User:Mathstat/Annuities section

Definitions in the first part of the article:
Let:
 * $$r$$ = the yearly nominal interest rate.
 * $$t$$ = the number of years.
 * $$m$$ = the number of periods per year.
 * $$i$$ = the interest rate per period.
 * $$n$$ = the number of periods.

Note:
 * $$ i = \frac{r}{m} $$


 * $$ n = t\cdot m $$

Also let:
 * $$P$$ = the principal (or present value).
 * $$S$$ = the future value of an annuity.
 * $$R$$ = the periodic payment in an annuity (the amortized payment).


 * $$S \,=\,R\left[\frac{\left(1+i\right)^n-1}{i}\right] \,=\,R\cdot s_{\overline{n}|i}$$ (annuity notation)

(deleted derivation) ... Hence:


 * $$S = R \frac{(1+i)^n-1}{i} $$.

Annuity-due

 * $$ S \, = \, R \left[ { (1+i)^{n+1} - 1 \over i } \right] - R\,=\,R\cdot \ddot{s}_{\overline{n|}i}$$

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

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
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



70000 = R a_{\overline{3}|.15} = R \left( \frac{1 - 1.15^{-3}}{0.15/1.15} \right), $$ 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



P = R \ddot a_{\overline{32}|.0125} = R \left( \frac{1 - 1.0125^{-32}}{0.0125/1.0125} \right), $$

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

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.



1600000 = R \ddot s_{\overline{3}|.09} = R \left( \frac{1.09^3 - 1}{0.09/1.09} \right), $$

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



55000 = R s_{\overline{36}|.0075} = R \left( \frac{1.0075^{36} - 1}{0.0075} \right), $$

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