CFA Level 1 · Module 02 Quantitative Methods · Chapter 5
A single year’s return tells you almost nothing about an investment. What matters is the whole pattern of returns it produces: where they cluster, how widely they scatter, whether the surprises tend to be pleasant or nasty, and whether they move in step with other assets. This reading is the toolkit for compressing that pattern into a handful of numbers that a decision can actually be built on.
These are the statistics that the rest of the curriculum assumes you can produce without thinking. Portfolio construction leans on variance, covariance, and correlation. Risk management leans on the shape of the tails. Performance evaluation leans on measures that put return and risk in the same sentence. None of the arithmetic is hard, but there is a lot of it, and the exam tests whether you can compute each measure cleanly and, just as importantly, say what it means for an investment.
Throughout, keep one distinction in mind. When the numbers you hold are the entire population, you divide by the count. When they are a sample drawn from a larger population, which is almost always the case with returns, you divide by the count minus one. Getting that denominator right is the difference between two answers the exam will happily offer you side by side.
Picture the annual returns of a fund plotted on a line. They form a cloud: some bunched near a typical value, some stretched out to the extremes. To describe that cloud you need three kinds of number. A measure of central tendency says where the cloud is centered. A measure of dispersion says how spread out it is. And measures of shape, skewness and kurtosis, say whether it is lopsided and whether its tails are unusually heavy. Add covariance and correlation to describe how one cloud relates to another, and you can characterize almost any set of returns.
Everything in this reading uses one small, invented return series so the arithmetic stays transparent. Suppose the Aegis Fund posted annual returns of 10%, 4%, 8%, 10%, and negative 2% over five years. Five clean numbers are enough to demonstrate every measure, and because they are round, you can follow each calculation in your head and check the result.
Three measures locate the center of a distribution, and they do not always agree. The mean is the arithmetic average, the sum divided by the count. The median is the middle value once the numbers are sorted, splitting the data into two equal halves. The mode is the value that occurs most often. The mean uses every observation and is the workhorse, but it is pulled toward extreme values, so when a distribution is lopsided the median often describes the typical experience better.
where x̄ is the arithmetic mean, x1 through xn are the observations, and n is the number of observations.
Setup. For the Aegis Fund returns of 10%, 4%, 8%, 10%, and negative 2%, find the mean, the median, and the mode, and read what they say together.
Answer: mean 6%, median 8%, mode 10%. The mean sits below the median, a first hint that the distribution leans to the left, dragged down by the negative 2% year. When the three measures diverge like this, the gap itself is information about the shape.
One extension of the mean is worth knowing, because it reappears in portfolio work. A weighted mean lets some observations count for more than others, multiplying each value by a weight and summing. A portfolio’s expected return is exactly this: the return of each holding weighted by the fraction of money invested in it. The simple mean is just the special case where every observation carries the same weight. Whenever the pieces of a total are not equal in size, the weighted mean, not the simple average, is the correct summary.
Beyond the center, you often want to describe a particular position in the data. Quantiles do this by cutting a sorted distribution into equal-frequency slices. Quartiles cut it into four parts, so the first quartile marks the value below which a quarter of the observations fall, and the third quartile marks the three-quarter point. Percentiles cut it into a hundred parts, so the tenth percentile is the value below which a tenth of the data lies. The distance between the first and third quartiles, the interquartile range, is a compact measure of spread that ignores the extremes.
Quantiles matter in practice because risk is often stated as a position rather than an average. A statement that a portfolio should not lose more than a certain amount in the worst 5% of years is a statement about the fifth percentile of its return distribution. Learning to read a distribution by its quantiles, not only by its mean, is what lets you talk about typical outcomes and bad outcomes separately.
