An Introduction to Market Risk Measurement - CiteSeerX

Consequently, Linsmeier and Pearson continue, "Perhaps the best answer begins: "Value at risk is. Recipients can also specify the horizon period - next day, next week, month, quarter, etc.

INTENDED READERSHIP

ETL is the loss we can expect if we get a loss in excess of VaR. Fortunately, it turns out that we can always estimate ETL if we can estimate VaR.

USING THIS BOOK

My advice to those who may use the book for teaching purposes is the same: cover the tools first, then do the risk assessment. 2The user must copy the An Introduction to Market Risk Measurement(IMRM) folder to his or her MATLAB works folder and activate the path to the IMRM folder thus created (so that MATLAB knows that the folder is there).

OUTLINE OF THE BOOK

CONTRIBUTORY FACTORS

A Volatile Environment
Growth in Trading Activity
Advances in Information Technology

One factor behind the rapid development of risk management was the high level of volatility in the economic environment within which firms operated. A third contributing factor to the development of risk management was the rapid advancement in the state of information technology.

RISK MEASUREMENT BEFORE V A R

Gap Analysis
Duration Analysis
Scenario Analysis
Portfolio Theory
Derivatives Risk Measures

hedges are therefore inaccurate against yield changes that include shifts in the slope of the yield curve. The moral of the story is that the extent to which a new asset contributes to portfolio risk depends on.

VALUE AT RISK

The Origin and Development of VaR
Attractions of VaR
Criticisms of VaR

It is simply a way of describing the magnitude of potential losses in the portfolio. Another characteristic of VaR is that it takes into account the correlations between different risk factors.

THE MEAN–VARIANCE FRAMEWORK FOR MEASURING FINANCIAL RISK

The Normality Assumption
Limitations of the Normality Assumption
Traditional Approaches to Financial Risk Measurement .1 Portfolio Theory

Equation (2.2b) is the normal quantile corresponding to the confidence levelcl (i.e. the lowest value we can expect at the stated confidence level) and allows us to answer quantity questions. The third advantage of the normal distribution is that it only requires estimates of two parameters – the mean and the standard deviation (or variance) – because it is completely described by these two parameters alone.

Figure 2.1 The normal probability density function.

VALUE AT RISK

VaR Basics
Choice of VaR Parameters
Limitations of VaR as a Risk Measure

The VaR of the diversified portfolio is therefore much greater than the VaR of the undiversified one. The VaR of the combined position is therefore greater than the sum of the VaRs of the individual positions; and the VaR is not sub-additive (Table 2.1).

EXPECTED TAIL LOSS

Coherent Risk Measures
The Expected Tail Loss

This ETL can also be viewed as the cohesive measure of risk associated with one distribution function. The ETL therefore has the various advantages of sub-additivity, while the VaR does not.

CONCLUSIONS

The prices of the securities in each scenario are then generated by a suitable model (eg black model) and the risk measure is the maximum loss incurred, using the full loss for the first 14 scenarios and the 35% loss for the last two extreme scenarios. This risk measure can be interpreted as the maximum expected loss under each of 16 different probability measures, and is therefore a coherent measure of risk.

DATA

Proﬁt/Loss Data
Loss/Proﬁt Data
Arithmetic Returns Data
Geometric Returns Data

When using arithmetic returns, we implicitly assume that the intermediate payment Dt does not earn its own return. One answer is that the difference between the two will be negligible if the returns are small, and the returns will be small if we are dealing with a short time horizon.

ESTIMATING HISTORICAL SIMULATION V A R

With arithmetic returns, a low realized return - or a high loss - implies that the asset valuePt is negative, and a negative asset price rarely makes economic sense; on the other hand, a very low geometric return implies that the asset price falls to zero but is still positive. So, if our data is an array called 'Loss data', our VaR is given by the Excel command 'Large (Loss data,6)'.

ESTIMATING PARAMETRIC V A R

Estimating VaR with Normally Distributed Proﬁts/Losses
Estimating VaR with Normally Distributed Arithmetic Returns
Estimating Lognormal VaR

Since the data are in P/L form, VaR is indicated by the negative of the cut point between the lower 5% and upper 95% of P/L observations. In this case, VaR is given by the breakpoint between the upper 5% and lower 95% of the L/P observations.

