Subjects finance

Present Value Annuity 751Fe2

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1. **State the problem:** A government employee wants to retire in one year and receive P25,000 annually for 15 years. We need to find the amount to deposit now, assuming a 6% annual effective interest rate, so the fund is depleted after 15 years. 2. **Formula used:** This is a present value of an annuity problem. The present value $PV$ of an annuity paying $R$ per year for $n$ years at interest rate $i$ is given by: $$PV = R \times \frac{1 - (1+i)^{-n}}{i}$$ 3. **Identify values:** - $R = 25000$ - $n = 15$ - $i = 0.06$ 4. **Calculate the present value:** $$PV = 25000 \times \frac{1 - (1+0.06)^{-15}}{0.06}$$ 5. **Calculate $(1+0.06)^{-15}$:** $$1.06^{-15} = \frac{1}{1.06^{15}}$$ 6. **Calculate $1.06^{15}$:** $$1.06^{15} \approx 2.39656$$ 7. **Calculate $1.06^{-15}$:** $$1.06^{-15} = \frac{1}{2.39656} \approx 0.41727$$ 8. **Substitute back:** $$PV = 25000 \times \frac{1 - 0.41727}{0.06} = 25000 \times \frac{0.58273}{0.06}$$ 9. **Simplify fraction:** $$\frac{0.58273}{0.06} = 9.7122$$ 10. **Calculate final present value:** $$PV = 25000 \times 9.7122 = 242805$$ 11. **Round to nearest ten cents:** $$PV = 242805.00$$ **Answer:** The employee needs to deposit approximately P242,805.00 now to receive P25,000 annually for 15 years at 6% interest.