You are planning to save for retirement over the next 30 years. To save for reti

Question

You are planning to save for retirement over the next 30 years. To save for retirement, you will invest $1,300 per month in a stock account in real dollars and $555 per month in a bond account in real dollars. The effective annual return of the stock account is expected to be 12 percent, and the bond account will earn 8 percent. When you retire, you will combine your money into an account with an effective return of 9 percent. The returns are stated in nominal terms. The inflation rate over this period is expected to be 4 percent. How much can you withdraw each month from your account in real terms assuming a 25-year withdrawal period? What is the nominal dollar amount of your last withdrawal?

Explanation / Answer

Find the monthly interest rate, which is the APR divided by 12. The Fisher equation uses the effective annual rate, so, the real effective annualinterest rates, and the monthly interest rates for each account are

Stock account =>

=> (1 +R) = (1 +r)(1 +h)

1 + 0.12 => (1 +r)(1 + 0.04)

r => 7.70%

APR =m[(1 + EAR)1/m1]

APR = >12[(1 + 0.077)1/121]

APR => 7.44%

Monthly Rate => 7.44 /12 => 0.62%

DEBT

Bond account:(1 +R) = (1 +r)(1 +h)

1 + 0.08 = (1 +r)(1 + 0.04)

r=> 3.85%

APR =m[(1 + EAR)1/m1]

APR = >12[(1 + 0.0385)1/121]

APR => 3.78%

Monthly Rate => 3.78 / 12 => 0.32%

Future value of the retirement account in real terms. The future value of each account will be

Stock account

FVA => C{(1 +r)t1] /r}

FVA => $1300{[(1 + 0.0062)3601] / 0.0062]}

FVA => $1730781

DEBT

FVA => C{(1 +r)t1] /r}

FVA => $555{[(1 + 0.0032)3601] / 0.0032]}

FVA => $374400

The total future value of the retirement account will be the sum of the two accounts

Account Value => 1730781 + 374400

Account Value => $2105181

Now we need to find the monthly interest rate in retirement.

=> (1 +R) = (1 +r)(1 +h)

1 + 0.09 => (1 +r)(1 + 0.04)

r => 4.81%

APR =m[(1 + EAR)1/m1]

APR = >12[(1 + 0.0481)1/121]

APR => 4.71%

Monthly Rate => 4.71 / 12 => 0.39%

Now we can find the real monthly withdrawal in retirement. Using the present value of an annuity equation

PVA =C({1[1/(1 +r)]t} /r)

$2105181 =C({1[1/(1 + 0.0039)]300} / 0.0039)

Nominal dollar amount of your last withdrawal=> $11917.40

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