Chapter 4: Value-at-risk and Expected Shortfall

Question 1

What is VaR?

Answer 1

• VaR is the widely used risk measure in finance: it has an attractive feature of being an aggregate risk measure that is easy to understand.
  
  • Aggregate risk measure: different risk/instruments can be combined to obtain a single measure of risk which allows one to get an overall view of the risk that one is exposed is
  • Easy to understand: the risk measure gives you some maximum loss amount and is thus intuitive: also for non-technically skilled people
• Need to make a choice concerning the likelihood and the horizon

Question 2

How can we define VaR?

Answer 2

• VaR summarizes the loss (in value or return) over an horizon T that will only be exceeded in the rare case as determined by the confidence level c (significance level 1-c)

Question 3

How should we interpret VaR?

Answer 3

• If a bank sets a 99% 1-day VaR limit of 1mio
  
  • There is only 1% chance that the bank will lose more than 1 mio over the next day.
  • Alternatively: over the next 100 days, there will be 1 day on which the bank’s loss exceeds 1mio.
• VaR can be computed in value terms or in return terms:
  
  • Return: better for the riskiness of different portfolios
  • Value: Height of a buffer to counter losses

Question 4

What is the reference point of VaR?

Answer 4

• Absolute VaR: the reference point is the initial portfolio value, it implicitly assumes zero expected P&L or return over the horizon T.
• Relative VaR: the reference point is the expected future portfolio value: it analyzes deviations from the expected P&L or return over the horizon T.
  
  • Preference for relative VaR since it is more conservative by accounting for the time-value of money.
  • For a short horizon T (a single day), relative and absolute VaR are almost identical.

Question 5

What are the distributional assumptions?

Answer 5

• To calculate the VaR of a portfolio, we need to estimate the (uncertain) future portfolio values or returns. We simulate future portfolio values by imposing distributional assumptions:
• Two main approaches:
  
  1. Historical simulation: changes in the past to estimate the probability distribution of changes in the portfolio values observed in the future:
     
     • Replay history on current portfolio values.
  2. Model-building approach: assume a model for the joing distribution of changes in the portfolio values, often combined with historical data to estimate the model parameters.

Question 6

What is historical VaR?

Answer 6

• Historical simulation relies on past data and assumes that what happened in the past might happen in the future, it uses no economic theory or model to draw the distribution.

Question 7

What is the model building VaR?

Answer 7

• Model-building VaR approac assumes that the portfolio are generated by a parametric distribution:
  
  • eg. normal distribution
  • More generally: Monte Carlo simulated distribution
• Main advantages:
  
  • Past is not necessarily a good predictor of the guture = you do not rely on the historical track record.
  • simulate portfolio values, consistent with theoretical priors.
• Main disadvantage: appropriateness of model depends on how well underlying assumptions are valid.
  
  • If there is skewness or fat tails = normal distribution is problematic and leads to an inaccurate judgment of the risk you’re exposed to.

Question 8

What is the Monte Carlo simulated VaR model?

Answer 8

• Most flexible of all VaR methods: you can any distributional assumptions, incorporating both theoretical and empirical facts.
• Combination of historical data, models and parametric distributions, one simulates future portfolio values Vt.
• VaR is then, the best portfolio value (lowest loss) such that the probability to do worse is 1-c.

Question 9

How can we evaluate VaR?

Answer 9

• Widely used because of its many advantages:
  
  • Intuitive, easy to understand, also by non-technical investors or managers.
  • Easy to compute
  • Measure of risk that is uniform across trading unites and allows for comparison
  • Can be translated into capital buffer that one needs to hold to protect against losses.
• Disadvantages:
  
  • Only as good as its distributional assumptions
  • Not binding enough in limiting risks

Question 10

Why is the ES a good alternative?

Answer 10

• VaR is not very informative about the size of losses that could be incurred in the event that VaR is exceeded: this is a serious limitation.
  
  • Need a more detailed description of what happens in the tail beyond VaR.
• Expected shortfall: measures average level of loss, given that VaR is exceeded.
  
  • Also labelled conditional VaR, conditional tail expectation, expected tail loss.
  • ES = probability weighted average of gains that are below the threshold VaR, divided by the probability of exceeding VaR.

Question 11

How can we evaluate expected shortfall?

Answer 11

• Advantages:
  
  • Much more binding in limiting risks
  • More informative in terms of the actual extreme risks one is exposed to
  • Coherent risk measure
• Disadvantage:
  
  • More complicated to compute
  • Hard to backtest, you need to validate a model on realizations in the tail that did not occur.

Question 12

What are the key properties for a risk measure Q to be acceptable?

