## Quantitative Methods - Net Present Value and the Internal Rate of Return

This section applies the techniques and formulas first presented in the time value of money material toward real-world situations faced by financial analysts. Three topics are emphasized: (1) capital budgeting decisions, (2) performance measurement and (3) **Net Preset Value**

NPV and IRR are two methods for making capital-budget decisions, or choosing between alternate projects and investments when the goal is to increase the value of the enterprise and maximize shareholder wealth. Defining the NPV method is simple: the present value of cash inflows minus the present value of cash outflows, which arrives at a dollar amount that is the net benefit to the organization.

To compute NPV and apply the NPV rule, the authors of the reference textbook define a five-step process to be used in solving problems:

1.Identify all cash inflows and cash outflows.

2.Determine an appropriate discount rate (r).

3.Use the discount rate to find the present value of all cash inflows and outflows.

4.Add together all present values. (From the section on cash flow additivity, we know that this action is appropriate since the cash flows have been indexed to t = 0.)

5.Make a decision on the project or investment using the NPV rule: Say yes to a project if the NPV is positive; say no if NPV is negative. As a tool for choosing among alternates, the NPV rule would prefer the investment with the higher positive NPV.

Companies often use the weighted average cost of capital, or WACC, as the appropriate discount rate for capital projects. The WACC is a function of a firm's capital structure (common and preferred stock and long-term debt) and the required rates of return for these securities. CFA exam problems will either give the discount rate, or they may give a WACC.

Example:

To illustrate, assume we are asked to use the NPV approach to choose between two projects, and our company's weighted average cost of capital (WACC) is 8%. Project A costs $7 million in upfront costs, and will generate $3 million in annual income starting three years from now and continuing for a five-year period (i.e. years 3 to 7). Project B costs $2.5 million upfront and $2 million in each of the next three years (years 1 to 3). It generates no annual income but will be sold six years from now for a sales price of $16 million.

For each project, find NPV = (PV inflows) - (PV outflows).Project A: The present value of the outflows is equal to the current cost of $7 million. The inflows can be viewed as an annuity with the first payment in three years, or an ordinary annuity at t = 2 since ordinary annuities always start the first cash flow one period away.

PV annuity factor for r = .08, N = 5: (1 - (1/(1 + r)^{N})/r = (1 - (1/(1.08)^{5})/.08 = (1 - (1/(1.469328)/.08 = (1 - (1/(1.469328)/.08 = (0.319417)/.08 =3.99271

Multiplying by the annuity payment of $3 million, the value of the inflows at t = 2 is ($3 million)*(3.99271) = $11.978 million.

Discounting back two periods, PV inflows = ($11.978)/(1.08)^{2}= $10.269 million.

NPV (Project A) = ($10.269 million) - ($7 million) = $3.269 million.Project B: The inflow is the present value of a lump sum, the sales price in six years discounted to the present: $16 million/(1.08)^{6}=$10.083 million.

Cash outflow is the sum of the upfront cost and the discounted costs from years 1 to 3. We first solve for the costs in years 1 to 3, which fit the definition of an annuity.PV annuity factor for r = .08, N = 3: (1 - (1/(1.08)

^{3})/.08 = (1 - (1/(1.259712)/.08 = (0.206168)/.08 = 2.577097.PV of the annuity = ($2 million)*(2.577097) = $5.154 million.

PV of outflows = ($2.5 million) + ($5.154 million) = $7.654 million.

NPV of Project B = ($10.083 million) - ($7.654 million) = $2.429 million.

Applying the NPV rule, we choose Project A, which has the larger NPV: $3.269 million versus $2.429 million.

Exam Tips and TricksProblems on the CFA exam are frequently set up so that it is tempting to pick a choice that seems intuitively better (i.e. by people who are guessing), but this is wrong by NPV rules. In the case we used, Project B had lower costs upfront ($2.5 million versus $7 million) with a payoff of $16 million, which is more than the combined $15 million payoff of Project A. Don\'t rely on what feels better; use the process to make the decision! |

**The Internal Rate of Return**

The IRR, or internal rate of return, is defined as the discount rate that makes NPV = 0. Like the NPV process, it starts by identifying all cash inflows and outflows. However, instead of relying on external data (i.e. a discount rate), the IRR is purely a function of the inflows and outflows of that project. The IRR rule states that projects or investments are accepted when the project's IRR exceeds a hurdle rate. Depending on the application, the hurdle rate may be defined as the weighted average cost of capital.

Example:

Suppose that a project costs $10 million today, and will provide a $15 million payoff three years from now, we use the FV of a single-sum formula and solve for r to compute the IRR.

IRR = (FV/PV)^{1/N }-1 = (15 million/10 million)^{1/3 }- 1 = (1.5)^{ 1/3 }- 1 = (1.1447) - 1 = 0.1447, or 14.47%

In this case, as long as our hurdle rate is less than 14.47%, we green light the project.

**NPV vs. IRR**

Each of the two rules used for making capital-budgeting decisions has its strengths and weaknesses. The NPV rule chooses a project in terms of net dollars or net financial impact on the company, so it can be easier to use when allocating capital.

However, it requires an assumed discount rate, and also assumes that this percentage rate will be stable over the life of the project, and that cash inflows can be reinvested at the same discount rate. In the real world, those assumptions can break down, particularly in periods when interest rates are fluctuating. The appeal of the IRR rule is that a discount rate need not be assumed, as the worthiness of the investment is purely a function of the internal inflows and outflows of that particular investment. However, IRR does not assess the financial impact on a firm; it only requires meeting a minimum return rate.

The NPV and IRR methods can rank two projects differently, depending on thesize of the investment. Consider the case presented below, with an NPV of 6%:

Project |
Initial outflow |
Payoff after one year |
IRR |
NPV |

A | $250,000 | $280,000 | 12% | +$14,151 |

B | $50,000 | $60,000 | 20% | +6604 |

By the NPV rule we choose Project A, and by the IRR rule we prefer B. How do we resolve the conflict if we must choose one or the other? The convention is to use the NPV rule when the two methods are inconsistent, as it better reflects our primary goal: to grow the financial wealth of the company.**Consequences of the IRR Method**

In the previous section we demonstrated how smaller projects can have higher IRRs but will have less of a financial impact. Timing of cash flows also affects the IRR method. Consider the example below, on which initial investments are identical. Project A has a smaller payout and less of a financial impact (lower NPV), but since it is received sooner, it has a higher IRR. When inconsistencies arise, NPV is the preferred method. Assessing the financial impact is a more meaningful indicator for a capital-budgeting decision.

Project |
Investment |
Income in future periods |
IRR |
NPV |
||||

t_{1} |
t_{2} |
t_{3} |
t_{4} |
t_{5} |
||||

A | $100k | $125k | $0 | $0 | $0 | $0 | 25.0% | $17,925 |

B | $100k | $0 | $0 | $0 | $0 | $200k | 14.9% | $49,452 |