Notes: Find the present value of the decreasing annuity necessary to fund the withdrawals. $ 1530 per quarter for 25 years, if the annuity earns 6% per...

Thursday, 27 July 2017

To solve, apply the formula of present value of annuity.

$\displaystyle{P}{V}=\frac{{{P}{M}{T}{\left[{1}-{{\left({1}+\frac{{r}}{{n}}\right)}}^{{-{n}{t}}}\right]}}}{{\frac{{r}}{{n}}}}$

where

PV is the present value

PMT is the periodic payment

r is the rate

n is the number of deposits/withdrawals in a year, and

t is the number of years.

Since $1530 per quarter is to be withdrawn for 25 years, then PMT=1530, n=4 and t=25. And the given rate is s r=6%.

Plugging them to the formula yields:

$\displaystyle{P}{V}=\frac{{{1530}\cdot{\left[{1}-{{\left({1}+\frac{{0.06}}{{4}}\right)}}^{{-{4}\cdot{25}}}\right]}}}{{\frac{{0.06}}{{4}}}}$

$\displaystyle{P}{V}=\ldots$

To solve, apply the formula of present value of annuity.

$\displaystyle{P}{V}=\frac{{{P}{M}{T}{\left[{1}-{{\left({1}+\frac{{r}}{{n}}\right)}}^{{-{n}{t}}}\right]}}}{{\frac{{r}}{{n}}}}$

where

PV is the present value

PMT is the periodic payment

r is the rate

n is the number of deposits/withdrawals in a year, and

t is the number of years.

Since $1530 per quarter is to be withdrawn for 25 years, then PMT=1530, n=4 and t=25. And the given rate is s r=6%.

Plugging them to the formula yields:

$\displaystyle{P}{V}=\frac{{{1530}\cdot{\left[{1}-{{\left({1}+\frac{{0.06}}{{4}}\right)}}^{{-{4}\cdot{25}}}\right]}}}{{\frac{{0.06}}{{4}}}}$

$\displaystyle{P}{V}=\frac{{{1530}{\left({1}-{{1.015}}^{{-{100}}}\right)}}}{{0.015}}$

$\displaystyle{P}{V}={78985.79661}$

Rounding off to nearest hundredths, it becomes 78985.80.

Therefore, the present value is $78985.80 .

Notes