Coefficient of Variation and Skewness of Severity

Why Summarize Severity With CV and Skewness?

The mean and standard deviation describe important features of a severity distribution, but they do not tell the whole story. The standard deviation is measured in the same units as the loss, so its size depends on whether losses are measured in dollars or thousands of dollars. The coefficient of variation puts spread on a relative scale. Skewness describes the direction and degree of asymmetry around the mean.

These summaries are useful for comparing hypothetical severity models, but neither replaces the mean, standard deviation, or a fuller description of the distribution. In particular, CV is sensitive to a change of origin, while skewness is not. Both measures require appropriate moments to exist.

Definition: Let $X$ be a nonnegative severity with $0<E[X]<\infty$ and finite, positive variance. Its coefficient of variation is $\operatorname{CV}(X)=\sqrt{\operatorname{Var}(X)}/E[X]$. If, in addition, $E[|X-E[X]|^3]<\infty$, its skewness is $\gamma_X=E[(X-E[X])^3]/(\operatorname{Var}(X))^{3/2}$. Both measures are dimensionless. CV is nonnegative; skewness can be positive, negative, or zero.

Write $m_r=E[X^r]$ for the $r$th raw moment, when it exists. The mean is $m_1$, and the variance is $m_2-m_1^2$. If the third raw moment is finite, the third central moment can be found without separately listing every deviation from the mean:

$$ E[(X-E[X])^3]=m_3-3m_1m_2+2m_1^3. $$

The sign of skewness describes the direction of the distribution's asymmetry, not the direction of a loss. Positive skewness indicates a longer or more influential upper side relative to the mean; negative skewness indicates a longer or more influential lower side. A value of zero means the third central moment is zero, but does not by itself prove that the whole distribution is symmetric.

Calculating the Measures From Moments

The following expansion justifies the raw-moment formula. Let $\mu=E[X]=m_1$. Expanding the cube and taking expectations gives:

\begin{aligned} E[(X-\mu)^3] &=E[X^3-3\mu X^2+3\mu^2X-\mu^3]\\ &=m_3-3\mu m_2+3\mu^2m_1-\mu^3\\ &=m_3-3m_1m_2+2m_1^3, \end{aligned}

because $m_1=\mu$. The coefficient of variation and skewness then follow by dividing by the appropriate powers of the mean and standard deviation. Using the variance $\sigma^2=m_2-m_1^2$, the calculations are:

Check the moment conditions.
For CV, the mean must be finite and positive and the variance must be finite and positive. For skewness, the variance must be finite and positive and the third absolute central moment must be finite.

Find the spread.
Calculate $\sigma^2=m_2-m_1^2$, then $\sigma=\sqrt{\sigma^2}$.

Standardize.
Divide $\sigma$ by the mean for CV. For skewness, divide the third central moment by $\sigma^3$.

Worked Example: CV and Skewness From a Discrete Severity Model

A hypothetical claim severity $X$, measured in thousands of dollars, has probability $1/3$ at each of $0$, $1$, and $3$. Find its mean, standard deviation, CV, and skewness. Give the final numerical summaries to four decimal places. Use raw moments, and check that the required moments are finite.

Method. Calculate the first three raw moments from the probability table, then obtain the variance and third central moment. The distribution has finite support, so all its moments are finite. Its mean and variance will also be positive.

First raw moment: $m_1=(0+1+3)/3=4/3$ thousand dollars.

Second raw moment: $m_2=(0^2+1^2+3^2)/3=10/3$ thousand dollars squared. Therefore, $\operatorname{Var}(X)=10/3-(4/3)^2=10/3-16/9=14/9$, and $\sigma=\sqrt{14}/3\approx1.2472$ thousand dollars.

Coefficient of variation: $\operatorname{CV}(X)=(\sqrt{14}/3)/(4/3)=\sqrt{14}/4\approx0.9354$. The standard deviation is about 93.54% of the mean.

Third raw and central moments: $m_3=(0^3+1^3+3^3)/3=28/3$. Hence \[ E[(X-E[X])^3]=\frac{28}{3}-3\left(\frac{4}{3}\right)\left(\frac{10}{3}\right)+2\left(\frac{4}{3}\right)^3 =\frac{252-360+128}{27}=\frac{20}{27}. \] The third central moment is in thousand-dollars cubed.

Skewness: Since $\sigma^3=(\sqrt{14}/3)^3=14\sqrt{14}/27$, $\gamma_X=(20/27)/(14\sqrt{14}/27)=10/(7\sqrt{14})\approx0.3818$. Thus the distribution has modest positive skewness. Check: the mean, variance, and third central moment were computed from the same three probabilities, and the final measures are dimensionless.

