Present Value: Questions

1. You have an expected liability (cash outflow) of $500,000 in 10 years, and you use a discount rate of 10%.

a. How much would you need right now as savings to cover the expected liability?

b. How much would you need to set aside at the end of each year for the next 10 years to cover the expected liability?

2. You are examining whether your savings will be adequate to meeting your retirement needs. You saved $1500 last year, and you expect your annual savings to grow 5% a year for the next 15 years. If you can invest your money at 8%, how much would you expect to have at the end of the fifteenth year?

3. You have just taken a 30-year mortgage loan for $200,000. The annual percentage rate on the loan is 8%, and payments will be made monthly. Estimate your monthly payments.

4. You are planning to buy a car worth $20,000. Which of the two deals described below would you choose:

• the dealer offers to take 10% off the price, and lend you the balance at the regular financing rate (which is an annual percentage rate of 9%)

• the dealer offers to lend you $20,000 (with no discount) at a special financing rate of 3%

5. A company is planning to set aside money to repay $100 million in bonds that will be coming due in 10 years. If the appropriate discount rate is 9%,

a. how much money would the company need to set aside at the end of each year for the next 10 years to be able to repay the bonds when they come due?

b. how would your answer change if the money were set aside at the beginning of each year?

6. What is the value of 15-year corporate bonds, with a coupon rate of 9%, if current interest rates on similar bonds is 8%? How much would the value change if interest rates increased to 10%? Under what conditions will this bond trade at par (face value)?

7. What is the value of stock in a company that currently pays out $1.50 per share in dividends and expects these dividends to grow 6% a year forever? (You can assume that investors require a 13% return on stocks of equivalent risk.)

8. What is the value of stock in a company that currently pays out $1.00 per share in dividends, and expects these dividends to grow 15% a year for the next 5 years, and 6% a year forever after that? (You can assume that investors require a 12.5% return on stocks of equivalent risk and that the dividend payout ratio will double after the fifth year.)

9. You buy a 10-year zero-coupon bond, with a face value of $1000, for $300. What is the rate of return you will make on this bond?

10. You are reviewing an advertisement by a finance company offering loans at an annual percentage rate of 9%. If the interest is compounded weekly, what is the effective interest rate on this loan?

11. You have an relative who has accumulated savings of $ 250,000 over his working lifetime and now plans to retire. Assuming that he wishes to withdraw equal installments from these savings for the next 25 years of this life, how much will each installment amount to if he is earning 5% on his savings?

12. You are offered a special set of annuities by your insurance company, whereby you will receive $20,000 a year for the next 10 years and $30,000 a year for the following 10 years. How much would you be willing to pay for these annuities, if your discount rate is 9% and the annuities are paid at the end of each year? How much would you be willing to pay if they were at the beginning of each year?

13. A bill that is designed to reduce the nation’s budget deficit passes both houses of legislature. Congress tells us that the bill will reduce the deficit by $500 billion over 10 years. What it does not tell us is the timing of the reductions.

Year Deficit Reduction

1 $ 25 Billion

2 $ 30 Billion

3 $ 35 Billion

4 $ 40 Billion

5 $ 45 Billion

6 $ 55 Billion

7 $ 60 Billion

8 $ 65 Billion

9 $ 70 Billion

10 $ 75 Billion

If the federal government can borrow at 8%, what is the true deficit reduction in the bill?

14. New York State has a pension fund liability of $25 billion, due in 10 years. Each year the legislature is supposed to set aside an annuity to arrive at this future value. This annuity is based on what the legislature believes it can earn on this money.

a. Estimate the annuity needed each year for the next 10 years, assuming that the interest rate that can be earned on the money is 6%.

b. The legislature changes the investment rate to 8% and recalculates the annuity needed to arrive at the future value. It claims the difference as budget savings this year. Do you agree?

15. Poor Bobby Bonilla! The newspapers claim that he is making $5.7 million a year. He claims that this is not true in a present value sense and that he will really be making the following amounts for the next 5 years:

Year Amount

0 (now) $ 5.5 million (Sign up Bonus)

1 $ 4 million

2 $ 4 million

3 $ 4 million

4 $ 4 million

5 $ 7 million

a. Assuming that Bonilla can make 7% on his investments, what is the present value of his contract?

b. If you wanted to raise the nominal value of his contract to $30 million, while preserving the present value, how would you do it? (You can adjust only the sign up bonus and the final year’s cash flow.)

