Measuring Risk Review of statistical concepts Probability distribution Discrete and continuous probability distributions. Discrete: Probability mass function assigns a probability to each possible out-come. Probabilities are non-negative and add up to 1. probability 1 3/4 1/2 1/4 1 2 3 4 5 6 2 with a prob. of 1/2 4 with a prob. of 1/2 outcomes Review of statistical concepts Probability distribution Continuous: Probability density function describes the relative likelihood of specific values of the distribution. It is non-negative and the area under the curve is equal to 1. relative likelihood 1 2 3 4 5 6 outcomes 1
Review of statistical concepts - Mean Application 2:Actuarially fair premium for an insurance contract If represents the claim on an insurance contract, the actuarially fair premium is equal to the expected value of. Probability of accident: Size of the claim in case of accident:, 0 Basic rule #1 (one-period version): The first basic rule of risk management is that provisions should be (at least) equal to expected losses. Review of statistical concepts Variance and standard deviation Variance and standard deviation: Dispersion of outcomes around the mean relative likelihood σ outcomes Review of statistical concepts Variance and standard deviation Two simple risks: 110 105 90 95 Basicrule#2: The second basic rule of risk management is that a minimum level of capital should be kept as a buffer against unexpected losses. This minimum level (called economic capital) is equal to or larger than a multiple of the standard deviation of the portfolio of risks:.thenumber is called the safety coefficient and is chosen by the firm. 2
Review of statistical concepts Variance and standard deviation Can be justified by Chebyshev s inequality, which states that Var P Application: Use Chebyshev s inequality to determine the level of economic capital such that the shortfall probability does not exceed 1%. Do you think your estimate is practical? Better results are obtained if we assume losses to follow a normal distribution. The normal distribution The normal (or Gaussian) distribution is the most common continuous pro-bability distribution. She is the queen of distributions! The probability density function is given by 1. 2 Some information: Denoted by,.takes values on the entire real line. Mean, variance, standard deviation. If 0and 1, the distribution is called standard normal distribution or unit normal distribution. Sometimes it is informally called the bell curve. The normal distribution 3
The normal distribution Standardization The normal distribution can be standardized. Assume that has a normal distribution with mean and standard deviation. Look at Mean: 0. Variance: Var Var Var 1. follows a standard normal distribution. The normal distribution Standardization The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter Φ, is the integral In practice: Table Φ Excel: =NORM.DIST(x,μ,σ) 1 2 d. The normal distribution - Standardization 4
Review of statistical concepts Variance and standard deviation Assume that losses follow a Normal distribution with mean and standard deviation. What is the level of economic capital such that the shortfall probability does not exceed 1? P P 1Φ Need to find such that 1Φ 1. Solve for Φ. is called the quantile of the standard Normal distribution. Excel: NORM.INV(,, Review of statistical concepts Variance and standard deviation Application: Complete the following table Some quantiles of the Standard Normal Threshold 95% 99% 99.5% 99.9% quantile Φ Use it to find the level of economic capital such that the probability of shortfall does not exceed 1%. Compare your result to that obtained from the Chebyshev inequality. Which approach is better? Some new concepts Skewness Skewness is a measure of the symmetry of a distribution. It informs us whether more risk is on the upside or the downside of the distribution. Start with the deviations from the mean. Cube them. Multiply them with the probabilities and sum everything up. Divide by the cubed standard deviation. Skewness is unit-free and can be positive, zero or negative. If skewness is zero, the distribution is symmetric. There is as much risk on the upside as there is on the downside. If skewness is negative, we say the distribution is skewed to the left. There is more risk on the downside than on the upside. If skewness is positive, we say the distribution is skewed to the right. There is more risk on the upside than on the downside. 5
Some new concepts Skewness Some insights: The 1 st distribution is symmetric For the 2 nd we add risk to the upside For the 3 rd we add risk to the downside no skewness! positive skewness! negative skewness! Compare the 2 nd to the 3 rd : Variance stays the same. No information about where the risk is. Reducing skewness corresponds to moving risk from the upside to the downside of the distribution while keeping the overall level of risk the same. Behaviorally, people tend to like positive skewness ( prudence ). It is easier to cope with risk when outcomes are high. Speculation: Stocks that have some chance of very high returns are attractive! Insurance: Risks that have some chance of very bad outcomes are unattractive! Some new concepts Coefficient of variation Coefficient of variation: Measure of relative dispersion of outcomes. Example: probability 0.5 20.5 43 Var 0.5 23 0.5 43 1 11 1 3/4 1/2 1/4 1 2 3 4 5 6 outcomes Some new concepts Coefficient of variation Now what happens if we shift the entire distribution to the right? For example if we add $100 to each outcome 0.5 102 0.5 104 103 Var 0.5 102 103 0.5 104 103 1 11 probability 1 3/4 1/2 1/4 101 102 103 104 105 106 outcomes 6
