Question

5- Experts say that the baby boom generation (born 1946-1960) cannot count on a company pension or Social Security benefits to provide a comfortable retirement. It is recommended that they start to save regularly and early. Michael, a baby boomer, has decided to deposit $200 each quarter in an account that pays 8% compounded quarterly for 20 years.

a) How much money will be in the account at the end of the 20 years?

b) Suppose Michael has determined he needs to accumulate $130,000 from this annuity. What rate would achieve this goal?

c) If he cannot get the higher rate, what amount would his payments need to be in order to achieve the goal?

d) Suppose Michael cannot get the higher interest rate, nor increase his payments. How many months would he need to invest in order to achieve his goal?

Answer 1

a) No of quarters in 20 years =20*4 =80

Interest rate per quarter =8%/4 =2% or 0.02

So, the amount accumulated by the end of 20 years = Future value of all deposits

=200*1.02^79+200*1.02^78+......+200

=200/0.02*(1.02^80-1)

=$38754.39

  

b) To accumulate $130,000 from this annuity, quarterly interest rate (r) required is given by

Future value of deposits = 130000

=> 200/r*((1+r)^80-1) =130000

Using Excel's SOLVER tool,

r= 0.042953468

So, Interest rate required is 0.042953468*4 or 17.18% p.a. compounded quarterly

c) If higher rate is not possible, amount required (A) to achieve the goal is given by

A/0.08*(1.08^80-1) = 130000

=> A* 193.7719578 = 130000

=> A = $670.89

So, amount deposited to reach the goal is $670.89

d) If neither higher interest rate nor higher amount is possible, no of quarters required (n) is given by

200/0.02*(1.02^n-1) = 130000

=> 1.02^n = 14

Taking natural log of both sides

n = ln(14)/ln(1.02) =133.268

So, no of quarters for which deposits need to be made = 133.268 quarters or 134 quarters

or No of months = 133.268 *4 = 533 months apx