Present Value Annuity 0D038C

1. **State the problem:** We need to find the present value of an ordinary annuity with quarterly deposits of $12,546 for 7 years at an interest rate of 11.2% compounded quarterly. 2. **Formula for present value of an ordinary annuity:** $$PV = P \times \frac{1 - (1 + r)^{-n}}{r}$$ where: - $P$ is the payment per period - $r$ is the interest rate per period - $n$ is the total number of payments 3. **Identify values:** - $P = 12546$ - Annual interest rate = 11.2%, so quarterly rate $r = \frac{11.2}{4} \% = 2.8\% = 0.028$ - Number of years = 7, so total payments $n = 7 \times 4 = 28$ 4. **Calculate present value:** $$PV = 12546 \times \frac{1 - (1 + 0.028)^{-28}}{0.028}$$ 5. **Calculate $(1 + r)^{-n}$:** $$ (1 + 0.028)^{-28} = \frac{1}{(1.028)^{28}} $$ Calculate $(1.028)^{28}$: $$ (1.028)^{28} \approx 2.123 $$ So, $$ (1 + 0.028)^{-28} = \frac{1}{2.123} \approx 0.471 $$ 6. **Substitute back:** $$PV = 12546 \times \frac{1 - 0.471}{0.028} = 12546 \times \frac{0.529}{0.028}$$ 7. **Simplify fraction:** $$ \frac{0.529}{0.028} \approx 18.89 $$ 8. **Final multiplication:** $$PV = 12546 \times 18.89 \approx 237,000.54$$ **Answer:** The present value of the ordinary annuity is approximately $237,000.54$.