108 Brooks ◼ Financial Management: Core Concepts, 4e
32. Challenge problem. Each holiday season, Michael received a U.S. savings bond from
his grandmother. Michael eventually received twelve savings bonds. The bonds vary in
their rates of interest and their face value. Assume that today is December 31, 2011.
What is the value of this portfolio of U.S. savings bonds? On what dates does each of
the individual bonds reach their face value or maturity date (note that the price is half
the face value)? Estimate to the nearest month and year for each bond. Note: the bonds
continue earning interest past their maturity dates.
Issue Date
Price
Face Value
Interest Rate
Maturity Date
12/31/1990
$50
$100
6.0%
12/31/1991
$50
$100
6.0%
12/31/1992
$25
$50
5.0%
12/31/1993
$25
$50
4.0%
12/31/1994
$25
$50
4.0%
12/31/1995
$50
$100
5.0%
12/31/1996
$25
$50
5.0%
12/31/1997
$25
$50
4.0%
12/31/1998
$25
$50
4.0%
12/31/1999
$50
$100
4.0%
12/31/2000
$25
$50
4.0%
12/31/2001
$25
$50
3.0%
TOTAL
$400
ANSWER
Chapter 4 ◼ The Time Value of Money (Part 2) 109
© 2018 Pearson Education, Inc.
FV#11 (11 years) = $25 × 1.042011–2000 = $25 × 1.5395 = $ 38.49
FV#12 (10 years) = $25 × 1.032011–2001 = $25 × 1.3439 = $ 33.59
TOTAL VALUE $885.42
Maturity Dates of Each Bond: n is the length of time between the issue date and the
maturity date.
N#12 = ln (2 /1) / ln (1.03) = 0.6931/0.0296 = 23.45 years or 23 years and 5 months
110 Brooks ◼ Financial Management: Core Concepts, 4e
Solutions to Advanced Problems for Spreadsheet Application
1. Future value with an annuity.
a. They will have accumulated $2,000,000 after approximately twenty-nine years.
Money market G overnment Bonds Large Cap Small Cap Real Estate Portfolio
G rowth rate 2.5% 5.5% 9.5% 12.0% 4.0%
Year
1 (4,000.00)$ (4,000.00)$ (4,000.00)$ (4,000.00)$ (4,000.00)$ (20,000.00)$
2 $8,100.00 $8,220.00 $8,380.00 $8,480.00 $8,160.00 41,340.00$
3 $12,302.50 $12,672.10 $13,176.10 $13,497.60 $12,486.40 64,134.70$
4 $16,610.06 $17,369.07 $18,427.83 $19,117.31 $16,985.86 88,510.13$
5 $21,025.31 $22,324.36 $24,178.47 $25,411.39 $21,665.29 114,604.83$
11 $49,933.87 $58,333.99 $72,154.07 $82,618.33 $53,945.41 316,985.67$
12 $55,182.21 $65,542.36 $83,008.71 $96,532.53 $60,103.22 360,369.04$
13 $60,561.77 $73,147.19 $94,894.54 $112,116.44 $66,507.35 407,227.29$
14 $66,075.81 $81,170.29 $107,909.52 $129,570.41 $73,167.64 457,893.67$
15 $71,727.71 $89,634.65 $122,160.92 $149,118.86 $80,094.35 512,736.49$
23 $122,337.71 $176,447.39 $297,411.43 $418,411.58 $146,471.55 1,161,079.65$
24 $129,396.15 $190,151.99 $329,665.51 $472,620.96 $156,330.42 1,278,165.04$
25 $136,631.06 $204,610.35 $364,983.73 $533,335.48 $166,583.63 1,406,144.26$
26 $144,046.83 $219,863.92 $403,657.19 $601,335.74 $177,246.98 1,546,150.66$
27 $151,648.00 $235,956.44 $446,004.62 $677,496.03 $188,336.86 1,699,441.95$
Chapter 4 ◼ The Time Value of Money (Part 2) 111
b. They will be able to retire two years earlier. If they retire after twenty-nine years, they
will have saved $428,274.30 more than the amount saved in Part (a)
Money Mkt. Gvt. Bonds Large Cap Small Cap Real Estate Portfolio
Growth Rate 2.5% 5.5% 9.5% 12.0% 4.0%
Year
1 (2,000.00)$ (4,000.00)$ (5,000.00)$ (6,000.00)$ (3,000.00)$ (20,000.00)$
2 $8,100.00 $8,220.00 $10,475.00 $12,720.00 $6,120.00 45,635.00$
3 $6,151.25 $12,672.10 $16,470.13 $20,246.40 $9,364.80 64,904.68$
4 $8,305.03 $17,369.07 $23,034.79 $28,675.97 $12,739.39 90,124.24$
5 $10,512.66 $22,324.36 $30,223.09 $38,117.08 $16,248.97 117,426.16$
6 $12,775.47 $27,552.20 $38,094.29 $48,691.13 $19,898.93 147,012.02$
7 $15,094.86 $33,067.58 $46,713.24 $60,534.07 $23,694.88 179,104.63$
8 $17,472.23 $38,886.29 $56,151.00 $73,798.16 $27,642.68 213,950.36$
9 $19,909.04 $45,025.04 $66,485.35 $88,653.94 $31,748.39 251,821.74$
10 $22,406.76 $51,501.42 $77,801.45 $105,292.41 $36,018.32 293,020.36$
11 $24,966.93 $58,333.99 $90,192.59 $123,927.50 $40,459.05 337,880.07$
12 $27,591.11 $65,542.36 $103,760.89 $144,798.80 $45,077.42 386,770.57$
13 $30,280.88 $73,147.19 $118,618.17 $168,174.66 $49,880.51 440,101.42$
14 $33,037.91 $81,170.29 $134,886.90 $194,355.61 $54,875.73 498,326.44$
15 $35,863.85 $89,634.65 $152,701.15 $223,678.29 $60,070.76 561,948.71$
23 $61,168.85 $176,447.39 $371,764.28 $627,617.36 $109,853.67 1,346,851.55$
24 $64,698.08 $190,151.99 $412,081.89 $708,931.45 $117,247.81 1,493,111.22$
25 $68,315.53 $204,610.35 $456,229.67 $800,003.22 $124,937.72 1,654,096.49$
Chapter 4 ◼ The Time Value of Money (Part 2) 113
Solutions to Mini-Case
Fitchminster Injection Molding, Inc.: Rose Climbs High
This case illustrates the important role that present and future values of annuities play in
business decisions, and emphasizes the structuring of loan payments as an important
practical application of time value of money computations.
