Question

In computing your answers to the cases below, you can round your answer to the nearest dollar. Present value tables are provided on the next page.

Use the following information in answering Cases 1 and 2 below:

On January 1, 2011, Gray Company sold $1,000,000 of 10% bonds, due January 1, 2021. Interest on these bonds is paid on July 1 and January 1 each year. According to the terms of the bond contract, Gray must establish a sinking fund for the retirement of the bond principal starting no later than January 1, 2019. Since Gray was in a tight cash position during the years 2011 through 2016, the first contribution into the fund was made on January 1, 2017.

Case 1: Assume that, starting with the January 1, 2017 contribution, Gray desires to make a total of four equal annual contributions into this fund. Compute the amount of each of these contributions assuming the interest rate is 8% compounded annually.

Case 2: Assume, instead, that starting with the January 1, 2019 contribution, Gray desires to make a total of five equal semiannual contributions into this fund. Compute the amount of each of these contributions assuming the annual interest rate is 12%, compounded semiannually.

Case 3: On January 2, 2017, Nelson Company loaned $200,000 to Holt Company. The terms of this loan agreement stipulate that Holt is to make 5 equal annual payments to Nelson at 10% interest compounded annually. Assume the payments are to begin on December 31, 2017. Compute the amount of each of these payments.

Case 4: Jim Marsh, a lawyer contemplating retirement on his 65th birthday, decides to create a fund on an 8% basis which will enable him to withdraw $80,000 per year beginning June 30, 2020, and ending June 30, 2024. To provide this fund, he intends to make equal contributions on June 30 of each of the years 2015 through 2019.

How much must the balance of the fund equal after the last contribution on June 30, 2019 in order for him to satisfy his objective?

What are each of his contributions to the fund?

Table 1

Future Value of 1

Periods	6%	8%	9%	10%	12%
1	1.06000	1.08000	1.09000	1.10000	1.12000
2	1.12360	1.16640	1.18810	1.21000	1.2544
3	1.19102	1.25971	1.29503	1.33100	1.4049
4	1.26248	1.36049	1.41158	1.46410	1.5735
5	1.33823	1.46933	1.53862	1.61051	1.7623

Table 2

Present Value of 1

Periods	6%	8%	9%	10%	12%
1	0.94340	0.92593	0.91743	0.90909	0.8928
2	0.89000	0.85734	0.84168	0.82645	0.7971
3	0.83962	0.79383	0.77218	0.75132	0.7117
4	0.79209	0.73503	0.70843	0.68301	0.6355
5	0.74726	0.68058	0.64993	0.62092	0.5674

Table 3

Future Value of an Ordinary Annuity of 1

Periods	6%	8%	9%	10%	12%
1	1.00000	1.00000	1.00000	1.00000	1.00000
2	2.06000	2.08000	2.09000	2.10000	2.12000
3	3.18360	3.24640	3.27810	3.31000	3.3744
4	4.37462	4.50611	4.57313	4.64100	4.7793
5	5.63709	5.86660	5.98471	6.10510	6.3528

Table 4

Present Value of an Ordinary Annuity of 1

Periods	6%	8%	9%	10%	12%
1	0.94340	0.92593	0.91743	0.90909	0.8928
2	1.83339	1.78326	1.75911	1.73554	1.6900
3	2.67301	2.57710	2.53130	2.48685	2.4018
4	3.46511	3.31213	3.23972	3.16986	3.0373
5	4.21236	3.99271	3.88965	3.79079	3.6047

Table 5

Present Value of an Annuity Due of 1

Periods	6%	8%	9%	10%	12%
1	1.00000	1.00000	1.00000	1.00000	1.00000
2	1.94340	1.92593	1.91743	1.90909	1.8928
3	2.83339	2.78326	2.75911	2.73554	2.6900
4	3.67301	3.57710	3.53130	3.48685	3.4018
5	4.46511	4.31213	4.23972	4.16986	4.0373

Answer 1

Case 1 Solution:

$1,000,000 is the amount of an 8% annuity due for 4 periods. Use the table factor for the future value of an 8% ordinary annuity for 4 periods, and multiply by (1.08):

4.50611 × (1.08) = 4.86660.

Periodic payments = $1,000,000 ÷ 4.86660 = $205,482

Case 2 Solution:

Since interest is compounded semiannually, divide the 12% annual interest rate by 2, and use the table factor for the future value of a 6% ordinary annuity for 5 periods.

Periodic payments = $1,000,000 ÷ 5.63709 = $177,396

Case 3 Solution:

$200,000 is the present value of a 10% ordinary annuity for 5 periods. Use the table factor for the Present value of a 10% ordinary annuity for 5 periods.

Periodic payments = $200,000 ÷ 3.79079 = $52,759

Case 4 Solution:

1.At June 30, 2019, the balance in the fund is the present value of an 8% ordinary annuity of $80,000 for 5 periods.

Balance in the fund = $80,000 × 3.99271 = $319,417

2.At June 30, 2019, $319,417 is the future value of an 8% ordinary annuity for five periods.

Periodic payments = $319,417 ÷ 5.86660 = $54,447