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# Time Value of Money

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IRA Investments develops retirement programs for individuals. You are 30 years old and plan to retire on your 60th birthday. You want to establish a plan with IRA that will require a series of equal, annual, end-of-year deposits into the retirement account. The first deposit will be made 1 year from today on your 31st birthday. The final payment on the account will be made on your 60th birthday. The retirement plan will allow you to withdraw \$120,000 per year for 15 years, with the first withdrawal on your 61st birthday. Also at the end of the 15th year, you wish to withdraw an additional \$250,000. The retirement account promises to earn 12 percent annually.

What periodic payment must be made into the account to achieve your retirement objective?

#### Solution Preview

"IRA Investments develops retirement programs for individuals. You are 30 years old and plan to retire on your 60th birthday. You want to establish a plan with IRA that will require a series of equal, annual, end-of-year deposits into the retirement account. The first deposit will be made 1 year from today on your 31st birthday. The final payment on the account will be made on your 60th birthday. The retirement plan will allow you to withdraw \$120,000 per year for 15 years, with the first withdrawal on your 61st birthday. Also at the end of the 15th year, you wish to withdraw an additional \$250,000. The retirement account promises to earn 12 percent ...

#### Solution Summary

The expert calculates the value of an annuity to achieve retirement objective.

\$2.19

## Time value of money - calculator operations

1. Find the annual interest rate. (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Present Value Future Value Time Period Annual Interest Rate
\$ 100 \$ 122.01 4 years %

200 281.82 5 %

100 113.28 6 %

2. If you take out an \$8,300 car loan that calls for 48 monthly payments starting after 1 month at an APR of 6%, what is your monthly payment? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Monthly payment \$

b. What is the effective annual interest rate on the loan? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Effective annual interest rate %

3. If the interest rate this year is 8.6% and the interest rate next year will be 10.6%, what is the future value of \$1 after 2 years? What is the present value of a payment of \$1 to be received in 2 years? (Do not round intermediate calculations. Round your answers to 4 decimal places.)

Future value \$
Present value \$

4. Compute the future value of a \$120 cash flow for the same combinations of rates and times: (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Future Value
a. r = 12%, t = 8 years \$
b. r = 12%, t = 16 years
c. r = 6%, t = 8 years
d. r = 6%, t = 16 years

5. A factory costs \$420,000. You forecast that it will produce cash inflows of \$130,000 in year 1, \$190,000 in year 2, and \$320,000 in year 3. The discount rate is 10%.

a. Calculate the PV of cash inflows. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Present value \$

b. Is the factory a good investment?

No
Yes

6. Investments in the stock market have increased at an average compound rate of about 5% since 1911. It is now 2012.

a. If you invested \$1,000 in the stock market in 1911, how much would that investment be worth today? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Investment \$

b. If your investment in 1911 has grown to \$1 million, how much did you invest in 1911? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Present value \$

7. Sure Tea Co. has issued 4.2% annual coupon bonds that are now selling at a yield to maturity of 6% and current yield of 5.8000%. What is the remaining maturity of these bonds? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Remaining period years

8. A 6-year Circular File bond pays interest of \$80 annually and sells for \$986. What are its coupon rate and yield to maturity? (Do not round intermediate calculations. Round "Coupon rate" to 1 decimal place and "Yield to maturity" to 2 decimal places.)

Coupon rate %
Yield to maturity %

9. A 2-year maturity bond with face value of \$1,000 makes annual coupon payments of \$86 and is selling at face value. What will be the rate of return on the bond if its yield to maturity at the end of the year is (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Rate of Return
a. 6% %

b. 8.6% %

c. 10.6 %

10. A 25-year Treasury bond is issued with face value of \$1,000, paying interest of \$56 per year. If market yields increase shortly after the T-bond is issued, what is the bond's coupon rate? (Round your answer to 1 decimal place.)

