Question

Larry is 58 and wants to retire by age 65. He expects that he will live to age 95. He currently has a salary of $140,000 and expects that he will need about 72% 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 because he will adjust his asset allocation so that his portfolio is more conservative. He expects inflation to continue at 3 percent. Larry currently has $800,000 invested for his retirement. His Social Security benefit in today’s dollars is $24,000 per year, assuming a retirement of age 65. He just calculated what he needs to save each year and it is more than he can afford. How will the following alternative affect his retirement plans (show your work):

a. Modify his portfolio during retirement to achieve an 8% rate of return instead of a 7% rate of return.

b. Delay his retirement by 2 years.

c. Reduce his expected needs by $10,000 in today’s dollars.

d. None of the above will help him achieve his goals.

Answer 1

First we need to calcualte the amt. he need to save every year with multiple steps as follows:

Annual amt. required after 7 years,ie. From end of 8 th year(age 65) to end of age 95th year-- for a period of 30 years =140000*72%= $ 100800

So, total amt. to be accumulated at end of 65 years(7 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)=100800/(7%-3%)*(1-((1+3%)/(1+7%))^30)

1716467

PV of funds available with him ,now

2..Retiremet investment = 800000

PV of social security benefits of $ 24000 p.a.

for a period of 7 years(65-58)

at an opportunity cost- rate of 9% with inflation at the same 3% as above,

Using the same (above) formula,

we get the PV of social benefits as

3.....PV=24000/(9%-3%)*(1-((1+3%)/(1+9%))^7)

130887

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

800000+130887=

930887

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

Amts. got above as per 1-4

ie. 1716467-930887=

785580

So, this $ 785580 is the FV of growing annuity

at end of 7 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.785580=P*(((1+9%)^7-(1+3%)^7)/(9%-3%))

solving for P, the annual savings, we get,

$78,799

So

he need to save $ 78799   each year to meet his retirement goals

Taking the options one by one:

(Changed STEPS are done again---others remain the same as the base solution)

a. Modify his portfolio during retirement to achieve an 8% rate of return instead of a 7% rate of return.

Now, we find the PV at end of year 65 , with 8% reqd. return as & other inputs same as above to be :

1…...ie.PV(GA)=100800/(8%-3%)*(1-((1+3%)/(1+8%))^30)

1529711

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

Amts. got above as per 1-4

ie. 1529711-930887=

598824

So, this $ 598824 is the FV of growing annuity

at end of 7 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.598824=P*(((1+9%)^7-(1+3%)^7)/(9%-3%))

solving for P, the annual savings, we get,

$60,066

So

he need to save $ 60066   each year to meet his retirement goals

b. Delay his retirement by 2 years.

30 yrs. Retirement yrs. Becomes 30-2=28 yrs. So,recalculating the 1st step,

1…...ie.PV(GA)=100800/(7%-3%)*(1-((1+3%)/(1+7%))^28)

1652844

With revised no.of years-in-working ,from 7 yrs. To 9 yrs.

we get the PV of social benefits as

3.....PV=24000/(9%-3%)*(1-((1+3%)/(1+9%))^9)

159698

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

800000+159698=

959698

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

Amts. got above as per 1-4

ie. 1652844-959698=

693146

So, this $ 693146 is the FV of growing annuity

at end of 9 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.693146=P*(((1+9%)^9-(1+3%)^9)/(9%-3%))

solving for P, the annual savings, we get,

$47,962

So

he need to save $ 47962   each year to meet his retirement goals

c. Reduce his expected needs by $10,000 in today’s dollars.

ie. (140000*72%)-10000 = $ 90800 very year during retirement

So, recalculating,

Annual amt. required after 7 years,ie. From end of 8 th year(age 65) to end of age 95th year-- for a period of 30 years = $ 90800

So, total amt. to be accumulated at end of 65 years(7 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)=90800/(7%-3%)*(1-((1+3%)/(1+7%))^30)

1546182

PV of funds available with him ,now

2..Retiremet investment = 800000

PV of social security benefits of $ 24000 p.a.

for a period of 7 years(65-58)

at an opportunity cost- rate of 9% with inflation at the same 3% as above,

Using the same (above) formula,

we get the PV of social benefits as

3.....PV=24000/(9%-3%)*(1-((1+3%)/(1+9%))^7)

130887

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

800000+130887=

930887

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

Amts. got above as per 1-4

ie. 1546182-930887=

615295

So, this $ 615295 is the FV of growing annuity

at end of 7 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.615295=P*(((1+9%)^7-(1+3%)^7)/(9%-3%))

solving for P, the annual savings, we get,

$61,718

So

he need to save $ 61718 each year to meet his retirement goals

Summary	Amt. to saved each year	Ranking (least amt. set aside)
a. Modify his portfolio during retirement to achieve an 8% rate of return instead of a 7% rate of return.	$60,066	2
b. Delay his retirement by 2 years.	$47,962	1
c. Reduce his expected needs by $10,000 in today’s dollars.	$61,718	3
Based on the above summary , he can choose as per his current affordability & retirement needs.