Question

Mr. Coleman is retiring today and he is expected to receive pension income annually over the next 25 years. His first pension income will be at the end of the first year of the 25-year period and this will be $70,000 times one plus the expected rate of inflation of 2% if the pension income will be fully indexed to inflation. The annual rate of inflation is expected to be 2% throughout the 25-year period and the relevant nominal rate of interest will be 6% per year. Calculate the present values of all the pension incomes under each of the following two situations:

i. There is zero indexation to inflation, that is his annual pension income will stay at $70,000 throughout

ii. Pension incomes will be fully indexed to inflation. This means that every year pension income will rise by the rate of inflation

Answer 1

Answer (i):

Annual income (at the end of the year) = $70,000

Time period = 25 years

Rate of interest = 6% per year.

There is zero indexation to inflation.

PV = Annual Income * (1 - 1/ (1 + Interest rate) ^{Number of years} ) / Interest rate

= 70000 * (1 - 1/ (1 + 6%) ²⁵) / 6%

= $894,834.93

Present values of all the pension incomes = $894,834.93

Answer (ii):

Pension incomes will be fully indexed to inflation

Annual rate of inflation is expected to be = 2%

Present Value growing annuity = (P / (r - g)) * (1- ((1 + g) / (1 + r)) ⁿ)

Where P = First payment

r = Interest rate

g = Growth = Inflation rate

n = Number of years

First payment = 70,000 * (1 +2%) = $71,400

Present value = (71400 / (6% - 2%)) * (1 - ((1 +2%) / (1 + 6%)) ²⁵)

= $1,102,667.77

Present values of all the pension incomes (fully indexed to inflation) = $1,102,667.77