Figure 3.2 shows the VaR at the 95% conﬁdence level for a normally distributed P/L with mean 0 and standard deviation 1

ESTIMATING EXPECTED TAIL LOSS

To give an idea of what this might be, Table 3.2 lists some alternative ETL estimates obtained using this procedure, with varying values ofn. These results show that the estimated ETL rises and gradually converges to a value close to 2.062.

Table 3.1 Estimating ETL as a weighted aver- aver-age of tail VaRs

SUMMARY

To avoid these problems, we need to ensure that our risk factors are not too closely linked, and this requires us to select an appropriate set of risk factors and tailor our instruments accordingly. The final stage is to calculate VaR and/or ETL using the charted instruments (i.e. the synthetic substitutes) instead of the actual instruments we own.

A3.1 SELECTING CORE INSTRUMENTS OR FACTORS

Furthermore, since the principal components are independent of each other, the only non-zero elements of the variance–covariance matrix will be the principal volatility components, further reducing the number of parameters we would need. But if you were to use principal component analysis, you could probably replace a huge proportion of bond price movements with three principal components, and the only variance-covariance parameters needed would be the volatilities of the three principal components.

A3.2 MAPPING POSITIONS AND V A R ESTIMATION

Basically, we map equity risk to the market index and use equation (A3.5) to estimate the VaR of the mapped equity return. Once our bond has been mapped, we estimate its VaR by estimating the VaR of the mapped bond (i.e., the portfolio of zeros at 5 and 7 years, with weights ω and 1−ω).

A3.3 RECOMMENDED READING

COMPILING HISTORICAL SIMULATION DATA

The first task is to construct an appropriate P/L series for our portfolio, and this requires a set of historical P/L or return observations on the positions in our current portfolio. Instead, the historical simulation P/L is the P/L we would have earned on our current portfolio if we had held it for the entire historical sample period.1.

ESTIMATION OF HISTORICAL SIMULATION V A R AND ETL

Basic Historical Simulation
Estimating Curves and Surfaces for VaR and ETL

It is more difficult to construct curves that show how VaR or ETL changes with holding period. In short, there is no theoretical problem per se with estimating HS VaR or ETL over any investment period.

Figure 4.2 Plots of HS VaR and ETL against conﬁdence level.

ESTIMATING CONFIDENCE INTERVALS FOR HISTORICAL SIMULATION V A R AND ETL

A Quantile Standard Error Approach to the Estimation of Conﬁdence Intervals for HS VaR and ETL
An Order Statistics Approach to the Estimation of Conﬁdence Intervals for HS VaR and ETL
A Bootstrap Approach to the Estimation of Conﬁdence Intervals for HS VaR and ETL

We then generate a large number of such datasets and estimate the VaR of each. It is also interesting to compare the VaR and ETL conﬁdence intervals obtained by the two methods.

WEIGHTED HISTORICAL SIMULATION

Age-weighted Historical Simulation
Volatility-weighted Historical Simulation
Filtered Historical Simulation

Therefore, our core information – the information fed into the HS estimation process – is the paired set of P/L values and associated weights w(i), rather than the traditional paired set of P/L values and associated equal weights 1/ n. . Because λnw(1) is probably less than 1/n for any reasonable value of λandn, the shock—the ghost effect—will be smaller than it would be under equal-weighted HS.

ADVANTAGES AND DISADVANTAGES OF HISTORICAL SIMULATION

Advantages
Disadvantages

For example, if there is a permanent change in exchange rate risk, it will usually take time for the HS VaR or ETL estimates to reflect the new exchange rate risk. In the most sophisticated versions of HS, such as FHS, or those proposed by HW, Duffie and Pan, or Holton (see Box 4.1), this constraint takes a softer form, and in periods of high volatility (or, depending on the approach used) , high correlation, etc.) we can get VaR or ETL estimates that exceed our largest historical loss.

Fortunately, we can deal with most of these problems (except the last) by suitable modifications of HS - especially by age weighting to deal with distortions, ghosting and novelty effects. However, we can estimate these conﬁdence intervals by supplementing PCA or FA methods with order statistics or bootstrap approaches.

CONCLUSIONS

These include the functions 'hsvar' and 'hsetl', which estimate HS VaR and ETL, 'hsvardfperc' and 'hsetldfperc', which estimate percentiles from VaR and ETL distribution functions using the theory of order statistics, 'hsvargure' and 'hsetlgure', which produce pdf figures with HS VaR and ETL,. The IMRM toolbox also includes the quantile standard error functions discussed in Section 4.3.1 and the bootstrap VaR and ETL functions discussed in Section 4.3.3.

