Risk management runs on three measured quantities: how much an asset earns, how much it swings, and how its swings line up with others. This chapter defines each precisely, and, just as importantly, shows where the convenient assumptions break. Returns can be measured two ways that disagree for large moves. Volatility can be looked up from the past or read from option prices about the future. And the tidy world of the normal distribution, symmetric, thin-tailed, and fully described by a mean and a variance, is not the world financial returns actually live in.
The through-line is a warning about normality. Real returns are skewed and fat-tailed, so the first two moments miss the tail risk that matters most, and ordinary correlation captures only the straight-line part of how assets move together, missing the joint crashes that define a crisis. The chapter builds the tools that see past those limits: the Jarque-Bera test and the power law for non-normal tails, and rank-based correlation measures for nonlinear dependence.
Key Takeaways
- Simple returns add across assets (portfolios); log returns add across time. They agree for small moves and diverge for large ones, with the log return always smaller.
- Volatility is the standard deviation of returns and scales with the square root of time; implied volatility is backed out of option prices and is forward-looking.
- The normal is fixed by its mean and variance, but returns are skewed and fat-tailed, so two moments understate tail risk.
- The Jarque-Bera test uses sample skewness and kurtosis to test normality; a large statistic (above 5.99 at 5 percent) rejects it.
- Power-law tails decay slowly as x to the negative tail index; a smaller index means heavier tails and more frequent extremes.
- Correlation measures only linear dependence; rank correlation and Kendall’s tau capture monotonic, nonlinear dependence and resist outliers.
Simple and Log Returns
There are two standard ways to measure a return, and knowing when to use each is a genuine exam skill. The simple return is the price change over the starting price. The log return, also called the continuously compounded return, is the difference in the natural logarithms of the two prices, equivalently the log of one plus the simple return.
where Rt is the simple return (uppercase) and rt the log return (lowercase). Convert between them with 1 + R = exp(r); the log return is always the smaller of the two.
The two additivity properties are the reason both survive. Log returns add across time: a multi-period log return is the sum of the single-period log returns, which makes them convenient for volatility scaling and multi-day calculations. Simple returns add across assets: a portfolio’s simple return is the weighted average of its holdings’ simple returns, which makes them the right choice for portfolio math. They agree closely for small moves but diverge for large ones, and the log return can fall below minus 100 percent while the simple return cannot, since a price can lose at most all of its value.
A stock rises from 100 to 110. Compute the simple and log returns, and confirm the conversion.
Step 1. Simple return: the change over the start.
Step 2. Log return: the log of one plus the simple return.
Answer: a simple return of 10% and a log return of 9.53%. The log return is smaller, as always. For a move this size the gap is modest, but it widens sharply for large moves: a 50% simple gain is only a 40.5% log return, and a 50% simple loss is a 69.3% log loss.
A stock goes 100 → 90 → 99 over two periods. Show that log returns add across time while simple returns multiply.
Step 1. Simple returns compound (multiply the growth factors).
Step 2. Log returns add.
Answer: both give about a 1% total loss, but by different arithmetic. The simple returns had to be multiplied as growth factors; the log returns simply added. That additivity is why log returns are the standard choice whenever returns are aggregated over time, such as in volatility and value-at-risk calculations.
Volatility, Variance Rate, and Implied Volatility
Three related measures describe how much a return swings. Volatility is the standard deviation of returns, in the same units as the returns. The variance rate is simply the variance, the square of the volatility. Both are usually estimated from a history of returns, so they are backward-looking. The third, implied volatility, is different in kind: it is the volatility value that, plugged into an option pricing model such as Black-Scholes-Merton or the VIX methodology, reproduces the option’s market price. Because option prices reflect what the market expects, implied volatility is forward-looking.
