In Week 6, you learned four tools for evaluating capital investments: NPV, IRR, payback period, and profitability index. When you're looking at a single, independent project with conventional cash flows (an outflow followed by a series of inflows), all four tools tend to point in the same direction. NPV is positive, IRR exceeds the required return, and you accept the project. Life is simple.
This week, we complicate things — because the real world is complicated. You'll encounter situations where the IRR rule and the NPV rule give you opposite recommendations. When that happens, you need to understand why they disagree and which one to trust. The short answer: always trust NPV. The longer answer is worth understanding, because it reveals something important about what these tools actually measure.
Recall from Week 6 that the internal rate of return (IRR) is the discount rate that makes a project's NPV equal to zero. The IRR rule says: accept if IRR exceeds the required return. The net present value (NPV) rule says: accept if NPV is positive at the required return. For a single independent project with conventional cash flows, these two rules will always agree. The problems arise in three specific situations:
1. Mutually exclusive projects with different scales. If you're choosing between a small project and a large project — and you can only pick one — the small project might earn a higher percentage return (higher IRR) while the large project creates more dollar value (higher NPV). IRR tells you about return rates; NPV tells you about wealth creation. When you can only pick one, you want the project that makes you the most money, not the one with the best percentage.
2. Mutually exclusive projects with different timing. Even when two projects require the same initial investment, their cash flow patterns can differ. A project that returns cash early will tend to have a higher IRR, while a project with larger but later cash flows might have a higher NPV. This happens because the IRR calculation implicitly assumes that intermediate cash flows can be reinvested at the IRR itself — an assumption that's often unrealistic when the IRR is high.
3. Non-conventional cash flows. When a project's cash flows change sign more than once — for example, an initial investment followed by revenues followed by cleanup costs — there can be multiple IRRs or no IRR at all. The IRR rule simply breaks down in these cases.
Let's work through each of these in detail.
Imagine Red Acre Industries is evaluating two mutually exclusive equipment upgrades. Project A is a small automation module costing $20,000 that generates cash flows of $15,000, $12,000, and $8,000 over three years. Project B is a full production line overhaul costing $200,000 that generates $90,000 per year for three years. Red Acre's required return is 10%.
The intuition here is straightforward: IRR is a rate — it doesn't know or care about the size of the investment. A 100% return on $1 is a dollar of value. A 10% return on $1,000,000 is $100,000 of value. When you can only take one project, you care about dollars, not percentages.
The second conflict arises when two mutually exclusive projects require the same initial investment but return cash at different speeds. A project that pays off quickly will tend to look better under the IRR rule, while a project with larger but more distant payoffs can have a higher NPV.
The deeper issue is the reinvestment rate assumption. When you compute a project's IRR, you're implicitly assuming that all intermediate cash flows can be reinvested at the IRR. If Project C has an IRR of 21%, the IRR calculation assumes every dollar of cash flow that comes in during the project's life earns 21% until the end. That's often unrealistic — especially for projects with unusually high IRRs. The NPV calculation, by contrast, discounts at the required return, which is a more reasonable assumption about what intermediate cash flows actually earn.
There's an elegant way to see why the two rules conflict here. If you plot the NPV of each project as a function of the discount rate, the two NPV curves cross at a rate called the crossover rate. For rates below the crossover, one project has a higher NPV; for rates above it, the other does. In this example, the crossover rate is about 14.29%. Since the required return (10%) is below the crossover, the back-loaded project has the higher NPV even though the front-loaded project has the higher IRR.
The most fundamental problem with the IRR rule arises when a project's cash flows switch signs more than once. A conventional cash flow pattern has one sign change: you invest money (negative) and then receive money (positive). A non-conventional cash flow pattern has multiple sign changes — for example, an investment followed by revenues followed by significant cleanup or decommissioning costs.
By Descartes' rule of signs, the number of possible IRRs is equal to the number of sign changes in the cash flow sequence. Two sign changes means up to two IRRs. Three sign changes means up to three. When there are multiple IRRs, which one do you use? The IRR rule gives no guidance.
The NPV of this project is positive for discount rates between 10% and 20%, and negative outside that range. If you plotted the NPV profile, it would be an inverted-U shape rather than the usual downward-sloping curve. This is a case where the IRR rule simply cannot help you — you'd have to know which IRR to use, and there's no principled way to choose. NPV, by contrast, gives a clear and unambiguous answer.
After three examples of IRR failing, it's worth emphasizing that IRR is a perfectly useful tool in many situations. When you're evaluating a single, independent project with conventional cash flows, the IRR rule and the NPV rule will always agree. Many practitioners like IRR because a rate of return is intuitive — it's easy to communicate and compare against hurdle rates. The trouble only starts when projects are mutually exclusive, when cash flows are non-conventional, or when you're comparing projects of very different sizes or timing.