Answer 12

1. Monotonicity:
   
   • If V2 has weak stochastic dominance over V1, V2 should not be judged more risky than V1.
2. Translation invariance: adding an amount of risk-free capital K reduces the net capital at risk accordingly.
3. Homogeneity: scaling up the bet, scales up the risk.
4. Subadditivity
   
   • The risk of a diversified portfolio is no greater than the weighted average of the risks of the constituents.
   • Without subadditivity, there is no incentive to hold portfolio.

An acceptable risk measure is called a coherent risk measure. In general: VaR is not coherent, but ES is.

Question 13

What is the subadditivity of normal VaR?

Answer 13

• Wile in general VaR is not subadditive, under specific assumptions, VaR is subadditive.
• This is when the returns are normally distributed: mean)variance trade-off of Markowitz in which the portfolio risk is smaller than the average risk.
• Since normal VaR is a function of volatility, this result implies that normal VaR is a coherent risk measure.

Question 14

How do you choose the quantitive factors of VaR parameters?

Answer 14

• VaR is only determined for a particular horizon T and a particular level c

1. VaR/ES as a potential loss measure:
   
   • choice of c is arbitrary, because VaR is never the worst loss, it is only the loss corresponding with a particular significance level
   • Choice of horizon T is determined by the period over which the portfolio is assumed static = determined by its liquidity.
2. Capital cushion: dpeend on the kind of risk:
   
   • c is set high to capture tail risk
   • Choice of T is determined by the period over which the portfolio is assumed static (10 days for market risk - 1 year for credit risk).
3. Backtesting: allow to observe frequent breaches.
   
   • c is set rather low as to allow VaR to be breached frequently.
   • T is chosen rather short as to observe independent VaR breaches

Question 15

What is the goal of backtesting?

Answer 15

• When using a model to estimate the risk we are exposed to, we need to validate the model: reality check of whether the VaR model is adequate and tests whether the actual losses are in line with project losses.
• Compares the VaR forecasts with actual portfolio returns: is the number of times VaR is breached acceptable.
• Goal:
  
  1. Identify if the VaR limits are too loose: allocate too few capital to buffer the losses.
  2. Identify VaR limits which are too strict: allocate too much capital to buffer losses which is inefficient = capital is expensive and scarce.

Question 16

How do you set up a backtest?

Answer 16

1. Identify number of times VaR was breached?
2. Compare with the expected number of VaR breaches?

• In a typical backtest, actual returns are compared with the VaR implied returns: such comparison of actual returns is not innocent.
• In computing VaR we implicitly assume that the portfolio remains frozen over the VaR-horizon.
  
  • But: actual portfolio composition is often changed, leading to portfolio values/returns that deviate from these fixed VaR simulated values/returns
• A complete backtest should compare not only to actual returns, but also the hypothetical fixed VaR returns.

Question 17

What are the types of backtests?

Answer 17

1. One-tailed Bernoulli test
2. Two-tailed Kupiec test
3. Clustered breaches:
   
   • Observe that VaR breaches are serially dependent even though in 1 and 2 it is assumed tat breaches are not clustered in time.
   • Extend the Kupiec test to account for serial correlation: VaR breaches conditional on what happened the day before.

Question 18

What is the trade-off in backtesting?

Answer 18

• Backtesting is a balancing operation:
  
  1. You want to identify understatements of risk.
  2. But you do not want to penalize bad luck.
• Formally: you need to balance:
  
  • Type I error: rejecting a correct model
  • Type II error: accepting a faulty model
• You need to create powerful test: allowing to observe sufficient breaches:
  
  • Keep the time horizon short
  • Choose a low confidence level.

Question 19

What are portfolio VaR tools?

Answer 19

• One of the key attractive properties of VaR is that it allows for aggregation of various isks into a single measure.
  
  • You can have an overall view on the degree of risk you are exposed to.
  • Disadvantage: no clear view on the underlying risk drivers or dynamics.
• 3 complementary VaR tools:
  
  1. Marginal VaR: MVaR
  2. Incremental VaR
  3. Component VaR

Question 20

What is the marginal VaR?

Answer 20

• Measure of the sensitivity of VaR to the amount V invested in component i.
• Goal of this marginal VaR is to rank trades based on VaR sensitivities: the higher marginal VaR, the higher the sensitivity of the portfolio to a given change in the amount invested / traded.

Question 21

What is the incremental VaR?

Answer 21

• Incremental VaR is a measrue of the incremental impact on VaR of adding a position X to the existing portfolio.
• When deciding on a new trade, such an incremental VaR allows us to get an idea of the impact of the trade on portfolio risk.
  
  • IVaRx> 0: adding compontent X increases the risk
  • IVaRx< 0: adding compontent X decreases the risk = hedge
• Drawback: time-consuming so approximation based on the Taylor series expansion.

Question 22

What is component VaR?

Answer 22

• Measure of importance of the different risks in a portfolio.
• It is additive and it reflects benefits of diversification.
• If CVaR < 0 the component i decreases the portfolio risk = hedge
• If CVaR > 0 the component i increases the portfolio risk