TIME VALUE MONEY – ANSWER KEY TO PRACTICE PROBLEMS ©
Written by Professor Gregory M. Burbage, MBA, CPA, CMA, CFM
tvm-key.doc
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1. For each of the following cases, indicate (a) to what number of periods and (b) to what interest rate columns
you would refer in looking up the table factor.
Number of
Years Invested
Annual
Rate
Compounded
Future value of 1 table:
Case A
Case B
8
3
6%
8%
Annually
Semiannually
= 1.5938 n=8; i=6
= 1.2653 n=6; i=4
Future value of an annuity:
Case C
Case D
2
5
12%
8%
Monthly
Quarterly
= 26.9735 n=24; i=1
= 24.2974 n=20; i=2
Present value of 1:
Case E
1
48%
Bi-monthly
= 0.6217 n=24; i=2
2 times per month
Present value of an annuity:
Case F
2
Interpolation using the table factors:
18%
Monthly
(21.2434 + 18.9139)/2 = 20.07865 n=24; i=1.5
2. You invested $5,000 at 8% annual interest leaving the money invested without withdrawing any of the
interest. How much would you have at the end of 12 years assuming the investment earns interest compounded
annually?
Future value of amount
n=12; i=8
5,000 * 2.5182 = 12,591
3. You borrowed $20,000 on July 1, 2005. This amount plus accrued interest at 12% per year, compounded
quarterly, is to be repaid on July 1, 2010. How much will you have to repay on July 1, 2010?
Future value of amount
n=20; i=3
20,000 * 1.8061 = 36,122
4. You agreed to make annual sinking fund deposits (savings account deposits) of $60,000 per year because
you borrowed $800,000 by issuing some bonds. Your deposits are made at the end of each year into an account
paying 5% annual interest. What amount will be in the sinking fund at the end of 10 years? Will you have enough
to pay back the bonds?
Future value of annuity
n=10; i=5
60,000 * 12.5779 = 754,674
not quite enough
5. When your daughter Betty-Joe-Bob was born, you and your spouse invested $7,000 in a savings account
paying 8% interest. Each year, starting with her first birthday and ending on her eighteenth birthday, you plan to
deposit $1,000 into the same account. How much will be in the savings account on your daughter’s 18th birthday
(after the last deposit)?
Future value of amount plus Future value of annuity
n=18; i=8
(7,000 * 3.9960) + (1,000 * 37.4502) = 65,422.20
6. What is the present value of $10,000 due in 8 periods discounted at 10%?
Present value of amount
n=8; i=10
10,000 * .4665 = 4,665
7. You want to earn 18% per year, compounded semiannually, on an investment that will return $200,000 eight
years from now. What is the amount you should invest now to earn this rate of return?
Present value of amount
n=16; i=9
200,000 * .2519 = 50,380
8. What is the present value of $10,000 to be received at the end of each 6 months, for 5 years, discounted at
8% per year?
Present value of annuity
n=10; i=4
10,000 * 8.1109 = 81,109.00
9. You are considering investing in an annuity contract that will return $2,000 at the end of each month for 3
years. What amount should you pay for this investment if it earns a 12% annual return?
Present value of annuity
n=36; i=1
2,000 * 30.1075 = 60,215
10. Your company is about to issue $100,000 of 6-year bonds paying a 10% interest rate (contract rate) with
interest payable annually. The discount rate (market rate) for these securities is 10%. How much can your
company expect to receive for the sale of these bonds?
Present value of amount plus present value of annuity
n=6; i=10
(100,000 * .5645) + (10,000 * 4.3553) = 100,003.00
When the stated rate and market rate are equal the bonds will sell at 100 or face or par.
Because of the use of the 4 digit TVM tables there is a rounding error of $3.00
The correct sales price for this bond is $100,000.00
11. Six months ago your company could have issued $100,000 of 6-year bonds paying a 10% interest rate
(contract rate) with interest payable annually. The discount rate (market rate) for such securities was 9%. How
much could your company have received for the sale of these bonds?
Present value of amount plus present value of annuity
n=6; i=9
(100,000 * .5963) + (10,000 * 4.4859) = 104,489.00
When the stated rate is greater than the market rate the bonds will sell at a premium.
Because of the use of the 4 digit TVM tables there is a rounding error of $3.08.
The correct sales price for this bond is $104,485.92.
12. If you wait for six months from now you could issue $100,000 of 6-year bonds paying a 10% interest rate
(contract rate) with interest payable annually. You think the discount rate (market rate) for such securities will be
11%. How much could you expect your company to receive for the sale of these bonds?
Present value of amount plus present value of annuity
n=6; I=11
(100,000 * .5346) + (10,000 * 4.2305) = 95,765.00
When the stated rate is less than the market rate the bonds will sell at a discount.
Because of the use of the 4 digit TVM tables there is a rounding error of $4.46.
The correct sales price for this bond is $95,769.46.
Supplemental Answer Key for TVM Problems 10-12:
The fair market value (sales price) of a bond is based upon the present value of the future cash flows using the
current market interest rate as the discount rate. A bond has two kinds of cash flows, periodic interest payments
and a final lump sum payment. The periodic interest payments usually occur annually or semiannually, and is
determined by multiplying the stated interest rate times the face value and adjusted for other than annually. In
effect, Principal times the annual interest rate times the time in relation to one year. The final payment is called
the maturity value or face value and is paid on the maturity date which also corresponds to the last periodic
interest payment.