A concrete reading makes quantiles less abstract. Suppose a fund’s monthly returns are sorted and the fifth percentile comes out at negative 7%. That single number says that in the worst one month out of twenty, the fund lost at least 7%, which is a far more useful sentence for a nervous client than the average monthly return. This is the idea behind value at risk, a headline risk measure built directly on a low percentile of the return distribution. Quantiles turn a vague sense of “bad months happen” into a specific, testable statement about how bad and how often.
Dispersion measures how far returns scatter from the center, and it is the statistical face of risk. The crudest measure is the range, the largest value minus the smallest, which uses only two observations and is easily distorted by a single outlier. The mean absolute deviation averages the absolute distances of every observation from the mean, using all the data but with awkward mathematical properties. The measures that matter most are the variance, the average of the squared deviations from the mean, and its square root the standard deviation, which brings the number back into the original units of return.
Squaring the deviations does two things: it removes the signs so positive and negative departures do not cancel, and it penalizes large deviations more than small ones, which matches the intuition that a huge miss is worse than several small ones. The standard deviation is the single most used measure of risk in all of finance, so its calculation must be automatic.
where s2 is the sample variance, s is the sample standard deviation, xi are the observations, x̄ is the sample mean, and n is the number of observations. For a full population, divide by n rather than n − 1.
Setup. Find the sample variance and sample standard deviation of the Aegis returns 10%, 4%, 8%, 10%, and negative 2%, whose mean is 6%.
Answer: sample variance 26 (in percent squared), sample standard deviation 5.10%. So Aegis returned about 6% on average, give or take roughly 5 percentage points in a typical year. Had these five numbers been the entire population, dividing by 5 would give a variance of 20.8 and a standard deviation of 4.56%.
| Return | Deviation from mean | Squared deviation |
|---|---|---|
| 10% | 4 | 16 |
| 4% | −2 | 4 |
| 8% | 2 | 4 |
| 10% | 4 | 16 |
| −2% | −8 | 64 |
| Sum | 0 | 104 |
Dividing by n when the data is a sample. Return histories are samples of all the returns an asset could produce, so the sample variance divides the sum of squared deviations by n − 1, not n. Using n understates the spread. The exam frequently offers both answers, and the smaller one is the trap.
A standard deviation only means something once you can picture it. If returns were normally distributed, roughly two-thirds of yearly outcomes would land within one standard deviation of the mean, and about ninety-five percent within two. For Aegis, that rough rule places most years between about 1% and 11%, with unusual years reaching from about negative 4% to 16%. This is why the standard deviation is quoted alongside the mean so often: the pair together sketches the whole range of plausible outcomes, where the mean alone sketches none of it. The caution, developed later, is that real returns are not perfectly normal, so these bands are a guide, not a guarantee.
A mean without a dispersion measure is half a description. Two funds can share an average return and offer completely different experiences, one steady and one wild. Always read the standard deviation next to the mean, because risk lives in the spread, not in the center.
Standard deviation treats an upside surprise and a downside surprise as equally undesirable, which is odd, because investors do not lose sleep over unexpectedly large gains. Semi-deviation, also called downside deviation, fixes this by measuring dispersion using only the observations that fall below a chosen target, usually the mean. It answers a more honest question: when returns disappoint, how badly do they disappoint?
where the sum runs only over observations below the target, and n is the total number of observations. The target is often the mean, but it can be any threshold such as zero or a minimum acceptable return.
Setup. Using the Aegis returns and a target equal to the mean of 6%, find the semi-deviation.
Answer: a semi-deviation of 4.12%, below the full standard deviation of 5.10%. The difference is the part of total variability that comes from upside moves, which downside risk measures deliberately exclude. For an investor who only fears losses, semi-deviation is the more relevant risk number.
A standard deviation of 5% means one thing for an asset that averages 6% and something very different for one that averages 30%. To compare risk fairly across investments of different sizes, scale the risk by the return. The coefficient of variation is the standard deviation divided by the mean, giving risk per unit of return. A lower coefficient means less risk taken for each unit of average return, which is generally better.
where CV is the coefficient of variation, s is the standard deviation, and x̄ is the mean. It is a pure number with no units, which is exactly what makes it comparable across assets.
