What amount would have to be invested today to fund semiannual annuity payments of $50,000 for 15 years? The first payment will be made in six years and the investment will earn 7.8% compounded semiannually.

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Quizplus · Accepted Answer

To find the present value of the annuity payments, we first calculate the present value of the annuity as if the first payment were made today and then discount that amount back to the present. The formula for the present value of an ordinary annuity is PV = PMT * [(1 - (1 + r)^-n) / r], where PMT is the payment amount, r is the interest rate per period, and n is the total number of payments. Since the interest rate is compounded semiannually and the payments are also semiannual, we use a rate of 7.8% / 2 = 3.9% per period, and the total number of payments over 15 years with semiannual payments is 15 * 2 = 30. The formula becomes PV = 50,000 * [(1 - (1 + 0.039)^-30) / 0.039]. This calculates the present value of the annuity as if the first payment were today. To find the present value of this amount as of today, considering the first payment will be made in 6 years (or 12 periods since it's semiannual), we discount it back using the formula PV = FV / (1 + r)^n, where FV is the future value (the result from the first calculation), r is the semiannual rate (3.9%), and n is the number of periods until the first payment (12). This two-step calculation results in the present value required to fund the annuity payments, which matches the value given in option B.

What Amount Would Have to Be Invested Today to Fund