Volatility scales with time in a specific way, inherited from the iid results of earlier chapters. The mean and the variance both scale linearly with the holding period, so the variance rate scales with time; but volatility, being the square root of the variance, scales with the square root of time. This is the rule used to annualize a volatility measured over a shorter interval.
where 252 is the approximate number of trading days in a year; monthly data would use a factor of the square root of 12. The variance rate, being a variance, scales by the full number of periods rather than its square root.
A stock’s daily return has a standard deviation of 1.2%. What is its annualized volatility, assuming 252 trading days?
Step 1. Scale the daily volatility by the square root of the number of days.
Answer: an annualized volatility of about 19%. Risk scales with the square root of the horizon, not the horizon itself, so the annual figure is far below 252 times the daily one. The variance rate, by contrast, would scale by the full 252, since variance adds across independent days.
The distinction between backward- and forward-looking is not academic. Implied volatility tends to sit above the volatility that is later realized, a gap known as the volatility risk premium: option buyers pay up for protection against uncertainty, which bids implied volatility above the market’s true expectation. Implied volatility also jumps ahead of realized volatility around known events, an earnings release, a central-bank decision, because the market prices the coming turbulence before it arrives. Watching implied against realized volatility is therefore a standard read on how much fear is embedded in option prices.
| Measure | What it is | Direction | Scaling |
|---|---|---|---|
| Volatility | Standard deviation of returns | Backward-looking (historical) | √time |
| Variance rate | Variance of returns (σ²) | Backward-looking (historical) | Linear in time |
| Implied volatility | Volatility backed out of option prices | Forward-looking (expected) | Already annualized |
Why Two Moments Are Not Enough
A normal distribution is fully described by just two numbers, its mean and its variance, because its skewness is exactly zero and its kurtosis is exactly three. That is the source of its convenience, and the source of its danger. Financial returns are not normal: they are typically skewed (often negatively, with large drops more common than equally large jumps) and fat-tailed (with kurtosis well above three, so extreme moves of either sign happen more often than a normal allows). Much of this comes from time-varying volatility, which mixes calm and turbulent periods into a heavy-tailed whole.
The consequence for risk is direct. A model that captures only the first two moments treats the tails as thin, and so it understates the probability of large losses, precisely the events risk management exists to guard against. The third and fourth moments, skewness and kurtosis, carry the information about asymmetry and tail heaviness that the mean and variance discard, which is why they must be measured, not assumed away.
The Jarque-Bera Test
The Jarque-Bera test makes the non-normality diagnosis formal. It tests whether the sample skewness and kurtosis are jointly consistent with a normal distribution. The null hypothesis is that skewness equals 0 and kurtosis equals 3; the test statistic combines the squared sample skewness and the squared excess kurtosis, scaled by the sample size, and follows a chi-squared distribution with two degrees of freedom.
where S is the sample skewness, κ (kappa) the sample kurtosis, and T the sample size. The statistic is small when the data look normal (skewness near 0, kurtosis near 3) and large when they do not. The chi-squared critical values are about 5.99 at 5 percent and 9.21 at 1 percent.
A sample of 100 daily returns has a skewness of −0.5 and a kurtosis of 5. Compute the Jarque-Bera statistic and test normality at 5%.
Step 1. Plug into the formula with S = −0.5, κ = 5, T = 100.
Step 2. Compare with the chi-squared critical value.
Answer: a Jarque-Bera statistic of about 20.6, far above the 5% critical value of 5.99 (and even the 1% value of 9.21), so normality is firmly rejected. The excess kurtosis contributes most of the statistic here, which is typical: fat tails, more than skewness, are what usually sink the normality assumption for daily returns.
Power Laws and the Tail Index
Rejecting normality raises the question of what the tails actually look like, and the most useful answer is a power law. A power law describes how fast the probability of an extreme outcome shrinks as the outcome grows: the chance of exceeding a large value is proportional to that value raised to a negative power, the tail index.
where α (alpha) is the tail index and k a constant. A smaller tail index means the probability falls off more slowly, so the tail is heavier and extreme events are more likely; a larger index means a thinner tail.
