Present Value Annuities F47C85

1. **State the problem:** We need to find the present value of an annuity with payments of $7000 made at the beginning of each month for 5 years, with an interest rate of 3.75% compounded monthly. 2. **Identify the formula:** Since payments are at the beginning of each period, this is an annuity due. The present value of an annuity due is given by: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)$$ where: - $P$ = payment amount per period - $r$ = interest rate per period - $n$ = total number of payments 3. **Calculate parameters:** - Annual interest rate = 3.75% = 0.0375 - Monthly interest rate $r = \frac{0.0375}{12} = 0.003125$ - Number of payments $n = 5 \times 12 = 60$ - Payment $P = 7000$ 4. **Calculate the present value:** First calculate the term inside the fraction: $$1 - (1 + r)^{-n} = 1 - (1 + 0.003125)^{-60}$$ Calculate $(1 + 0.003125)^{-60}$: $$ (1.003125)^{-60} = \frac{1}{(1.003125)^{60}} $$ Calculate $(1.003125)^{60}$: $$ (1.003125)^{60} = e^{60 \times \ln(1.003125)} $$ Calculate $\ln(1.003125) \approx 0.003120$: $$ e^{60 \times 0.003120} = e^{0.1872} \approx 1.2059 $$ So: $$ (1.003125)^{-60} = \frac{1}{1.2059} \approx 0.8297 $$ Now: $$ 1 - 0.8297 = 0.1703 $$ Now calculate the fraction: $$ \frac{0.1703}{0.003125} = 54.496 $$ Multiply by $(1 + r) = 1.003125$: $$ 54.496 \times 1.003125 = 54.666 $$ Finally multiply by $P = 7000$: $$ PV = 7000 \times 54.666 = 382,662 $$ 5. **Answer:** The present value of the payments is approximately **382,662.00** dollars.