Investment Choice 91Fe90

1. **State the problem:** We want to find the accumulated value of two different investment plans: - Plan A: Deposit $1500$ monthly for $2$ years at $6\%$ annual interest compounded monthly. - Plan B: Deposit $2500$ monthly for $1.5$ years at $4\%$ annual interest compounded monthly. 2. **Formula used:** The future value $FV$ of an ordinary annuity (deposits at the end of each period) compounded monthly is given by: $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ where: - $P$ = monthly deposit - $r$ = monthly interest rate (annual rate divided by 12) - $n$ = total number of deposits (months) 3. **Calculate for Plan A:** - Annual interest rate = $6\% = 0.06$ - Monthly interest rate $r = \frac{0.06}{12} = 0.005$ - Number of months $n = 2 \times 12 = 24$ - Monthly deposit $P = 1500$ Calculate: $$FV_A = 1500 \times \frac{(1 + 0.005)^{24} - 1}{0.005}$$ Calculate $(1 + 0.005)^{24}$: $$1.005^{24} \approx 1.12749$$ So: $$FV_A = 1500 \times \frac{1.12749 - 1}{0.005} = 1500 \times \frac{0.12749}{0.005}$$ Simplify fraction: $$\frac{0.12749}{0.005} = 25.498$$ Therefore: $$FV_A = 1500 \times 25.498 = 38247$$ 4. **Calculate for Plan B:** - Annual interest rate = $4\% = 0.04$ - Monthly interest rate $r = \frac{0.04}{12} = 0.0033333$ - Number of months $n = 1.5 \times 12 = 18$ - Monthly deposit $P = 2500$ Calculate: $$FV_B = 2500 \times \frac{(1 + 0.0033333)^{18} - 1}{0.0033333}$$ Calculate $(1 + 0.0033333)^{18}$: $$1.0033333^{18} \approx 1.06168$$ So: $$FV_B = 2500 \times \frac{1.06168 - 1}{0.0033333} = 2500 \times \frac{0.06168}{0.0033333}$$ Simplify fraction: $$\frac{0.06168}{0.0033333} = 18.504$$ Therefore: $$FV_B = 2500 \times 18.504 = 46260$$ 5. **Conclusion:** - Plan A accumulates to approximately $38247$ - Plan B accumulates to approximately $46260$ Plan B yields a higher accumulated value despite the lower interest rate because of the larger monthly deposit and still significant compounding period. **Final answer:** Plan B is the better choice with an accumulated value of approximately $46260$.