NORMAL V A R AND ETL

General Features
Disadvantages of Normality

If our VaR over a 1-day holding period is VaR(1,cl), then the VaR over a holding period of hpdays is VaR(hp,cl), given by:. It therefore never turns down and the VaR surface rises upwards as the confidence level and holding period approach their maximum values.

Figure 5.1 VaR and ETL for normal loss/proﬁt.

THE STUDENT t -DISTRIBUTION

If our distribution is not normal, but the departures from normality are 'small', we can approximate our non-normal distribution using the Cornish-Fisher expansion, which tells us how to adjust the standard normal variate to accommodate non-normal skewness and kurtosis. When using the Cornish-Fisher approximation, we must bear in mind that it will only provide a 'good' approximation if our distribution is 'close' to being normal, and we cannot expect it to be of much use , if we have a distribution that is too abnormal.

THE LOGNORMAL DISTRIBUTION

The log-normal VaR was previously illustrated in Figure 3.5, and the typical log-normal VaR area – that is, the VaR area with positive µR – is shown in Figure 5-6. This is shown in Figure 5.7, which is directly analogous to the normal zero mean case shown in Figure 2.9.

EXTREME VALUE DISTRIBUTIONS

The Generalised Extreme Value Distribution
The Peaks Over Threshold (Generalised Pareto) Approach

The relationship of the location and scale parameters to the mean and variance is explained in Tool No. These decisions will have a critical effect on the VaR and on the shape of the VaR surface.

THE MULTIVARIATE NORMAL VARIANCE–COVARIANCE APPROACH

The explanation is that the shocks to the two returns perfectly offset each other, so the portfolio return is safe; the standard deviation of the portfolio return and VaR are therefore both zero. Apart from the special case where ρ =1 - in which case σpis alwaysσ - the standard deviation of the portfolio decreases as much as possible.

CONCLUSIONS

The curves in the figure are the standard deviations of the portfolio taking into account the effects of diversification divided by the standard deviation of the portfolio if the effects of diversification are not taken into account (ie, and invested entirely in one asset). Furthermore, given the parameters assumed in Figure 5.10, these same curves also give us the ratio of diversified to undiversified VaR - giving the VaR with diversification taken into account divided by the VaR without taking diversiﬁcation into account (ie the VaR we would get). if we invested in only one asset).

A5.1 DELTA–NORMAL APPROACHES

As a result, the price/volatility behavior of a callable or convertible bond can be quite different from that of a corresponding 'straight' bond. The rule of zero arbitrage must ensure that the price of any instrument with an embedded option is the same as the price of the corresponding straight instrument plus or minus the price of the embedded option.

A5.2 DELTA–GAMMA APPROACHES

14 A good example is the option position just considered, since the delta-gamma estimate of VaR is actually worse than the delta-normal one. Equation (A5.7) implies that the delta-gamma-normal procedure gives an estimate of VaR that is even higher than the delta-normal estimate, and the delta-normal estimate is already too large.

A5.3 CONCLUSIONS

This method also gives the portfolio's marginal VaRs, and is therefore very useful for risk decomposition. The second-order approximation approach used to handle nonlinearity in option positions can also be used to handle nonlinearity in bonds.

A5.4 RECOMMENDED READING

OPTIONS V A R AND ETL

Preliminary Considerations
Reﬁning MCS Estimation of Options VaR and ETL

It is therefore important to use all possible refinements – to use control variables, stratified sampling or whatever other refinements are appropriate, and to choose good values for MandN – to speed up the calculations and/or reduce the number Reduce. number of calculations required.3 6.1.2 An example: estimating the VaR and ETL of a US well. To illustrate, suppose we want to estimate the VaR and ETL of a US well.

Basic Principal Components Simulation
Scenario Simulation

These findings suggest that we may want to focus on the first three principal components. Jamshidian and Zhu suggest that we can allow seven states for the first component, five for the second, and three for the third.

FIXED-INCOME V A R AND ETL

General Considerations
A General Approach to Fixed-income VaR and ETL

So we need a projected future term structure to price our instruments at the end of the holding period so that we can then estimate future earnings and thereby estimate VaR and ETL. For convenience, we assume that the term structure is unchanged at 5%, so all spot rates are 5%.

ESTIMATING V A R AND ETL UNDER A DYNAMIC PORTFOLIO STRATEGY

We also assume that when the value of the portfolio increases (or decreases), α% of the increase (or decrease) in the portfolio value is invested in (or withdrawn from) shares of the venture asset. The effect of the filter rule strategy on VaR is shown in Figure 6.2, which shows an illustrative VaR against different values for the participation rate.

