How Banks Manage Risk. A look at Value-at-Risk, Expected... | by Jeppe Andersen | Towards Data Science

Roughly, there are three types of risk that financial institutions are exposed against and that regulators try to regulate. First there is market risk, which includes stock prices, interests, FX, volatility etc. Then there is credit risk/counterparty risk or funding risk and lastly operational risk. The risk measures we will look at here primarily focus on the first two. Value-at-Risk (VaR) and Expected Shortfall (ES/CVaR) are developed as an attempt to quantify these types of risk.
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First, we will be interested in looking at scenarios with big losses, i.e. we are interested in the tail of the distribution of possible losses. The most common risk measure in finance after volatility is VaR. VaR is a single measure of market risk, meaning changes in asset value, and is conceived to help the actual decision about taking a risk. It is a measure for what the maximum expected loss is, given some 1 — α% probability and some time horizon T. Risk managers use VaR to measure and control the level of financial risk exposure, which can be within the firm, a portfolio consisting of any type of asset or a specific position and thus be used to gauge the asset value needed to cover possible losses. In the 90s, the SEC ruled that firms had to include a quantitative measure of market risk in their financial statements, and VaR became the main tool for doing so. The Basel Committe even said that banks could rely on their own internal VaR calculation to set their capital requirements. This widespread reliance on VaR has shown to be a mistake. The risk that VaR measured did not include the biggest risk of all; the possibility of a financial crisis.
If L is a loss distribution or potential loss with some horizon T and a continous distribution function F_L, then the VaR(α, T) measure is given by:
This is just a fancy way of saying that the VaR measure is the biggest lower boundary c, where the probability that the loss L is bigger than this c is less than or equal to 1 — α. For a continuous loss distribution this is equivalent to:
which simply means that we, with probability α, won’t lose more than VaR(α).
When computing VaR, three steps have to be considered. First, the quantile α has to be specified. Common levels are 1%-5%, but higher quantiles might be used on trading floors of investment banks for daily risk assessments, while the lower quantiles might be applied in long-run risk analysis for pension funds. Second, the holding period T. For active traders it will mostly be a one-day VaR. Lastly, the probability distribution has to be identified. This is the most essential part and I will return to this later on. A typical conclusion is then that “with 95% probability, the investor will not lose more than VaR(α = 0.95)”.
Value-at-Risk has pros and cons. The biggest con is the uncertainty of what will happen if the VaR measure is exceeded — if we end up “in the last 5%” — and this makes VaR a questionable metric for risk management. It is a quantile measure and actually a value depicting the best of the worst case scenarios, thus underestimating the potential losses. That seems inappropriate for a risk measure. Another way of thinking of this is to say that VaR is the boundary between “normal” market days and extreme events, making it terrible to use during or close to a financial crisis. VaR is often measured daily, and because it is a very short-term measure, it assumes that tomorrow will be more or less like today.
In 2008 in Global Association of Risk Professionals Review, Einhorn compared VaR to “an airbag that works all the time, except when you have a car accident” and that it is potentially catastrophic when its use creates a false sense of security among executives. A guy called Nassim Taleb also once testified in Congress asking for the banning of VaR.
So why is it even used?Well first, VaR is so appealing because it is the only commonly used risk measure that can be applied to almost any type of asset and it takes into account a lot of variables, e.g. diversification, leverage and volatility, which make up the kind of risk firms see every day. Second, it is an advantage that it can measure both individual risk (a single portfolio) and firm-wide risk, which is a net number. Top executives usually know the firm’s VaR within minutes of market’s close.
The simple illustration above shows a loss distribution with a red VaR threshold. If we for example have a time horizon T in one week, a confidence coefficient of 95% (i.e. α = 0.05) and that VaR(α = 0.05) = $5 million, then there is only a 5% chance of the loss being bigger than $5 million over the next week or a 95% probability that we won’t lose more than $5 million. Note that it always represents a loss, but the VaR value is reported as a positive number. A negative VaR will imply that the portfolio has a high probability of making profit. It tells us nothing about how far out on the right tail of the distribution we will land and thus nothing about how bad the losses will get.
So in this risky setup, is there any way to improve the situation?
