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 $1,450 a month in a stock account in real dollars and $570 a month in a bond account in real dollars. The effective annual return of the stock account is expected to be 10 percent, and the bond account will earn 6 percent. When you retire, you will combine your money into an account with a 7 percent effective return. The inflation rate over this period is expected to be 3 percent. How much can you withdraw each month from your account in real terms assuming a 25-year withdrawal period? a) How much can you withdraw each month from your account in real terms assuming a 25-year withdrawal period? b) What is the nominal dollar amount of your last withdrawal?

Explanation / Answer

a) How much can you withdraw each month from your account in real terms assuming a 25-year withdrawal period?

Solution:

The value of the stock account at retirement is

= $1450 x ((((1 + (0.10 / 12)) ^ (30 x 12)) - 1) / (0.10 / 12))

=$1450 x (1 + 0.0083)360 - 1 / 0.0083

=$1450 x (1.0083)360 - 1 / 0.0083

=$1450 x 19.60 - 1 / 0.0083

= $1450 x 2,241

= $3,249,450

The value of the bond account at retirement is

= $570 x ((((1 + (0.06 / 12)) ^ (30 x 12)) -1) / (0.06 / 12))

=$570 x ((((1 + (0.005))360 -1) / 0.005

= $570 x (1.005)360 - 1 / 0.005

= $570 x 6.023 - 1 / 0.005

= $570 x 5.023 / 0.005

= $570 x 1004.5

= $572,622

The value of the two accounts combined is

$3,249,450 + $572,622 = $3,822,072

The monthly withdrawal from the combined account will be

= $3,822,072 / ((1 - (1 / ((1 + (0.07 / 12)) ^ (25 x 12)))) / (0.07 / 12))

= $3,822,072 / ((1 - (1 / ((1 + (0.0058)300)))) / 0.0058

= $3,822,072 / ((1 - (1 / ((1.0058)300)) / 0.0058

= $3,822,072 / ((1 - (1 / 5.6688 / 0.0058

= $3,822,072 / ((1 - 0.176 / 0.0058

= $3,822,072 /0.824 / 0.0058

= $3,822,072 /142.07

= $26,902.74 is the monthly withdrawal from the combined account

b) What is the nominal dollar amount of your last withdrawal?

Solution: The nominal amount of the final withdrawal would be that amount multiplied by the MONTHLY inflation rate which will have compounded 300 times(i.e, 25years * 12months). So:

= $26,902.74*(((1+(.03/12))300)

= $26,902.74*(((1+(0.0025))300)

= $26,902.74*(((1.0025))300)

= $26,902.74 * 2.1150

= $56.899.30 is the NOMINAL withdrawal in the final month of year 25.

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