Financial Accounting

Appendix A

Compound Interest and the Time-Value of Money

EXERCISESEA-1. Future value, 4%, 6 years: $4,000 x 1.26532 = $5,061.28

Future value, 6%, 6 years: $4,000 x 1.41852 = $5,674.08Future value, 8%, 6 years: $4,000 x 1.58687 = $6,347.48

EA-2. Future value, 12%, 4 years: $7,500 x 1.57352 = $11,801.40Future value, 3%, 16 quarters: $7,500 x 1.60471 = $12,035.33Future value, 1%, 48 months: $7,500 x 1.61223 = $12,091.70The last calculation relies on a future value factor that is not in the table at the end of Appendix A. The solution can be determined using a financial calculator or using Excel. The Excel calculation is as follows:

©Cambridge Business Publishers, 2014Solutions Manual, Appendix A A-1

EA-3. Present value, 1% per quarter, 8 quarters: $14,000 x 0.92348 = $12,929.

EA-4. Present value, 6%, 4 years: $24,000 x 0.79209 = $19,010.

©Cambridge Business Publishers, 2014A-2 Financial Accounting, 4th Edition

EA-5. PV annuity, 1%, 36 months: $12,500/30.10751 = $415.18

EA-6. The interest for the first month would be $12,500 x .01 = $125. The remaining $290.18 would go to reduce the loan balance.

EA-7. PV annuity, 2%, 24 quarters: $40,000 / 18.91393 = $2,114.84.


EA-8. This problem requires computation as the future value of an annuity. This can be calculated from Table 1 as follows:

Future value, 6%, 5 years ………. 1.33823





Total ………………………………... 5.97533

Annual deposit x Future value factor = Future valueDeposit x 5.97533 = $20 million; $20 million / 5.97533 = $3,347,095.5 per year.

EA-9. PV of an Annuity, 9%, 7 years: $60,000 x 5.03295 = $301,977.


EA-10. First, we must compute the payment due on December 1, 2016:

Future value, 5%, 3 years: $10,000 x 1.15763 = $11,576.30.

Next, we compute the present value of that payment at the market interest rate of 8%:

Present value, 8%, 3 years: $11,576.30 x 0.79383 = $9,189.61.

Finally, we compute the interest from December 1 through December 31, 2013:

$9,189.61 x 0.08 = $735.17.

Thus, in 2013, Rex Corporation would recognize sales revenue of $9,189.61 and interest income of $735.17.

EA-11. Present value, 12%, 10 years: $50,000 x 0.32197 = $16,099.

PV annuity, 12%, 10 years: $200,000 x 5.65022 = $1,130,044.

Maximum price Rye Company should be willing to pay is $1,130,044 + $160,985 = $1,146,143.

Using Excel:


EA-12. If Debra Wilcox chooses 26 payments, she will receive $7 million/26 = $269,230.77 per year for 26 years, with the first payment due today. This is an annuity due. Equivalently, this can be structured as an annuity in arrears by recognizing that the present value factor for the first payment is 1.0. Thus, the annuity factor is 1.0 plus the present value of an annuity for 25 payments.

a. PV annuity due, 8%, 25+1 payments: $269,230.77 x (1 + 10.67478) = $3,143,210.

b. PV annuity due, 4%, 25+1 payments: $269,230.77 x (1 + 15.62208) = $4,475,175.40.

c. If her opportunity cost of capital is 8%, Debra Wilcox should choose the lump sum of $3,500,000 rather than take the annuity. On the other hand, if her discount rate is 4%, the annuity gives a higher present value. One way to examine this problem is to ask what opportunity cost of capital (or discount rate) would make her indifferent between the two options. This is answered by solving for the rate at which the present value of the annuity is equal to $3,500,000. Using the rate function in Excel:

So, the rate at which Debra Wilcox would be indifferent between $3,500,000 as a lump sum payment and an annuity due of $269,230.77 for 26 payments is approximately 6.68% per year.


EA-13. The answer depends on Ms. Reed’s opportunity cost of capital (discount rate). To determine the rate at which she would be indifferent between these two options, we divide the present value by the future value and consult Table 2: $60,000/$100,000 = 0.60000. From Table 2, the present value of $1 received five years in the future discounted at 10% is 0.62092. At 11%, the present value is 0.59345. Thus the discount rate at which she would be indifferent is between 10% and 11%.

