Question

At an annual effective interest rate, i, where i > 0, a 40-year annuity-due with quarterly payments of $6 has the same present value as a 20-year annuity-due with quarterly payments of $8.

Determine i. (ALSO please draw any time diagrams that would be helpful for this problem)

Answer 1

Let the applicable quarterly interest rate be R

Annuity 1: Annuity due with a tenure of 40 years and quarterly payments of $ 6 each. Interest Rate = R

Present Value = p1 = 6 x (1/R) x [1-{1/(1+R)^(40 x 4)}] x (1+R)

Annuity 2: Annuity due with a tenure of 20 years and quarterly payments of $ 8 each. Interest Rate = R

Present Value = p2 = 8 x (1/R) x [1-{1(1+i)^(20 x 4)}] x (1+i)

As p1 = p2, we have:

6 x (1/R) x [1-{1/(1+R)^(40 x 4)}] x (1+R) = 8 x (1/R) x [1-{1(1+R)^(20 x 4)}] x (1+R)

6 x [1-{1/(1+R)^(160)}] = 8 x [1-{1/(1+R)^(80)}]

6 - 6 / (1+R)^(160) = 8 - 8 / (1+R)^(80)

Replacing 1/(1+R)^(80) with K we have:

6 - 6K^(2) = 8 - 8K

6K^(2) - 8K + 2 = 0

K = 1/3 or 1

1/(1+R)^(80) = 1/3

R = [1/3]^(1/80) - 1 = 0.01383 or 1.383 %

Effective Annual Rate = i = (1+R)^(4) - 1 = 1.01383)^(4) - 1 = 0.05648 or 5.648 % ~ 5.65 %