51.2
2. Your company is considering a new product that
is expected to generate profits of $500,000 per quarter
for the next 10 years, at which point it will be obsolete. If
your discount rate for this project is 16%, compounded quarterly,
what would you be willing to pay right now to buy the equipment
to start producing this product? Assume that the only investment
you need to make is to buy the equipment and that the first
$500,000 profit will come one quarter from now.
This question is simply asking you what an annuity of $500,000 per quarter for 40 quarters (10 years) is worth at a discount rate of 4% per quarter (16%APR, compounded quarterly). To see this, remember that for any financial calculation, the periodicity of your interest rate MUST match the periodicity of your cash flows. Thus, you must convert your annual rate into a quarterly rate to match the fact that you have quarterly cash flows. 16%, comp. quarterly means r =16%/4=4%. The cash flows look like this:
| 0 | 1 | 2 | 3 | 4 | … | 40 |
| 500,000 | 500,000 | 500,000 | 500,000 | … | 500,000 |
\( \large PV = P3 \left [ \frac{1}{r} - \frac{1}{r \left(1+r \right )^{n} } \right ] = 500{,}000 \left [ \frac{1}{.04} - \frac{1}{.04 \left(1+.04 \right )^{40} } \right ] = 9{,}896{,}386.94 \).
You should be willing to invest up to $9,896,386.94 in the equipment.