An annuity pays $10 per year for 50 years. What is the present value (PV) of this annuity, given that the discount rate is 5%?

Question

BlackTom AI · Accepted Answer

Restating the problem in my own words: we have an ordinary annuity that pays 10 dollars every year for 50 years, and we want its present value at a 5% discount rate.

Option a states: $182.56.

Now, step by step, I evaluate the present value formula for an ordinary annuity: PV = PMT * [1 - (1 + i)^(-n)] / i, where PMT = 10, i = 0.05, n = 50.

First, compute the discount factor term (1 + i)^(-n) = (1.05)^(-50). Estimating powers, (1.05)^{50} is about 11.5, so its reciprocal (1.05)^{-50} is roughly 0.087. This gives the bracketed expression: 1 - 0.087 ≈ 0.913.

Next, divide by the interest rate i = 0.05: 0.913 / 0.05 ≈ 18.26.

Finally, multiply by the annual payment PMT = 10: 18.26 * 10 ≈ 182.6.

Rounding considerations: carrying a few extra decimal places during calculation yields PV ≈ 182.56 when computed precisely, which matches option a.

Conceptual check: the result makes sense because a level cash flow of 10 over 50 years discounted at 5% yields a present value somewhat less than 10 * 50 = 500, since future cash is discounted. The long horizon (50 years) and modest rate (5%) together produce a PV around the low hundreds, consistent with about 182.56.

If there were other options, I would compare them by applying the same PV formula and checking which one matches the computed value, but with only option a provided, the computation above confirms the numerical result aligns with that choice.

An annuity pays $10 per year for 50 years. What is the present value (PV) of this annuity, given that the discount rate is 5%?

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