Figure 6.1 VaR with a stop-loss portfolio management strategy.

ESTIMATING CREDIT-RELATED RISKS WITH SIMULATION METHODS

We also need to consider the rate of recovery and how this may change over time. We can examine the impact of recovery rate by plotting VaR against a range of recovery rates, as in Figure 6.3.

ESTIMATING INSURANCE RISKS WITH SIMULATION METHODS

We then adjust losses for the deductible and pricing policy and determine the VaR and ETL from the sample of adjusted losses. A parameterized example is given in Figure 6.4, which shows a histogram of simulated L/P values and associated VaR and ETL estimates – in this case, the L/P histogram has a long right tail (which is actually lognormal), and VaR and ETL are equal to 17.786 or 24,601.

ESTIMATING PENSIONS RISKS WITH SIMULATION METHODS

Estimating Risks of Deﬁned-beneﬁt Pension Plans
Estimating Risks of Deﬁned-contribution Pension Plans

In this context, pension risk is the risk that the pension provider's assets do not meet its obligations. The programming strategy is to model the pension fund's terminal value during each simulation.

Figure 6.5 VaR of deﬁned-beneﬁt pension scheme.

CONCLUSIONS

Running these calculations, we find that the pension ratio has a sample mean of 0.983 and a sample standard deviation of 0.406 - which should indicate that DC schemes can be very risky even without looking at any VaR analysis. The pension VaR - the likely worst pension outcome at the relevant (and in this case, 95%) confidence level - is 0.501, and this suggests that there is a 95%.

INCREMENTAL V A R

Interpreting Incremental VaR
Estimating IVaR by Brute Force: The ‘Before and After’ Approach

If it is a 'small' relative peak, we can approximate the VaR of our new portfolio (ie VaR(p+a)) by . Once we have the portfolio's VaR and delVaRs, we can then take any candidate trade, map the trade, and use the mapped trade and delVaRs to estimate the IVaR associated with that candidate trade.

Figure 7.1 Incremental VaR and relative position size.

COMPONENT V A R

Properties of Component VaR
Uses of Component VaR .1 ‘Drill-down’ Capability

This is restrictive because it implies that the component VaR is proportional to the position size: if we change the size of the position byk%, the component VaR will also change by k%. However, it also allows us to break down our risks into multiple levels, and at each level the component risks will properly add up to the total risk of the entity at the next level up.

CONCLUSIONS

Comparing implied insights about returns with actual insights is a useful tool to help understand how portfolios can be improved. A good example, suggested by Litterman (1997b, p. 41), occurs in large financial institutions whose portfolios are influenced by large numbers of traders operating in different markets: at the end of each day, the implicit views of the portfolio are estimated and compared with the actual views of, for example, internal forecasters.

LIQUIDITY AND LIQUIDITY RISKS

The bid-ask spread also has an associated risk because the spread itself is a random variable. However, if our position is large relative to the market, our activities will have a noticeable effect on the market itself, and can affect both the 'market' price and the bid-ask spread.

ESTIMATING LIQUIDITY-ADJUSTED V A R AND ETL

A Transactions Cost Approach
The Exogenous Spread Approach
The Market Price Response Approach
Derivatives Pricing Approaches
The Liquidity Discount Approach
A Summary and Comparison of Alternative Approaches

The second term gives the effect of the bid-ask spread on transaction costs, increased by the amount redeemed at the end of the holding period. In such cases, they suggest that we think about liquidity risk in terms of the spread between supply and demand and its volatility.

Figure 8.1 The impact of holding period on LVaR.

ESTIMATING LIQUIDITY AT RISK (L A R)

The second point is that positions that may be similar from a market risk perspective (eg, such as a futures hedge and an options hedge) may have very different cash flow risks. The choice of engine will depend on the types of cash flow risks we have to face.

ESTIMATING LIQUIDITY IN CRISES

In such circumstances, it is often better to build a model for measuring liquidity risk from scratch, and we can begin to outline the basic types of cash flows to consider. They suggest that we should use a method of estimating the worst-case losses – they actually suggest a Greece-based approach – and then derive the cash flows by applying margin requirements to the loss involved.

