`r = 6%, t = 40` Find the principal P that must be invested at a rate r, compounded monthly, so that $1,000,000 will be available for...

Sunday, August 5, 2012

The formula in compounding interest is

$\displaystyle{A}={P}{{\left({1}+\frac{{r}}{{n}}\right)}}^{{{n}\cdot{t}}}$

where

A is the accumulated amount

P is the principal

r is the annual rate

n is the number of compounding periods in a year, and

t is the number of years.

Plugging in the values A = 1000000, r = 0.06 and t = 40 , the formula becomes:

$\displaystyle{1000000}={P}{{\left({1}+\frac{{0.06}}{{n}}\right)}}^{{{n}\cdot{40}}}$

Since the r is compounded monthly, the value of n is 12.

$\displaystyle{1000000}={P}{{\left({1}+\frac{{0.06}}{{12}}\right)}}^{{{12}\cdot{40}}}$

Simplifying the right side, it becomes

$\displaystyle{1000000}={P}{{\left({1}+{0.005}\right)}}^{{480}}$

$\displaystyle{1000000}={P}{{\left({1.005}\right)}}^{{480}}$

Isolating the P, it yields

$\displaystyle\frac{{1000000}}{{{1.005}}^{{480}}}=\frac{{{P}{{\left({1.005}\right)}}^{{480}}}}{{{1.005}}^{{480}}}$

$\displaystyle{91262.08}={P}$

Therefore, the principal amount that should be invested is $91,262.08 .

Blog