Question

a) You decide that your family would be comfortable living on an annual income of $150,000, growing at 4% per year. You’d also like to continue generating this cash flow for your descendants, forever. With interest rates of 8%, how much wealth would you need today to provide this income starting with $150,000 one year from now?

b) Suppose all income is taxed at 20%. How would this (if at all) change your answer to a)? Assume that you still feel 4% growth and $150,000 (after-tax) income is sufficient. Show a calculation and briefly explain.

Answer 1

a)

We have following information-

interest rate (i) = 8%

Annual Income = $ 150,000

Constant growth rate of Income (g) = 4%

Maturity = perpetual (never ending)

The value of perpetual investment with constant growth rate can be calculated in following manner

$V = \frac{I}{i-g}$

Where,

Value = value of perpetual investment

I = Annual Income

i = interest rate

g = constant growth rate

Therefore,

$V = \frac{150,000}{.08-.04}$

$V = \frac{150,000}{0.04}$

$V = $ 3,750,000$

Thus, amount required today is $ 3,750,000

b)

Tax rate = 20%

After tax Annual Income = $ 150,000

Therefore, Annual Income before tax would be = 150000/(1-0.2)

= $ 187,500

So now, we need to earn $ 187,500 per year to have $ 150,000 income in hand after paying taxes.

We can calculate value of investment today using above formula -

$V = \frac{187,500}{.08-.04}$

$V = \frac{187,500}{0.04}$

$V = 4,687,500$

Thus, to earn after tax $ 150,000 with 4% constant growth, we need to invest $ 4,687,500 today.