Lifetime Income Analysis: Determining The Value Of Professor's Annuity Corp.

Professor’s Annuity Corp. offers a lifetime annuity to retiring professors. For a payment of $80,000 at age 65, the firm will pay the retiring professor $600 a month until death. a. If the professor’s remaining life expectancy is 20 years, what is the monthly interest rate on this annuity? b. What is the effective annual interest rate? c. If the monthly interest rate is .5%, what monthly annuity payment can the firm offer to the retiring professor?

Solution:

We will solve each part step by step.

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Part (a): Finding the Monthly Interest Rate

We are given:

• Present Value (PV) = $80,000 (initial payment)
• Monthly Payment (C) = $600
• Number of Months (t) = $20 \times 12 = 240$ months
• Interest Rate per Month (r) = ??? (to be found)

The annuity formula for present value is:

$$PV = C \times \left(1 - \frac{1}{(1 + r)^t} \right) \div r$$
$$80,000 = 600 \times \left(1 - \frac{1}{(1 + r)^{240}} \right) \div r$$

This equation needs to be solved for $r$. We will solve it numerically.

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Part (b): Effective Annual Interest Rate (EAR)

Once we have the monthly interest rate $r$, the EAR is given by:

$$EAR = (1 + r)^{12} - 1$$

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Part (c): Monthly Payment with $r = 0.5\%$

If the monthly interest rate is given as $0.5\% = 0.005$, we use the annuity formula again:

$$C = PV \times \frac{r}{1 - \frac{1}{(1 + r)^t}}$$ $$C = 80,000 \times \frac{0.005}{1 - \frac{1}{(1.005)^{240}}}$$

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Solving Numerically

Now, let’s compute these values.

Results

(a) The monthly interest rate on the annuity is 0.5479% (or 0.00548 in decimal form).
(b) The effective annual interest rate (EAR) is 6.78%.
(c) If the monthly interest rate is 0.5%, the monthly annuity payment would be $573.14 instead of $600.