Time Value of Money Problems Answers

1.

Solution

P = 100(P/F, 0.1, 1) + 50(P/A, 0.1, inf)(P/F, 0.1, 1) = 545.45

 

The P/A factor for infinity is 1/i. The factor moves the uniform series back to a single payment one period before the first payment.

2. Use a period of 3 months. The interest per 3 months is 0.16/4 = 0.04.

P = 500(P/A, 0.04, 40) = 9896.39. This is what you should be willing to pay.

This illustrates the situation of a compounding period other than one year. The 9896.39 is an investment that saves expenditures in the future. The value of P is the investment that makes the net present worth equal to zero.

3. The solution is the interest rate at which the net present worth is zero.

At 1%, P = ­200 + 14.44(P/A, 0.01, 16) = 12.53.

At 2%, P = ­200 + 14.44(P/A, 0.02, 16) = ­3.94.

Interpolating: i = 0.01 + 0.01[12.53/(12.53 + 3.94)]= 0.01760 or 1.76%.

The effective rate is = (1.0176)52 ­ 1 = 2.4774 ­ 1 = 1.4774 or 148%/year

4. Annual payment on loan = 1000(A/P, 0.05, 10) = 129.50.

Find the value of A that makes the present worth of the cash flow equal to zero.

PW = 1000 ­ 129.50(P/A, 0.05, 6) + 1000(P/F, 0.05, 6) ­ A(P/A, 0.05, 8)(P/F, 0.05, 8) = 0

or

A = [1000 ­ 129.50(P/A, 0.05, 6) + 1000(P/F, 0.05, 6)] / [(P/A, 0.05, 8)(P/F, 0.05,8)]

A = [1000 ­ 129.50(5.076) + 1000(0.7462)] / [6.463 ¥ 0.6768]

A = [342.66 + 746.2] / [4.372] = 249.1

5. Find the uniform annual equivalent: A = 1000 - 20(A/G, 0.1, 25) = 850.840

6. Find out how much you would have to invest to save 20,000 in 40 years at 10% interest. A = 20000(A/F, 0.1, 40) = 45.19. Then you are paying 170 ­ 45.19 = 124.81 per year for protection.

7. The cash flows associated with the problem are the bond investment at time 0, returns of $50 at the end of each year and $1000 at the end of 20 years. The investment should equal the present worth of the receipts computed at 10% interest.

PW = 50(P/A, 0.1, 20) + 1000(P/F, 0.1, 20)

= 50(8.5135) + 1000(0.1486)

= 425.68 + 148.64 = 574.32

8. The interest rate per quarter is 0.1/4 = 0.025 per quarter. The effective rate is the annual rate that yields the same annual return as when the interest is compounded quarterly. 1 + i_eff = (1 + 0.025)4 . Therefore, i_eff = 10.38%.

9. Since the loan is paid monthly, the interest rate is 0.12/12 = 0.01 per month.

 

a. The monthly payment is: A = 2000(A/P, 0.01, 24) = 211.16.

b. The interest in any period is the money still owed times the interest rate. In the first period the money owed is 2000 so the interest in the first payment is

2000(0.01) = $20.