Answer to Question #316284 in Finance for Helen

Question

8.) The following terms of payment for an annuity are as follows:

Periodic payment = P20,000

Payment interval = 1 month

Interest rate = 18% compounded monthly

Terms = 15 years

1. Find the present worth paid of all the payments if it is paid at the end of each month.

2. Find the difference between the sums of an annuity due and an ordinary annuity on these payments.

3. Find the difference between the present values of an annuity due and an ordinary annuity based on these payments.

Ans. 1. P1,214,911.246 2. P271,687.35 3. P18,628.67

Answer 1

Given

Periodic payment = P20,000

Payment interval, m = 1 month

Interest rate, r= 18% compounded monthly

Number of years, n=15 years

A) The present value of an ordinary annuity is given by the formula:

$PV=pmt[\frac{(1-(1+\frac{r}{m})^{-(nm)}}{\frac{r}{m}}]$

$\therefore PV=P20000[\frac{(1-(1+\frac{18\%}{12})^{-(15×12)}}{\frac{18\%}{12}}]$

$PV=P1241911.25$

B)

i) The future value of an annuity due is given as

$FV=pmt [\frac{(1+\frac{r}{m})^{nm}-1}{\frac{r}{m}}](1+\frac{r}{m})$

$\therefore FV=P20000[\frac{(1+\frac{18\%}{12})^{15×12}-1}{\frac{18\%}{12}}](1+\frac{18\%}{12})$

$FV=P18384177.61$

ii) The future value of an ordinary annuity is given as:

$FV=pmt [\frac{(1+\frac{r}{m})^{nm}-1}{\frac{r}{m}}]$

$\therefore FV=P20000[\frac{(1+\frac{18\%}{12})^{15×12}-1}{\frac{18\%}{12}}]$

$FV=P18112490.25$

The difference between the future values of the annuity due and ordinary annuity is:

P18384177.61-P18112490.25

=P271687.36

C) The present value of an annuity due is

$PV=pmt[\frac{(1-(1+\frac{r}{m})^{-(nm)}}{\frac{r}{m}}]+pmt$

$PV=P20000[\frac{(1-(1+\frac{18\%}{12})^{-(15×12)}}{\frac{18\%}{12}}]+P20000$

$PV=P1261911.25$

The difference between the present values of the annuity due and ordinary annuity is

$P1261911.25-P1241911.25$

$=P20000$