12 Risk management
The last major topic of this course is risk management. There are many types of risk in finance, such as credit risk, operational risk and market risk. Here, we focus on market risk. A common definition of market risk1 is the risk of losses in positions arising from movements in market variables like prices and volatility. A typical situation is to quantify the risk of a portfolio which consists of several financial assets such as stocks, fixed-income securities, currencies, and derivatives. It is also of interest to aggregate the positions of smaller units (e.g. trading desks) and compute the risk of a larger unit. In this section, we focus on Value at Risk and Expected Shortfall. We follow here Chapter 7 of Tsay (2010) and Chapter 14 of Linton (2019).
12.1 Value at risk
We begin by recalling the definition of Value at Risk (see Definition 3.2) which is a very popular (but not perfect) measure of risk. In Definition 3.2, we defined the VaR when the underlying random variable denotes the loss of the portfolio. In this subsection, we give the definition in terms of the value of the portfolio. Clearly the two definitions are equivalent (upon some sign changes) but we have to be careful about the convention. For example, what we call the 1% VaR here may be called the 99% VaR elsewhere.
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Definition 12.1 (Value at risk). Let X be a random variable interpreted as the value of a portfolio over some specified horizon. The value at risk (VaR) at level α ∈ (0, 1) is the α-quantile of X:
provided QX is well-defined. Thus, we have
Proposition 12.3. Let X ∼ N(μ,σ2). Then, we have F −1(α) = μ + σzα,
VaRα(X) = QX (α) = F −1(α), X
P(X ≤ VaRα(X)) = α.
In other words, with probability 1 − α, the value of the portfolio will not be smaller than VaRα(X). Usually X is a continuous random variable whose density (exists and) is strictly positive. Thus its quantile function is well defined. Typical values of α are 0.05 and 0.01.
Exercise 12.2. Show that for a>0 and b∈R, we have, for any α∈(0,1). we have VaRα(aX + b) = aVaRα(X) + b, Var1−α(−X) = −Varα(−X).
When X is normally distributed, there is an explicit formula of the VaR. For this reason (as well as the popularity of the normal distribution), this formula is often applied. However, the normality assumption is a dangerous one as we have seen that the distribution of financial returns are typically fat-tailed. Thus, the normal VaR underestimates the value at risk.
where zα is the α-quantile of the standard normal distribution.
1Taken from https://en.wikipedia.org/wiki/Market_risk 99
Proof. UsethefactthatX=μ+σZwhereZ∼N(0,1). For example, if α = 0.01, then zα ≈ −2.33.
Example 12.4. Consider daily (simple) returns of FTSE 100 from Jan 1, 2016 to Dec 31, 2021. Using the historical returns, we fit a normal distribution whose mean is ≈ 0.00016 and standard deviation is ≈ 0.011. The 1%-quantile of the fitted distribution is about −0.024. So, if we model the next return by this historical distribution, with probability 99% we do not expect to lose more than 2.4%. If our current investment in the index is 100000 (as of Dec 31, 2021), then the 1% VaR of X = 100000(1 + R) is
VaR0.01(X) = 100000 × (1 − 0.024) ≈ 97567. For later use, we also note the 5% VaR:
VaR0.05(R) = −0.017. Daily returns of FTSE 100
0 500 1000 1500
Figure 12.1: Daily returns of FTSE 100 (from 2016 to 2021). We show the normal fit in blue and the 1% VaR in red.
We visualize the situation in Figure 12.1. Note that if the normal assumption was valid, we’d expect about 15 ≈ 0.01 × 1516 (where 1516 is the sample size) observations that are less than −0.024. However, in the data, there are 31 such observations. Moreover, the minimum return is −0.1087 which is more than 10 standard deviations below the mean. Such an observation is extremely unlikely under the normal assumption. In fact, for a sample of size 1516, the probability is practically 0. These results suggest that the normal VaR probably underestimates the risk of the return.
Note that the VaR (and other measures of risk) depends on the time horizon which is implicit in the definition of X. Here is another computational advantage of the normal assumption. Suppose the log returns r of the portfolio are modelled by i.i.d. N(μ,σ2). Then the q-period log return satisfies
r[q]=r1+···+rq ∼N(qμ,qσ2).
Consequently, it is immediate to derive the q-period VaR (or a fixed α) as a function of
the 1-period VaR. We leave it as an exercise for the reader to derive such a formula. 100
−0.10 −0.05 0.00 0.05
12.2 Quantile estimation
Since the Value at Risk is nothing but a quantile of a distribution, estimating the VaR is the same as estimating the quantiles of a distribution. Other than assuming a parameteric model (such as the normal distribution), one may adopt a nonparametric approach. In this subsection, we consider nonparameteric estimation of quantiles from historical data.
Let r1, . . . , rn be the returns of a portfolio, say, in the sample period. Sorting the observations in increasing order gives the ordered statistics of the sample:
r(1) ≤ r(2) ≤ ··· ≤ r(n). We have the following asymptotic result.
