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User:Mathstat/Annuities section

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Trying to clean up the examples in the Annuity article.

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Definitions in the first part of the article:

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Let:

= the yearly nominal interest rate.
= the number of years.
= the number of periods per year.
= the interest rate per period.
= the number of periods.

Note:

Also let:

= the principal (or present value).
= the future value of an annuity.
= the periodic payment in an annuity (the amortized payment).
(annuity notation)

(deleted derivation) ... Hence:

.

Annuity-due

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The part to be cleaned up

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Equations relating the periodic payment (R) of an annuity with n level payments, present value (P), and periodic effective interest rate i:

Annuity immediate Annuity due

Note that v=1/(1+i), and d = i/(1+i) = 1-v. When the interest is quoted as a nominal annual rate r convertible m-thly, theni=r/m, so v=1/(1+r/m) and d=(r/m)/(1+r/m).

Examples

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1. Finding the periodic payment of an annuity.

1(a). Find the periodic payment of an annuity due of $70,000, payable annually for 3 years at 15% compounded annually.

In equation (2) n=3, v=1/1.15=0.869565 and d=0.15/1.15=0.130435, so

so the annual payment amount is R = $70,000 / 2.62571 = $26,659.47.

1(b) Find the periodic payment of an annuity due of $250,700, payable quarterly for 8 years at 5% compounded quarterly.

In equation (2) P=250700, there are n=8(4)=32 payments, r=0.05 and i=0.05/4=0.0125 per quarter, so

and the quarterly payment amount (R) is $250,700 / 26.5692901 = $9,435.71. </math>

2. Finding the Periodic Payment(R), Given S:


2(a). Find the periodic payment of an accumulated value of $1,600,000, payable annually at the beginning of each year for 3 years at 9% compounded annually.

Use equation (4) with n=3, S=1600000, d=i/(1+i)= 0.09/1.09.

so the level annual payment amount is R=$1,600,000 / 3.573129 = $447,786.80.

2(b). Find the periodic payment of an accumulated value of $55000, payable monthly at the end of each month for 3 years at 9% compounded monthly.

In equation (3) the monthly effective rate is r/12=0.09/12=0.0075, S=55000, and the total number of payments n is 3(12)=36, so

and the monthly payment amount is R= $55,000 / 41.15271612 = $1,336.49.