THE AMORTIZATION SCHEDULES SHOWN BELOW FOR PROBLEMS 10, 11 AND 12 ARE BASED UPON
USING THE “EFFECTIVE INTEREST METHOD” FOR CALCULATING THE INTEREST EXPENSE AND
AMORTIZING THE DISCOUNT OR PREMIUM, IF ANY. THIS METHOD IS PREFERRED BY GAAP UNLESS
THE STRAIGHT-LINE METHOD IS NOT MATERIALLY DIFFERENT.
Problem 10:
In the case where the stated interest rate is equal to the current market rate the present value of the future cash
flows will be equal to the face value of the bond, and the bond would have a fair market value of (sell for) face
value as shown in the answer to problem 10. As a result the “interest payment” is also equal the “interest
expense”.
Compound Period: Annual
Effective Annual Rate (Market rate): 10.000 %
CASH FLOW DATA
1
2
3
Event
Loan
Payment
Payment
Date
12/31/2005
12/31/2006
12/31/2011
Amount
100,000.00
10,000.00
100,000.00
Number
1
6
1
Period
End Date
Annual
12/31/2011
AMORTIZATION SCHEDULE
Loan
Payment 1
Payment 2
Payment 3
Payment 4
Payment 5
Payment 6
Payment 7
Grand Totals
Date
12/31/2005
12/31/2006
12/31/2007
12/31/2008
12/31/2009
12/31/2010
12/31/2011
12/31/2011
Payment
Interest
Principal
10,000.00
10,000.00
10,000.00
10,000.00
10,000.00
10,000.00
100,000.00
10,000.00
10,000.00
10,000.00
10,000.00
10,000.00
10,000.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
100,000.00
160,000.00
60,000.00
100,000.00
Balance
100,000.00
100,000.00
100,000.00
100,000.00
100,000.00
100,000.00
100,000.00
0.00
Problem 11:
In the case where the stated interest rate is greater than the current market rate the present value of the future
cash flows would be greater than the face value of the bond, and the bond would have a fair market value of (sell
for) more than face value (a premium) as shown in the answer to problem 11 above. As a result the “interest
payment” of $10,000.00 will be more than the “interest expense” as shown in the amortization schedule.
Compound Period: Annual
Effective Annual Rate (Market rate): 9.000 %
CASH FLOW DATA
1
2
3
Event
Loan
Payment
Payment
Date
12/31/2005
12/31/2006
12/31/2011
Amount
104,485.92
10,000.00
100,000.00
Number
1
6
1
Period
End Date
Annual
12/31/20011
AMORTIZATION SCHEDULE
Using the Time Value Money Tables to determine the present value (sales price) of the bond results in
an error of $3.08 because the tables are rounded to four significant digits. Therefore, the beginning balance of
the loan (sales price of the bond and present value of the cash flows discounted at 9%) has been recalculated
using a financial calculator, which does not result in a rounding error.
Loan
Payment 1
Payment 2
Payment 3
Payment 4
Payment 5
Payment 6
Payment 7
Grand Totals
Date
12/31/2005
12/31/2006
12/31/2007
12/31/2008
12/31/2009
12/31/2010
12/31/2011
12/31/2011
Payment
Interest
Principal
10,000.00
10,000.00
10,000.00
10,000.00
10,000.00
10,000.00
100,000.00
9,403.73
9,350.07
9,291.57
9,227.82
9,158.32
9,082.57
0.00
596.27
649.93
708.43
772.18
841.68
917.43
100,000.00
160,000.00
55,514.08
104,485.92
Balance
104,485.92
103,889.65
103,239.72
102,531.29
101,759.11
100,917.43
100,000.00
0.00
Problem 12:
In the case where the stated interest rate is less than the current market rate the present value of the future cash
flows would be less than the face value of the bond, and the bond would have a fair market value of (sell for) less
than face value (a discount) as shown in the answer to problem 12 above. As a result the “interest payment” of
$10,000.00 will be more than the “interest expense” as shown in the amortization schedule below.
Compound Period: Annual
Effective Annual Rate: 11.000 %
Using the Time Value Money Tables to determine the present value (sales price) of the bond results in a
error of $4.46 because the tables are rounded to four significant digits. Therefore, the beginning balance of the
loan (sales price of the bond and present value of the cash flows discounted at 11%) has been recalculated
using a financial calculator, which does not result in a rounding error.
CASH FLOW DATA
1
2
3
Event
Loan
Payment
Payment
Date
12/31/2005
12/31/2006
12/31/20011
Amount
95,082.67
10,000.00
100,000.00
Number
1
6
1
Period
End Date
Annual
12/31/2011
AMORTIZATION SCHEDULE
Loan
Payment 1
Payment 2
Payment 3
Payment 4
Payment 5
Payment 6
Payment 7
Grand Totals
Date
12/31/2005
12/31/2006
12/31/2007
12/31/2008
12/31/2009
12/31/2010
12/31/2011
12/31/2011
Payment
Interest
Principal
10,000.00
10,000.00
10,000.00
10,000.00
10,000.00
10,000.00
100,000.00
10,534.64
10,593.45
10,658.73
10,731.19
10,811.62
10,900.91
0.00
-534.64
-593.45
-658.73
-731.19
-811.62
-900.91
100,000.00
160,000.00
64,230.54
95,769.46
Balance
95,769.46
96,304.10
96,897.55
97,556.28
98,287.47
99,099.09
100,000.00
0.00