Question

Suppose a man is 25 years old and would like to retire at age 60. ​Furthermore, he would like to have a retirement fund from which he can draw an income of ​$50,000 per year​forever! How can he do​ it? Assume a constant APR of ​8%.

Answer 1

$26421.76

Explanation:

We have to calculate final value i.e. balance to earn $50,000 annually from interest.

`= (50,000)/(0.08) = \$625,000`

Now, N = n × y = 12 × 25 = 300

I = 8% = APR = 0.08

PV = 0 = PMT = 0

FV = 625,000 = A

`\text{A}=\frac{\text{PMT}*[(1+\frac{\text{apr}}{\text{n}})^{\text{ny}}-1 }{\frac{\text{apr}}{\text{n}} }`

`\text{PMT}=\frac{\text{A}*(\frac{\text{APR}}{\text{n}}) }{[(1+\frac{\text{APR}}{\text{n}})^{\text{ny}}-1 }`

`\text{PMT}=(625,000*((0.08)/(12)) )/([(1+(0.08)/(12))^(12*25)-1] )`

`\text{PMT}=(625,000*(0.006667) )/([(1+(0.08)/(12))^(12*25)-1] )`

`\text{PMT}=(625,000*(0.006667) )/([(1+0.006667)^(300)-1] )`

`\text{PMT}=((33335)/(8) )/([1.006667^(300)-1])`

`\text{PMT}=((33335)/(8) )/(6.34090515)`

Monthly payment (PMT) = $26421.7591469 ≈ $26421.76

$26421.76 is required monthly payment in order to $50,000 interest.

Answer 2

Explanation:

0.08*X >= 100000, i.e. X >= 100000%2F0.08 = 1,250,000 dollars.

Then each year he will draw 100000, but the bank will return equal or even greater amount than 0.08*1250000 = 100000,

so his balance will only increase from year to year.

It is a rough estimation.

If we want to consider more realistic case, when he withdraws 100000 at the first day of the year and the bank compounds 8% at the end of the year,

then the unknown starting amount X must satisfy inequality

0.08*(X-100000) >= 100000, which gives

0.08X - 0.08*100000 >= 100000,

0.08X > (1+0.08)*100000

x >= %28%281%2B0.08%29%2A100000%29%2F0.08 = 1350000.

Answer. Having $1,350,000 or more initially on the account provides the sough condition.