# Use the information to answer the following two questions You plan to retire in 30 years. .You project that you will live another 20 years after retirement. Your interest rate is 5 percent During...

Use the information to answer the following two questions You plan to retire in 30 years. .

You project that you will live another 20 years after retirement. Your interest rate is 5 percent

During these retirement years, you will need an income of $50,000 at the end of each year. To achieve your goal, you intend to make equal annual deposits before your retirement into an account. The first deposit will be made at the

end of the year. How much will each of these deposits have to be?

a. $8,932

b. $9,379

c. $9,848

d. $8,507

e. None of the above

The answer is b, but can someone tell me how to resolve this? If possible, with a financial calculator?

N I/YR PMT FV PV

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### 2 Answers

I disagree... You are right that B is correct. Here's how to go about it.

First, you need to figure out how much you will need at the beginning of your retirement (so that you can draw money out at $50,000 per year and not run out until you die, 20 years after retiring).

The formula for that is:

PVA = ANN* {1-[1/(1+i)^n]}/i

ANN is your annuity -- the $50,000 that you want each year, i is your interest rate and n is how many years you need the annuity.

So: PVA = 50,000*{1-[1/(1.05)^20]}/.05

1.05^20 is 2.653298; 1 divided by that is .376889. When you subtract that from 1 you get .623111. Divide that by .05 and you have 12.46221.

So then 50,000*12.46221 = 623,110.5171

So that's the lump sum you need when you retire.

And now you have to figure out how much you need to invest each year (at 5% compounded annually) to get to that $623,110.52 number.

The formula for that is:

ANN = FVA*i/(1+i)^n - 1

Again, i is .05 and n is the number of years you have to save, in this case 30.

By plugging in info you gave and the number we got above:

ANN = 623110.5171*.05/(1.05^30)-1

which gets us

31155.52586/4.321942375-1

which gets you

9378.7, which rounds to 9379

Sorry, don't have a financial calculator, but that's the thought process and the formulas...

We assume that an amount of P is deposited each year . The value amount after 30 from first year investment =P(1.05)^30 in compound interest or P(1+30*0.05) if it i s in simple interest.

Similarly the amount of P invested in 2nd year becomes of value P(1.05)^2 or P(29.0.5) in compound or simple interest.

So, after 30 years his recurring investment of P each year will have the value A given by:

A = P(1.05^30+1.05^29+1.05^28+....1.05)=P(1.05)(1.05^30 - 1)/(0.05) .

So at the end of each year if gets $50000 after retitrement , it must be the 5% interest over the amount A , which is P(1.05)(1.05^n-1).

Therefore , 1.05(1.05^30 - 1)P = $50000.

Therefore, P = $50000/{1.05(1.05^30-1)}

=$14334.47 **, **if he has invested in compound interest.

If he has ivested in simple interest, the value A of his deposits at the end of 30 years is given by:

A = 30P+P{30(0.05)+29(0.05)+28(0.05)+....+2(0.05)+1(0.5)}

30P+P(0.05)(31*30/2) = 53.25P. So the interest at 5% on this should bring an annual sum of $50000 during the retirement.

Therefore,

53.25P(5/100) = $50000 or

P = 50000/(53.25*0*05) = $18779.34 should be invested each month.

No choice is correct.