Question

A couple received a $108,000 inheritance the year they turned 48 and in-vested it in a fund that earns 6.5% compounded semiannually (every six months). They leave the money in the account for 12 years (until they retire), and then want to get regular payments from the account, so that all the money is paid during the next 20 years. How much will the couple receive in these regular payments?

Answer 1

First we need to find annual effective interest rate

Effective interest rate = (1 + i/m) ^m -1

Effective interest rate = (1 + 0.065/2) ^2 - 1

                                              = (1 + 0.0325) ^2 - 1

                                              = (1.0325) ^2 - 1

                                              = 1.06606 - 1

                                              = 0.0661

So annual effective interest rate is 6.61% per year

FV = Present value *(1 + r)^n

Where,

               Present value = $108000

               Time (n) = 12

               Interest rate [r] = 6.61%

FV = 108000*(1 + 0.0661)^12

      = 108000*(1.0661)^12

      = 108000*(2.1556354655)

      = 232808.63

So after 12 year the amount will grow to $232808.63

Now we need to find the annual payment that the couple will receive.

PV of annuity = P * [1 - (1 + r) ^-n]/ r

Where,

                 Present value of annuity = $232808.63

                 Interest rate (i) = 6.61%

                 Time (n) = 20

Let's put all the values in the formula

232808.63 = P* [1 - (1 + 0.0661) ^-20]/ 0.0661

232808.63 = P* [1 - (1.0661)^-20]/ 0.0661

232808.63 = P* [1 - 0.2779976553]/ 0.0661

232808.63 = P* [0.7220023447]/ 0.0661

232808.63 = P* 10.9228796475

P = 232808.63/ 10.9228796475

P = 21313.85

So annual amount that couple will receive for 20 years is $21313.85

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