CHAPTER 10

Overcoming VaR's Limitations

INTRODUCTION

• While VaR is the single best way to measure risk, it does have several limitations. The most pressing limitations are the following:– It assumes that the variances and correlations

between the market-risk factors are stable.– It does not give a good description of extreme

losses beyond the 99% level.– It does not account for the additional danger o

f holding instruments that are illiquid

INTRODUCTION

• One approach to addressing VaR's limitations is to measure the risk both with VaR and with other, completely different methods, such as stress and scenario testing, as discussed in Chapter 5

• In this chapter, we discuss approaches that can be used to augment the standard VaR methods

• These methods allow the risk manager to improve the measurement of VaR, thereby improving the ability to set capital, measure performance, and identify excessive risks

INTRODUCTION

• This chapter has three sections– An approach to letting variances change over

time– several approaches for assessing extreme

events– several approaches to quantify liquidity risk.

ALLOWING VARIANCE TO CHANGE OVER TIME

• The usual approach to constructing the covariance matrix is to calculate the variance of the risk factors over the last few months

• Then we assume that tomorrow's changes in the risk factors will come from a distribution that has the same variance as experienced historically

ALLOWING VARIANCE TO CHANGE OVER TIME

• We could write this as an equation by saying that the expected variance tomorrow is the variance of changes in the factor over the last few months:

the change of risk factor on day t

We assume that the mean change is relatively small and therefore neglect it from the equation

ALLOWING VARIANCE TO CHANGE OVER TIME

• Although this approach is simple and robust

• it is well known by practitioners that the volatility of the market changes over time

• Sometimes the market is relatively calm, then a crisis will happen, and the volatility will jump up

ALLOWING VARIANCE TO CHANGE OVER TIME

• GARCH is an approach that allows the estimation of variance to vary quickly with recent market moves

• GARCH stands for Generalized Autoregressive Conditional Heteroskedasticity

• The variance on one day is a function of the variance on the previous day

ALLOWING VARIANCE TO CHANGE OVER TIME

• GARCH assumes that the variance is equal to a constant plus a portion of the previous day's change in the risk factor, plus a portion of the previous day's estimated variance

ALLOWING VARIANCE TO CHANGE OVER TIME

• We can also use GARCH to estimate the covariance between two factors, x and y:

APPROACHES FOR ASSESSING EXTREME EVENTS

• The usual implementations of Parametric and Monte Carlo VaR assume that the risk factors have a Normal probability distribution.

• As discussed in the statistics chapter, most markets, especially poorly developed markets, exhibit many more extreme movements than would be predicted by a Normal distribution with the same standard deviation

• The term used to describe probability distributions that have a kurtosis greater than that of the Normal distribution is leptokurtosis

• Leptokurtosis can be considered to be a measure of the fatness of the tails of the distribution

• Measuring the effects of leptokurtosis is important because risk factors with a high kurtosis pose greater risks than factors with the same variance but a lower kurtosis

(a)PDF of Dow Jones Index Return Shock: Linear Model

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-5 -4.7 -4.4 -4.1 -3.8 -3.5 -3.2 -2.9 -2.6 -2.3 -2 -1.7 -1.4 -1.1 -0.8 -0.5 -0.2 0.12 0.42 0.72 1.02 1.32 1.62 1.92 2.22 2.52 2.82 3.12 3.42 3.72 4.02 4.32 4.62 4.92

(a)PDF of Dow Jones Index Return Shock: Linear Model

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-5 -4.7 -4.4 -4.1 -3.8 -3.5 -3.2 -2.9 -2.6 -2.3 -2 -1.7 -1.4 -1.1 -0.8 -0.5 -0.2 0.12 0.42 0.72 1.02 1.32 1.62 1.92 2.22 2.52 2.82 3.12 3.42 3.72 4.02 4.32 4.62 4.92

APPROACHES FOR ASSESSING EXTREME EVENTS

• We will describe four techniques that are used to assess the additional risk caused by leptokurtosis– Jump Diffusion– Historical Simulation– Adjustments to Monte Carlo Simulation– Extreme Value Theory

Jump Diffusion

• The jump-diffusion model assumes that tomorrow's random change in the risk factor can come from one of two Normal distributions

• One distribution describes the typical market movements

• the other describes crisis movements. In simplified form

• there is a probability of P that the sample will come from the typical distribution and a small probability of (1 — P) that it will come from the crisis distribution

Jump Diffusion

•The main problem to this approach is that it is difficult to determine the parameter values

•In the model above, five parameters must be determined

Jump Diffusion

• Normal +Normal=Normal?

• The difference between simple plus and mixing

• The mixing distribution – some observations from Dist. 1– other observations from Dist. 2

Jump Diffusion

0.00

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-5 -4.2 -3.5 -2.7 -2 -1.2 -0.5 0.27 1.02 1.77 2.52 3.27 4.02 4.770.00

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-5 -4.2 -3.4 -2.6 -1.8 -1 -0.2 0.62 1.42 2.22 3.02 3.82 4.62

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0.01

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-5 -4.2 -3.5 -2.7 -2 -1.2 -0.5 0.271.021.772.523.274.024.77

x11,x12,x13,x14,..

x21

x22

x23

…

x21

x22

x23

x11,x12, .……………… x13,x14

-----Distribution 2: A high Volatility

Distribution

_____Distribution 1: A Low Volatility

Distribution

---- Distribution 2___ Distribution 1

Historical Simulation

• Historical simulation does not require an assumption for the form of the probability distribution

• It simply takes the price movements that have occurred and uses them to revalue the portfolio directly

• However, historical simulation is strongly backward looking because the changes in the risk factors are determined by the last crisis, not the next crisis.