Coupon rate %

11. A bond's credit rating provides a guide to its risk. Long-term bonds rated Aa currently offer yields to maturity of 7.9%. A-rated bonds sell at yields of 8.2%. Assume a 10-year bond with a coupon rate of 7.4% is downgraded by Moody's from Aa to A rating.

a. Calculate the initial price. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Initial price \$

b. Calculate the new price. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

New price \$

12. Steady As She Goes, Inc., will pay a year-end dividend of \$2.90 per share. Investors expect the dividend to grow at a rate of 6% indefinitely.

a. If the stock currently sells for \$29 per share, what is the expected rate of return on the stock? (Do not round intermediate calculations.)

Expected rate of return %

b. If the expected rate of return on the stock is 18.5%, what is the stock price? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Stock price \$

13. Arts and Crafts, Inc., will pay a dividend of \$7 per share in 1 year. It sells at \$70 a share and firms in the same industry provide an expected rate of return of 14%. What must be the expected growth rate of the company's dividends? (Do not round intermediate calculations.)

Expected growth rate %

14. No-Growth Industries pays out all of its earnings as dividends. It will pay its next \$6 per share dividend in a year. The discount rate is 21%.

a. What is the price-earnings ratio of the company? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

P/E ratio

b. What would the P/E ratio be if the discount rate were 20%? (Round your answer to 2 decimal places.)

P/E ratio

15. Grandiose Growth has a dividend growth rate of 20%. The discount rate is 12%. The end-of-year dividend will be \$4 per share.

a. What is the present value of the dividend to be paid in year 1? Year 2? Year 3? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Present Value
Year 1 \$
Year 2
Year 3

16. Waterworks has a dividend yield of 9.25%. If its dividend is expected to grow at a constant rate of 6.25%, what must be the expected rate of return on the company's stock? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Expected rate of return %

17. Computer Corp. reinvests 50% of its earnings in the firm. The stock sells for \$70, and the next dividend will be \$3.50 per share. The discount rate is 15%. What is the rate of return on the company's reinvested funds? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Rate of return %

The following are the cash flows of two projects:

Year Project A Project B
0 −\$350 −\$350
1 180 250
2 180 250
3 180 250
4 180
________________________________________

18. What is the payback period of each project? (Round your answers to 2 decimal places.)

Project Payback Period
A years
B years

19. The following are the cash flows of two projects:

Year Project A Project B
0 −\$280 −\$280
1 160 180
2 160 180
3 160 180
4 160

a. If the opportunity cost of capital is 10%, what is the profitability index for each project? (Do not round intermediate calculations. Round your answers to 4 decimal places.)

Project Profitability Index
A

B

b. Does the profitability index rank the projects correctly?

20. The following are the cash flows of two projects:

Year Project A Project B
0 −\$220 −\$220
1 100 120
2 100 120
3 100 120
4 100

a. Calculate the NPV for both projects if the discount rate is 10%? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Project NPV
A \$
B

b. Suppose that you can choose only one of these projects. Which would you choose?

Project B
Project A

The following table presents sales forecasts for Golden Gelt Giftware. The unit price is \$40. The unit cost of the giftware is \$20.

Year Unit Sales
1 34,000
2 42,000
3 16,000
4 10,000
Thereafter 0
________________________________________

21. It is expected that net working capital will amount to 20% of sales in the following year. For example, the store will need an initial (year-0) investment in working capital of .20 × 34,000 × \$40 = \$272,000. Plant and equipment necessary to establish the Giftware business will require an additional investment of \$212,000. This investment will be depreciated using MACRS and a 3-year life. After 4 years, the equipment will have an economic and book value of zero. The firm's tax rate is 30%. What is the net present value of the project? The discount rate is 20%. (Do not round intermediate calculations. Round your answer to the nearest dollar amount.)

Net present value \$

22. Canyon Tours showed the following components of working capital last year:

Beginning End of Year
Accounts receivable \$27,400 \$24,700
Inventory 13,700 15,900
Accounts payable 16,200 19,900
________________________________________

a. What was the change in net working capital during the year? (Negative amount should be indicated by a minus sign.)

Change in net working capital \$

b. If sales were \$37,700 and costs were \$25,700, what was cash flow for the year? Ignore taxes.

