Net Present Value and Internal Rate of Return for Actuaries

Net Present Value Fundamentals

The net present value (NPV) of a project is the present value of all cash inflows minus the present value of all cash outflows, discounted at the required rate of return (cost of capital). NPV = sum of CF_t / (1+i)^t for t = 0, 1, ..., n. A positive NPV means the project adds value; a negative NPV means it destroys value. The decision rule: accept if NPV > 0.

The discount rate i used in NPV calculations represents the opportunity cost of capital. It reflects the return the investor could earn on a comparable investment of similar risk. Choosing the appropriate discount rate is crucial because NPV is sensitive to this input. A project with NPV > 0 at one discount rate may have NPV < 0 at a higher rate.

Example: An investment costs $10,000 today and generates $3,000 per year for 5 years. At i = 8%, NPV = -10,000 + 3,000 * a-angle-5(0.08) = -10,000 + 3,000 * 3.9927 = $1,978. Since NPV > 0, accept the project. At i = 12%, NPV = -10,000 + 3,000 * a-angle-5(0.12) = -10,000 + 3,000 * 3.6048 = $814. Still positive, but the margin is smaller.

NPV Profile

The NPV profile is a graph of NPV as a function of the discount rate. For conventional cash flows (an initial outflow followed by inflows), the NPV profile is a decreasing function that crosses the horizontal axis at the IRR. At i = 0, NPV equals the simple sum of all cash flows. As i increases, NPV decreases and eventually becomes negative.

The NPV profile is useful for understanding how sensitive a project's attractiveness is to the discount rate assumption. A steep profile means NPV is highly sensitive to rate changes. A flat profile means the project is robust to different rate assumptions.

Internal Rate of Return

The internal rate of return (IRR) is the discount rate that makes NPV = 0. It represents the rate of return earned on the invested capital. The decision rule: accept if IRR > cost of capital. For the example above, solve -10,000 + 3,000 * a-angle-5(IRR) = 0, giving a-angle-5 = 3.333, so IRR is approximately 15.2%.

Computing IRR on Exam FM typically requires interpolation. Find two rates that bracket the solution (one giving a positive NPV, the other giving a negative NPV), then interpolate linearly. For the example: at 15%, NPV = -10,000 + 3,000 * 3.3522 = $56.60. At 16%, NPV = -10,000 + 3,000 * 3.2743 = -$177.10. IRR is approximately 15% + 56.60/(56.60 + 177.10) * 1% = 15.24%.

Multiple IRRs and Non-Conventional Cash Flows

When cash flows change sign more than once, there may be multiple IRRs. Descartes' rule of signs tells us that the number of positive real roots of the NPV equation is at most equal to the number of sign changes in the cash flow sequence. A project with cash flows -100, +230, -132 (two sign changes) could have zero or two positive IRRs.

In such cases, NPV is the more reliable criterion because it always gives a unique answer for a given discount rate. The IRR rule can give conflicting signals when multiple IRRs exist. On Exam FM, most problems have conventional cash flows (one initial outflow followed by inflows, giving exactly one sign change and one unique positive IRR), but you should be aware of the multiple-IRR possibility.

NPV vs. IRR for Mutually Exclusive Projects

NPV and IRR usually agree on accept/reject decisions for independent projects with conventional cash flows. However, for mutually exclusive projects (where you must choose one or the other), they may conflict. This occurs because IRR ignores the scale of investment and the timing of cash flows.

Example: Project A costs $1,000 and returns $1,500 in one year (IRR = 50%). Project B costs $10,000 and returns $13,000 in one year (IRR = 30%). At a cost of capital of 10%, NPV_A = -1,000 + 1,500/1.10 = $363.64 and NPV_B = -10,000 + 13,000/1.10 = $1,818.18. Project B has the higher NPV and should be preferred, even though Project A has the higher IRR.

The resolution is to use the incremental IRR approach: compute the IRR of the incremental cash flows (B minus A). If the incremental IRR exceeds the cost of capital, choose the larger project. Alternatively, just use NPV directly, which is the recommended approach when NPV and IRR conflict.

Profitability Index

The profitability index (PI) is the ratio of the present value of future cash flows to the initial investment: PI = PV(inflows) / |PV(outflows)|. Accept if PI > 1. The PI is equivalent to NPV for accept/reject decisions (PI > 1 if and only if NPV > 0) but provides a measure of value per dollar invested, which is useful for capital rationing.

For the original example: PI = (3,000 * 3.9927) / 10,000 = 11,978 / 10,000 = 1.198. The project generates $1.198 in present value for every $1.00 invested. On Exam FM, the profitability index appears occasionally in comparison problems.

Study Tips and Common Pitfalls

The most common mistake is using the wrong sign convention for cash flows. Outflows should be negative and inflows positive (or vice versa, as long as you are consistent). Another frequent error is confusing NPV with accumulated value. NPV discounts all cash flows to time 0, while accumulated value compounds all cash flows to time n.

For IRR interpolation, make sure your two trial rates bracket the solution (one gives a positive NPV, the other gives a negative NPV). If both give the same sign, your bracket is wrong and you need to adjust. Practice at least 5 NPV problems and 5 IRR problems, including one or two mutually exclusive project comparisons. These are among the most practical topics on Exam FM and appear frequently.