# investment with varying interest rates

Step-by-step solution in equation form for all cost analysis sections

1. What sum of money will be accumulated in 10 years if:

a) $100 is invested at the end of each month at a 15% rate of return compounded annually.

b) The same investment as above for a 3% rate of return compounded monthly. What is the effective rate?

c) $1,000 is invested at the end of the first year, $1,100 is invested at the end of the second year and each succeeding years investment is $100 higher for 10 years at 15% compounded annually.

d) What single investment at time zero would generate the part (a) future worth?

e) What will the present worth of investments in part (c) be if the investments are started at the end of year 5 for 10 years?

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#### Solution Preview

1. What sum of money will be accumulated in 10 years if:

a) $100 is invested at the end of each month at a 15% rate of return compounded annually.

PV = 0

payment = 100

monthly interest rate is r= 15%/12=1.25%

Number of months is N = 120

By a financial calculator,

compute FV = $27,521.71

b) The same investment as above for a 3% rate of return compounded monthly. What is the effective rate?

PV = ...

#### Solution Summary

This job explores investments with varying interest rates are provided.

PV calculations: single cash flow, mixed stream, FV, annuities, EBIT, leverage

See attached file.

P4-4 For each of the cases shown in the following table, calculate the future value of the single cash flow deposited today that will be available at the end of the deposit period if the interest is compounded annually at the rate specified over the given period.

Case Signgle Cash Flow Interest Rate Deposit Period

A $200.00 5% 20

B 4,500 8 7

C 10,000 9 10

D 25,000 10 12

E 37,000 11 5

F 40,000 12 9

P4-9 Present value calculation

Without referring to tables or to the preprogrammed function on your financial calculator, use the basic formula for present value, along with the given opportunity cost, i, and the number of periods, n, to calculate the present value interest factor in each of the cases shown in the accompanying table. Compare the calculated value to the table value.

Case Opportunity cost, i Number of periods, n

A 2% 4

B 10% 2

C 5 3

D 13 2

P4-25 Value of a mixed stream

For each of the mixed streams of cash flows shown in the following table, determine the future value at the end of the final year if deposits are made at the beginning of each year into an account paying annual interest of 12%, assuming that no withdrawals are made during the period.

Cash flow stream

Year A B C

1 $ 900 $30,000 $1,200

2 1,000 25,000 1,200

3 1,200 20,000 1,000

4 10,000 1,900

5 5,000

P4-31 Relationship between future value and present value?Mixed stream

Using only the information in the accompanying table, answer the questions that follow.

Year (t) Cash flow Future value interest factor at 5% (FVIF5%,n)

1 $ 800 1.050

2 900 1.102

3 1,000 1.158

4 1,500 1.216

5 2,000 1.276

A) Determine the present value of the mixed stream of cash flows using a 5% discount rate.

B) How much would you be willing to pay for an opportunity to buy this stream, assuming that you can at best earn 5% on your investments?

C) What effect, if any, would a 7% rather than a 5% opportunity cost have on your analysis? (Explain verbally.)

P4-37 Annuities and compounding

Janet Boyle intends to deposit $300 per year in a credit union for the next 10 years, and the credit union pays an annual interest rate of 8%.

A) Determine the future value that Janet will have at the end of 10 years, given that end-of-period deposits are made and no interest is withdrawn, if

a. $300 is deposited annually and the credit union pays interest annually.

b. $150 is deposited semiannually and the credit union pays interest semiannually.

c. $75 is deposited quarterly and the credit union pays interest quarterly.

B) Use your finding in part a to discuss the effect of more frequent deposits and compounding of interest on the future value of an annuity.

P4-43 Loan amortization schedule

Joan Messineo borrowed $15,000 at a 14% annual rate of interest to be repaid over 3 years. The loan is amortized into three equal, annual, end-of-year payments.

A) Calculate the annual, end-of-year loan payment.

B) Prepare a loan amortization schedule showing the interest and principal breakdown of each of the three loan payments.

C) Explain why the interest portion of each payment declines with the passage of time.

P4-51 Interest rate for an annuity

Anna Waldheim was seriously injured in an industrial accident. She sued the responsible parties and was awarded a judgment of $2,000,000. Today, she and her attorney are attending a settlement conference with the defendants. The defendants have made an initial offer of $156,000 per year for 25 years. Anna plans to counteroffer at $255,000 per year for 25 years. Both the offer and the counteroffer have a present value of $2,000,000, the amount of the judgment. Both assume payments at the end of each year.

A) What interest rate assumption have the defendants used in their offer (rounded to the nearest whole percent)?

B) What interest rate assumption have Anna and her lawyer used in their counteroffer (rounded to the nearest whole percent)?

C) Anna is willing to settle for an annuity that carries an interest rate assumption of 9%. What annual payment would be acceptable to her?

Chapter 11:

P11-6 EBIT sensitivity

Stewart Industries sells its finished product for $9 per unit. Its fixed operating costs are $20,000, and the variable operating cost per unit is $5.

A) Calculate the firm's earnings before interest and taxes (EBIT) for sales of 10,000 units.

B) Calculate the firm's EBIT for sales of 8,000 and 12,000 units, respectively

C) Calculate the percentage changes in sales (from the 10,000-unit base level) and associated percentage changes in EBIT for the shifts in sales indicated in part b.

D) On the basis of your findings in part c, comment on the sensitivity of changes in EBIT in response to changes in sales.

P11-7 Degree of operating leverage

Grey Products has fixed operating costs of $380,000, variable operating costs of $16 per unit, and a selling price of $63.50 per unit.

1. Calculate the operating breakeven point in units.

2. Calculate the firm's EBIT at 9,000, 10,000, and 11,000 units, respectively.

3. With 10,000 units as a base, what are the percentage changes in units sold and EBIT as sales move from the base to the other sales levels used in part b?

4. Use the percentages computed in part c to determine the degree of operating leverage (DOL).

5. Use the formula for degree of operating leverage to determine the DOL at 10,000 units.

P11-10 Degree of financial leverage

Northwestern Savings and Loan has a current capital structure consisting of $250,000 of 16% (annual interest) debt and 2,000 shares of common stock. The firm pays taxes at the rate of 40%.

A) Using EBIT values of $80,000 and $120,000, determine the associated earnings per share (EPS).

B) Using $80,000 of EBIT as a base, calculate the degree of financial leverage (DFL).

C) Rework parts a and b assuming that the firm has $100,000 of 16% (annual interest) debt and 3,000 shares of common stock.