Common mistake avoided: Reporting the standard deviation, approximately $1.2472$, as if it were CV would leave a quantity in thousands of dollars. CV is the ratio of standard deviation to mean, approximately $0.9354$, and has no units.

Interpreting CV and Skewness

CV compares the standard deviation with the mean. For example, a CV of $0.5$ means that the standard deviation is half the mean. This is a relative-spread statement; it does not mean that half of all losses are close to the mean. CV is especially useful for comparing positive severities with different scales, provided the means are positive and the models meet the moment conditions.

Skewness uses the third power of deviations from the mean. Large positive deviations therefore contribute strongly to positive skewness, while large negative deviations contribute strongly to negative skewness. This is why skewness can be sensitive to extreme observations. It is not a direct measure of tail probability, nor does a given skewness value uniquely identify a distribution.

The effect of changing units or shifting all losses can be derived directly. Let $Y=aX+b$, where $a>0$. Then $E[Y]=aE[X]+b$, $\operatorname{SD}(Y)=a\operatorname{SD}(X)$, and the third central moment of $Y$ is $a^3$ times that of $X$. Consequently,

\operatorname{CV}(Y)=\frac{a\sigma_X}{aE[X]+b}, \qquad \gamma_Y=\gamma_X,

but the CV expression is a coefficient of variation only if $Y$ itself satisfies the definition: it must be a nonnegative severity with finite, positive mean and finite, positive variance. For example, $a>0$ and $b\geq0$ preserve nonnegativity when $X$ is nonnegative. A shift $b$ changes the mean without changing the standard deviation, so it generally changes CV. If $b$ makes the mean zero or negative, the ratio is not a severity CV. A positive change of units, such as converting dollars to thousands of dollars, has $b=0$ and leaves CV unchanged.

Skewness is unchanged by adding a constant because deviations from the mean are unchanged. A positive scaling multiplies the third central moment and $\sigma^3$ by the same factor $a^3$, so their ratio is unchanged. For a negative scaling the sign would reverse, but the resulting variable would generally not be a nonnegative severity.

Worked Example: Scaling and Shifting an Exponential Severity

A hypothetical severity $X$ has an exponential distribution with rate $\lambda=1/100$ per dollar, so its mean is $\theta=1/\lambda=\$100$. Define a modified quantity $Y=2X+\$50$. Find the mean, standard deviation, CV, and skewness of $Y$, and explain why the CV formula is valid here. Give final summaries to four decimal places.

Method. Use the exponential raw moments $E[X^r]=r!\theta^r$, then apply the positive-scale and shift relationships. Since $X\geq0$, $Y=2X+50\geq50$; it is a nonnegative severity with positive mean. Its variance is finite and positive, so its CV is defined.

For $X$, $E[X]=100$, $E[X^2]=2(100)^2=20{,}000$, and $E[X^3]=6(100)^3=6{,}000{,}000$. Thus $\operatorname{Var}(X)=20{,}000-100^2=10{,}000$, so $\sigma_X=\$100$. Its third central moment is $6{,}000{,}000-3(100)(20{,}000)+2(100)^3=2{,}000{,}000$, and its skewness is $2{,}000{,}000/100^3=2$.

For $Y=2X+50$, $E[Y]=2(100)+50=\$250$, and $\operatorname{SD}(Y)=2(100)=\$200$. Therefore, $\operatorname{CV}(Y)=200/250=0.8000$. Its third central moment is $2^3(2{,}000{,}000)=16{,}000{,}000$, while its standard deviation cubed is $200^3=8{,}000{,}000$. Thus $\gamma_Y=16{,}000{,}000/8{,}000{,}000=2.0000$.

Check: the shift raises the mean from $\$100$ to $\$250$ but does not change the standard deviation from its scaled value of $\$200$. CV therefore changes from $1.0000$ for $X$ to $0.8000$ for $Y$, while skewness remains $2.0000$. The common mistake is to say that every affine transformation preserves CV; only a pure positive rescaling does so in general.

Worked Example: Same Mean and Spread, Different Skewness

Compare two hypothetical severity models, with all values measured in thousands of dollars. Model A assigns probability $1/2$ to each of $0$ and $4$. Model B assigns probabilities $1/3$, $1/2$, and $1/6$ to $0$, $2$, and $6$, respectively. For each model, calculate the mean, standard deviation, CV, and skewness, rounding the final summaries to four decimal places.

