Determine the principal P that must be invested at the rate

Question

Determine the principal P that must be invested at the rate r=9%, compounded monthly, so that $500000 will be available for retirement in t=13 years. (Round to the nearest cent)

Answer 1

5,000,000 = P(1+.09/12)^(12x13) is the formula for interest compounded monthly on a principal amount.

P = $1,558,624.34

Answer 2

We need to use the compound interest formula: A=P(1+r/n)^nt

which can be rearranged to solve for P as follows:

P=A/((1+r/n)^nt)

where P=principal, A=total amount (principal+interest), r=interest rate (decimal), n=number of times compounded per t, t=time (years)

We want a total of $500,000, which is A.

So P=500,000/((1+0.09/12)^12*13)

Simplify:

P=500,000/(1+0.0075)^156

P=500,000/(1.0075^156)

P=500,000/(3.20795)

P=155,863 rounded up to the nearest whole number

So a principal amount of $155,862 must be invested.