13.7
7. Imagine that it is now 1995 and some guy named Dave Matthews comes to you and tells you he needs money to cut a demo tape with a band he’s forming. He promises to pay you $5000 per year for 3 years (starting next year) if you could loan him $7500 today. What rate of return is he offering? Your answer can be expressed as between 2 rates, not more than 5% apart: for example, 20-25%. You must show your work. Try 40% as an initial guess. (8 pts)
There is no way to solve for r directly in an annuity formula, so we have to do 2 trials. First try P3=5000, n=3, and r=40%:
\( \Large PV = P3 \left [ \frac{1}{r} - \frac{1}{r \left(1+r \right )^{n} } \right ] = 5000 \left [ \frac{1}{.40} - \frac{1}{.40 \left(1+.40 \right )^{3} } \right ] = 7{,}944.61 \).
The PV using r=40% is too high so we know that we need a higher r, because PV and r move in opposite directions. Try 45%:
\( \Large PV = 5000 \left [ \frac{1}{.45} - \frac{1}{.45 \left(1+.45 \right )^{3} } \right ] = 7{,}466.48 \).
which is too low. Thus, the true rate must lie between 40 and 45%.