Question

We are going to retire at age 65. To do this we wish to have some money set aside to help us survive the rest of our life. Let’s suppose that we wish for an annuity to be set up so that we can take out $6000 every quarter for 15 years (until age 80). An annuity that you receive money from is sometimes referred to as a payout annuity. Let’s suppose further that this annuity will get 6% interest compounded quarterly.

1. What type of problem is this? {Future Value (FV), Present Value (PV), Sinking Fund (S), or Amortization(A)} _________________

How much must we invest (at age 65) for this to happen? ___________________

Now in order to have this money to invest we needed to start saving for it a long time before we are 65. Let’s start saving at age 25. (Let’s set up a sinking fund)

2. How much money would we need to put away monthly into an ordinary annuity that earns 12% compounded monthly in order to get to our goal? (12% is a somewhat realistic return for a good IRA depending, of course, on how the stock market does) ________________

3. What would be our monthly value if we had started saving at age 35 instead of 25? _______________

How much are we earning???

We will compare what we get out to what we put in.

4. First, how much are we getting out? At age 65 we will start to receive payments until age 80. What is the total of all of these payments coming out of the first annuity described in this problem? ___________

5. Second calculate the total invested into the sinking funds; Starting at age 25 ___________

Starting at age 35___________

The difference is how much you are earning (in interest) 6. Calculate how much interest is earned starting at age 25. _________

Calculate how much interest is earned starting at age 35.__________

What is the difference by starting at an earlier age? _______________

Answer 1

1• Since $6000 are to be taken out quarterly for 15 years.

Therefore , Total amount = $ 6000 × 15 × 4 = $ 360000

interest rate (R) = 6% = 0.06

Time (T) = 15 years . ; P = ?.

n = 4 (since compounded quarterly)

we know that , Amount of compound interest is given by ,

A = P { 1 + (R/n) }^nT

=> 360000 = P {1+(0.06/4)}^(4×15)

=> 360000 = P { 4.06 / 4 }^60

=> 360000 = P (1.015)^60

=> P = 360000/{(1.015)^60}

=> P = $ 147347.74

Thus , money should be invested (at age 65) = $147347.74.

2• If started saving at age 25 , then upto age 65 , time (t) = 40 years.

n = 12 ( since compounded monthly )

r = 12% = 0.12

A = $147347.74

By formula of compound interest,

A = P { 1 + (r/n) }^nt

=> 147347.74 = P {1+(0.12/12)}^(12×40)

=> 147347.74 = P {1.01}^480

=> 147347.74 = P 118.647

=> P = $1241.90

Therefore principal ( if started at age 25) = $1241.90

3• If started saving at age 35 upto 65 , time (t) = 30 years

n = 12. ; r = 12% = 0.12. ; A = $147347.74

and ,

A = P {1 + (r/n)}^nt

=> 147347.74 = P {1 + (0.12/12)}^(12×30)

=> 147347.74 = P {1.01}^360

=> 147347.74 = P 35.94

=> P = $4099.80

therefore , money invested (if started at age 35) = $4099.80

4• Total amount taken out after 15 years if $6000 taken out quarterly ,

= $6000 × 15 × 4 = $360000