You are planning to save for retirement over the next 25 years. To do this, you will invest $880...

Question:

You are planning to save for retirement over the next 25 years. To do this, you will invest $880 per month in a stock account and $480 per month in a bond account. The return of the stock account is expected to be 10.8 percent, and the bond account will earn 6.8 percent. When you retire, you will combine your money into an account with an annual return of 7.8 percent. Assume the returns are expressed as APRs. How much can you withdraw each month from your account assuming a 20-year withdrawal period?

IRA:

Various individual retirement accounts (IRAs) exist that allow individuals to allocate their savings to different investments depending on their tolerance for risk. The riskier the investment, the higher its return on average,

Answer and Explanation: 1

Let r be the interest rate, and n = 25 * 12 = 300 be the number of periods.

The interest rate for the stock account is 10.8% / 12 = 0.9%. The future value after 25 years of the $880 payments in the stock account is:

{eq}FV=Payment*\frac{(1+r)^{n}-1}{r}\\ FV=880*\frac{(1+0.009)^{300}-1}{0.009}\\ FV=\$1,339,663\\ {/eq}

The interest rate on the bond account is 6.8% / 12 = 0.57%. The future value of the $480 payments is:

{eq}FV=Payment*\frac{(1+r)^{n}-1}{r}\\ FV=480*\frac{(1+0.0057)^{300}-1}{0.0057}\\ FV=\$379,131\\ {/eq}

Combining the two amounts,

Total amount = 1,339,663 + 379,131 = $1,718,794

This amount must be equal to the present value of the withdrawals over the next 20 years at an interest rate of 7.8% / 12 = 0.65%:

{eq}PV=Withdrawal*\frac{1-(1+r)^{-n}}{r}\\Withdrawal=\frac{PV*r}{1-(1+r)^{-n}}\\Withdrawal=\frac{1,718,794*0.0065}{1-(1+0.0065)^{-240}}\\PMT=14,163\\ {/eq}

The monthly withdrawals are $14,163.


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How to Calculate the Present Value of an Annuity

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Chapter 8 / Lesson 3
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Learn how to find present value of annuity using the formula and see its derivation. Study its examples and see a difference between Ordinary Annuity and Annuity Due.


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