Theorem 12.5. Suppose X1,X2,… are i.i.d. with density f. Let p ∈ (0,1) and suppose xp = F−1(p) is the p-quantile and f(xp) > 0. Then, as n → ∞, the order statistic X(l) of X1,…,Xn, where l = np, is asymptotically normal with mean xp and variance p(1 − p)/(nf2(xp)).
Proof. Omitted.
Example 12.6. We illustrate Theorem 12.5 empirically by a simulation. Suppose X1, X2, . . . are i.i.d. standard normal, and p = 0.05. Then xp ≈ −1.64. Suppose n = 5000. Then l = np = 5000 · 0.05 = 250. We simulate B = 10000 batches of samples (each with n data points), and record the empirical distribution of xˆp = X(l). As shown in Figure 12.2, the empirical distribution (shown here by its density estimate) is very close to the asymptotic distribution.
Empirical distribution of quantile
−1.75 −1.70 −1.65 −1.60 −1.55 −1.50
N = 10000 Bandwidth = 0.004276
Figure 12.2: Result of simulation in Example 12.6.
Remark 12.7. If np is not an integer, let l1 and l2 = l1 + 1 be integers such that l1 <
np < l2. Define pi = li , i = 1, 2. Then, we may estimate the quantile xp, via linear
interpolation, by
xˆp = p2 − p r(l1) + p − p1 r(l2). (12.3)
0 2 4 6 8 10 12
Example 12.8. Consider the FTSE data as in Example 12.4. Using (12.3), the empirical 5% quantile is ≈ −0.0151 which is a bit larger than the normal VaR −0.017. The empirical 1% quantile is −0.0334, which is smaller (more extreme) than the normal VaR −0.024.
We note that while the empirical quantiles do not depend on the parameteric form of the density (but it still assumes that the observations are i.i.d.), the empirical quantiles are noisy when α is close to 0 or 1. Moreover, by construction, the predicted loss cannot be more than the historical loss, since this approach assumes that the future distribution is equal to the past.
12.3 Riskmetrics model
In this subsection, we consider the Riskmetrics model (introduced briefly in Section 11.3) applied to VaR calculations. This approach was first developed by JP Morgan in the 1990s.
For concreteness, we consider daily frequency. Let rt be the log return over the time interval [t−1, t] and let Ft−1 be the information set available at time t−1. The Riskmetrics model assumes that rt|Ft−1 ∼ N(μt,σt2), where
μt = 0, σt2 = ασt2−1 + (1 − α)rt2−1, (12.4)
and α ∈ (0,1) is a fixed parameter. Write rt = at where at = σtεt satisfies a GARCH(1, 1) model described by (12.4). Then the log price pt = log Pt satisfies pt = pt−1 + at. Tsay (2010) notes that α is typically in the range (0.9, 1) and a typical value is α = 0.94. A simulated path of this process is provided in Figure 12.3.
Simulated log returns
ACF of r^2
0 100 200 300 400 500 0 5 10 15 20 25
Figure 12.3: A simulated path of the model 12.4. We assume here the initial values σ02 = 0.0152 and r0 = −0.005, and α = 0.94. Also shown in red is ±σt where σt is the conditional volatility.
Example 12.9. Using the package rugarch, we fit a RiskMetrics model to the FTSE log returns (note that in Figure 12.1 we used instead the simple return). The estimated value of α is about 0.946. In Figure 12.9 we plot the data series together with the fitted conditional standard deviations. With these estimates, we can construct, using (12.2), the VaR for the next daily return because the conditional distribution is normal.
−0.03 −0.01 0.01 0.02
0.0 0.2 0.4 0.6 0.8 1.0
Series with 2 Conditional SD Superimposed
−0.10 −0.05 0.00 0.05
GARCH model : iGARCH
2016 2017 2018 2019 2020 2021 2022
Figure 12.4: Fitted conditional standard deviation for the FTSE log return using the RiskMetrics model.
12.4 Risk measures
We considered the Value at Risk which is a common measure of risk. In the literature, there are many other measures of risk. Let X be a random variable (interpreted as the payoff or portfolio value at a particular time) and let ρ(X) be a quantity which depends on the distribution of X. For ρ(·) to be a quantification of risk, the functional ρ(·) should satisfy some reasonable properties. Here are some desirable properties:
(Normalized) ρ(0) = 0. That is, the risk when holding no assets should be zero.
(Monotonicity) Let X1 and X2 be random variables. If P(X1 ≤ X2) = 1, then ρ(X1) ≥ ρ(X2). Intuitively, this means that a portfolio with greater future returns has less risk.
(Sub-additivity) For any X1 and X2, we have
ρ(X1 + X2) ≤ ρ(X1) + ρ(X2). (12.5)
Here, the random variables X1 and X2 are defined on the same probability space so that the distribution of X1 +X2 (which depends on the joint distribution of (X1, X2) makes sense). Then X1 + X2 is the combined payoff if we hold both positions. Then (12.5) states that the risk of X1 + X2 is less than or equal to the sum of the individual risks ρ(X1) and ρ(X2). Intuitively, this is because if X1 and X2 are not perfectly positively correlated, the total risk should be smaller due to some diversification.
