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# Calculations of the Time Value of Money

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1) You invest \$20,000 today, at a rate of 10% compound quarterly. What will the investment be worth at the end of year twenty?
2) You are offered an annuity that will pay you \$9,000 at the end of each of the next 10 years. What is the maximum amount you would be willing to pay today for this annuity? (Assume you require a 15% rate of return on an investment of this nature.)
3) You have \$15,000 to put down on a new house that cost \$200,000, and you have been quoted the following finance terms by your local banker: 6% Annual Percentage Rate, for 30 years. If you decide to purchase this home, what will your monthly payment be? Additionally, over the life of the loan what would your total interest expense be?
4) You want to start saving for your child's education. You project that your child will need \$170,000 to attend school 15 years from now. If you can earn a rate of return of 10% compounded semi-annually on a given investment, what dollar amount will you need to invest today to ensure your child can attend college?
5) Steaks Galore has \$190,000 in excess cash that it wishes to invest. Bank One offers a certificate of deposit that is paying 10%, compounded monthly. Bank Two offers a certificate of deposit paying 9.5%, compounded daily. In which Bank should the firm opt to invest its' surplus cash? (You must use the EAR formula to solve this problem. In addition, you must show all of your work.) Additionally, what is the nominal and period rate of interest offered by Bank One?
6) You plan on depositing \$3,000 in an account at the end of each of the next 5 years. If the account is paying interest at an annual rate of 10% per year, what will the total value of your investment be at the end of the 10th year?
7) Your Life Insurance Agent is trying to sell you an investment that will pay you \$5,000 a year forever. If your required rate of return is 11% on an investment of this nature, what would you be willing to pay your agent today for this investment opportunity?
8) It is forecasted that you will receive the following cash inflows at the end of the next four years: Year 1 \$1,000, Year 2 \$2,000, Year 3 \$4,000 and Year 4 \$1,000. If upon receipt of these cash inflows, you can re-invest the amounts received at a rate of 10%, what will the total future value of this investment be?
9) While Bob Jones was a student at Tiffin University, he borrowed \$43,063 in student loans at an annual rate of 7 percent. If Bob repays \$500 per month, how long, to the nearest year, will it take him to repay the loan?
10) Company XYZ plans to invest \$5 million to clear a tract of land and to set out some young trees. These trees will mature in 12 years, at which time XYZ plans to sell all the trees at an expected price of \$10 million. What is XYZ's expected rate of return? In addition, given this rate of return, would you recommend that XYZ proceed with the plan? Why or why not.

#### Solution Summary

10 problems regarding time value of the money.

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## Time Value of Money Problems

Please help me understand what type of specific TVM problem each of the problems from 1 to 10 are and give the formulas required to solve each. l will work out the calculations.

For problems 11 and 12, please help me see the work for the complete problem including the calculations and everything.

1. First Bank and Trust of Winnetka offers a five-year Certificate of Deposit (CD) with an APR (Annual Percentage Rate) of 3.6%, compounded monthly? Cook County Federal Savings and Loan offers a five-year CD with an APR of 3.58%, compounded continuously. Both accounts are insured by the government, so which institution offers the better deal? (document your decision)

2. Angelo Geraldis, the general manager of a stumbling professional football team signed a six-year contract on September 1, 2008. The contract called for an initial salary payment of \$1,200,000 at the contract's signing and annual payments thereafter increasing at an annual rate of 4%. On August 31, 2010, the owners of the team informed him that his services were no longer required. Of course, "a contract is a contract," so he threatened to sue unless he was paid the present value of the remaining contractual payments. At September 1st, 2010, what is the present value of the remaining payments at 8% per annum? Would a 10% rate be better for him?

3. You are saving for your retirement. You have decided that one year from today you will deposit 5% of your salary (which is \$70,000) into an account that will earn 7% per year. You expect your salary will increase at 2.5% per year throughout your career. How much money will you have in your retirement account at the end of your working career, thirty-five years from now, if you continue to make annual deposits equal to 5% of your salary?

4. Toni Nuckly wishes to have \$40,000 at the end of twelve years. She is considering an investment that promises a six percent return each year. Suppose that she made equal payments for the first five years under the assumption that her funds would earn six percent every year. Suddenly the return fell from six percent to four percent per year for the final seven years. This caused her to increase her (equal) annual payments in years six through twelve. What equal payments would she have to make for the first five years and then for the final seven years in order to amass \$40,000 at the end of the twelfth year?

