Today Is Your Birthday And You Are Now 37
Today Is Your Birthday And You Are Now 37 You Are Planning Your Retir
Today is your birthday and you are now 37! You are planning your retirement and have decided that you can save $8,000.00 per year to go toward your retirement. The plan is to make your first deposit one year from today. You found a mutual fund that is expected to provide a return of 7.5% per year. You plan to retire at the age of 65, exactly 28 years from today. It is your expectation that you will live for 25 years after your retirement. Under these assumptions, how much can you spend each year after you retire? Your first withdrawal will be made at the end of your first retirement year.
Paper For Above instruction
The goal of this financial planning scenario is to determine the annual retirement income based on current savings, investment returns, and projected lifespan. This involves calculating the future value of the accumulated savings at retirement and then determining the fixed annual withdrawal amount that sustains the account balance over the post-retirement period.
Accumulation Phase (Saving Period)
Initially, the individual will deposit $8,000 annually into a mutual fund starting one year from today. Given that the first deposit occurs after one year, this is an ordinary annuity situation. The mutual fund's expected return of 7.5% per year influences the growth of these contributions.
The future value (FV) of the savings at the retirement age of 65 can be calculated using the future value formula for an ordinary annuity:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
where:
- \( P = 8,000 \) dollars (annual deposit),
- \( r = 0.075 \) (annual interest rate),
- \( n = 28 \) years (number of deposits).
Substituting the values gives:
\[
FV = 8,000 \times \frac{(1 + 0.075)^{28} - 1}{0.075}
\]
Calculating inside the formula:
\[
(1.075)^{28} \approx 6.876
\]
Thus,
\[
FV \approx 8,000 \times \frac{6.876 - 1}{0.075} = 8,000 \times \frac{5.876}{0.075}
\]
\[
FV \approx 8,000 \times 78.347 \approx \$626,776
\]
This sum represents the total accumulated savings at retirement.
Decumulation Phase (Retirement Spending)
Post-retirement, the goal is to determine the annual withdrawals that deplete the accumulated fund over 25 years, assuming a 7.5% return during retirement. This is a standard amortization problem where the present value (the accumulated fund) is converted into an annuity of fixed payments.
The annual withdrawal, \( W \), can be calculated using the present value of an annuity formula:
\[
PV = W \times \frac{1 - (1 + r)^{-t}}{r}
\]
where:
- \( PV = 626,776 \) dollars,
- \( r = 0.075 \),
- \( t = 25 \) years.
Rearranged to solve for \( W \):
\[
W = PV \times \frac{r}{1 - (1 + r)^{-t}}
\]
Calculating:
\[
(1 + 0.075)^{25} \approx 6.126
\]
then
\[
(1 + r)^{-t} = \frac{1}{6.126} \approx 0.163
\]
so,
\[
W = 626,776 \times \frac{0.075}{1 - 0.163} = 626,776 \times \frac{0.075}{0.837}
\]
\[
W \approx 626,776 \times 0.0896 \approx \$56,186
\]
Therefore, the individual can withdraw approximately $56,186 per year during retirement, starting at the end of the first retirement year, to deplete their savings over 25 years.
Conclusion
By saving $8,000 annually at an expected return of 7.5%, the individual will accumulate approximately $626,776 by age 65. This amount can support annual withdrawals of roughly $56,186 for 25 years of retirement, assuming the same rate of return during retirement.
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