The Good Life Insurance Co. wants to sell you an annuity which will pay you $770 per quarter for 25 years. You want to earn a minimum rate of return of 6.2 percent. What is the most you are willing to pay as a lumpsum today to buy this annuity?

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BlackTom AI · Accepted Answer

We’re evaluating the maximum lump-sum you’d pay today for an annuity that pays 770 every quarter for 25 years, with a minimum required return of 6.2% per year compounded quarterly.

First, convert the annual rate to a quarterly rate because payments occur quarterly. The nominal annual rate is 6.2%, so the quarterly rate i is 0.062 / 4 = 0.0155 (1.55% per quarter).

Next, determine the number of payments. 25 years with quarterly payments equals n = 25 × 4 = 100 payments.

The present value of an ordinary annuity (payments at the end of each period) is given by PV = P × [1 − (1 + i)^(−n)] / i, where P is the payment per period. Here, P = 770, i = 0.0155, n = 100.

Compute the discount factor (1 + i)^(−n). (1 + 0.0155) = 1.0155. We need 1.0155^(-100). A rough evaluation uses the natural log: ln(1.0155) ≈ 0.015387, so −100 × ln(1.0155) ≈ −1.5387. Exponentiating gives (1.0155)^(-100) ≈ e^(−1.5387) ≈ 0.2147 (around 0.215).

Now the bracket term is [1 − 0.2147] ≈ 0.7853. Divide by i = 0.0155: 0.7853 / 0.0155 ≈ 50.66 (this is the annuity factor for 100 periods at 1.55% per period).

Finally, multiply by the payment: 770 × 50.66 ≈ 38,978. Yields around the mid-to-high 38k region. If a more exact calculation is performed (using a precise power calculation for (1.0155)^100 and then 1 − that reciprocal, divided by 0.0155, times 770), you converge to a value very close to the provided option in the list.

When comparing to the given answer choices, the calculation aligns most closely with the figure "$39,007.37". The other options represent notably different PV values that would arise from miscomputing the rate per period, the number of periods, or misapplying the annuity formula, such as using an incorrect i, an incorrect n, or treating the cash flows as due (beginning-of-period) rather than ordinary end-of-period payments.

- If we accidentally used a different quarterly rate (for example, using 6.2% as a decimal per quarter or treating it as an effective annual rate without conversion), the PV would diverge from the 39,007 range and likely land in one of the other provided numbers.
- If the number of periods were miscounted (e.g., 100 vs. 96 vs. 120), the annuity factor would change significantly, pushing the present value away from 39k.
- If the payment timing were assumed to be at the beginning of the period (an annuity due), the PV would be larger than the end-of-period case, again altering which option is correct.

Thus, the process of converting the annual rate to a quarterly rate, using the correct number of quarterly payments, and applying the ordinary annuity present-value formula leads to a present value near 39,007.37, which matches one of the listed choices.

The Good Life Insurance Co. wants to sell you an annuity which will pay you $770 per quarter for 25 years. You want to earn a minimum rate of return of 6.2 percent. What is the most you are willing to pay as a lumpsum today to buy this annuity?

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