Question:
Good news: You will almost certainly be a millionaire by the time you retire in 40 years.
Bad news: The inflation rate over your lifetime will average about 3.2%
a) What will be the real value of $1 million by the time you retire in terms of today's dollars?
b) What real annuity (in today's dollars) will $1 million support if the real interest rate at retirement is 2.2% and the annuity must last for 10 years.
Fisher Equation:
The Fisher Equation, named after economist Irving Fisher, is an economic theory that relates nominal interest rate to real interest rate and the expected rate of inflation. When these rates are relatively small, the equation takes a simple form: nominal interest rate = real interest rate + expected rate of inflation.
Answer and Explanation:1
a) The real value in today's dollar is $283,669.15.
The value of the $1 million today is the value of $1 million discounted at the inflation rate of 3.2% for 40 years, i.e.,
- {eq}\frac{1,000,000}{(1 + 3.2\%)^{40}} = 283,669.15{/eq}
b) The present value of the annuity is $37,453.45.
We can first use the following formula to compute the nominal value of the annuity:
{eq}\frac{rF}{1 - (1 + r)^{-T}}{/eq}, where
- F is the nominal value of the endowment
- r is the nominal interest rate
- T is the duration of the annuity
In this question, the nominal value of the endowment is $1,000,000. Applying the Fisher Equation, the nominal interest rate = real interest rate + rate of inflation = 3.2% + 2.2% = 5.4%, the duration of the annuity is 10 years. Applying the formula, the nominal amount of the annuity is
- {eq}\frac{5.4\%*1,000,000}{1 - (1 + 5.4\%)^{-10}} = 132,032.16{/eq}.
The present value of this annuity is {eq}\frac{132,032.16}{(1 + 3.2\%)^{40}} = 37453.45{/eq}