The lesson from Part A is not that IRR is a bad tool. It's that IRR has specific limitations, and NPV doesn't share them. Here's the summary:
| Situation | IRR Rule | NPV Rule |
|---|---|---|
| Independent, conventional CFs | Works ✓ | Works ✓ |
| Mutually exclusive (scale difference) | Can mislead ✗ | Works ✓ |
| Mutually exclusive (timing difference) | Can mislead ✗ | Works ✓ |
| Non-conventional cash flows | Breaks down ✗ | Works ✓ |
NPV is the universal rule. It always works. We'll formalize the connection between NPV and the cost of capital when we study WACC in Weeks 9 and 10.
We now shift gears. Part A completed our coverage of project evaluation (Chapter 9). Part B begins a new thread that will carry us through the rest of the course: understanding risk and return in capital markets. The question is deceptively simple: if you invested money in the past, what kind of returns did you earn, and how much risk did you take? The answers provide the empirical foundation for everything that follows — portfolio theory, the CAPM, and the cost of capital.
This material draws from Chapter 12 of your textbook, which presents historical return data for major U.S. asset classes. The numbers tell a consistent story: higher average returns come with higher volatility. Understanding exactly what we mean by "average return" and "volatility" — and how to calculate them — is the goal of this section.
When you invest, your return comes in two forms. A dollar return is the absolute amount you gain or lose — the change in market value plus any income (dividends or interest) you received. A percentage return expresses that gain or loss as a fraction of what you invested. Percentage returns let you compare investments of different sizes on equal footing, which is why they're the standard unit for discussing historical performance.
If you bought a stock for $50, received a $1 dividend, and sold it for $54, your dollar return is $5 ($4 capital gain + $1 dividend) and your percentage return is 10% ($5 ÷ $50).
Given a series of annual returns, the simplest summary is the arithmetic average return: add them up and divide by the number of years. This gives you the return you earned in a "typical" year.
The arithmetic average is useful and intuitive, but it has a subtle problem: it can overstate how much your money actually grew. To see why, we need the geometric average.
The geometric average return tells you the single, constant rate of return that would have produced the same cumulative growth over the entire period. It accounts for the fact that losses hurt more than gains help — a 50% loss followed by a 50% gain doesn't get you back to where you started (you end up at 75% of your original value).
Notice the geometric average (7.10%) is lower than the arithmetic average (7.80%). This is always the case when returns vary — the more volatile the returns, the larger the gap. The geometric average is the return that, compounded steadily, reproduces your actual ending wealth. The arithmetic average overstates growth because it ignores the drag of volatility.
Which average should you use? It depends on the question. If you want to know what to expect in any single future year, the arithmetic average is appropriate — it's the unbiased estimate of next year's return. If you want to know the rate at which your wealth actually grew (or the rate to use for compounding over multiple years), use the geometric average. We'll see this distinction matter when we discuss risk premiums.
Averages tell you the center of the distribution, but they say nothing about how spread out the returns are. An investment that returns 10% every year is very different from one that returns +40% one year and −20% the next, even if they have similar averages. To measure this spread — this volatility — we use the variance and its square root, the standard deviation.
Variance measures the average squared deviation from the mean. We square the deviations so that negative deviations (returns below the mean) and positive deviations (returns above the mean) don't cancel out. Standard deviation brings us back to the same units as the returns themselves, making it directly interpretable: a standard deviation of 13% means that in a typical year, the actual return is about 13 percentage points away from the average.
The reason we divide by T − 1 rather than T is a statistical adjustment called Bessel's correction. When you estimate a population parameter from a sample, dividing by T would systematically understate the true variance. Dividing by T − 1 corrects this bias. For large samples the difference is negligible, but for the small samples you'll work with in this course, it matters.
A standard deviation of 13.57% means that this fund's annual returns typically deviate from the 7.80% average by about 13.57 percentage points. If returns are roughly normally distributed, about two-thirds of annual returns will fall within one standard deviation of the average (between −5.77% and 21.37%), and about 95% will fall within two standard deviations.
Now that you can compute average returns and standard deviations, let's look at what history actually shows us. The table below summarizes arithmetic average annual returns and standard deviations for five major U.S. asset classes, computed from data going back to the late 1920s. The broad patterns are remarkably consistent:
| Asset Class | Avg. Annual Return | Standard Deviation |
|---|---|---|
| Small-company stocks | 16.0% | 32.0% |
| Large-company stocks | 12.0% | 19.1% |
| Long-term corporate bonds | 6.9% | 7.6% |
| Long-term government bonds | 4.8% | 7.9% |
| U.S. Treasury bills | 3.3% | 3.0% |
Sources: Equity returns from Kenneth French Data Library (size-decile portfolios, value-weighted, 1927–2025). Bond returns from Damodaran Online (10-year T-bond and Baa corporate bond total returns, 1928–2025). T-bill returns from the Fama-French risk-free rate series.
The important message is the pattern, not the exact numbers to the decimal. Higher-risk asset classes have delivered higher average returns over long horizons.
This is worth pausing on. The risk-return tradeoff is not just an interesting empirical fact — it's the logical foundation for how markets price risk. If stocks didn't offer higher expected returns than bonds, why would anyone bear the additional volatility? If small stocks didn't offer higher returns than large stocks, why would anyone accept the wilder ride? Investors demand compensation for bearing risk, and the historical data confirms that they've received it — on average, over long periods.