PRELIMINARY DATA ISSUES

Obtaining Data

Given the number of observations (200) and the VaR confidence levels, we would expect 10 positive and 10 negative exceptions, and in reality we get 10 positive and 17 negative exceptions. The number of negative exceptions (or tail losses) is well above what we would expect, and the risk expert would do well to investigate this further.

STATISTICAL BACKTESTS BASED ON THE FREQUENCY OF TAIL LOSSES

The Basic Frequency-of-tail-losses (or Kupiec) Test
The Time-to-ﬁrst-tail-loss Test
A Tail-Loss Conﬁdence-interval Test
The Conditional Backtesting (Christoffersen) Approach

4 The Kupiec test also provides useful information about the pattern of tail losses over time. We can construct a confidence interval for the number of tail losses using the inverse of the tail loss binomial distribution (eg, using the 'binofit' function in MATLAB).

Figure 9.2 Probabilities for the time of ﬁrst tail loss.

STATISTICAL BACKTESTS BASED ON THE SIZES OF TAIL LOSSES

The Basic Sizes-of-tail-losses Test
The Crnkovic–Drachman Backtest Procedure
The Berkowitz Approach

The main difference between these tests is that the tail loss size test considers loss sizes that exceed VaR, while the Kupiec test does not. This means that the tail loss size test can be considered as a generalization of the (first) CD test.

Figure 9.3 Predicted and empirical tail-loss distribution functions.

FORECAST EVALUATION APPROACHES TO BACKTESTING

Basic Ideas
The Frequency-of-tail-losses (Lopez I) Approach
The Size-adjusted Frequency (Lopez II) Approach
The Blanco-Ihle Approach
An Alternative Sizes-of-tail-losses Approach

This 'Lopez I' loss function is intended for the user concerned (solely) with the frequency of tail losses. This loss function gives each tail loss observation a weight equal to the tail loss divided by the VaR.

OTHER METHODS OF COMPARING MODELS

ASSESSING THE ACCURACY OF BACKTEST RESULTS

In this case, we can be at least 95% sure that the true prob value is above the critical level, and we can pass the model with confidence. In this case, we can be at least 95% sure that the true prob value is below the critical level, and we can confidently reject the model.

BACKTESTING WITH ALTERNATIVE CONFIDENCE LEVELS, POSITIONS AND DATA

Backtesting with Alternative Conﬁdence Levels
Backtesting with Alternative Positions
Backtesting with Alternative Data

If we take a large number of such samples, we can get an idea of the distribution of these probable value estimates. We can also backtest at any company level we want, down to the level of the individual trader or asset manager, or the individual instrument we hold.

Figure 9.5 A general backtesting framework.

SUMMARY

Therefore, we need to bootstrap our posterior tests to get a better idea of the distribution of the prob value estimates. The IMRM toolbox also includes functions for a modified version of the Crnkovic-Drachman test assuming normal P/L, 'modiﬁednormalCDbacktest', Kupiec backtest, 'kupiecbacktest', Lopez prediction estimation backtest, 'lopezbacktest', tail sizes - The last statistical loss test for normal P/L, the 'normaltaillossesbacktest', discussed in Section 9.3.1, and the result of estimating the loss prediction of tail sizes presented in Section 9.4.5, the 'tailloss-FEbacktest'.

BENEFITS AND DIFFICULTIES OF STRESS TESTING

Beneﬁts of Stress Testing
Difﬁculties with Stress Tests

Even stress testing is completely dependent on the selected scenarios and thus the judgment and experience of the people performing the stress testing. The result of this process can be thought of as a set of P/L numbers and their associated probabilities.

SCENARIO ANALYSIS

Choosing Scenarios
Evaluating the Effects of Scenarios

We may also want to consider the impact of other factors, such as changes in the slope or shape of the yield curve, a change in correlations and a change in credit spreads (eg a jump or dip in the TED spread). Alternatively, we can focus on the mean of the conditional loss distribution, in which case our stress loss is X1tR1t+X2tΣ21Σ−111R1t.

MECHANICAL STRESS TESTING

Factor Push Analysis
Maximum Loss Optimisation

The solution to this last problem is to look for losses that occur for intermediate and extreme values of the risk variables. If the portfolio consists of straight positions, each of which takes its maximum loss at extreme values of the underlying risk factors, then FP and MLO will give exactly the same results, and we can also use the simpler computational approach of FP .

CONCLUSIONS

We can obtain comparable expressions for multi-option portfolios provided we can model the relationship—and more specifically, the variance-covariance matrix—between the underlying variables.

11 Model Risk

MODELS AND MODEL RISK

Models