A woman plans to withdraw $1800 18-months from now,and $2400 36-months from now.How much does she need to deposit today if the interest rate is j₁₂ = 12% for the first 2 years and j₁₂ = 6% thereafter?

Question

Quizplus · Accepted Answer

To find the present value of the future withdrawals, we need to discount them back to the present using the given interest rates. For the first $1800 withdrawal at 18 months, we use the 12% annual rate, and for the $2400 withdrawal at 36 months, we first apply the 12% rate for the initial 24 months, then the 6% rate for the remaining 12 months.1. For the $1800 withdrawal at 18 months (1.5 years), using the 12% annual rate:$$ PV = \frac{1800}{(1 + 0.12)^{1.5}} $$2. For the $2400 withdrawal at 36 months (3 years), we need to discount it back to 24 months using the 6% rate for the last year, and then discount that amount back to the present using the 12% rate for the first 2 years:$$ PV_{24} = \frac{2400}{(1 + 0.06)^1} $$$$ PV = \frac{PV_{24}}{(1 + 0.12)^2} $$Calculating these:1. $ PV_{1800} = \frac{1800}{(1 + 0.12)^{1.5}} = \frac{1800}{(1.12)^{1.5}} \approx 1504.96 $2. $ PV_{24} = \frac{2400}{(1 + 0.06)^1} = \frac{2400}{1.06} \approx 2264.15 $$$ PV_{2400} = \frac{2264.15}{(1 + 0.12)^2} = \frac{2264.15}{(1.12)^2} \approx 1780.22 $$Adding these two present values together gives the total amount she needs to deposit today:$$ PV_{total} = PV_{1800} + PV_{2400} \approx 1504.96 + 1780.22 = 3285.18 $$Therefore, the correct answer is $3285.18.

A Woman Plans to Withdraw $1800 18-Months from Now,and $2400