Question

a debt of $4,000 due five years from now and $4,000 due ten years from now is to be repaid by a payment of $2,000 in two years, a payment of $3,000 in four years, and a final payment at the end of six years. If the interest rate is 3% compounded annually, how much is the final payment?

Answer 1

Solution:-

Substitute r = 0.03, n = 3, P = $2000 in the equation

S = 2000 (1 + 0.03)^3
= 2000 * (1.03)^3
= 2000 * 1.092727
= 2185.45

The contribution of the amount $3000 in four years, it is equal to the amount after one year in the account
Substitute r = 0.03, n = 1, P = $3000
S = $3000 (1 + 0.03)^1
= $3000 (1.03)
= $3090

The amount in the account after repaying the first $4000 with the above amounts is

= $3090 + $2185.45 - 4000

= $1275.45

So the amount contributed the above amount to pay the second $4000 after five years from now is

S = 1275.45 (1 + 0.03)^5
= 1275.45 * 1.1592740743
= 1478.60

The required amount to repay the total amount is 4000 - 1478.60 = 2521.4

The amount before 4 years to get the above amount can be found as below suppose that the principal is $P, that must be invested at the periodic rate of r for n interest periods, so that the compound amount is S is given by P = S(1 + r) ^-n

Substitute, r = 0.03, n = 4, S = 2521.4, in the equation (1) then

P = 2521.4 * (1 + 0.03) ^-4
= 2521.4 * (1.03) ^-4
= 2521.4 * 0.9709
= 2448.027

Therefore the amount required in the sixth year is 2448.027