Adjustments to Monte Carlo Simulation

• Usually, Monte Carlo simulation uses Normal distributions

• However, it is also possible to carry out Monte Carlo evaluation using leptokurtic distributions, such as jump diffusion or the Student's T distribution

• It is relatively easy to create such distributions for single risk factors, but more difficult to ensure that the correlations between the factors are correct

Extreme Value Theory

• Extreme Value Theory (EVT) takes a different approach to calculating VaR

• EVT concentrates on estimating the shape of only the tail of a probability distribution

• Given this shape, we can find estimates for losses associated with very small probabilities, such as the 99.9% VaR

• A typical shape used is the Generalized Pareto Distribution that has the following form:

Extreme Value Theory

• Here, a, b, and c are variables that are chosen so the function fits the data in the tail

• The main problem with the approach is that it is only easily applicable to single risk factors

• It is also, by definition, difficult to parameterize because there are few observations of extreme events

LIQUIDITY RISK

• The Importance of Measuring Liquidity Risk– Liquidity risks can increase a bank's losses; therefore,

they should be included in the calculation of VaR and economic capital

– There are two kinds of liquidity risk: liquidity risk in trading, and liquidity risk in funding (also known as funding risk)

– The funding risk is the possibility that the bank will run out of liquid cash to pay its debts

– This funding risk is usually considered in the framework for asset liability management and will be discussed in later chapters

– This chapter discusses liquidity risk in trading

LIQUIDITY RISK

– The liquidity risk in trading is the risk that a trader will be unable to quickly sell a security at a fair price. This could happen if few people normally trade the given security e.g., if it was the equity for a small company

– It could also happen if the general market is in crisis and few people are interested in buying new securities

LIQUIDITY RISK

• We can view these two possible loss mechanisms as two extreme manifestations of the same problem– In one extreme, the trader sells immediately at an

unusually low price– In the other extreme, the trader slowly sells at the

current fair price, but risks suffering additional losses• It is important to recognize the liquidity risk

because it can add significantly to losses• Furthermore, if liquidity risk is not included in the

risk measurement, it gives incentives to traders to buy illiquid securities

Quantifying Liquidity

• The close-out time– is the time required to bring the position to a state wh

ere the bank can make no further loss from the position

– It is the time taken to either sell or hedge the instrument

– The number of days can be based on the size of the position held by the trader compared with the daily traded volume

Quantifying Liquidity

• F is a factor that gives the percentage of the daily volume that can be sold into the market without significantly shifting the price

• If F were set equal to 10%, it would imply that 10% of the daily volume can be sold each day without significantly shifting the market

• The Daily Volume can be the average daily volume or the volume in a crisis period

• The volume in a crisis period could be approximated as the average volume minus a number of standard deviations

Quantifying Liquidity

• Another alternative to quantifying the liquidity risk is to measure the average bid-ask spread relative to the mid price

• The bid is the price that investors are willing to bid (or pay) to own the security.

• The ask is the price that owners of the security are asking to sell the security

• The mid is halfway between the bid and ask• If the bid and ask are close to the mid, it implies that

there are many market participants who agree on the fair value of the security and are willing to trade close to that price

Quantifying Liquidity

• If the bid-ask spread is wide, it means that few investors are willing to buy the security at the price the sellers think is fair

• If a trader wanted to sell the security immediately, the trader would have to lower the ask price to equal the bid rather than wait for some investor to agree that the high ask price was fair

• Both the close-out time and the bid-ask spread can be used to quantify liquidity risk

• We will explore how in the following sections.

Using Close-Out Time to Quantify Liquidity Risk

• The most common approach to assessing the liquidity risk is to use the "square-root-of-T” adjustment for VaR

• This is also known as "close-out-adjusted VaR“• The result of the approach is that the VaR for a

position that takes T days to close is assumed to equal the VaR for an equivalent liquid position that could be closed out in one day times the square root of T

Using Close-Out Time to Quantify Liquidity Risk

• The approach assumes that the position will be held for T days

• then on the last day, it will be sold completely• It uses the reasoning that the losses over T

days will be the sum of losses over the individual days

is the cumulative loss over T days is the loss on day 2

Using Close-Out Time to Quantify Liquidity Risk

• We can assume with reasonable accuracy that losses are independent and identically distributed (IID)

• meaning losses are not correlated day to day, and the standard deviation of losses is the same each day

• The variance of the loss over T days is therefore the sum of the variance of the losses on the individual days

Using Close-Out Time to Quantify Liquidity Risk

• If we assume that the variance of the losses on each day is the same, then the sum equals T times the variance on the first day

Using Close-Out Time to Quantify Liquidity Risk

• A slightly refined approach is to assume that the position is closed out linearly over T days

• In this case, the variance of the loss decreases linearly each day

Using Close-Out Time to Quantify Liquidity Risk

Using Close-Out Time to Quantify Liquidity Risk

• To illustrate the difference between this and the simple square-root-of-T adjustment

• consider a closeout period of 10 days

• The square-root-of-T method gives:

Using Close-Out Time to Quantify Liquidity Risk

• Whereas the linear close-out gives a measure of VaR that is significantly smaller

Using the Bid-Ask Spread to Assess Liquidity Risk

• The closeout adjustments discussed above assumed that the trader was taking one extreme course of action by gradually closing out the position at the mid price and refusing to give any discount

• The other extreme is to assume that the trader will sell out immediately by giving a discount that brings the price down to the bid price. This discount is an additional loss

• Please refer to Page 163