A family is planning their finances for the next two years, considering two main savings strategies: a "Home Down Payment Fund" and a "Child's Education Fund." They use an annual effective interest rate of [math]6\%[/math] for their financial projections.

Home Down Payment Fund:

• They currently plan to deposit [math]$10,000[/math] into this fund today.
• They plan to deposit an additional [math]$5,000[/math] one year from today.
• They aim to withdraw [math]$16,000[/math] two years from today to cover a down payment on a house.

Child's Education Fund:

• They want to make an initial deposit of [math]$Y[/math] into this fund today.
• They plan to deposit an additional [math]$3,000[/math] one year from today.
• They anticipate needing to withdraw [math]$12,000[/math] two years from today for tuition costs.

The family's financial advisor recommends that the accumulated value of both funds at the end of two years should be identical, indicating balanced planning for both goals. (Assume deposits are cash outflows for the family and withdrawals are cash inflows.) Calculate [math]Y[/math].

• $8,060.70
• $8,321.43
• $8,326.81
• $8,487.01
• $11,886.79

Step 1: Calculate the Accumulated Value of the Home Down Payment Fund

The Home Down Payment Fund has the following cash flows:

• Initial deposit at [math]t=0[/math]: [math]$10,000[/math]
• Deposit at [math]t=1[/math]: [math]$5,000[/math]
• Withdrawal at [math]t=2[/math]: [math]$16,000[/math]

The annual effective interest rate is [math]i = 6\% = 0.06[/math]. We need to accumulate these cash flows to [math]t=2[/math]. Deposits are treated as positive contributions to the fund, and withdrawals as negative. The accumulated value (AV) of the Home Down Payment Fund at [math]t=2[/math] is calculated as:

[[math]]AV_{Home} = 10,000(1+0.06)^2 + 5,000(1+0.06)^1 - 16,000[[/math]]

[[math]]AV_{Home} = 10,000(1.1236) + 5,000(1.06) - 16,000[[/math]]

[[math]]AV_{Home} = 11,236 + 5,300 - 16,000[[/math]]

[[math]]AV_{Home} = 16,536 - 16,000[[/math]]

[[math]]AV_{Home} = 536[[/math]]

Step 2: Express the Accumulated Value of the Child's Education Fund in terms of Y

The Child's Education Fund has the following cash flows:

• Initial deposit at [math]t=0[/math]: [math]$Y[/math]
• Deposit at [math]t=1[/math]: [math]$3,000[/math]
• Withdrawal at [math]t=2[/math]: [math]$12,000[/math]

Using the same annual effective interest rate [math]i = 0.06[/math], the accumulated value of the Child's Education Fund at [math]t=2[/math] is:

[[math]]AV_{Child} = Y(1+0.06)^2 + 3,000(1+0.06)^1 - 12,000[[/math]]

[[math]]AV_{Child} = Y(1.1236) + 3,000(1.06) - 12,000[[/math]]

[[math]]AV_{Child} = 1.1236Y + 3,180 - 12,000[[/math]]

[[math]]AV_{Child} = 1.1236Y - 8,820[[/math]]

Step 3: Equate the Accumulated Values and Solve for Y

The family's financial advisor recommends that the accumulated value of both funds at the end of two years should be identical. Therefore, we set [math]AV_{Home} = AV_{Child}[/math]:

[[math]]536 = 1.1236Y - 8,820[[/math]]

Now, we solve for [math]Y[/math]:

[[math]]1.1236Y = 536 + 8,820[[/math]]

[[math]]1.1236Y = 9,356[[/math]]

[[math]]Y = \frac{9,356}{1.1236}[[/math]]

[[math]]Y \approx 8,326.80669[[/math]]

Rounding to two decimal places, [math]Y \approx \$8,326.81[/math].

Key Insights

• When calculating accumulated values, each cash flow must be moved to the valuation date using the appropriate compounding factor [math](1+i)^n[/math], where [math]n[/math] is the number of periods.
• Deposits increase the fund's value, so they are added, while withdrawals decrease it, so they are subtracted. Careful attention to the sign convention of cash flows is crucial.
• Equating the accumulated values of two distinct financial strategies allows for determining an unknown initial investment that balances the two goals.
• Intermediate calculations for compound interest factors, such as [math](1.06)^2 = 1.1236[/math] and [math](1.06)^1 = 1.06[/math], should be accurate to avoid significant rounding errors in the final result.

Answer was verified by AI

Answer was verified numerically