# Present Value, Compounding Interest, Effective Annual Rate

1. Your subscription to Jogger's World Monthly is about to run out and you have the choice of renewing it by sending in the $10 a year regular rate or of getting a lifetime subscription to the magazine by paying $100. Your cost of capital is 7 percent. How many years would you have to live to make the lifetime subscription the better buy? Payments for the regular subscription are made at the beginning of each year. (Round up if necessary to obtain a whole number of years.)

a. 15 years b. 10 years c. 18 years d. 7 years e. 8 years

2. Steaks Galore needs to arrange financing for its expansion program. One bank offers to lend the required $1,000,000 on a loan that requires interest to be paid at the end of each quarter. The quoted rate is 10 percent, and the principal must be repaid at the end of the year. A second lender offers 9 percent, daily compounding (365-day year), with interest and principal due at the end of the year. What is the difference in the effective annual rates (EFF%) charged by the two banks?

a. 0.31% b. 0.53% c. 0.75% d. 0.96% e. 0.25%

3. You want to borrow $1,000 from a friend for one year, and you propose to pay her $1,120 at the end of the year. She agrees to lend you the $1,000, but she wants you to pay her $10 of interest at the end of each of the first 11 months plus $1,010 at the end of the 12th month. How much higher is the effective annual rate under your friend's proposal than under your proposal?

a. 0.00% b. 0.45% c. 0.68% d.0.89% e. 1.00%

4. Today, Bruce and Brenda each have $150,000 in an investment account. No other contributions will be made to their investment accounts. Both have the same goal: They each want their account to reach $1 million, at which time each will retire. Bruce has his money invested in risk-free securities with an expected annual return of 5 percent. Brenda has her money invested in a stock fund with an expected annual return of 10 percent. How many years after Brenda retires will Bruce retire?

a. 12.6 b. 19.0 c. 19.9 d. 29.4 e. 38.9

5. You have $2,000 invested in a bank account that pays a 4 percent nominal annual interest with daily compounding. How much money will you have in the account at the end of July (in 132 days)? (Assume there are 365 days in each year.)

a. $2,029.14 b. $2,028.93 c. $2,040.00 d. $2,023.44 e. $2,023.99

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#### Solution Preview

1. We need to calculate the present value (PV) of the $10 each year, and add them up. When, after t years, the sum equals or is greater than 100, it's no more worthwhile to pay by regular rate annually.

The PV for year t is PVt = S/(1+r)^(t-1) where S = 10 and r = 7%

Then SUM(PVt) = S[1 - 1/(1+r)^t] (1+r)/r = 10[1-1/(1+0.07)^t](1+0.07)/0.07 =1070/7(1-1/1.07^t)

Let SUM(PVt) = 100, i.e. 1070/7(1-1/1.07^t) = 100 or 1.07^t = 107/37=2.892 or t = log1.07(2.892)

We can calculate t = 15.7. Therefore (a) is the right choice

2. For every compounding interest plan there is an effective annual rate. This effective annual rate is an imagined rate of simple interest that would yield the same final value as the compounding plan over one year. Formula:

i Effective annual rate

r Annual ...

#### Solution Summary

The Solution addresses several questions about Present and Future Values.