= 0.08 = 8% = 0.0816 = 8.16%

(3) ieff = (1 + 0.08/365)365 – 1 (4) ieff = (ei – 1)

= (1 + 0.000219)365 – 1 = (e0.08 – 1)
= (1.0833) – 1 = (1.0833 – 1)
= 0.0833 = 8.33% = 0.0833 = 8.33%

c. Compounding continuously will result in $0.39 more dollars at the end of the 10-
year period than compounding annually.

d. The more frequent the compounding, the larger the future value. This result is shown
in part (a) by the fact that the future value becomes larger as the compounding period
moves from annually to continuously. Since the future value is larger for a given
fixed amount invested, the effective return also increases directly with the frequency
of compounding. In part (b) we see this fact as the effective rate moved from 8% to
8.33% as compounding frequency moved from annually to continuously.

4-48 LG 3, 5: Annuities and compounding

𝑃𝑀𝑇
FVAn = 𝑖 [(1 + 𝑖)𝑛 − 1]

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a. (1) Annual (2) Semi-annual (3) Quarterly
$300 10 $150 20 $75
FVA10 [(1 + 0.08) − 1] [(1 + 0.04) − 1] [(1 + 0.02)40 − 1]
0.08 0.04 0.02
=$4,345.97 = $4,466.71 = $4,530.15

b. The sooner a deposit is made, the sooner the funds will be available to earn interest
and contribute to compounding. Thus, the sooner the deposit and the more frequent
the compounding, the larger the future sum.

4-49 LG 6: Deposits to accumulate future sums

𝑃𝑀𝑇
FVA = [(1 + 𝑖)𝑛 − 1]
𝑖

FVA×i
PMT=
(1+𝑖)𝑛 −1

Case Terms
$5,000×0.12
A 12%, 3 yrs. PMT = = $1,481.74
(1+0.12)3 −1

$100,000×0.07
B 7%, 20 yrs. PMT = = $2,439.29
(1+0.07)20 −1

$30,000×0.10
C 10%, 8 yrs. PMT = = $2,623.32
(1+0.10)8 −1

$15,000×0.08
D 8%, 12 yrs. PMT = = $790.43
(1+0.08)12 −1

4-50 LG 6: Creating a retirement fund

$220,000×0.08 $600
a. PMT = b. FVA42 = [(1 + 0.08)42 − 1]
(1+0.08)42 −1 0.08
= $723.11 = $182,546.11

4-54 LG 6: Loan payment

𝑃𝑀𝑇
PVA = [1 − (1 + 𝑖)−𝑛 ]
𝑖

8
 PVA×i
PMT=
1−(1+𝑖)−𝑛

Loan:
$12,000×0.08
A PMT = = $4,656.40
1−(1+0.08)−3

$60,000×0.12
B PMT = = $10,619.05
1−(1+0.12)−10
$75,000×0.10
C PMT = = $7,955.94
1−(1+0.10)−30
$4,000×0.15
D PMT = = $1,193.26
1−(1+0.15)−5

4-55 LG 6: Loan amortisation schedule

$15,000×0.14
a. PMT = = $6,459.95
1−(1+0.14)−3

b. 1 2 3 4=3x0.14 5=2-4 6=3-5

End of Loan Beginning of Payment End of Year
Year Payment Year Principal Interest Principal Principal
1 $6,459.95 $15,000.00 $2,100.00 $4,359.95 $10,640.05
2 6,459.95 10,640.05 1,489.61 4,970.34 5,669.71
3 6,459.95 5,669.71 793.76 5,666.19 0
(The difference between the last year’s beginning principal, and the principal
component of the last year’s repayment, is due to rounding.)

c. Because of the effect of compound interest, each time a payment is made the amount
owing at the end of the year is reduced, which reduces the interest accrued in the
following year. Since the payments are remaining constant, but the interest
component is falling, the principal component of each payment is increasing.

4-57 LG 6: Monthly loan payments

a.
i=0.12/12=0.01, n=24

$4,000×0.01
PMT = = $188.29
1−(1+0.01)−24

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b.
i=0.09/12=0.0075, n=24

$4,000×0.0075
PMT = = $182.74
1−(1+0.0075)−24

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