User:Mathstat/Annuities section

Trying to clean up the examples in the Annuity article.

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:

Annuity immediate	Annuity due
$(1)\quad P=Ra_{{\overline {n}}|i}=R\left({\frac {1-v^{n}}{i}}\right)$	$(2)\quad P=R{\ddot {a}}_{{\overline {n}}|i}=R\left({\frac {1-v^{n}}{d}}\right)$
$(3)\quad S=Rs_{{\overline {n}}|i}=R\left({\frac {(1+i)^{n}-1}{i}}\right)$	$(4)\quad S=R{\ddot {s}}_{{\overline {n}}|i}=R\left({\frac {(1+i)^{n}-1}{d}}\right)$

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

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

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. </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.

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

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

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

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