Some new concepts Coefficient of variation Variance and standard deviation stay the same. We might feel that the $1 coin flip is less of an issue when we have $103 on average rather than $3 on average. Coefficient of variation:. First example:. Second example:. The coefficient of variation measures the dispersion in the distribution of outcomes relative to the mean. It is a measure of relative riskiness. Mutualization, law of large numbers and central limit theorem Let us have a look at the collective management of risks. Primitive forms of insurance involved mutuals covering a posteriori the losses of their unlucky members (i.e., the ones that had been injured). Modern insurance: Started in the 17 th century (invention of statistical methods and actuarial calculus) Calculation in advance of the contribution of each individual agent Management of mutuality: The price of a risk does not exist without reference to the mutuality to which this risk belongs. More formally: Let us assume that we have a group of individuals, each of which is exposed to a risk. These individual risks are assumed to be identical in terms of their distribution and independent. We denote the risks by,,. In probability theory this situation is called i.i.d., which is short for independent identically distributed risks. Specifically, all risks have the same mean and the same variance Var Var. We want to understand the effects of pooling on the aggregate risk and the average risk 7
: Aggregate risk Effects of pooling on aggregate risk: Mean Variance Var Var Var Standard deviation Coefficient of variation : Aggregate risk Conclusion:As we include more and more risks in the pool aggregate risk increases. You can see this because the expected value of aggregate risk becomes larger and larger, the variance and standard deviation become larger and larger. However, the standard deviation increases more slowly than the mean. As a result, the coefficient of variation decreases. Pooling reduces relative risk. Average risk Effects of pooling on average risk: Mean Variance 1 Var 1 1 Var Standard deviation Coefficient of variation 8
Average risk Conclusion:As we include more and more risks in the pool, we see that the level of average risk decreases. Notice that the expected value of the average risk is always the same, the variance and standard deviation become smaller and smaller. the coefficient of variation becomes smaller and smaller, too. The Law of Large Numbers (LLN): The average loss from a portfolio of in-dependent and identically distributed risks with finite mean converges with probability one to the expectation of these risks when the number of risks,, becomes arbitrarily large. Strong law of large numbers P 1 lim 1 As the number of trials goes to infinity, the probability that the average of the observations is equal to the expected value will be equal to one. Justifies the interpretation of the expected value as long-term average. Weak law of large numbers lim P 1 0 for any 0 For any nonzero margin specified, no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin. Application: Basis of traditional insurance activities By having a large portfolio of independent risks, an insurance company decreases the uncertainty on its average liability. Mutualization principle: A collective management of risks (i.e., increasing ) reduces the variance of average losses and is therefore (in general) an efficient tool for risk management. Thus, when economic capital is proportional to the standard deviation of losses (e.g., when risks are normally distributed), it is reduced by pooling. 9
Pooling makes risks more predictable because average losses are less volatile than individual losses. Surprisingly, pooling also improves our knowledge of the distribution. We can standardize to obtain ; has a mean of 0 and a standard deviation of 1. Central limit theorem (CLT): The distribution of converges to the standard normal distribution N(0,1) as. Mathematically: lim P 1 Φ Strong result! Even if we only know mean and variance of the distribution of individual risks, pooling allows us to use the normal distribution as an approximation., for the aggregate risk., for the average risk. Most loss distributions are not normal. According to the CLT we can, however, still use the normal distribution as an approximation if the number of exposures is large and exposures are independent. Worker injury losses for firms with many employees Auto accidents for firms with a large fleet Application: Insurer wants to guarantee survival with probability (say, 99%) Net deficit at the end of period: are the individual claims (i.e., losses), which are i.i.d., and is the premium for each risk Failure occurs whenever Exercise: Assume the premium is actuarially fair; what is the mean and standard deviation of the insurer s net deficit? Use the CLT to determine the level of economic capital that allows the insurer to reduce its shortfall probability to 1 or less. How does the economic capital depend on the safety level, the pool size and the riskiness of each individual claim? 10
Pitfalls In the sequel, we discuss several cases where mutualization, LLN and/or the CLT work only imperfectly or do not work at all. Assume that risks are no longer independent; does mutualization still work? Var 1 Cov 1, 1 Cov,, Each covariance term is bounded by the common variance: Cov, 1 Cov, As a result: Var. Pitfalls Lack of independence Example: Geographic concentration (Hurricane Andrew in 1992) Claims generated by different individual insurance contracts cannot be considered as independent Example: Pharmaceutical industry (generic products) Assessment of Catastrophic Risks (see next slide) Pitfalls A distribution is heavy-tailed if the probability of large events (i.e., P for large) converges to zero very slowly when tends to infinity, for example like with 2. Example: Cauchy distribution, 0, Expected value and variance are undefined Distribution is stable: If and have independent, identical Cauchy distributions with dispersion parameter, then also has a Cauchy distribution with the same dispersion parameter. Pooling is ineffective! 11