1. Rose could probably borrow the money to purchase the shares outright because
the shares would serve as collateral and dividends would cover a good part of the
loan payments. The interest rate is 7%, and the loan will be amortized with a
series of equal payments. What are the annual payments if the bank amortizes the
114 Brooks ◼ Financial Management: Core Concepts, 4e
Calculator Solution
N
i/y
PV
PMT
FV
5
7
10,000,000
–2,438,906.94
0
10
7
10,000,000
–1,423,775.03
0
20
7
10,000,000
–943,929.26
0
2. Repeat Question 1, but assume that Rose makes payments at the beginning of each
year.
3. Complete the amortization schedule below for a $10,000,000 loan at 7% with five
equal end-of-year payments.
Year
Beginning
Principal
Annual
Payment
Interest
Expense
Principal
Reduction
Remaining
Principal
1
$10,000,000.00
$2,438,906.94
$ 700,000.00
$1,738,906.94
$ 8,261,093.06
2
$ 8,261,093.06
$2,438,906.94
$ 578,276.51
$1,860,630.43
$ 6,400,462.63
3
$ 6,400,462.63
$2,438,906.94
$ 448,032.38
$1,990,874.56
$ 4,409,588.06
4
$ 4,409,588.06
$2,438,906.94
$ 308,671.16
$2,130,235.78
$ 2,279,352.28
5
$ 2,279,352.28
$2,438,906.94
$ 159,554.66
$2,279,352.28
$
4. Sam has offered to finance the purchase with a ten-year, interest-only loan. How
much is Rose’s annual payment? Describe the pattern of payments over the ten
5. Assume that Rose accepts Sam’s offer to finance the purchase with a ten-year,
interest-only loan. If Sam can reinvest the interest payments at a rate of 7% per
year, how much money will he have at the end of the tenth year?
PMT
–9,671,513.57
Chapter 4 ◼ The Time Value of Money (Part 2) 115
Additional Problems with Solutions
1. Present value of an annuity due. Julie has just been accepted into Harvard, and her
father is debating whether he should make monthly lease payments of $5,000 at the
beginning of each month on her flashy apartment or to prepay the rent with a one-time
payment of $56, 662. If Julie’s father earns 1% per month on his savings, should he pay
by month or take the discount by making the single annual payment?
ANSWER (Slides 4-45 to 4-46)
Use the TVM Keys from a Texas Instrument BAII Plus Calculator and round to two
2. Future value of uneven cash flows. If Mary deposits $4,000 a year for three years,
starting a year from today, followed by three annual deposits of $5000, into an account
that earns 8% per year, how much money will she have accumulated in her account at
the end of ten years?
ANSWER (Slides 4-47 to 4-49)
116 Brooks ◼ Financial Management: Core Concepts, 4e
© 2018 Pearson Education, Inc.
ALTERNATIVE METHOD:
Using the Cash Flow (CF) key of the calculator, enter the respective cash flows.
That is, CF0 = 0; CF1 = –$4000; CF2 = –$4000; CF3 = –$4000; CF4 = –$5000;
CF5 = –$5000; CF6 = –$5000
Next calculate the NPV using I = 8%; ➔NPV = $20,537.30133;
Finally, using PV = –$20,537.30133; n = 10; I = 8%; PMT = 0; CPT FV➔$44,338
3. Present value of uneven cash flows. Jane Bryant has just purchased some equipment
for her beauty salon. She plans to pay the following amounts at the end of the next five
years: $8,250, $8,500, $8,750, $9,000, and $10,500. If she uses a discount rate of 10
percent, what is the cost of the equipment that she purchased today?
ANSWER (Slides 4-50 to 4-51)
2 3 4 5
$8, 250 $8,500 $8,750 $9,000 $10,500
PV (1.10) (1.10) (1.10) (1.10) (1.10)
$7,500 $7,024.79 $6,574 $6,147.12 $6,519.67
$33,765.58
= + + + +
= + + + +
=
4. Computing annuity payment. The Corner Bar & Grill is in the process of taking a five-
year loan of $50,000 with First Community Bank.
The bank offers the restaurant owner his choice of three payment options:
1) Pay all of the interest (8% per year) and principal in one lump sum at the end of five
years;
2) Pay interest at the rate of 8% per year for four years and then a final payment of
interest and principal at the end of the fifth year;
3) Pay five equal payments at the end of each year inclusive of interest and part of the
principal.
Under which of the three options will the owner pay the least interest and why? Hint:
Calculate the total amount of the payments and the amount of interest paid under each
alternative.
ANSWER (Slides 4-52 to 4-56)
Chapter 4 ◼ The Time Value of Money (Part 2) 117
Under option 2: Interest-only loan