Question

The present values of the following three annuities are equal: (i) perpetuity-immediate paying 1 each year, calculated at an annual effective interest rate of 7.84%. (ii) 26-year annuity-immediate paying 1 each year, calculated at an annual effective interest rate of j%. (iii) n-year annuity-immediate paying 1 each year, calculated at an annual effective interest rate of (j−1)%. Calculate n.

Answer 1

i)

Present value oof perpetuity immediate = Perpetual payments/Interest rate

Present value = 1/0.0784 = 12.7551

ii)

Present value of annuity immediate = Annuity amount*{1-(1+r)^-n}/r

12.7551 = 1*{1-(1+j)^-26}/j

Solving above, j = 6.20%

iii)

Interest rate = 6.20% - 1 = 5.20%

12.7551 = $1*(1-1.0520^-n)/0.0520

12.7551*0.0520 = 1-1.0520^-n

1.0520^-n = 1-0.6633 = 0.3367

1.0520ⁿ = 1/0.3367 = 2.9697

n * log 1.0520 = log 2.9697

n * 0.022016 = 0.47271

n = 0.47271/0.022016 = 21.47

Hence, n = 21.47 years