Question

Matt inherited as a trust a fifteen-year annuity-immediate with annual payments. He has been told that the annuity payments earn compound interest at a level rate and that at the end of fifteen years, their accumulated value will be $37,804.39. He has further been assured that figured at this same rate of interest, the value of his inheritance was $15,077.10. The trust executor will not reveal the amount of the annual payments. Determine this amount and also the annual effective interest rate earned by the annuity payments.

Answer 1

effective annual interest rate = 6.32%

annual payment = $1,585

Step-by-step explanation:

I believe that this is an ordinary annuity, so we can use the future and present value of an ordinary annuity formula:

FV = annual payment x FV annuity factor, so annual payment = FV / FV annuity factor

PV = annual payment x PV annuity factor, so annual payment = PV / PV annuity factor

we can equal both equations:

PV / PV annuity factor = FV / FV annuity factor

FV / PV = FV annuity factor / PV annuity factor

$37,804.39 / $15,077.10 = FV annuity factor / PV annuity factor

2.5074 = FV annuity factor / PV annuity factor

the easiest way to solve this is to use an annuity table since we already know that there are 15 periods (I used an excel spreadsheet):

%,15 periods FV annuity factor PV annuity factor FV/PV

1 16.097 13.865 1.1609

2 17.293 12.849 1.34586

3 18.599 11.938 1.55797

4 20.024 11.118 1.80104

5 21.579 10.380 2.07890

6 23.276 9.7122 2.3966

7 25.129 9.1079 2.7590

8 27.152 8.5595 3.1721

9 29.361 8.0607 3.6425

10 31.772 7.6061 4.4112

The interest rate must be between 6 and 7%:

%,15 periods FV annuity factor PV annuity factor FV/PV

6 23.276 9.7122 2.3966

6.1 23.45404 9.6461 2.43145

6.2 23.63369 9.5858 2.46549

6.3 23.81491 9.52467 2.50034

6.31 23.83312 9.51851 2.50387

6.32 23.85135 9.51236 2.5074

6.4 23.99773 9.46337 2.53585

effective interest rate = 6.32% per year

annual payment = $37,804.39 / 23.85135 = $1,585