Setup. The Aegis Fund averages 6% with a standard deviation of 5.10%. The Zephyr Fund averages 10% with a standard deviation of 12%. Which took on less risk per unit of return?
Answer: Aegis at 0.85 is more efficient than Zephyr at 1.20, even though Zephyr earned the higher raw return. The coefficient of variation is how you compare a steady, modest earner against a higher, more volatile one on the same footing.
| Fund | Mean return | Standard deviation | Coefficient of variation |
|---|---|---|---|
| Aegis | 6% | 5.10% | 0.85 |
| Zephyr | 10% | 12% | 1.20 |
One caution keeps the coefficient of variation honest. It only behaves sensibly when the mean return is clearly positive. If the mean is near zero the ratio explodes, and if the mean is negative the coefficient turns negative and loses its meaning, since you cannot sensibly speak of risk per unit of a return that is itself a loss. In those cases fall back on the standard deviation directly, or on a downside measure, rather than forcing a ratio that the numbers cannot support.
Skewness measures whether a distribution is symmetric or lopsided. A symmetric distribution has zero skewness, with the mean, median, and mode all together. Positive skewness means a long tail to the right: most outcomes are modest, but there is a chance of a large gain, and the rare big number pulls the mean above the median. Negative skewness means a long tail to the left: most outcomes are fine, but there is a chance of a severe loss, and that pulls the mean below the median.
The direction matters for investors because the two are not equally comfortable. Negative skewness is the dangerous kind, since it hides the risk of an occasional sharp loss behind a run of ordinary returns. Many financial assets, and especially strategies that sell insurance or take on credit risk, exhibit negative skewness: they earn a little most of the time and lose a lot rarely. Recall from the worked example that the Aegis mean of 6% sat below its median of 8%, the signature of a mild left skew produced by that single negative year.
Skew is not random; it usually traces to the shape of an investment’s payoffs. Strategies that collect a small, steady premium in exchange for absorbing a rare large loss, such as selling insurance, writing options, or lending to weaker borrowers, build negative skew into their returns by design. Strategies with capped losses and open-ended upside, such as buying options or holding early-stage ventures, tend toward positive skew. Recognizing where a return series sits on this spectrum tells you what kind of surprise to expect before it arrives, which is far more useful than discovering the skew only after a bad year has already happened.
Compare the mean and the median to read skew at a glance. Mean above median points to positive skew and a right tail; mean below median points to negative skew and a left tail. Standard deviation alone cannot see this, because it treats both tails the same, which is why shape measures are needed alongside it.
Kurtosis measures how heavy a distribution’s tails are relative to a normal distribution, which is the benchmark with a kurtosis of 3. Because the normal is the reference, results are usually quoted as excess kurtosis, kurtosis minus 3, so that a normal distribution has an excess kurtosis of zero. Positive excess kurtosis, called leptokurtic, means fatter tails and a sharper peak: extreme outcomes, both good and bad, happen more often than a normal distribution would predict. Negative excess kurtosis, called platykurtic, means thinner tails and fewer extremes.
This is not an academic nicety. Real asset returns are famously leptokurtic, so market crashes and spikes occur far more often than a tidy normal model suggests. An investor who assumes normal returns will systematically underestimate the chance of an extreme move, which is one reason risk models that ignore fat tails fail precisely when they are needed most. Skewness tells you which tail to worry about; kurtosis tells you how fat that tail is.
| Measure | Value | What it means |
|---|---|---|
| Skewness | Positive | Long right tail; mean above median; chance of large gains |
| Skewness | Negative | Long left tail; mean below median; chance of large losses |
| Excess kurtosis | Positive | Fat tails; extreme outcomes more likely than normal |
| Excess kurtosis | Negative | Thin tails; extreme outcomes less likely than normal |
So far every measure has described one asset. Portfolios need a measure of how two assets move in relation to each other, because that relationship, not the individual risks alone, drives the risk of holding them together. Covariance captures the direction of that co-movement. It multiplies the paired deviations of the two assets from their own means and averages the products. A positive covariance means the two tend to be above or below their means at the same time; a negative covariance means when one is up the other tends to be down.
where xi and yi are paired observations of the two assets, x̄ and ȳ are their means, and n is the number of pairs. As with variance, divide by n for a population.
