Questions: A couple will retire in 40 years; they plan to spend about 31,000 a year in retirement, which should last about 20 years. They believe that they can earn 7% interest on retirement savings. a. If they make annual payments into a savings plan, how much will they need to save each year? Assume the first payment comes in 1 year. (Do not round intermediate calculations. Round your answer to 2 decimal places.) Annual savings 1,645.07 b. How would the answer to part (a) change if the couple also realize that in 15 years they will need to spend 61,000 on their child's college education? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Annual savings 1,534.33

Solution Steps

To solve part (a), we need to determine the annual savings required to fund the couple's retirement. This involves calculating the present value of the retirement spending and then determining the annuity payment needed to reach that present value over 40 years with a 7% interest rate.

For part (b), we need to adjust the savings plan to account for the additional $61,000 expense in 15 years. This involves calculating the present value of the $61,000 expense and adjusting the annual savings accordingly.

To determine the amount needed at the start of retirement, we calculate the present value of the retirement spending. The couple plans to spend \$31,000 annually for 20 years with an interest rate of 7%. The present value \( PV \) is given by the formula for the present value of an annuity:

\[ PV = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]

where \( C = 31000 \), \( r = 0.07 \), and \( n = 20 \).

\[ PV = 31000 \times \left( \frac{1 - (1 + 0.07)^{-20}}{0.07} \right) \approx 328414.4416 \]

Next, we calculate the annual savings required to accumulate the present value calculated in Step 1 over 40 years. The formula for the future value of an annuity is used:

\[ FV = \frac{PV}{\left( \frac{1 - (1 + r)^{-n}}{r} \right)} \]

where \( PV = 328414.4416 \), \( r = 0.07 \), and \( n = 40 \).

\[ \text{Annual Savings} = \frac{328414.4416}{\left( \frac{1 - (1 + 0.07)^{-40}}{0.07} \right)} \approx 24634.08 \]

The couple also needs to account for a \$61,000 expense in 15 years for their child's college education. We calculate the present value of this future expense:

\[ PV_{\text{college}} = \frac{F}{(1 + r)^t} \]

where \( F = 61000 \), \( r = 0.07 \), and \( t = 15 \).

\[ PV_{\text{college}} = \frac{61000}{(1 + 0.07)^{15}} \approx 22109.2072 \]

Add the present value of the college expense to the present value of the retirement spending:

\[ PV_{\text{adjusted}} = 328414.4416 + 22109.2072 \approx 350523.6488 \]

Finally, calculate the new annual savings required to reach the adjusted present value over 40 years:

\[ \text{Adjusted Annual Savings} = \frac{350523.6488}{\left( \frac{1 - (1 + 0.07)^{-40}}{0.07} \right)} \approx 26292.48 \]

Final Answer