Question

You currently have $620k in savings. You plan on living off of a growing perpetuity which grows at 3% per year (and passing it on to your kids). You plan on retiring exactly 12 years from today, at which point you collect your first cash payment. You will not contribute any more money to the savings account before retirement, but your current $620k in savings will grow at the discount rate until you retire. How much will the first payment of the growing perpetuity be if the appropriate discount rate is 6%?

Answer 1

Our current savings are $620,000.

The appropriate discount rate given is 6%. The question says that our current savings of $620k will grow at the discount rate of 6% per annum until we retire.

We will retire 12 years from today, so the Future Value(FV) of our current savings after 12 years will be $620,000 x(1.06)¹²  

(Since our current savings will grow by 6% next year, and then 6% again the year after that...and so on for 12 years. Basically it's the compounding formula : FV = PV x (1 + r)ⁿ )

So, $620,000 x (1.06)¹² = $ 1,247,561.81. This is the value of our savings when we'll retire after 12 years.

Now, $ 1,247,561.81 will also be the Present Value (PV) of the growing perpetuity that we will live off of after retiring.

The formula for the PV of a growing perpetuity is: PV = C₁  / (r - g) ,

where C₁ is the First Cash flow or Payment of the perpetuity, r is the appropriate discount rate, and g is the growth rate of the perpetuity which is given as 3%.

1,247,561.81 = C₁  / (0.06 - 0.03)

or C₁ = 1,247,561.81 x 0.03 = 37,426.85

Hence, the first payment of the growing perpetuity will be $37,426.85 or $37,427 (rounded off).