Method. Calculate $m_1,m_2,m_3$ for each model and use the raw-moment formulas. Both distributions have finite support and positive mean and variance, so both summaries are defined.

For Model A, $m_1=(0+4)/2=2$, $m_2=(0+16)/2=8$, and $m_3=(0+64)/2=32$. Its variance is $8-2^2=4$, so $\sigma=2$ and $\operatorname{CV}=2/2=1.0000$. Its third central moment is $32-3(2)(8)+2(2^3)=32-48+16=0$, so its skewness is $0/2^3=0.0000$.

For Model B, $m_1=0(1/3)+2(1/2)+6(1/6)=1+1=2$. Also, $m_2=0+4(1/2)+36(1/6)=2+6=8$, so the variance is $8-2^2=4$, the standard deviation is $2$, and the CV is $1.0000$. The third raw moment is $m_3=0+8(1/2)+216(1/6)=4+36=40$. Its third central moment is $40-3(2)(8)+2(2^3)=40-48+16=8$, giving skewness $8/2^3=1.0000$.

Interpretation and check: both models have the same mean, standard deviation, and CV, but their skewness differs. Model A has zero skewness; Model B has positive skewness. The summaries therefore distinguish aspects of shape that the mean and standard deviation alone do not. The common mistake is to infer that equal means and standard deviations imply equal distribution shapes.

Existence Conditions and Exam Traps

A calculated ratio is meaningful only if its defining quantities exist. The earlier material on when severity moments exist gives ways to assess moment existence for particular models. For this topic, keep the requirements explicit: CV needs a finite positive mean and finite positive variance; skewness needs a finite positive variance and a finite third absolute central moment. For a nonnegative severity, a finite third raw moment is sufficient for the third absolute central moment to be finite.

A particularly important trap occurs when a severity is shifted. Consider $X$ exponential with mean and standard deviation both equal to $1$, and set $Y=X-1$. The algebra gives $E[Y]=0$ and $\operatorname{SD}(Y)=1$. The CV is undefined: its definition requires a positive mean, and $Y$ is not nonnegative. The expression $\sigma_Y/E[Y]$ has a zero denominator.

If instead $Y=X-2$, then $E[Y]=-1$, and the formal ratio is $1/(-1)=-1$. That negative number is not a coefficient of variation. The formula $\operatorname{CV}(aX+b)=a\sigma_X/(aE[X]+b)$ for $a>0$ is usable as a CV only when the transformed variable meets the severity CV conditions. Without those conditions it is merely an algebraic ratio, not the defined summary.

Key takeaway: CV describes standard deviation relative to a positive severity mean; skewness describes the standardized third central moment. Both are dimensionless, but only skewness is invariant to adding a constant. Always check moment existence and the transformed severity's domain before interpreting a result.

Check Your Understanding

Work through these original problems in order. Each answer includes the requested summaries so you can check the calculations.

A hypothetical severity $X$ equals $1$ with probability $1/2$ and $3$ with probability $1/2$, in thousands of dollars. Find its mean, standard deviation, CV, and skewness. (Answer: $E[X]=2$, $\operatorname{SD}(X)=1$, $\operatorname{CV}(X)=0.5000$, $\gamma_X=0.0000$.)
A hypothetical severity is $0$ with probability $0.2$ and $5$ with probability $0.8$, in dollars. Find its mean, variance, CV, and skewness. (Answer: mean $4$, variance $4$, CV $0.5000$, skewness $-1.5000$.)
A nonnegative severity has raw moments $m_1=10$, $m_2=125$, and $m_3=2{,}000$, in consistent units. Find its standard deviation, CV, and skewness. (Answer: standard deviation $5$, CV $0.5000$, skewness $2.0000$.)
A nonnegative severity $X$ has mean $\$80$, standard deviation $\$48$, and skewness $1.2$. For $Y=3X$, find the mean, standard deviation, CV, and skewness. (Answer: mean $\$240$, standard deviation $\$144$, CV $0.6000$, skewness $1.2000$.)
A hypothetical exponential severity $X$ has mean and standard deviation $\$50$. Let $Y=X-\$50$. Is the CV of $Y$ defined as a severity coefficient of variation? What are the mean and standard deviation of $Y$? (Answer: No; $E[Y]=\$0$, $\operatorname{SD}(Y)=\$50$, and CV is undefined because the mean is not positive and $Y$ is not nonnegative.)

What You'll Learn