Note the emphasis on "on average" and "over long periods." In any given year, stocks can (and do) underperform bonds or even lose money. The higher average return is compensation for the possibility of bad outcomes, not a guarantee of good ones. We'll formalize this idea when we study portfolio theory and the CAPM in Weeks 8 and 9.
Historical U.S. asset class returns. Higher risk (standard deviation) has been rewarded with higher average returns. This positive relationship is the empirical foundation for the risk-return tradeoff — investors demand compensation for bearing volatility.
The difference between the average return on stocks and the average return on a risk-free investment (typically T-bills) is called the equity risk premium (ERP). It measures the extra return that investors have historically earned — and that they presumably require — for bearing the risk of holding stocks instead of risk-free assets.
The equity risk premium matters enormously for practice. When we compute the cost of equity capital in Weeks 9 and 10, we'll need an estimate of the ERP as an input to the Capital Asset Pricing Model (CAPM). The historical average is a starting point, though there's active debate among practitioners and academics about whether the future ERP will be as large as the historical one.
Why might investors demand this premium? Two reasons. First, stocks are risky — in the short run, your portfolio might lose 30% or more of its value. That potential for loss is psychologically and financially painful, so investors need to be compensated for bearing it. Second, stocks are claims on real economic activity, which is inherently uncertain. Recessions, competitive disruptions, and management failures all create risk that bondholders largely avoid because they have a contractual claim on cash flows. Stockholders bear this residual risk and are compensated through higher expected returns.
Standard deviation is our primary tool for comparing the riskiness of different investments. Two investments with similar average returns but different standard deviations present a clear choice — and two investments with similar risk but different returns present an equally clear one.
This example illustrates a key principle: average returns alone don't tell you enough. Two investments can have the same average return while offering very different experiences. Fund F is a smooth ride. Fund G is a roller coaster. A risk-averse investor — which describes most people — would choose Fund F without hesitation. To accept Fund G's volatility, you'd need it to offer a meaningfully higher average return as compensation.
This is exactly the logic behind the risk-return tradeoff we saw in the historical data. Investors don't just chase returns — they evaluate returns relative to the risk taken. We'll make this idea precise in Week 8 when we study portfolios and diversification, and in Week 9 when we introduce the CAPM.
| Formula | Expression |
|---|---|
| Net Present Value | NPV = −C₀ + Σ [Ct ÷ (1 + r)t] |
| IRR | 0 = −C₀ + Σ [Ct ÷ (1 + IRR)t] |
| Arithmetic Average Return | Avg = (R₁ + R₂ + ⋯ + RT) ÷ T |
| Geometric Average Return | Geo = [(1+R₁)(1+R₂)⋯(1+RT)]1/T − 1 |
| Sample Variance | σ² = Σ(Rt − Avg)² ÷ (T − 1) |
| Standard Deviation | σ = √σ² |
| Equity Risk Premium | ERP = Avg. Stock Return − Avg. Risk-Free Return |
Red Acre Industries must choose between two mutually exclusive projects. Project P requires a $80,000 investment and generates cash flows of $50,000 in year 1, $40,000 in year 2, and $20,000 in year 3. Project Q requires a $200,000 investment and generates cash flows of $90,000 per year for three years. The required return is 10%. Compute the NPV and IRR of each project. Which should Red Acre choose, and why?
A stock had the following annual returns over five years: 15%, −10%, 25%, 5%, and −3%. Compute both the arithmetic and geometric average return. If you invested $5,000 at the start of this period, how much would you have at the end?
A bond fund reported the following annual returns: 6%, −2%, 14%, 8%, and −4%. Compute the arithmetic average return, the sample variance, and the standard deviation.
Rank the following asset classes from lowest to highest expected return, based on the historical evidence: large-company stocks, long-term government bonds, small-company stocks, U.S. Treasury bills, long-term corporate bonds. Then explain in 2–3 sentences why this ordering makes sense from an economic perspective.
Suppose the average annual return on large-company stocks has been 11.5%, the average on long-term corporate bonds has been 5.8%, and the average on T-bills has been 3.2%. Compute the equity risk premium relative to T-bills and relative to corporate bonds. Which comparison is more standard, and why?
Green Acre Industries is evaluating a project with the following cash flows: an initial investment of $50,000, a cash inflow of $115,000 in year 1, and a required environmental cleanup cost of $66,000 in year 2. (a) How many IRRs does this project have? Find them. (b) Compute the NPV at a 15% required return. (c) Explain why the IRR rule is unreliable for this project.
A portfolio generated the following annual returns over six years: 20%, −12%, 8%, 15%, −5%, and 18%. Compute: (a) the arithmetic average return, (b) the geometric average return, (c) the sample variance, and (d) the standard deviation. If you invested $10,000 at the start, what would you have after six years?
Explain in your own words why the geometric average return is always less than or equal to the arithmetic average return (they're equal only when returns are identical every year). Then explain when you would use each average and why. Give a concrete example of a situation where using the wrong average would lead to a misleading conclusion.