Question

Parker is 50 and wants to retire in 15 years. His family has a history of living well into their 90s. Therefore, he estimates that he will live to age 95. He currently has a salary of $120,000 and expects that he will need about 65% of that amount annually if he were retired. He can earn 9 percent in his portfolio while he is working. However, he expects that he will only earn 7 percent in his portfolio during retirement. He expects inflation to continue at 3 percent. Parker currently has $350,000 invested for his retirement. His Social Security benefit in today’s dollars is $30,000 per year at normal age retirement of age 67. His Social Security benefit will be reduced by 6 2/3 percent for each year he begins collecting before full age retirement. How much does he need to save each year to meet his retirement goals?

Answer 1

Annual amt. required after 15 years,ie. From end of 16 th year(age 65) to end of age 95th year-- for a period of 30 years = 120000*65%= $ 78000

So, total amt. to be accumulated at end of 65 years(15 years from now) ,

at a rate of 7% with an inflation rate (growing annuity)of 3%

Using PV of gowing annuity formula,

we find the PV at end of year 65 , with the above inputs:

PV(GA)=P/(r-g)*(1-((1+g)/(1+r))^n)

1…...ie.PV(GA)=78000/(7%-3%)*(1-((1+3%)/(1+7%))^30)

1328218

PV of funds available with him ,now

2..Retiremet investment = 350000

PV of social security benefits of $ 30000 p.a.

for a period of 17 years(67-50)

at an opportunity cost- rate of 9% at a decreasing annuity of

6&2/3%-3%= 3.67% net of inflation

Using the same (above) formula,

we get the PV of decreasing social benefits as

3.....PV(Decr.A)=30000/(9%-3.67%)*(1-((1-3.67%)/(1+9%))^17)

493972

4…..Total PV of funds available with him now,

350000+493972=

843972

5..Total PV of amt. to be saved to meet retirement goals=

Amts. got above as per 1-4

ie. 1328218-843972=

484246

So, this $ 484246 is the FV of growing annuity

at end of 15 years

at an opportunity cost- rate of 9% & inflation growth rate of 3%

Using the formual

FV(GA)=P*(((1+r)^n-(1+g)^n)/(r-g))

ie.484246=P*(((1+9%)^15-(1+3%)^15)/(9%-3%))

solving for P, the annual savings, we get,

$13,938

So the ANSWER is:

he need to save $ 13938 each year to meet his retirement goals