# An introduction to risk measures

This post is an introduction to measures of risk. Examples of risk measures are considered. The post also focuses on coherent risk measures, which possess several desirable properties for risk measures.

In actuarial applications, an important focus is on developing loss distributions for insurance products. When the loss distribution is a probability based model, we can also use the model to derive a description of risk. Often times, a measure of risk is given by one number that describes the level of exposure to risk. One example of a measure of risk is value-at-risk (VaR), discussed here, which can quantify the likelihood of an adverse outcome. VaR can then be used to determine the amount of capital required to withstand such adverse outcomes. Actuaries and risk managers along with investors, regulators and rating agencies are particularly interested in using VaR and other measures of risk to quantify the ability of an enterprise to withstand various potential adverse events.

Two important examples of measures of risk – value-at-risk and tail-value-at-risk – have been discussed in this previous post.

Examples of Risk Measures

A measure of risk (or a risk measure) is a mapping that maps a given loss distribution to the real numbers. In a general discussion, we use the Greek letter $\rho$ to denote the mapping. For example, if $X$ is the random variable that describes the losses, then $\rho(X)$ is a number that is intended to quantify the risk exposure. We first look at some examples.

Some of the early risk measures in actuarial science were based on the so called premium principles. The purpose is to develop an appropriate premium to charge for a given risk. The following gives several risk measures based on premium principles.

$\rho(X)=E[X]$ ………………………………………………….(Equivalence Principle)

$\rho(X)=(1+k) \ E[X]$ ……………………………………….(Expected Value Principle)

$\rho(X)=E[X]+ k \ Var(X)$…………………………………(Variance Principle)

$\rho(X)=\mu+k \ \sqrt{Var(X)}$…………………………………..(Standard Deviation Principle)

Each case is a rule for assigning a premium to an insurance risk. In each of the last three cases, $k$ is a fixed constant where $k \ge 0$. The fixed constant $k$ reflects an adjustment that may include expenses and profit and/or a risk loading. The equivalence principle does not account for any risk loading; it simply charges the expected losses, i.e. the pure premium.

Example 2 (Standard Deviation Principle)
This example highlights the standard deviation premium principle:

$\rho(X)=\mu+k \ \sigma$

where $\mu=E[X]$ and $\sigma^2$ is the variance of $X$.

To shed light on the standard deviation principle, suppose that the loss $X$ has a normal distribution. The $\rho(X)$ resembles a quantile in the normal distribution where $k$ is a z-score. With $k=1.645$, $P(X>\mu+1.645 \ \sigma)=0.05$. This means that $\mu+1.645 \ \sigma$ is a threshold that indicates the level of adverse outcome such that the probability of exceeding this threshold is 5%. With $k=2.326$, $P(X>\mu+2.326 \ \sigma)=0.01$. The higher threshold $\mu+2.326 \ \sigma$ is the level of adverse outcome so that there is a 1% chance of exceeding it.

If the loss distribution $X$ is not a normal distribution, the coefficient $k$ would not be the same as the ones given for normal distribution. Whatever the loss distribution is, the fixed constant $k$ is usually chosen such that the probability of losses exceeding the risk measure equals to some pre-determined small probability level.

Example 3 (Value-at-Risk)
When the loss $X$ does not have a normal distribution, the same constants $k$ in Example 2 cannot be used. Instead, we can start with a small probability level $p$ (5%, 1%, 0.5% etc) and determine the $100p$th percentile. The corresponding $k$ can then be determined. This is the basic idea behind the risk measure of value-at-risk.

The risk measure of value-at-risk (VaR) is mathematically a percentile of the loss distribution. Given that $X$ represents the losses, $\text{VaR}_p(X)$ is the $100p$th percentile of $X$. The value-at-risk (VaR) discussed here is for gauging the exposure of risk with respect to insurance losses such that the probability of exceeding the threshold is $p$, the pre-determined security level. See this previous post for a more detailed discussion.