Setup. Over four periods, Northbank returns 10%, negative 5%, 15%, and 0% (mean 5%), while Corvus returns negative 2%, 8%, 4%, and 10% (mean 5%). Find the covariance of their returns.
Answer: a covariance of about negative 33.33. The negative sign is the useful part: these two assets tend to move in opposite directions, so holding them together should smooth a portfolio. The magnitude alone is hard to interpret, which is exactly the weakness correlation fixes next.
The sign of a covariance is easy to reason about once you look at what the products are doing. When both assets are above their means in the same period, two positive deviations multiply to a positive number; when both are below, two negatives also multiply to a positive. Periods where the assets agree therefore push the covariance up. Periods where one is above its mean while the other is below produce a negative product and push the covariance down. The average of all these products is positive when agreement dominates and negative when disagreement does, which is exactly what “move together” and “move apart” should mean numerically.
Covariance has a serious drawback: its size depends on the units and the volatility of the assets, so a covariance of negative 33 means nothing until you know the scale. Correlation solves this by dividing the covariance by the product of the two standard deviations, which strips out the scale and forces the result into a fixed range from negative 1 to positive 1. A correlation of positive 1 is perfect same-direction movement, negative 1 is perfect opposite movement, and 0 is no linear relationship.
where ρXY is the correlation coefficient, CovXY is the covariance, and sX and sY are the standard deviations of the two assets. The result always lies between negative 1 and positive 1.
Setup. Continue with Northbank and Corvus. Their covariance is negative 33.33. Northbank’s sample standard deviation is 9.13% and Corvus’s is 5.29%. Find the correlation.
Answer: a correlation of about negative 0.69. Now the number speaks for itself: the two assets move against each other most of the time, which is precisely the property that reduces portfolio risk. Correlation, unlike covariance, can be compared directly across any pair of assets.
Correlation has two limits worth remembering. First, it captures only linear relationships, so two assets can be strongly related in a curved way and still show a correlation near zero. Second, correlation is not stable over time. In calm markets assets can look comfortably diversified, only for their correlations to jump toward positive 1 during a crisis, when almost everything falls together and diversification helps least. A single historical correlation is therefore a starting point, not a promise, and prudent risk management assumes correlations can rise at the worst possible moment.
Covariance gives the direction of co-movement; correlation gives its strength on a universal scale. Because correlation is bounded between negative 1 and positive 1, it is the number you compare across asset pairs, and it is the input the portfolio risk formulas actually use.
Skewness and kurtosis are usually estimated together, and read together they give a fuller warning than either alone. A distribution that is both negatively skewed and fat-tailed is the most treacherous of all: it earns steadily, hides its risk in a long left tail, and delivers extreme losses more often than a normal model expects. Many popular strategies have exactly this profile, which is why a track record of calm, positive months is not by itself evidence of low risk. The shape measures exist to catch the danger that the mean and standard deviation, on their own, politely ignore.
Risk is often estimated from short-period data, such as daily or monthly returns, but quoted on an annual basis. Under the common assumption that returns are independent from one period to the next, variance grows in proportion to time, which means standard deviation grows with the square root of time. To annualize a volatility, multiply the period volatility by the square root of the number of periods in a year.
where σperiod is the standard deviation measured over one sub-period, T is the number of such sub-periods in a year (12 for months, about 252 for trading days), and σannual is the annualized standard deviation.
Setup. A strategy has a monthly return standard deviation of 5%. Estimate its annual volatility.
