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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.

Respuesta :

jepessoa

Answer:

effective annual interest rate = 6.32%

annual payment = $1,585

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

           

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