Example 4 (Tail-Value-at-Risk)
We now briefly discuss Tail-value-at-risk (TVaR). Suppose that $X$ is the random variable that models losses. As before we assume that $X$ takes on positive real numbers. The tail-value-at-risk at the $100p$% security level, denoted by $\text{TVaR}_p(X)$, is the expected loss on the condition that loss exceeds the 100pth percentile of $X$. The following is a more succinct way of describing it.

$\text{TVaR}_p(X)=E(X \lvert X > \pi_p)$

where $\pi_p=\text{VaR}_p(X)$. Tail-value-at-risk is a risk measure that is in many ways superior than VaR. The risk measure VaR is a merely a cutoff point and does not describe the tail behavior beyond the VaR threshold. TVaR reflects the shape of the tail beyond VaR threshold. See this previous post for a more detailed discussion.

Desirable Properties of Risk Measures

One important consideration when choosing risk measures concerns the combining of risk entities in a given company or enterprise. For example, an insurance company may have different divisions specializing in different insurance products and services. The results of the risk measure applied individually to the different divisions should be consistent to the results of the risk measure applied to the entire company. For example, it is desirable for the risk measure for two risks combined be no greater than the sum of the two results applied to two risks individually. The following is a list of four desirable properties for a risk measure. Any risk measure that satisfies these four properties is said to be a coherent risk measure.

Coherent Risk Measure
Let $X$ and $Y$ be two loss random variables. The risk measure $\rho$ is a coherent risk measure if it satisfies the following four properties.

1. $\rho(X+Y) \le \rho(X)+\rho(Y)$ ……………………………………….(Subadditivity)
2. If $X \le Y$ with probability 1, then $\rho(X) \le \rho(Y)$ ………………..(Monotonicity)
3. For any constant $c >0$, $\rho(c X)=c \ \rho(X)$ ………………………(Positive Homogeneity)
4. For any constant $c >0$, $\rho(X+c)=\rho(X)+c$ ……………….(Translation Invariance)

The first one, subadditivity, requires that for any two random loss variables $X$ and $Y$, the risk measure for $X+Y$ be no greater than the risk measures for $X$ and $Y$ separately. This is a sensible requirement for a risk measure. This reflects the notion that combining risks leads to diversification and thus to a reduction of total overall risk. Otherwise, a large enterprise would simply be broken into smaller entities in order to reduce risk.

The property of monotonicity is another sensible requirement. If the random loss $X$ is always less than the random loss $Y$, then it makes sense for the risk measure of $X$ be no greater than the risk measure of $Y$. For example, if the risk measure represents the surplus or reserve that is required to cover a random loss, then we would want the reserve for the lesser loss to be no greater than the reserve for a larger potential loss.

The property of positive homogeneity specifies that the risk measure of a constant multiple of the random loss should be the constant multiple of the risk measure. For example, expressing the loss random variable in another currency should indicate the same level of reserve or surplus.

Translation invariance says that the risk measure of combining a random loss and a fixed loss should be the risk measure of the random loss plus the fixed loss. The reserve to cover a fixed loss should be just the fixed loss.

A Closer Look at Examples

Now that we have a list of desirable properties a risk measure should have, which of the risk measures discussed above possess these properties? We comment on these risk measures one by one.

Example 5 (Equivalence Principle)
The risk measure based on the equivalence principle is a coherent risk measure since it satisfies all four properties. The properties of expected value derive the four properties of coherent measure.

• $\rho(X+Y)=E(X+Y)=E(X)+E(Y)=\rho(X)+\rho(Y)$
• If $X \le Y$ always, then $\rho(X)=E(X) \le E(Y)=\rho(Y)$.
• $\rho(cX)=E(c X)=c \ E(X)=c \ \rho(X)$
• $\rho(X+c)=E(X+c)=E(X)+c=\rho(X)+c$

Example 6 (Expected Value Principle)
For the four measures based on the premium principles, equivalence principle is the only one that is coherent. The risk measure based on the expected value principle fails translation invariance. Here’s the derivation.