Alternatively, we can use the rate function in Excel:

The rate at which she would be indifferent is 10.76%. So, if her discount rate is higher than that, she should accept the $60,000 today. If her rate is less than 10.76%, she should choose the deferred payment of $100,000.


EA-14. This calculation (and the calculation in A13) can be done with either the present value or future value tables, depending on which value is placed in the numerator and which is in the denominator. Using the present value tables:

$400,000/$955,000 = 0.41885

The present value factor for 11%, 8 years is 0.43393.

The present value factor for 12%, 8 years is 0.40388.

Hence, the required rate of return on the investment is between 11% and 12%. Using Excel, we can determine the exact rate:


EA-15. This question can be answered by adding the future values at 12%:

Date:Amount

Deposited

Future Value Factor

Future Value

July 1, 2013 …………….. $360,000 1.57352 $566,467

June 30, 2014 ………….. 60,000 1.40493 84,296

June 30, 2015 ………….. 60,000 1.25440 75,264

June 30, 2016 ………….. 60,000 1.12000 67,200

June 30, 2017 ………….. 60,000 1.00000 60,000

Value on June 30, 2017 $853,227

EA-16. a. Present value of an annuity, 10%, 20 years:

Payment = $10,000,000/8.51356 = $1,174,597

b. $10,000,000 x 10% = $1,000,000.

c. Present value of an annuity, 10%, 10 years:$1,174,597 x 6.14457 = $7,217,392


EA-17. a. (i) Present value 2%, 10 periods: $200,000 x 0.82035 = $164,070

PV annuity, 2%, 10 periods: ($200,000x.05/2) x 8.98259 = $44,913$164,070 + $44,913 = $208,983.

(ii) Present value 3%, 10 periods: $200,000 x 0.74409 = $148,818

PV annuity, 3%, 10 periods: ($200,000x.05/2) x 8.53020 = $42,651$148,818 + $42,651 = $191,469.

continued next page


EA-17. concludedb. Present value 4%, 6 periods: $200,000 x 0.79031 = $158,062

PV annuity, 4%, 6 periods: ($200,000x.05/2) x 5.24214 = $26,211

$158,062 + $26,211 = $184,273.


EA-18. a. PV annuity, 9%, 20 years: $450,000 x 9.12855 = $4,107,847.

b. PV annuity due, 9%, 19+1 years: $450,000 x (1+ 8.95011) = $4,477,551.


EA-19. a. A: 1 + PV annuity, 5%, 5 years: $8,000 x (1 + 4.32948) = $42,635.84

B: PV annuity, 5%, 5 years: $9,000 x 4.32948 = $38,965.32

PV annuity, 5%, 2 years: $2,000 x 1.85941 = $3,718.82

$38,965.32 + $3,718.82 = $42,684.14 at 5% interest.

Alternative A appears to be the cheaper alternative.

b. A: 1 + PV annuity, 7%, 5 years: $8,000 x (1 + 4.10020) = $40,801.60

B: PV annuity, 7%, 5 years: $9,000 x 4.10020 = $36,901.80

PV annuity, 7%, 2 years: $2,000 x 1.80802 = $3,616.04

$36,901.80 + $3,616.04 = $40,517.84.

Alternative B appears to be the cheaper alternative at 7% interest.

EA-20. A: PV annuity 2%, 24 quarters: $3,000 x 18.91393 = $56,741.79

B: PV annuity 8%, 5 years: $14,300 x 3.99271 = $57,095.75

Payment alternative A appears to be cheaper at an 8% discount rate. However, this is misleading because the effective discount rate is higher in alternative A. If the same discount rate (2% compounded quarterly) is used for both calculations, the annuity factor for alternative B would be lower. To compute the appropriate discount rate, add the present value of a single payment for each of the five payments as follows:

2%, 4 quarters 0.92385

2%, 8 quarters 0.85349

2%, 12 quarters 0.78849

2%, 16 quarters 0.72845

2%, 20 quarters 0.67297

Total 3.96725

$14,300 x 3.96725 = $56,731.68, hence, alternative B is slightly cheaper.


Documents

Financial Accounting