Cash flow \$

23. The owner of a bicycle repair shop forecasts revenues of \$236,000 a year. Variable costs will be \$69,000, and rental costs for the shop are \$49,000 a year. Depreciation on the repair tools will be \$29,000. The tax rate is 40%.

a. Calculate operating cash flow for the year by using all three methods: (a) adjusted accounting profits; (b) cash inflow/cash outflow analysis; and (c) the depreciation tax shield approach.

Method Operating Cash Flow
Cash inflow/cash outflow analysis
Depreciation tax shield approach

Quick Computing installed its previous generation of computer chip manufacturing equipment 3 years ago. Some of that older equipment will become unnecessary when the company goes into production of its new product. The obsolete equipment, which originally cost \$35.5 million, has been depreciated straight-line over an assumed tax life of 5 years, but it can be sold now for \$17.1 million. The firm's tax rate is 40%. What is the after-tax cash flow from the sale of the equipment? (Enter your answer in millions rounded to 2 decimal places.)

After-tax cash flow \$ million

25. Ilana Industries, Inc., needs a new lathe. It can buy a new high-speed lathe for \$1.3 million. The lathe will cost \$38,000 per year to run, but will save the firm \$138,000 in labor costs, and will be useful for 10 years. Suppose that for tax purposes, the lathe will be depreciated on a straight-line basis over its 10-year life to a salvage value of \$300,000. The actual market value of the lathe at that time also will be \$300,000. The discount rate is 7%, and the corporate tax rate is 35%. What is the NPV of buying the new lathe? (Negative amount should be indicated by a minus sign. Enter your answer in dollars not in millions. Do not round intermediate calculations. Round your answer to 2 decimal places.)

NPV \$

26. A new project will generate sales of \$73.4 million, costs of \$41.4 million, and depreciation expense of \$9.4 million in the coming year. The firm's tax rate is 40%.

a. Calculate cash flow for the year by using all three methods: (a) adjusted accounting profits; (b) cash inflow/cash outflow analysis; and (c) the depreciation tax shield approach. (Enter your answers in millions rounded to 2 decimal places.)

Method Cash Flow
Cash inflow/cash outflow analysis million
Depreciation tax shield approach million

b. Are the above answers equal?

27. Modern Artifacts can produce keepsakes that will be sold for \$50 each. Nondepreciation fixed costs are \$1,500 per year and variable costs are \$30 per unit.

a. If the project requires an initial investment of \$2,000 and is expected to last for 5 years and the firm pays no taxes. The initial investment will be depreciated straight-line over 5 years to a final value of zero, and the discount rate is 10%. What are the accounting and NPV break-even levels of sales? (Do not round intermediate calculations. Round your answers to the nearest whole number.)

Accounting break-even levels of sales units
NPV break-even levels of sales units

b. What will be the accounting and NPV break-even levels of sales, if the firm's tax rate is 40%? (Do not round intermediate calculations. Round your answers to the nearest whole number.)

Accounting break-even levels of sales units
NPV break-even levels of sales units

28. Dime a Dozen Diamonds makes synthetic diamonds by treating carbon. Each diamond can be sold for \$100. The materials cost for a standard diamond is \$50. The fixed costs incurred each year for factory upkeep and administrative expenses are \$180,000. The machinery costs \$1.3 million and is depreciated straight-line over 10 years to a salvage value of zero.

a. What is the accounting break-even level of sales in terms of number of diamonds sold?

Break-even sales

b. What is the NPV break-even level of sales assuming a tax rate of 30%, a 10-year project life, and a discount rate of 12%? (Do not round intermediate calculations. Round your answer to the nearest whole number.)

Break-even sales

29. A silver mine can yield 16,000 ounces of silver at a variable cost of \$34 per ounce. The fixed costs of operating the mine are \$56,000 per year. In half the years, silver can be sold for \$50 per ounce; in the other years, silver can be sold for only \$25 per ounce. Ignore taxes.

a. What is the average cash flow you will receive from the mine if it is always kept in operation and the silver always is sold in the year it is mined?

Average cash flow \$

b. Now suppose you can shut down the mine in years of low silver prices. Calculate the average cash flow from the mine.

Average cash flow \$

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