What if the cash flows occur mid-year?

Adjust the exponent $t$ to reflect the exact fraction of the period, such as $t=0.5$ for a cash flow six months from now.

What if the discount rate changes per period?

Discount each cash flow by the product of the specific rates up to that period: $PV = C_t / \prod_{i=1}^t (1+r_i)$.

Does NPV account for inflation?

Nominal cash flows must be discounted by a nominal rate, while real cash flows require a real discount rate. Mixing them yields incorrect results.

How to calculate net present value of a project

Net present value (NPV) is calculated by summing the present values of all future cash flows of a project and subtracting the initial investment. This method applies when evaluating mutually exclusive projects or determining the absolute profitability of an investment under a known, constant discount rate.

The setup

Identify the initial investment $C_0$ , the projected cash flows $C_t$ for each period $t$ , and the discount rate $r$ per period.

The steps

Identify the initial cash outflow at $t=0$ . 2. Calculate the present value of each future cash flow using the formula $PV = C_t / (1+r)^t$ . 3. Sum the present values of all future cash flows. 4. Subtract the initial investment from the sum of the discounted cash flows to find the NPV.

Checking the result

Verify that higher discount rates yield lower NPVs. Ensure that cash flows occurring further in the future contribute proportionally less to the total present value than near-term cash flows.

Common errors

Using the wrong discount rate, misaligning cash flows with their respective time periods, or forgetting to include the initial investment as a negative value at time zero.

Worked example

Calculate the NPV of a project requiring an initial investment of $\$ 10,000 $, yielding$ $3,000 $in year 1,$ $4,000 $in year 2, and$ $5,000 $in year 3. The discount rate is$ 10%$ per year.

$NPV = -C_0 + \sum_{t=1}^{3} \frac{C_t}{(1+r)^t}$ $NPV = -10000 + \frac{3000}{(1+0.10)^1} + \frac{4000}{(1+0.10)^2} + \frac{5000}{(1+0.10)^3}$ $NPV = -10000 + \frac{3000}{1.1} + \frac{4000}{1.21} + \frac{5000}{1.331}$ $NPV = -10000 + 2727.27 + 3305.79 + 3756.57$ $NPV = -10000 + 9789.63$ $NPV = -210.37$

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References: Principles of Corporate Finance (Brealey, Myers, Allen) · OpenStax Principles of Finance Chapter 9