Answer: about 17.32%. Note that volatility does not simply multiply by 12; it multiplies by the square root of 12, because it is variance, not standard deviation, that scales linearly with time. Confusing the two badly overstates annual risk.
The square-root-of-time rule is a favorite. Remember that variance scales with time and standard deviation with the square root of time, and that the rule assumes returns are independent across periods. If a question gives a daily or monthly volatility and asks for annual, reach for the square root, not a plain multiplication.
Watch the population-versus-sample denominator on every dispersion, covariance, and correlation question. Return data is treated as a sample, so divide by n − 1. The wrong denominator produces a plausible but incorrect answer that is deliberately placed among the choices.
These measures are not an end in themselves; they are the raw material for everything that follows in the module. The mean and standard deviation feed directly into the portfolio return and risk formulas of the next reading, covariance and correlation decide how much diversification a pair of assets provides, and the shape measures warn where a normal-distribution model will mislead. Master the mechanics here, because the later readings assume you can produce any of these numbers on demand and move straight to using them.
A return series is 6%, 2%, 6%, 10%. What are its mean, median, and mode?
Mean: (6 + 2 + 6 + 10) ÷ 4 = 24 ÷ 4 = 6%. Median: sorted 2, 6, 6, 10, the middle two average to (6 + 6) ÷ 2 = 6%. Mode: 6% appears twice, so the mode is 6%. Here all three coincide, suggesting a roughly symmetric set.
An asset averages 8% with a standard deviation of 20%. What is its coefficient of variation, and what does it say next to a bond averaging 4% with a standard deviation of 6%?
Asset: 20 ÷ 8 = 2.5. Bond: 6 ÷ 4 = 1.5. The bond has the lower coefficient of variation, so it delivers less risk per unit of average return, even though the asset has the higher raw return.
The mean return of a fund is 5% and its median is 9%. Is the distribution positively or negatively skewed?
Negatively skewed. The mean sits below the median, which happens when a long left tail of occasional large losses drags the average down. This is the riskier kind of skew for an investor.
Daily return volatility is 1%. Roughly what is the annualized volatility, assuming 252 trading days?
σannual = 1% × √252 = 1% × 15.87 = about 15.9%. Again, the square root of time, not a straight multiplication.
When the distribution is skewed or has outliers. The mean is pulled toward extreme values, so in a lopsided distribution the median often describes the typical outcome more faithfully. Comparing the two is itself a quick test of skew.
Because a sample tends to understate the true spread of the population it comes from, and dividing by n minus 1 rather than n corrects that bias. Return histories are samples, so this smaller denominator is almost always the right one.
Standard deviation counts upside and downside variability equally, but investors only fear the downside. Semi-deviation isolates variability below a target, so it measures the kind of risk people actually care about, at the cost of using only part of the data.
It puts risk in proportion to return, so assets of very different return levels can be compared on the same footing. A 10% standard deviation is small for a high-returning asset and large for a low-returning one, and the coefficient of variation captures that.
Negative skew. It hides the risk of an occasional severe loss behind a run of normal returns, so an investor can be lulled by steady performance right up until a large drawdown. Positive skew, with its chance of a big gain, is generally the more welcome shape.
Because real returns produce extreme moves, crashes and spikes, far more often than a normal distribution predicts. A model that ignores this positive excess kurtosis will understate the odds of a large loss, and it will fail in exactly the rare conditions where accuracy matters most.
They describe the same co-movement, but covariance keeps the scale of the assets while correlation removes it. Covariance tells you the direction; correlation tells you the strength on a fixed range from negative 1 to positive 1, which is why correlation is comparable across pairs and covariance is not.
Through calculation with small data sets: means and medians, sample variance and standard deviation, the coefficient of variation, covariance and correlation, and annualizing volatility with the square root of time. Interpretation questions on skewness and kurtosis appear alongside, so know both the numbers and their meaning.
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