## Calculating Conditional Value at Risk (CVaR) or Expected Shortfall – VaR and beyond

*in*Computational Finance

# Calculating Conditional Value at Risk (CVaR) or Expected Short fall using Historical returns.

Imagine a board meeting where you have just presented your Value at Risk (VaR) analysis and a board member asks a simple question. “So what are we talking about here? What is the expectation? What is the most that we can drop if we cross your Value at Risk threshold?” or “What lies beyond the barrier X? What would it cost us to give that risk away and insure it?” To answer this series of simple questions you need **Conditional Value at Risk** or **CVAR estimate.**

We all understand what Value at Risk is. A worst case loss, associated with a probability and a time horizon. CVaR or conditional Value at Risk is the expected loss, the average loss if that worst case threshold is ever crossed. It answers the what really lies beyond barrier X question. While VaR is an estimate that board members sometimes have difficulty quantifying, the insurance policy and premium analogy is much more easier to present, quantify and understand.

What is the expected loss if the Value at Risk threshold is breached? Sounds like a simple enough question with a simple enough answer.

But there is a problem. Since most financial models assume a normally distributed world, CVaR numbers produced by assuming the normal distribution grossly underestimate the actual risk. We therefore need a technique or tool that would supplement and correctly model the tail risk. While there is no one perfect solution, the historical returns method is one possible candidate for estimating more realistic CVaR estimates. We review that in the post below.

## Conditional VaR – Context and background

In the paper by Yamai and Yoshiba – Comparative analysis of expected shortfall & Value at risk under market stress – Expected Shortfall is defined as “*the conditional expectation of loss given that the loss is beyond the VaR level*“. You can also look at the following two additional sources for more background on CVaR.

http://www-iam.mathematik.hu-berlin.de/~romisch/SP01/Uryasev.pdf as well as the original BIS paper at http://www.bis.org/bcbs/ca/acertasc.pdf

The authors mention in their findings that though Expected Shortfall addresses some of the underestimation of risk of securities which have fat tailed distributions and a potential for larger losses the measure is still exposed to tail risk if losses are infrequent and large, especially when the market is stressed. However under more lenient conditions (such as normal market conditions) when the VaR measure would still be exposed to tail risk because it disregards any losses beyond the confidence level, expected shortfall would have no tail risk because it considers the conditional expectation of loss beyond the VaR level.

To review the calculation methodology of conditional VaR (CVaR) see our post, Calculating Value at Risk (VaR) – Comparing VaR models, methods & metrics , which we will revisit below when comparing results obtained from the Monte Carlo simulation using the normal distribution (MC –Normal), Monte Carlo simulation using the historical returns (MC- Hist) and Historical Simulation approaches.

To review the MC- Historical returns model and how it is used to calculate VaR please see:

- Monte Carlo Simulation – Simulating returns by replacing the normal distribution with historical returns
- Monte Carlo simulation and historical returns – Calculating Value at Risk (VaR)

## Estimating Conditional Value at Risk – CVaR for Gold

For Gold, assume that we have simulated a 365-day price path using the Monte Carlo simulation approaches and used a 365-day window for the Historical Simulation approach. For each approach we have generated a series of returns and used these returns to calculate 99% confidence level daily VaR %. We then compare the results to see which approach produced bigger and more realistic CVaR estimates.

Once the daily VaR metric is obtained, the calculation of CVaR follows the same process for all three VaR approaches. As an example we have used the daily VaR from the Historical Simulation approach as an input in our CVaR worksheet. After we review the CVaR methodology we will present the results from all three methods.

To determine the expectation of loss given that the loss is beyond the VaR level we first need to determine the loss incurred at the VaR level. Consider the following instance: Current Gold price is 1,657.50 and the daily VaR % using the Historical simulation approach is 4.149%. The loss at the VaR level or the price shock at the VaR level is 68.77.

Next we determine the loss amounts. For each of the 364 returns that we have calculated, we calculate the price shocks in monetary terms. For the losses column we will only consider the negative price shocks (i.e. price declines). For positive price shocks we will consider a loss of zero.

What is the conditional expectation of loss if the loss amount exceeds 68.77?

## Conditional Value at Risk – Calculation methodology review

The methodology followed here is the same as that used for determining the conditional expectation or expected value of a roll of a fair die given that the value rolled is greater than a certain number.

First let us consider the unconditional expectation of a six sided fair die. It is equal to the sum product of the value on the face of the die that turns up when rolled times the probability of that occurrence. For a fair die, as there are six possible occurrences, the probability of any value being rolled is 1/6. The unconditional expectation is then equal to 1*1/6 + 2*1/6 + 3*1/6 + 4*1/6 + 5*1/6 + 6*1/6 =21/6 =3.5

The conditional expectation is equal to the sum product of the value on face the die that turns up given that it is greater than a certain number times the probability of its occurrence. Supposing that it is given that the value rolled is greater than 3. There are three occurrences that meet this condition (4,5,6), each having an equal probability of occurrence, i.e. 1/3. The conditional expectation works out to 4*1/3 + 5*1/3 + 6*1/3 = 5

In a similar manner once we have determined the loss amounts for each data point, we need to factor in the condition that the loss amount exceeds 68.77, the VaR loss amount. This is factored in the worksheet as follows- we will only take losses that exceed 68.77 in the calculation, losses that are less than this amount are ignored. We may do this in one of two ways.

Calculate a separate column which will take the loss amount as is if it exceeds 68.77 or replace it with zero if it doesn’t.

We then apply the **AVERAGEIF** function to the array of these conditional losses so that only those instances where the loss exceeds zero are considered. Note that as we consider each return as a separate observation the probability of occurrence is 1/number of occurrences where the conditional loss is greater than zero. The conditional VaR amount or Expected Short fall works out to 83.65 for a confidence level of 99%.

The same result may be obtained by directly applying the AVERAGEIF function to the array of unconditional losses and resetting the criteria from greater than zero to greater than the VaR Amount, i.e. =**AVERAGEIF(F11:F374,CONCATENATE(“>”,I5))**.

The Conditional VaR % is then equal to the Conditional VaR Amount/ Current Value of the position = 83.65/1657.50 =5.047%. This CVaR% may be determined directly from the array of returns by applying the AVERAGEIF function to the array of returns and setting the criteria to the Daily VaR (%), specifically CVaR%=-**AVERAGEIF(array of returns, CONCATENATE(“<“,-Daily VaR%)**.

How do the results from the Monte Carlo simulation using Historical returns approach compare to those obtained using the historical simulation method and the original MC-Normal approach. The average CVaR%s over 25 simulation runs are given below:

*Also see Unexpected Loss (UL) and The Economic Capital Case Study if you are interested in extending the shortfall model for Economic Capital applications.*