• $\rho(X+Y)=(1+k) \ E(X+Y)=(1+k) \ E(X)+(1+k) \ E(Y)=\rho(X)+\rho(Y)$
• If $X \le Y$ always, then $\rho(X)=(1+k) \ E(X) \le (1+k) \ E(Y)=\rho(Y)$.
• $\rho(cX)=(1+k) \ E(c X)=c \ (1+k) \ E(X)=c \ \rho(X)$
• $\rho(X+c)=(1+k) \ E(X+c)=(1+k) \ E(X)+(1+k) \ c \ne \rho(X)+c$

Example 7 (Variance Principle)
Now, consider the risk measure based on the variance principle. Recall the risk measure based on the variance principle is of the form $\rho(L)=E(L)+k Var(L)$ where $L$ represents the random losses and $k$ is some positive constant. First we show that it does not satisfy the subadditivity property.

\displaystyle \begin{aligned} \rho(X+Y)&=E(X+Y)+k Var(X+Y) \\&=E(X)+E(Y)+k Var(X)+k Var(Y)+2 k Cov(X,Y) \\&=\rho(X)+\rho(Y)+2 k Cov(X,Y) \end{aligned}

When $Cov(X,Y)>0$, i.e. the risk $X$ and the risk $Y$ are positively correlated, we have $\rho(X+Y)>\rho(X)+\rho(Y)$. There is no benefit of risk reduction in combing risks that are positively correlated.

Next we demonstrate why the variance principle fails the monotonicity. Recall that $\rho$ satisfies the property of monotonicity if $\rho(X) \le \rho(Y)$ for any risks $X$ and $Y$ such that $X \le Y$ for all scenarios. For the risk measure $\rho(L)=E(L)+k Var(L)$, we show that for each $k>0$, we can find random variables $X$ and $Y$ such that $X \le Y$ always and $\rho(X)>\rho(Y)$. To this end, we define random variables $X_n$ and $Y$ for each positive integer $n$.

Let $n$ be any positive integer. Let $X_n$ be a random variable such that $P(X_n=-n)=0.1$ and $P(X_n=90)=0.9$. Let $Y$ be a constant value of 100, i.e. $P(Y=100)=1$. Then it is straightforward to verify that

$E(X_n)=81-0.1n$

$E(X_n^2)=7290+0.1n^2$

$Var(X_n)=7290+0.1n^2-(81-0.1n)^2=0.09 n^2+16.2 n+729$

$E(Y)=100$

$Var(Y)=0$

For $\rho(X_n)=81-0.1 n^2+k (0.09 n^2+16.2 n+729)>\rho(Y)=100$, the constant $k$ must satisfy the following.

$\displaystyle k>\frac{19+0.1 n}{0.09 n^2+16.2 n+729} \longrightarrow 0$ as $n \longrightarrow \infty$

Note that the quantity on the right approaches zero as $n$ becomes large. For each $k>0$ (however small), we can always choose an integer $n$ large enough such that the above inequality is satisfied. Then for the pair $X_n$ and $Y$, $X_n \le Y$ always and $\rho(X_n) >\rho(Y)$. Thus monotonicity fails for the variance principle.

The following derivation shows that positive homogeneity fails and translation invariance is satisfied.

$\rho(c \ X)=c \ E(x)+k \ c^2 \ Var(X) \ne c \ \rho(X)=c \ E(X)+k \ c \ Var(X)$

$\rho(X+c)=E(X+c)+k \ Var(X+c)=E(X)+c+k \ Var(X)=\rho(X)+c$

Example 8 (Standard Deviation Principle)
The risk measure according to the standard deviation principle is also not coherent even though it is an improvement to the variance principle. The standard deviation principle satisfies all properties except monotonicity.