(Positive homogeneity) For α ≥ 0, we have ρ(αX) = αρ(X). This property states that the risk of a position is directly proportional to its size.
(Translation invariance) If a ∈ R is a constant, then ρ(X + a) = ρ(X) − a. That is, adding cash to the portfolio reduces the risk by the same amount.
This provides an axiomatic framework to study quantification of risk. In particular, a functional ρ(·) which satisfies all the five properties above is called a coherent risk measure. This concept was first introduced in 1998.
Definition 12.10 (Coherent risk measure). A functional ρ(·) is said to be a coherent risk measure if it satisfies the five properties stated above.
One may ask whether Value at Risk is a coherent risk measure. To address this question, we need to fix a convention of the VaR which is consistent with the way ρ(·) is interpreted. Let X be again the value of the portfolio, so that −X is the potential loss. For a fixed significance level α ∈ (0, 1), we define here
ρ(X) = F−1 (1 − α). (12.6) −X
That is, we define here ρ(X) be the number such that the loss −X does not exceed ρ(X) with probability 1 − α. It is not difficult to show that the VaR thus defined is normalized, monotonic, positive homogeneous and translation invariant. However, it is not sub-additive:
Proposition 12.11. The VaR as defined by (12.6) is not subadditive. Hence, the VaR is not a coherent risk measure.
Proof. We give here a simple counter example (with discrete random variables) to show that ρ(·) is not subadditive. Suppose X and Y are i.i.d. with
P(X = 0) = 0.96 and P(X = −1000) = 0.04. Then,atα=0.05,theVaRofX andY areboth0,sothatρ(X)+ρ(Y)=0.
However, the distribution of X + Y is
P(X +Y = 0) = 0.962, P(X +Y = −1000) = 2·0.96·0.04, P(X +Y = −2000) = 0.042.
Since 2 · 0.96 · 0.04 = 0.0768 > 0.05, at the same significance level, the VaR of X + Y is 1000 which is larger than ρ(X) + ρ(Y ).
This means that using VaR as a measure of risk may discourage diversification – this is clearly undesirable. On the other hand, we will show that the expected shortfall (Definition 3.3) defines a coherent risk measure. We adopt the following convention:
Definition 12.12 (Expected shortfall). Let X be a random variable which represents the value of the portfolio. The expected shortfall at level α ∈ (0, 1) is defined by
ESα(X) = −α1 E[X1{X≤xα}], where xα is the α-quantile of X, i.e., P(X ≤ xα) = α.
This is equivalent to (3.3) after a sign change in X. This is because α1 E[X1{X≤xα}] = E [X|X ≤ xα] .
Theorem 12.13. The expected shortfall (for α fixed) is a coherent risk measure.
Proof. We only show that (12.7) is subadditive as the other properties can be easily verified.2
LetXandY begivenandletZ=X+Y.Then α (ESα(X) + ESα(Y ) − ESα(Z))
= E Z1{Z≤zα} − X1{X≤xα} − Y 1{Y ≤yα}
= E X 1{Z≤zα} − 1{X≤xα} + Y 1{Z≤zα} − 1{Y ≤yα}
≥ x(α)E 1{Z≤zα} − 1{X≤xα} + y(α)E 1{Z≤zα} − 1{Y ≤yα} = x(α)(α − α) + y(α)(α − α) = 0.
As an explicit example, we compute the expected shortfall of a normal random variable. Proposition 12.14. Suppose the portfolio value X follows the normal distribution N(μ,σ2).
Then the expected shortfall, defined by (12.7), is given by
ESα(X) = −μ + σφ(Φ−1(α)), (12.8)
where φ and Φ are respectively the density and cdf of the standard normal distribution.
Proof. Let α be given and let xα be the α-quantile of X whose distribution is xα = μ + σΦ−1(α).
Also, the density of X is
So, the expected shortfall of X is
−1xα 1x−μ ESα(X)= α xσφ σ dx
−∞−1 −1 μ+σΦ (α) 1 x−μ
=α xσφ σ dx. −∞
Making the substitution z = x−μ , we have σ
−1 Φ−1(α)
ESα(X) = α (μ + σz)φ(z)dz
fX(x)= 1φx−μ. σσ
= −μ + α1 σ
(−z)φ(z)dz.
−∞ Φ−1(α)
we have, as desired,
2This proof is taken from: Acerbi, C., & Tasche, D. (2002). On the coherence of expected shortfall. Journal of Banking & Finance, 26(7), 1487–1503.
dzφ(z) = dz √2πe 2 = −zφ(z), ESα(X) = −μ + ασ φ(Φ−1(α)).
In Figure 12.5 we illustrate the definition for the standard normal distribution. By construction, the expected shortfall is always greater than or equal to the value at risk.
VaR and ES for N(0, 1)
0.0 0.1 0.2
Figure 12.5: Value at risk and expected shortfall of N (0, 1), both as functions of α. Note that when α ↓ 0 both measures diverge to +∞ (not shown as the minimum α in this plot is 0.001).
0.3 0.4 0.5
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