Regarding VaR, most people are concerned about what happens “in the last 5%” (or 1%), at the extreme edge of the normal distribution curve from before. The fact that you are not likely to lose more than a certain amount 95% of the time tells you nothing about what could happen the other 5 percent of the time. You could lose $50 million — no big deal. It happens. You could also lose billions and go out of business. VaR has no way of measuring which it will be. The cause of the extreme losses is what Nassim Taleb calls “fat tail” or “black swan” events (he has written a few books on this topic). It means that we don’t know when or how a black swan might appear, a surprising event that the observer could not have anticipated; like the Dot-com bubble in the late 90s or the financial crisis in 2007–2008. An alternative to VaR is Expected Shortfall (ES).
Expected Shortfall is defined as the tail expected loss given that the α-VaR estimate is exceeded:
The latter is the average of VaR_γ(L) over all γ that are less than or equal to α. Related to the typical VaR conclusion from above, one could instead ask “What is the expected loss incurred in the α worst cases over a horizon T?”. This question implies that ES takes the entire tail distribution into consideration when computed. ES compliments VaR. If the two risk measures are combined, we can have a good estimate of an asset’s or a portfolio’s risk. VaR tells us where the boundary is and ES tells us how bad it will be if we reach this boundary, illustrated below.
Risk measures will rarely be known and instead they have to be estimated, which gives us more risk and insecurity. A normal loss distribution and a t-loss distribution could have the same VaR but different ES, implying that the tail is important. We will look at two approaches. First, a non-parametric approach (meaning not normal- or t distributed). Here we will have no assumptions about the return distribution. The observations of the historical returns are sorted from smallest to biggest and the VaR_α(L) are determined by an α-quantile of this sorted distribution. ES will be calculated as the average of all losses bigger than VaR:
S is the current size of the position in some asset and ^q(α) is the estimated upper α-quantile of the return distribution. The ES estimate is thus the average of all L_i that surpasses VaR. The non-parametric approach is very simple and it’s an advantage that we don’t have any distribution assumptions. However, this will only work if the future looks sufficiently like the past.
The parametric approach is based on the assumption of a certain distribution. A portfolio’s distribution can be determined by 3 methods: 1. AnalyticalSet up a model, find the simultaneous distribution and fit the model to the data. As an example, you could find VaR from the lognormal distribution function.2. SimulationSimulate the outcome from a model of the market, evaluate the portfolio in every outcome and estimate a distribution. This is often done with Monte Carlo simulation. The method is very slow, but it is probably the most powerful method. It is flexible enough to include historical observations and it can be improved by the so-called variance reduction methods.3. HistoricalSimulate outcomes based on the data, evaluate the portfolio in every outcome and estimate a distribution. Useful when the amount of data is not very large. It is time consuming, but its main advantage is that it catches recent market crashes, which can be very important for risk measurement.
Both the parametric and non-parametric approaches have criticisable assumptions. The nonparametric estimates are inaccurate for small α, but this can be improved by assuming a polynomial tale, which I won’t include here. In addition, α can be estimated using the Hill estimator, which is a Maximum Likelihood estimate:
where n(c) is the number of returns Y_i ≤ c. c is chosen, but it has a bias/variance trade-off. If n(c) is large we have more bias and if n(c) is small we have a high variance. We can use a Hill plot that plots ^a against n(c). We can then find a “stable region” and roughly asses the Hill estimate:
In this example, (b) is a close-up of (a) and we will choose the Hill estimate to be about 2.2 for n(c) between 60 and 100.
A VaR tail distribution example in R could look like this for an old S&P500 dataset:
data(SP500,package="Ecdat")library("fGarch")# number of observations : 2783# daily return S&P500 (change in log index)n = 2783## grab last 1000 obs.SPreturn = SP500$r500[(n-999):n]year = 1981 + (1:n)* (1991.25-1981)/nyear = year[(n-999):n]n = length(SPreturn)# fit t-distr to SPreturnfitt = fitdistr(SPreturn,"t")param = as.numeric(fitt$estimate)mean = param[1]df = param[3]sd = param[2]*sqrt( (df)/(df-2) )lambda = param[2]x=sort(SPreturn)q = quantile(x,.1)a= 1.975## vector of alphas for plotting VaRalpha = seq(.002,.2,.0001)qalpha = qt(alpha,df=df)VaR_par = -20000*(mean + lambda*qalpha)VaR_norm = -20000*(mean(x)+ sd(x)* qnorm(alpha))par(mfrow=c(1,1))plot(alpha, -20000*q * (.1/alpha)^(1/a),type="l",lwd=2,     xlab=expression(alpha),ylab=expression(paste("VaR(",alpha,")")),     ylim=c(0,1700),log="x" )a.hill=2.2lines(alpha, -20000*q * (.1/alpha)^(1/a.hill),lty=2,lwd=2)lines(alpha,VaR_par,lty=5,lwd=2)lines(alpha,VaR_norm,lty=3,lwd=2)legend("topright",c("polynomial tail: regression",                    "polynomial tail: Hill","t","normal"),lwd=2,lty=c(1,2,5,3))
and the plot is:
I won’t explain all the details here and I also decided not to explain polynomial tails, but it definitely shows the importance of the tail distribution when looking at different αs.