16. You are comparing houses in two towns in New Jersey. You have $100,000 to put as a down payment, and 30-year mortgage rates are at 8% –

Chatham South Orange

Price of the house $ 400,000 $ 300,000

Annual Property Tax $ 6,000 $ 12,000

The houses are roughly equivalent.

a. Estimate the total payments (mortgage and property taxes) you would have on each house. Which one is less expensive?

b. Are mortgage payments and property taxes directly comparable? Why or why not?

c. If property taxes are expected to grow 3% a year forever, which house is less expensive?

17. You bought a house a year ago for $250,000, borrowing $200,000 at 10% on a 30-year term loan (with monthly payments). Interest rates have since come down to 9%. You can refinance your mortgage at this rate, with a closing cost that will be 3% of the loan. Your opportunity cost is 8%. Ignore tax effects.

a. How much are your monthly payments on your current loan (at 10%)?

b. How would your monthly payments be if you could refinance your mortgage at 9% (with a 30-year term loan)?

c. You plan to stay in this house for the next 5 years. Given the refinancing cost (3% of the loan), would you refinance this loan?

d. How much would interest rates have to go down before it would make sense to refinance this loan (assuming that you are going to stay in the house for five years)?

18. You are 35 years old today and are considering your retirement needs. You expect to retire at age 65 and your actuarial tables suggest that you will live to be 100. You want to move to the Bahamas when you retire. You estimate that it will cost you $ 300,000 to make the move (on your 65th birthday) and that your living expenses will be $30,000 a year (starting at the end of year 66 and continuing through the end of year 100) after that.

a. How much will you need to have saved by your retirement date to be able to afford this course of action?

b. You already have $50,000 in savings. If you can invest money, tax-free, at 8% a year, how much would you need to save each year for the next 30 years to be able to afford this retirement plan?

c. If you did not have any current savings and do not expect to be able to start saving money for the next 5 years, how much would you have to set aside each year after that to be able to afford this retirement plan?

19. You have been hired to run a pension fund for TelDet Inc, a small manufacturing firm. The firm currently has $5 million in the fund and expects to have cash inflows of $2 million a year for the first 5 years followed by cash outflows of $ 3 million a year for the next 5 years. Assume that interest rates are at 8%.

a. How much money will be left in the fund at the end of the tenth year?

b. If you were required to pay a perpetuity after the tenth year (starting in year 11 and going through infinity) out of the balance left in the pension fund, how much could you afford to pay?

20. You are an investment advisor who has been approached by a client for help on his financial strategy. He has $250,000 in savings in the bank. He is 55 years old and expects to work for 10 more years, making $100,000 a year. (He expects to make a return of 5% on his investments for the foreseeable future. You can ignore taxes)

a. Once he retires 10 years from now, he would like to be able to withdraw $80,000 a year for the following 25 years (his actuary tells him he will live to be ninety years old.). How much would he need in the bank 10 years from now to be able to do this?

b. How much of his income would he need to save each year for the next 10 years to be able to afford these planned withdrawals ($80,000 a year) after the tenth year?

c. Assume that interest rates decline to 4% 10 years from now. How much, if any, would you client have to lower his annual withdrawal by, assuming that he still plans to withdraw cash each year for the next 25 years?

21. You have been asked to estimate the value of a 10-year bond with a coupon that will be low initially but it is expected to grow later in the bondís life. The coupon is expected to be 5% of the face value of the bond (which is $ 1000) for the first 5 years, and will increase by 1% every year for the next 5 years ñ the coupon rate will be 6% in year 6, 7% in year 7, 8% in year 8, 9% in year 9 and 10% in year 10. Estimate the value of this bond.

22. You are trying to assess the value of a small retail store that is up for sale. The store generated a cash flow to its owner of $ 100,000 in the most year of operation, and is expected to have growth of about 5% a year in perpetutity.

• If the rate of return required on this store is 10%, what would your assessment be of the value of the store?

• What would the growth rate need to be to justify a price of $ 2.5 million for this store?