Question

Leonard Hofstadter would like to retire in 30 years (1st withdrawal in year 31). He is told by Raj Koothrappali that he will need about $230,000 per year (in t= 31 dollars) to fund his retirement. Leonard wants to be able to maintain that level of purchasing power for 25 years (Assume inflation = 2% per year). Leonard plans to increase his savings by 5% per year and expects to earn 8% per year on his investments.

What is Leonard’s retirement number? That is, how much does Leonard need to have saved by the end of year 30?

How much does Leonard have to save the first year to fund his retirement goal?

Answer 1

PV of annuity for growing annuity
P = (PMT/(r-g)) x (1-((1+g)/(1 + r)) ^n)
Where:
P = the present value of an annuity stream	To be calculated
PMT = the dollar amount of each annuity payment	$         230,000
r = the effective interest rate (also known as the discount rate)	8%
n = the number of periods in which payments will be made	25
g= Growth rate	2%
PV of retirement corpus=	(PMT/(r-g)) x (1-((1+g)/(1 + r)) ^n)
PV of retirement corpus=	(230000/(8%-2%)) * (1-((1+2%)/(1 + 8%)) ^25)
PV of retirement corpus=	$ 2,915,028.24
FV of growing annuity
P = PMT x (((1 + r)^n-(1+g)^n)/(r-g))
Where:
P = the future value of an annuity stream	$ 2,915,028.24
PMT = the dollar amount of each annuity payment	To be computed
r = the effective interest rate (also known as the discount rate)	8%
n = the number of periods in which payments will be made	30
g= Growth rate	5%
P=	PMT x (((1 + r)^n-(1+g)^n)/(r-g))
$                                                                            2,915,028.24	=PMT * (((1 + 8%)^30-(1+5%)^30)/(8%-5%))
First payment	=2915028.24/ (((1 + 8%)^30-(1+5%)^30)/(8%-5%))
First payment	$      15,233.44