Question

You are planning your retirement investment plan. Assume you are now twenty-two years old and plan to make investments as follows:

Invest $6,000 at the end of each year for the first ten years.

Invest $12,000 at the end of each year for the second ten years.

Invest $18,000 at the end of each year for the third ten years.

Invest $24,000 at the end of each year for the fourth ten years.

Assume that you invest in a diversified stock portfolio for the entire forty years. You expect that the portfolio will return 12% per year.

a) How much will accumulate in your retirement fund by age sixty-two if your predictions are correct? $___________

b) Now that you are sixty-two, you decide to retire and purchase a guaranteed annuity for the thirty years you expect to live. You will earn 5% on the invested funds and you will receive your annuity payment at the beginning of each year. How much will you receive each year? $___________   

c) If you, instead, decide to leave the funds in the diversified portfolio (expected return 12%), how much could you take out of the fund each year and still leave your grandchildren a $1,000,000 inheritance. $___________

Answer 1

Future value of annual payments = PMT [((1 + r)^n - 1) / r]

Future value of an investment (which is not annual payment) = PV (1 + r)^n

a) How much will accumulate in your retirement fund by age sixty-two if your predictions are correct? $___________

FV of $6,000 invested at the end of each year for the first ten years at the end of 10th year (at the age of 32)

= PMT [((1 + r)^n - 1) / r]

= 6000[((1+.12)^10-1)/.12] =  $105,292.41

FV of this $105,292.41 at the age of 62 at 12% interest =  PV (1 + r)^n

= $105,292.41(1+.12)^30 = $3,154,552.42

here n = 62 - 32 = 30

___________________

FV of $12,000 at the end of each year for the second ten years at the end of 20th year(at the age of 42)

=PMT [((1 + r)^n - 1) / r]

12000[((1+.12)^10-1)/.12] = $210,584.82

FV of this $210,584.82 at the age of 62 at 12% interest =  PV (1 + r)^n

=$210,584.82 * (1+.12)^20 =$2,031,362.90

here n = 62 - 42 = 20

___________________

FV of $18,000 at the end of each year for the third ten years at the end of 30th year ( at the age of 52)

= PMT [((1 + r)^n - 1) / r]

18000[((1+.12)^10-1)/.12] = $315,877.23

Fv of this $315,877.23  at the age of 62 at 12% interest =  PV (1 + r)^n

= $315,877.23 * (1+.12)^10 = $981,066.73

here n = 62 - 52 =10

_________________

FV of $24,000 at the end of each year for the fourth ten years at the end of 40th year (at the age of 62)

= PMT [((1 + r)^n - 1) / r]

24000[((1+.12)^10-1)/.12] =$421,169.64

here no need to find accumulated value of $421,170 since it is at the age of 62

_________________

Total amount accumulated in retirement fund by age sixty-two

=$3,154,552.42+$2,031,362.90 + $981,066.73 +  $421,169.64 = $6,588,151.69

_________________________________________________________________________________________

b) Now that you are sixty-two, you decide to retire and purchase a guaranteed annuity for the thirty years you expect to live. You will earn 5% on the invested funds and you will receive your annuity payment at the beginning of each year. How much will you receive each year? $___________   

You can use the MS Excel to solve this using PMT function as follows

Rate = .05

Nper = 30

Pv = $6,588,151.69

Type = 1 (since payment is at the beginning of the year. If the payment is at the end of the year, give type as zero)

Answer is = $408,160.69

If the payment is at the end of the year, answer is $428,568.72 giving Type as zero

__________________________________________________________________________________________

c) If you, instead, decide to leave the funds in the diversified portfolio (expected return 12%), how much could you take out of the fund each year and still leave your grandchildren a $1,000,000 inheritance. $___________

In this case first find out the present value of $1,000,000 =

using Ms Excel, use PV function as follows

Rate = .12

Nper = 30

FV = 1,000,000

Answer is $33,377.92

Then deduct $33,377.92 from $6,588,151.69 to use this as Present value for this question.

=$6,588,151.69 - $33,377.92 = $6,554,773.77

Now use PMT function as follows

Rate = .12

Nper = 30

PV = $6,554,773.77

Type = 1 (since payment is at the beginning of the year. If the payment is at the end of the year, give type as zero)

Answer is $726,547.85

If the payment is at the end of the year, answer is $813,733.59 giving Type as zero