Assume that five months from today you plan to make the first of several quarter
ID: 2742299 • Letter: A
Question
Assume that five months from today you plan to make the first of several quarterly deposits into an account that pays an APR of 5.5% with monthly compounding. Your first deposit will equal $500, each of your subsequent quarterly deposits will grow by 1% each, and your final quarterly deposit will occur two years and eight months from today. From this account you plan to make semiannual withdrawals beginning three years and one month from today. Subsequent semiannual withdrawals will shrink by 2% each and your final withdrawal will occur five years and seven month from today.
a. What is the size your first withdrawal?
b. If your withdrawals were all the same size rather than shrinking, would your first withdrawal be larger or smaller than your answer in part a? Assume nothing else changes.
Explanation / Answer
a. This is a growing annuity
The future value of the investments 2 years, 8 months ( 32 months) from today is P *((1+r)^n - (1+g)^n)/(r-g))
where P = 500
r = 0.055/12
g = 0.01
n = 32
FV = 500*((1+0.055/12)^32-(1+0.01)^32)/(0.055/12-0.01)) = $20,064.0669
Now this amount will reamin invested for next 5 months So the value when the first withdrawal happens is = 20,064.0669*(1+0.055/12)^5 = 20,528.1057
Now there are 30 withdrwals from 37th month to 67th month
and the withdwaral shrink by 2% or -0.02
Now we use the PV of a growing annuity to find the size of the first withdrawal
Iniital withdrawal = PV * (r-g)/(1 -(1+g/1+r)^n)
Initial withdrawal = 20,528.1057*(0.055/12 +0.02)/(1-(0.98/1.00458)^30) = 962.258 or $962.26
So size of the first withdrawal = $962.26
b. If all the withdrawals were of the same size, the first withdrawal would be smaller than the answer in part a