I have a econ midterm coming up soon and stumbled upon this question. I know this is the math section but it appears not many use the finance one. My approach is:
$2C=\frac{800}{1.12^2}+\frac{1200}{1.12^6}=1245.71$ or $C=1245.71/2=622.85$
But I have a gut feeling this is wrong. I believe the answer is somewhere around $\$781$.
Consider the following cash flow diagram. What value of C makes the inflow series equivalent to the outflow series at an interest rate of 12% compounded annually?
I would approach this question by observing the present value of all cash flows (in and out) at every point in time. Lets first begin with the outflows then inflows.
$$PV_{\mathrm{Outflow}}=2C + \sum_{i=2}^{8}\frac{C}{(1+12\%)^i}.$$
$$PV_{\mathrm{Inflow}}=\sum_{i=1}^{4}\frac{800}{(1+12\%)^i}+\sum_{i=5}^{8}\frac{1200}{(1+12\%)^i}$$
Now: "What value of C makes the inflow series equivalent to the outflow series". Thus the equation I am trying to solve is
$$PV_{\mathrm{Inflow}} = PV_{\mathrm{Outflow}},$$
implying that
$$ \sum_{i=1}^{4}\frac{800}{(1+12\%)^i}+\sum_{i=5}^{8}\frac{1200}{(1+12\%)^i} = 2C + \sum_{i=2}^{8}\frac{C}{(1+12\%)^i}.$$
Furthermore
$$C = \frac{\sum_{i=1}^{4}\frac{800}{(1+12\%)^i}+\sum_{i=5}^{8}\frac{1200}{(1+12\%)^i}}{2 + \sum_{i=2}^{8}\frac{1}{(1+12\%)^i}}.$$
$$C = \$781.30\ (\mathrm{2dp.})$$