The power law matters because it decays far more slowly than the normal’s tail. A normal distribution’s tail probability drops off at a rate that makes truly large moves essentially impossible; a power-law tail keeps assigning meaningful probability to extreme moves, which is what real markets deliver. The Student’s t distribution is the familiar example of a power-law-tailed distribution, with its degrees-of-freedom parameter controlling the tail index: fewer degrees of freedom, a smaller index, and heavier tails. Estimating the tail index is a direct way to quantify how fat an asset’s tails really are.
Correlation Versus Dependence
The last theme is how assets move together, which drives diversification and portfolio tail risk. The starting tool is covariance, the average co-movement of two variables around their means, and its scale-free cousin correlation, the covariance divided by the two standard deviations, which lands between minus one and plus one. But correlation, the ordinary Pearson correlation, has a crucial limitation: it measures only linear dependence.
That limitation is the heart of the section. Independence is the strong condition that the joint distribution equals the product of the marginals, so no relationship of any kind exists. Zero correlation is much weaker: it rules out a straight-line relationship but not a curved one. Two variables can be strongly dependent, even deterministically linked, and still have zero correlation. Financial assets routinely have nonlinear dependence, most importantly in their tails, where markets crash together even when their day-to-day correlation looks moderate. Linear correlation simply cannot see this.
Reading a low correlation as “these assets are basically independent, so the portfolio is safe.” Correlation only measures the linear part of the relationship. Two assets can have a modest everyday correlation yet plunge together in a crisis, a form of tail dependence that leaves ordinary correlation unchanged but destroys diversification exactly when it is needed. Low linear correlation is not a guarantee of independence, least of all in the tails.
Correlation in a One-Factor Model
A powerful way to structure the correlations across many assets is a one-factor model, in which each asset’s return is driven by a single common factor plus its own idiosyncratic noise. The strength of an asset’s link to the factor is its factor loading. The elegant result is that the correlation between any two assets is simply the product of their two factor loadings.
where γi and γj are the two assets’ loadings on the common factor, each between minus one and plus one. Every pairwise correlation in the portfolio is generated by exposure to that single shared source of risk.
The practical value is enormous. Instead of estimating a full correlation matrix, with a separate number for every pair of assets, a one-factor model summarizes the whole thing with a single loading per asset. It also guarantees the resulting correlation matrix is internally consistent, or positive definite, a technical requirement that any real correlation matrix must satisfy and that a carelessly assembled matrix can violate. This factor structure is the correlation counterpart of the CAPM, where the market is the common factor.
In a one-factor model, stock A has a factor loading of 0.8 and stock B a loading of 0.6. What is the correlation between them?
Step 1. Multiply the two loadings.
Answer: a correlation of 0.48. Both stocks are positively linked to the common factor, so they are positively correlated with each other, but only moderately, because neither loads on the factor perfectly. If a third stock loaded 0.9, its correlation with A would be 0.8 × 0.9 = 0.72, all pairwise correlations flowing from the single set of loadings.
Rank Correlation and Kendall’s Tau
Because linear correlation misses nonlinear dependence, two rank-based measures are used alongside it. Spearman’s rank correlation is just the ordinary correlation computed on the ranks of the data instead of the values. Kendall’s tau measures dependence a different way, through the balance of concordant pairs (both variables move the same way) against discordant pairs (they move oppositely). Both lie between minus one and plus one, both are zero under independence, and both share two advantages over Pearson correlation.
| Measure | Captures | Robust to outliers? | Invariant to monotonic transforms? |
|---|---|---|---|
| Pearson (linear) | Linear dependence | No | No (only linear rescaling) |
| Spearman (rank) | Monotonic dependence | Yes | Yes |
| Kendall’s τ | Concordance of pairs | Yes | Yes |
The two advantages are worth stating plainly. First, the rank measures are robust to outliers, because they use only the ordering of the data, not the raw values, so a single extreme observation cannot dominate them the way it can dominate a Pearson correlation. Second, they are invariant to any monotonic transformation: taking logs, or any other order-preserving change, leaves them unchanged, whereas Pearson correlation is invariant only to linear rescaling. For jointly normal variables all three measures roughly agree, so a large gap between the Pearson correlation and the rank measures is itself a useful signal: it flags nonlinear dependence that the linear correlation is missing.