5. Suppose that Toni, in problem 2, wished to pay an equal amount each year for all twelve years and that she knew that the interest rate would decrease from six percent to four percent after the fifth year. What equal amount would she have to set aside at the end of each of the twelve years to amass her \$40,000 nest egg?

6. Suppose that you are considering a thirty-year mortgage with an APR of 3.9%, compounded monthly, with payments due at the end of each month. In how many months would sixty percent of the original principal be paid down?

7. William Chris opened a steak house a few years ago with his sister, Ruth. In going through their financial records they found an old amortization schedule that their lender had prepared when they took out a loan to start the business. Unfortunately, the amortization schedule had A-1 Sauce, Worcestershire Sauce, and other condiments spilled all over it. The only thing that William and Ruth could make out was that the loan was for thirty years, the monthly payments were at an APR of 6.6%, and the balance after twelve years was \$135,146.44. Not being too financially astute (nor having either good memories or good records), they ask you to determine the original amount that they borrowed.

8. A few years ago a subsidiary of Stanley Works offered to purchase another company for \$45.3 million in order to diversify its business. One way to value an entire company is to find the present value of the annual cash flows generated by the company.

a. How large would the annual cash flows (after-tax) have to be in order to justify Stanley's purchase price if they were discounted at 18% and continued indefinitely with no growth?

b. Assume that the first annual cash flow (one year after acquisition) was expected to be \$5.0 million, but it would then grow at a constant annual growth rate (indefinitely). If Stanley's required rate of return (discount rate) was 18%, how large would the growth rate have to be in order to justify the purchase price?

9. Suppose that just yesterday Black & Decker Company purchased and installed a made-to-order machine tool for fabricating parts for small appliances. The machine cost \$120,000. Today, Square D Company offers a similar machine tool that will do exactly the same work, but costs only \$75,000. The discount rate (also known as the hurdle rate) is 15% and both machines will last for five years at which time they will have no residual value. Black & Decker will depreciate either machine on a straight-line basis with no salvage value for income tax purposes. The relevant income tax rate is 30%, and Black & Decker earns sufficient income from its other operations so that it can utilize any annual operating losses or losses on disposal of equipment.

Required:
What is the minimum resale value of the "old machine tool" that would justify Black & Decker's purchase of the Square D machine tool at this time?

10. You are evaluating two different database management systems for The University of Chicago's admissions department. The Banner I costs \$420,000, has a three-year useful life, and has operating costs of \$68,000 per year. The Banner II costs \$640,000, has a five-year life, and has operating costs of only \$46,000 per year. The salvage value (received at the end of its useful life) of each machine is equal to ten percent (10%) of its initial cost. Since the useful lives are unequal, you decide to use equivalent uniform cost analysis. In addition, because Northwestern is a not-for-profit entity, you realize that income taxes are irrelevant to your analysis.

a. Assuming that the relevant interest rate is 14%, which system should you recommend?

b. Using net present value analysis, either support or reject your answer to part a.

11. Kerry Dean owns a very profitable ice cream stand. Based on the results of a \$6,000 marketing research project, Kerry is now considering adding a line of frozen yogurt to his product mix. Kerry expects that the machine necessary to produce the yogurt will last for five years and have a salvage value of \$10,000 at the end of the fifth year. For tax purposes, Kerry will depreciate the yogurt machine over the five years to a zero salvage value using the straight-line method. Kerry will incur extra annual fixed costs of \$2,500 per year if he acquires this machine, while his variable costs (the cone/cup, yogurt, wrapping paper, spoons, etc.) will be \$.50 per serving. He expects to sell 100 servings a day (365 days in a year) at \$1.50 each. The property taxes and insurance on his ice cream stand are \$18,000 per year. The relevant income tax rate for Kerry's business is 30% (both for ordinary income and capital gains/losses) and his cost of capital (same as MARR or "hurdle rate") is 18%.

Required:
What is the most that Kerry should pay for this machine so that the investment makes economic sense? Show your work!

12. Southern Illinois Publishing Company is trying to decide whether or not to revise its popular textbook, Football for Monday Morning Quarterbacks. The revision is expected to cost \$170,000. Contribution margin from the increased sales will be \$70,000 in the first year. These cash flows will increase at the inflation rate of five percent (5%) per year, until the book goes out of print. The book will go out of print five (5) years from now. Assume that the initial cost is paid now and that other cash flows are received at the end of each year. The company requires a ten percent (10%) real rate of return on such investments. (Ignore taxes)

Required:
a Using nominal cash flows, find the net present value of this investment; and
b. Using real cash flows, find the net present value of this investment.
c. For which set of cash flows, nominal or real, is this a better investment?

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