Artzner et al. (1999) defined how a risk measure should function in order to be considered coherent. Through four distinct axioms, they managed to formulate the concept of a coherent risk measure, using precise statements of the nature of risk. Thus translating complex reality into a simpler mathematical framework. Let X and Y be stochastic variables that describe the loss of two portfolios. Then a risk measure ρ(·) (requirement to capital or “riskiness”) is said to be coherent if it meets 4 constraints:1. Proportionality:
If you double your position you will double your risk.2. Monoticity:
If the loss Y is always smaller than or equal to the loss X, the risk for Y must be smaller than or equal to the risk for X.3. Translation:
Translation simply states that cash can be deducted from the capital requirement. 4. Subadditivity:
The risk measure for the two portfolios together should be smaller than or equal to the risk measure for the two portfolios separately, because the combination creates diversification. If you for example set capital requirements for a company and the subadditivity constraint, the company can reduce the capital requirements by splitting the assets. The above listed axioms are treated as guidelines for a proper risk measure. VaR does not respect subadditivity. As an example, assume a portfolio have the following loss distribution with VaR(0.5) = 50:
Then two independent portfolios will have loss distribution:
This will have VaR(0.5) = 1950 > 2×50, which is obviously in contradiction to subadditivity. The risk for the two portfolios together is larger than the risk for the two portfolios separately. For ES this is not the case, it will look like this:ES(P_1) = ES(P_2) = 1590 og ES(P_1 + P_2) = 2015.6 < 2×1590
ES is a coherent risk measure. As a consequence of ES fulfilling the subadditivity, you can say that it is impossible to build examples in which ES violates the definition of a coherent risk measure. It is also more sensitive to heavy tails, wrong estimates of the tail distribution and outliers and it is not as easy to interpret as VaR.
VaR sums up the risk of a portfolio with a single number and is thus more superficial than alternative risk measures (e.g. Greeks). If we combine VaR with ES, we will have a much more reliable risk measure and VaR and ES are very dependent on the choice of parameters. In spite of the theoretical advantage of ES over VaR, the latter has been preferred over ES. This could be because computing ES involves computing VaR first, thus potentially giving two sources of error that could invalidate the result. The Basel Committee proposed in 2012 moving from VaR to ES, because:“Unlike VaR, ES measures the riskiness of an instrument by considering both the size and likelihood of losses above a certain threshold ... In this way, ES accounts for tail risk in a more comprehensive manner”. And later: “Despite criticism focused on the complexity, computational burden, and backtesting issues associated with ES, the recent literature suggests that many issues have been resolved or have been identified as less severe than originally expected”.The key element is that ES will be able to capture tail events. However, today it is still not that simple. Just like VaR, ES will not be able to capture e.g. credit and liquidity risk because of its similarity to VaR. The validation procedure for ES is much more complicated and hence the model can be hard to interpret, making it difficult to know when and where the model fails. All in all, financial risk and regulation of it is incredibly complex, maybe too complex...
In regulation, the use of risk measures can help determine the capital requirements from worst-case scenarios for some portfolio or for a firm. In the credit business, losses occur all the time — there are always some borrowers that default on their obligations. The losses that are actually experienced in a particular year varies from year to year. The figure below shows how realised losses over time lead to a distribution of losses for a bank.
It is never possible for a bank to define their future losses, but they can forecast the average level of credit loss they expect to experience. In this setting we will define Expected Loss (EL) as being the loss you expect to get in a portfolio with n assets, as shown by the dashed line. Financial institutions view EL as a cost component of doing business. One of the functions of bank capital is to provide a buffer to protect a bank’s debt holders against peak losses that exceed EL. Such peaks are illustrated by the spikes above the dashed line in the figure above. Peak losses do not occur often, but when they occur, they can potentially be very large. Losses above EL are usually referred to as Unexpected Losses (UL). Institutions know that they will occur now and then, but they cannot know in advance their timing or severity. Capital is needed to cover the risk of such peak losses.