The three correlation measures are a diagnostic set, not competitors. When they agree, the dependence is essentially linear and Pearson correlation is a fine summary. When the rank measures are much larger than the Pearson correlation, the relationship is monotonic but curved, or distorted by outliers, and the linear number is understating the true association. Reporting all three, and watching the gaps, is how a careful analyst detects the nonlinear dependence that matters most in the tails.
A stock falls 50% in one day. What is its simple return and its log return, and why is the difference so large?
Show answer
The simple return is −50%. The log return is ln(1 − 0.50) = ln(0.5) ≈ −69.3%. The gap is large because the two measures agree only for small moves and diverge sharply for big ones. The log return can fall below minus 100%, while the simple return is floored at minus 100% (a total loss), which is why the log return of a large drop is so much more negative.
A return series of 200 observations has a skewness of 0 but a kurtosis of 6. Compute the Jarque-Bera statistic and state the conclusion at 1%.
Show answer
With S = 0 and κ = 6, only the kurtosis term contributes: JB = (200 − 1)(0 + (6 − 3)²/24) = 199 × (9/24) = 199 × 0.375 ≈ 74.6. This is far above the 1% critical value of 9.21, so normality is decisively rejected. Even with zero skewness, the excess kurtosis of 3 (a kurtosis of 6 versus the normal’s 3) is enough to reject normality on its own, which is the usual story for financial returns.
Two variables have a Pearson correlation near zero but a Spearman rank correlation of 0.85. What does this tell you?
Show answer
That the two variables have a strong monotonic but nonlinear relationship. The high rank correlation shows they move together consistently in the same direction, so one reliably rises as the other rises, but the near-zero Pearson correlation shows the relationship is not a straight line. Relying on the Pearson number alone would badly understate the dependence. The gap between the two measures is the tell-tale sign of nonlinearity.
In a one-factor model, three stocks have factor loadings of 0.9, 0.5, and 0.2. Which pair is most correlated, and what is that correlation?
Show answer
The most correlated pair is the two stocks with the highest loadings, 0.9 and 0.5, since each pairwise correlation is the product of the loadings. Their correlation is 0.9 × 0.5 = 0.45. The pair 0.9 and 0.2 gives 0.18, and 0.5 and 0.2 gives 0.10, both lower. Higher loadings on the common factor mean stronger co-movement, so the two stocks most exposed to the factor move together the most.
Chapter Summary
- Simple returns (price change over start) add across assets; log returns (difference of log prices) add across time. They agree for small moves; the log return is always smaller, and 1 + R = exp(r) converts between them.
- Volatility is the standard deviation of returns and scales with the square root of time; the variance rate is its square and scales linearly; implied volatility is backed out of option prices and is forward-looking.
- The normal is fixed by two moments, but returns are skewed and fat-tailed (kurtosis above 3), so two moments understate tail risk.
- The Jarque-Bera test, (T − 1)(S²/6 + (κ − 3)²/24), is chi-squared with 2 degrees of freedom; above 5.99 (5%) it rejects normality.
- Power-law tails decay as x to the negative tail index; a smaller index means heavier tails. The Student’s t has power-law tails.
- Correlation measures only linear dependence; zero correlation does not imply independence, and nonlinear tail dependence can hide behind a modest correlation.
- In a one-factor model, the correlation between two assets is the product of their factor loadings, which summarizes the whole matrix and keeps it positive definite.
- Spearman’s rank correlation and Kendall’s tau capture monotonic dependence, resist outliers, and are invariant to monotonic transforms; a gap from the Pearson correlation flags nonlinearity.
Frequently Asked Questions
What is the difference between a simple return and a log return?
