Question:
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Expert answer: 0 0
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Step: 1
Ans a.)
To calculate how much money she will have at 65, we can use the future value of an annuity formula:
FV = P * [(1 + r)^n - 1] / r
Where:FV = Future valueP = Annual paymentr = Interest rate per periodn = Number of periods
In this case, the annual payment (P) is ＄11,000, the interest rate (r) is 12% (0.12), and the number of periods (n) is 65 - 24
= 41.
Plugging in the values, we have:
FV = 11,000 *
[4.13159133526 - 1] / 0.12 =
34,447.51
11, 000 ∗ [(1 + 0.12)
41 − 1]/0.12 =
11, 000 ∗ 3.13159133526/0.12 =
Explanation
So, if she follows the advice and saves
34,447.51 at age 65.
Step: 2
Ans b.)
To calculate how much money she will have at 70, we can use the same future value of an annuity formula as before:
FV = P * [(1 + r)^n - 1] / r
The annual payment (P) and the interest rate (r) remain the same, which are ＄11,000 and 12% (0.12) respectively.
However, the number of periods (n) will now be 70 - 24 = 46.
Plugging in the values, we have:
FV = 11,000 *
[6.1917364224 - 1] / 0.12 =
474,309.7
Explanation
Therefore, if she continues to save
474,309.7 at age 70.
Step: 3
Ans c.)
11, 000peryearandinvestsitinthestockmarketwithanave
11, 000 ∗ [(1 + 0.12)
46 − 1]/0.12 = 11, 000 ∗ 5.1917364224/0.12 =
11, 000peryearandinvestsitinthestockmarketwithanave
To calculate the annual withdrawals after retirement, we need to use the future value of an annuity formula in reverse.
We'll solve for the payment amount (P) in the formula:
FV = P * [(1 + r)^n - 1] / r
For retirement at 65, the future value (FV) will be the amount she has at 65, which is ＄7,746,305.34 (as calculated
earlier). The interest rate (r) and the number of periods (n) will be 12% (0.12) and 20 years, respectively.
Plugging in the values, we have:
＄7,746,305.34 = P * [(1 + 0.12)^20 - 1] / 0.12 To solve for P, we rearrange the formula:
P =
7,746,305.34 * 0.12 / 3.20713594309 = ＄904,388.50
Step: 4
Therefore, if she retires at 65 and her investments continue to earn an average return of 12%, she will be able to
withdraw approximately ＄904,388.50 at the end of each year after retirement.
For retirement at 70, we'll use the same formula, but with a different number of periods. The future value (FV) remains
＄474,309.7 (as calculated earlier), and the number of periods (n) is now 15 years.
Plugging in the values, we have:
＄474,309.7 = P * [(1 + 0.12)^15 - 1] / 0.12 Solving for P, we get:
P = 474,309.7 *
0.12 / 2.76032652918 = ＄857,411.31
7, 746, 305.34 ∗ 0.12/[(1 + 0.12)
20 − 1] =
474, 309.7 ∗ 0.12/[(1 + 0.12)
15 − 1] =
Explanation
Therefore, if she retires at 70 and her investments continue to earn an average return of 12%, she will be able to
withdraw approximately ＄857,411.31 at the end of each year after retirement.
Final Answer
Summary:
a) If she follows the advice and saves
7,746,305.34 at age 65.
b) If she continues to save
474,309.7 at age 70.
c) If she retires at 65 and her investments continue to earn an average return of 12%, she will be able to withdraw approximately ＄904,388.50 at the end of each year after retirement.
If she retires at 70 and her investments continue to earn an average return of 12%, she will be able to withdraw
approximately ＄857,411.31 at the end of each year after retirement.
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