Banks must carefully balance the risks and rewards of holding capital. In the worst case, though highly unlikely, the bank lose their entire credit portfolio, but holding capital against this event is economically ineffecient. They also have an incentive to minimise capital because it frees up resources to be directed into profitable investments. However, the less capital a bank holds, the greater is the probability that it will not meet its own obligations. Now we can return to VaR.
This looks like something we have seen before. There is a bigger probability of seeing expected losses and the VaR threshold is as we already know. If capital is set according to the gap between EL and VaR and if EL is covered by revenues, then the likelihood that the bank will remain solvent over a one year horizon is equal to the confidence level (1-α). Expected Losses can be written as:EL = PD×EAD×LGDor if expressed as a percentage of the EAD as:EL = PD*LGDThe three factors here are probability of default (PD), exposure at default (EAD) and loss given default (LGD). PD is the average percentage of obligors that default in the course of one year, EAD is an estimate of the amount outstanding in case the borrower defaults and LGD is the percentage of exposure the bank might lose in case the borrower defaults, usually shown as a percentage of EAD.
Now we can look at the Risk-Weighted Assets (RWA). It is an expression for the value of assets that you can risk losing, weighted from the risk. It is used for the derivation of supervisory capital charges for UL and is based on a specific model developed by the Basel Committee. It is used to determine the minimum amount of capital that must be held by banks in order to reduce the risk of insolvency. During the financial crisis of 2008, many financial institutions lost large amounts of capital and some became insolvent. Basel III set some guidelines to avoid this in the future. Each bank must group its assets together by risk category so that the amount of required capital is matched with the risk level of each asset type. The riskier the asset, the higher the RWA and the greater the amount of regulatory capital required. It looks like this:RWA = 12.5×K×EADK is the capital requirement, given as:
where:
and it is multiplied by EAD and the reciprocal of a minimum capital ratio of 8%, i.e. by a factor of 12.5.
That is a lot... But it has four components:1. The Asymptotic Singe Risk Factor (ASRF) model, which is the first part up until -PD·LGD.It is derived from “ordinary” credit portfolio models by the law of large numbers and it can be shown that these ASRF models are portfolio invariant, meaning that capital required for any given loan should only depend on the risk of that loan and must not depend on the portfolio it is added to. This is a standard measure, banks are encouraged to use whatever credit risk models best fit their internal risk management needs. The LGD is here included as part of the ASRF model. N is the cumulative normal distribution and N^-1 its inverse. R is the correlation and it has a fixed confidence level of 99.9%.
2. Correlation, R. The asset correlations show how the asset value (e.g. sum of all asset values of a firm) of one borrower depends on the asset value of another borrower. The asset correlation function has been derived by supervisors to be:
It is built of two limit correlations of 12% and 24% for very high and very low PDs (100% and 0%, respectively). Correlations between these limits are modelled by an exponential weighting function that displays the dependency on PD. Its pace is set by a so-called “k-factor”, which is set to 50 for corporate exposures. It is also adjusted by firm size (the last part), which is measured by annual sales. It affects borrowers with annual sales between €5 mn and €50 mn. For borrowers above €50 mn, the size adjustment become zero and for borrowers with less than €5 mn it takes the value of 0.04, thus lowering asset correlation from 24% to 20% (very high PD) or from 12% to 8% (very low PD). It determines the shape of the risk weight formulas. Correlations are asset class dependent, because different borrowers and/or asset classes show different degrees of dependency on the overall economy.
3. Expected loss of a loan in percentage of EAD, which is -PD·LGD. LGD is here part of the Exptected Loss. It was decided in the Basel framework that banks were only required to hold capital against UL. In the above figure of VaR, EL and UL, the risk weights now relate to the distance between the VaR and the EL only. As the ASRF model delivers the entire capital amount from the origin to the VaR in the figure, the EL has to be taken out of the capital requirement. This was done by taking the product of PD and downturn LGD and yields a “UL-only” capital requirement.
4. And the maturity adjustment, which is the fraction after the last parenthesis. Note that PD is the average probability under normal circumstances and that LGD is under economic downturn. M is maturity.The capital requirement is increasing in maturity because, normally, long-term credits are riskier than short-term credits and downturns are more likely in the long term. b(PD) is a “smoothed” statistical regression model for maturity adjustment.
All right, that was a lot. Congratulations on making it through. Note that this is not meant to be an exhaustive or conclusive description. I have probably left out a lot of information. Feel free to correct me if I missed something or did something wrong (I probably did). Financial risk is a huge, complex topic and there is way more to risk management than in the scope of this article. I’m not a risk manager, this is purely gathered from cources I’ve had as a student.
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