A simple return is the price change divided by the starting price, while a log, or continuously compounded, return is the difference in the natural logs of the two prices. They are close for small moves but diverge for large ones, and the log return is always smaller than the simple return. Their key advantage differs: log returns add up across time, so a multi-period log return is the sum of the single-period log returns, whereas simple returns add up across assets, so a portfolio’s simple return is the weighted average of its holdings’ simple returns. Convert between them with one plus the simple return equals the exponential of the log return.
What is the difference between volatility, the variance rate, and implied volatility?
Volatility is the standard deviation of returns, in the same units as the returns themselves. The variance rate is simply the variance, the square of the volatility. Both are usually estimated from historical returns and are therefore backward-looking. Implied volatility is different: it is the volatility backed out of option prices using a pricing model such as Black-Scholes-Merton or the VIX, so it reflects the market’s forward-looking expectation of future volatility rather than a past average. Volatility scales with the square root of time, while the variance rate scales linearly with time.
Why are the mean and variance insufficient to describe financial returns?
A normal distribution is completely described by its mean and variance, because its skewness is zero and its kurtosis is exactly three. Financial returns, however, are typically skewed and fat-tailed, with kurtosis well above three, so their third and fourth moments carry real information that the first two moments miss. A model that captures only the mean and variance will therefore understate the probability of large losses, because it ignores the heavy tails where the worst outcomes live. This is why risk measurement must look beyond the first two moments.
What is the Jarque-Bera test?
The Jarque-Bera test is a formal test of whether a return series is normally distributed, based on its sample skewness and kurtosis. Its null hypothesis is that the skewness is zero and the kurtosis is three, the values a normal distribution has. The test statistic combines the squared sample skewness and the squared excess kurtosis, scaled by the sample size, and follows a chi-squared distribution with two degrees of freedom. A large statistic, above the chi-squared critical values of about 5.99 at 5 percent or 9.21 at 1 percent, rejects normality, which is the usual outcome for daily financial returns.
What is a power law and a tail index?
A power law describes how the probability of an extreme outcome decays as the outcome grows: the probability that a variable exceeds a large value x is proportional to x raised to the negative tail index. A smaller tail index means the probability falls off more slowly, so the tails are heavier and extreme events are more likely. Power-law tails decay much more slowly than the thin tails of a normal distribution, which is why they describe the frequent large moves seen in financial returns. The Student’s t distribution is a familiar example of a distribution with power-law tails.
Does zero correlation mean two variables are independent?
No. Correlation measures only linear dependence, so two variables can have zero correlation and still be strongly dependent through a nonlinear relationship. Independence is the stronger property: it requires the joint distribution to equal the product of the marginals, so no relationship of any kind exists. Financial assets frequently have nonlinear dependence, such as tails that move together in a crisis even when the ordinary correlation is modest, which is exactly the kind of dependence that linear correlation cannot see.
What is the correlation structure of a one-factor model?
In a one-factor model, each asset’s return is driven by a single common factor plus its own idiosyncratic noise, and the strength of each asset’s link to the factor is its factor loading. The correlation between any two assets is then simply the product of their two factor loadings, so all the pairwise correlations in the portfolio are generated by exposure to that one common source of risk. This structure is convenient because it summarizes an entire correlation matrix with a single loading per asset, and it guarantees the matrix is internally consistent, or positive definite.
What is the difference between Pearson, Spearman, and Kendall correlation?
Pearson correlation is the ordinary linear correlation, which measures the strength of a straight-line relationship and is sensitive to outliers. Spearman’s rank correlation is Pearson correlation applied to the ranks of the data rather than the values, so it captures any monotonic relationship and is robust to outliers. Kendall’s tau measures dependence through the proportion of concordant versus discordant pairs. All three lie between minus one and plus one and equal zero under independence, but the two rank-based measures are invariant to monotonic transformations and less affected by extreme values, which makes a large gap between them and the Pearson correlation a signal of nonlinear dependence.
