Net Present Value

Study guide: Exam FM-2

The net present value (NPV) is the sum of the present values of all cash flows associated with a project or investment, with inflows treated as positive and outflows as negative. Discounting each cash flow $C_t$ at effective rate $i$ using $v = (1+i)^{-1}$ :

$NPV = \sum_t C_t\, v^t$

$= PV(\text{inflows}) - PV(\text{outflows})$

A positive NPV means the investment creates value at rate $i$ — the inflows more than compensate for the outflows at the chosen discount rate.
A negative NPV means the opposite.
The internal rate of return (IRR) is the rate $i^*$ at which $NPV = 0$ .

Comparing Two Investment Projects Project A requires an outlay of $ $10{,}000$ today and returns $ $4{,}000$ at the end of each of the next 3 years. Project B requires $ $10{,}000$ today and returns $ $12{,}500$ at the end of year 3 only. Using $i = 6\%$ , which project has the higher NPV?

Answer Project A: $NPV_A = -10{,}000 + 4{,}000\,a_{\overline{3}|0.06}$ $a_{\overline{3}|} = \frac{1-(1.06)^{-3}}{0.06} = \frac{1-0.83962}{0.06} \approx 2.6730$ $NPV_A = -10{,}000 + 4{,}000 \times 2.6730 \approx -10{,}000 + 10{,}692 = \$692$

Project B: $NPV_B = -10{,}000 + 12{,}500\,(1.06)^{-3} = -10{,}000 + 12{,}500 \times 0.83962 \approx -10{,}000 + 10{,}495 = \$495$

Project A has the higher NPV ($ $692 > \$495$ ), so Project A is preferred at a 6% discount rate.