Question

Emman is indebted to Lisa the following obligations (a) $14,000 due at the end of 3 years with accumulated Interest at 4% compounded quarterly and (b) $16,000 due at the end of 6 years with accumulated Interest at 7% annually.

Emman is allowed to replace these obligations by two payments. The first payment due at the end of 4 years and the second payment due at the end of 5 years. If the second payment is twice as much as the other payment. Find the payments if the money is 8% compounded quarterly

Answer 1

a) $14,000 plus interest due at the end of 3 years

Annual interest rate = 4%

Quarterly interest rate = 4%/4 = 1%

Accumulation period = 3 years = 12 quarters

Principal + interest owed at end of 3 years = $14,000*(1+1%)¹² = $14,000*1.12682503 = $15,775.55

b) $16,000 plus interest due at the end of 6 years

Annual interest rate = 7%

Accumulation period = 6 years

Principal + interest owed at end of 6 years = $16,000*(1+7%)⁶ = $16,000*1.500730352 = $24,011.69

Now we can calculate the present value of a) and b)

Discount rate = 8% annually = 2% Quarterly

Number of quarters from the present to payment of a) = 3*4 = 12

Number of quarters from the present to payment of b) = 6*4 = 24

Present value, PV = Amount Due for a) / (1+2%)¹² + Amount Due for b) / (1+2%)²⁴

= $15,775.55 / (1+2%)¹² + $24,011.69 / (1+2%)²⁴

= $15,775.55 / 1.268241795 + $24,011.69 /1.608437249 = $12,438.91 + $14,928.58 = $27,367.49

Present value of amount due = $27,367.49 --------------------(1)

Let the new payments be P and 2P

P is paid at the end of 4 years

Number of quarters from the present to payment of P = 4*4 = 16

Present value of P = P/(1+2%)¹⁶ = P/1.372785705 = P*(1/1.372785705)

= P * 0.7284458137 = 0.7284458137P --------------(2)

2P is paid at the end of 5 years

Number of quarters from the present to payment of 2P = 5*4 = 20

Present value of 2P = 2P/(1+2%)²⁰ = 2P/1.485947396 = 2P*(1/1.485947396)

= 2P * 0.6729713331 = 1.345942666 P --------(3)

Combined present values of P and 2P = (2) + (3) = 0.7284458137 P + 1.345942666 P = 2.07438848 P ------------(4)

Equating (1) and (4)

2.07438848 P = $27,367.49

P = $27,367.49 / 2.07438848 = $13,193.04

2P = 2 * $13,193.04 = $26,386.08

Hence the two payments are $13,193.04 (to be paid after 4 years) and $26,386.08 (to be paid after 5 years)