To show subadditivity, note that the fact that $\sqrt{Var(X+Y)} \le \sqrt{Var(X)}+\sqrt{Var(Y)}$. This is derived by the following:

\displaystyle \begin{aligned} Var(X+Y)&=Var(X)+Var(Y)+2 Cov(X,Y) \\&=Var(X)+Var(Y)+2 Corr(X,Y) \sqrt{Var(X)} \sqrt{Var(Y)} \\&=Var(X)+Var(Y)+2 \sqrt{Var(X)} \sqrt{Var(Y)} \\&\le (\sqrt{Var(X)}+\sqrt{Var(Y)})^2 \end{aligned}

In the above derivation, $Corr(X,Y)$ refers to the correlation coefficient, which is always $\le 1$. The following shows the subadditivity.

\displaystyle \begin{aligned} \rho(X+Y)&=E(X+Y)+k \sqrt{Var(X+Y)} \\&\le E(X)+E(Y) +k (\sqrt{Var(X)}+\sqrt{Var(Y)}) \\&=\rho(X)+\rho(Y) \end{aligned}

The following derivation verifies positive homogeneity and translation invariance.

\displaystyle \begin{aligned} \rho(c \ X)&=E(c \ X)+k \sqrt{Var(c \ X)} \\&=c \ E(X)+ k \sqrt{c^ 2 \ Var(X)} \\&=c \ E(X)+ k \ c \ \sqrt{Var(X)} \\&=c \ \rho(X) \end{aligned}

\displaystyle \begin{aligned} \rho(X+c)&=E(X+c)+k \ \sqrt{Var(X+c)} \\&=E(X)+c+k \ \sqrt{Var(X)} \\&=\rho(X)+c \end{aligned}

We now show that the standard deviation principle does not satisfy monotonicity. The proof is similar to the one for variance principle in Example 7 but with a crucial adjustment. For the risk measure $\rho(L)=E(L)+k \sqrt{Var(L)}$, we show that for each $k>0$, we can find random variables $X$ and $Y$ such that $X \le Y$ always and $\rho(X)>\rho(Y)$. To this end, we define random variables $X_n$ and $Y$ for each positive integer $n$.

Let $n$ be any positive integer. Let $X_n$ be a random variable such that $P(X_n=- x_n)=10^{-n}$ and $P(X_n=90)=1-10^{-n}$ where $x_n$ is the positive number defined by $x_n=10^{n+1}-90$. Let $Y$ be a constant value of 100, i.e. $P(Y=100)=1$. Then it is straightforward to verify that

$E(X_n)=80$

$E(X_n^2)=10^{n+2}+6300$

$Var(X_n)=10^{n+2}+6300-80^2=10^{n+2}-100$

$E(Y)=100$

$Var(Y)=0$

For $\rho(X_n)=80+k \ \sqrt{10^{n+2}-100} > \rho(Y)=100$, the constant $k$ must satisfy the following inequality.

$\displaystyle k > \frac{20}{\sqrt{10^{n+2}-100}} \longrightarrow 0$ as $n \longrightarrow \infty$

For any $k >0$ (however small), we can always find an integer $n$ large enough so that the above inequality holds. Then the pair of random variables $X_n$ and $Y$ are such that $X_n \le Y$ always and $\rho(X_n)>\rho(Y)$. Thus monotonicity fails for the standard deviation principle.

Example 9 (Value-at-Risk)
The risk measure of VaR is not coherent despite its attractive quality. An example for the violation of subadditivity can be found in this article. Example 1 in this article gives two independent risks $X$ and $Y$ such that $\text{VaR}_{0.99}(X+Y)>\text{VaR}_{0.99}(X)+\text{VaR}_{0.99}(Y)$.

Example 10 (Tail-Value-at-Risk)
One advantage of TVaR is that it describes the tail behavior beyond the threshold of VaR. Another is that TVaR is a coherent risk measure. This fact has been verified in the article Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1997). Thinking Coherently. Risk, 10, 68-71.

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