Two partners agree to invest equal amounts in their business. One will contribute $10,000 immediately. The other plans to contribute an equivalent amount in 3 years. How much should she contribute at that time to match her partner's investment now, assuming an interest rate of 9% compounded quarterly?
Using the formula for future value of a single sum with quarterly compounding:
FV = PV * (1 + r/4)^n
Where FV is the future value, PV is the present value, r is the interest rate (9%), and n is the number of quarters.
For the first partner who invests $10,000 now, the future value in 3 years would be:
FV = $10,000 * (1 + 0.09/4)^(4*3) = $12,794.31
To find out how much the second partner needs to contribute in 3 years to match this investment, we need to solve for PV in the same formula:
PV = FV / (1 + r/4)^n
Where FV is the future value we just calculated, r is still 9%, n is the number of quarters in 3 years (12), and PV is what we need to find.
PV = $12,794.31 / (1 + 0.09/4)^12 = $9,341.03
Therefore, the second partner should contribute $9,341.03 in 3 years to match the first partner's investment of $10,000 now.