Pitfalls Example: Lévy distribution 2, 0, 0 Expected value and variance are undefined Distribution is stable: If and have independent, identical Lévy distributions with dispersion parameter, then also has a Lévy distribution with the dispersion parameter 2. Pooling hurts! The management of tail events What happens if a catastrophe (tail-event) occurs? Executives: Small likelihood, limited liability protects shareholders Might hurt public interest (externality) Need for collective risk management policy Example: Use of civil nuclear plants Main obstacle: allocation of risks between parties (designers of the plant, utility companies, government) Traditional risk carries: no appetite for this risk Nuclear accident could contaminate many countries Several governments worked together to design an interesting legal framework for the liability of the civil nuclear industry (early 1960s). The management of tail events Main ideas: Impose a strict liability on the utility company; Limit the amount of this liability; Limit the duration of the liability to the time needed by victims to claim damages Impose on the utility company to cover a first tranch of its liability through insur-ance contracts; Give governmental and multi country warranties above this threshold. Results of this regulation Allocation of risk is clearly determined No uncertainty around liabilities Creates conditions for the development of an insurance (and reinsurance) market for these risks W/o some guarantee provided by the government, very large or catastrophic risks would not be insurable 12
Value at Risk: Definition For any level (typically 95% or 99%), the at level is by definition the threshold such that losses on this portfolio over a specific time horizon remain below with probability : P. Definition includes: Specific financial position (portfolio, cash flows, revenues, value of the firm, etc.) Time horizon: day, week, month, etc. Safety level: 90%, 95%, 99%, etc. VaR: Normal distribution Obtaining value at risk when losses follow a Normal distribution is very similar to the economic capital calculations performed earlier: P P Φ Solution: Φ VaR: Heavy tails When tails are heavy, using a normal approximation can lead to catastrophes! 90% 95% 99% 99.9% Normal 1.28 1.64 2.33 3.09 Cauchy 3.08 6.31 31.82 318.31 13
VaR: Portfolios Value at risk is frequently used for investment positions. If we are looking at a distribution of returns, where are the values that we should be concerned about? When do investors get nervous? VaR: Portfolio Application Download the daily adjusted close for the S&P 500 in the period Mar 27, 2016 Mar 27, 2017. Determine the daily returns. Assume an investment of $10,000 and calculate the daily gains and losses. Find minimum, maximum, expected value, standard deviation and skewness (=SKEW) for daily returns and daily gains and losses. Visualize daily gains and losses in a suitable histogram. Find the daily at a confidence level of 90%, 95%, 99% and 99.9% based on the data. Fit a Normal distribution to the data and use it to find at the different confidence levels. Compare your results. VaR: Its use in practice VaR was used in elite quantitative trading groups at several financial institutions, notably Bankers Trust, before 1990. The VaR methodology and estimations of the necessary underlying parameters were published by J. P. Morgan (RiskMetrics Group) in 1994. In 1997, the U.S. Securities and Exchange Commission ruled that public corporations must disclose quantitative information about their derivatives activity. Major banks and dealers chose to implement the rule by including VaR information in the notes to their financial statements. Worldwide adoption of the Basel II Accord, beginning in 1999, gave further impetus to the use of VaR.VaR became the preferred measure of market risk. As early as in 1997, NassimTaleb (the author of The Black Swan ) openly criticized the use of VaR. 14
VaR: Its use in practice: Correlation This method attempts to calculate the distribution of the entire portfolio based on the mean and the variance of the individual assets in the portfolio and the correlation between them. 1) Use historical data to estimate mean, variance and correlations. 2) Apply portfolio weights to obtain mean and variance of the entire portfolio. 3) Use the normal distribution to obtain VaR. This is exactly what we did in the second part of the practice problem! Problem:What if asset returns do not follow a normal distribution? VaR: Its use in practice: Historical simulation Historical data are used directly to predict the returns of the assets. The assumption of a normal distribution is not required. 1) Gather data over the relevant period of time and calculate returns. 2) For each point in time calculate the value of the portfolio. If we know the return on Amazon and Facebook stock between two periods, we can calculate the portfolio s new value. 3) Rank the portfolio values you obtain and look at the (empirical) distribution. Determine the value that separates the 5% worst outcomes from the remaining 95%. This will be the corresponding VaR. This addresses potential non-normality. We still need the assumption of stationarity to attribute predictive power to VaR. VaR: Its use in practice: Monte Carlo simulation This approach is more rigorous than historical simulation because it takes into account the possibility of market shocks. 1) Use regression analysis to estimate the time series of returns. Example:. Based on market data you can estimate the model to find, and the variance of the random shock. 2) Use these parameters to generate (many!!) future scenarios for the returns. Value the portfolio under each scenario. 3) Rank the portfolio values you obtain and look at the (empirical) distribution. Determine the value that separates the 5% worst outcomes from the remaining 95%. This will be the corresponding VaR. The use of Monte Carlo simulation allows us to capture potential time trends. 15
VaR: Critique Value at risk does have some restrictions that are worth mentioning. VaR can be wrong: If errors occur in the estimation of VaR, it can be a misleading measure of risk exposure. Assumption about return distributions History may not be a good predictor Non-stationary correlations Narrow focus Type of risk: VaR ignores the upside potential; focus on market risk effects Short term: Hedging on a day-to-day basis and regulatory pressure Absolute value: VaR might not be appropriate to compare investments with very different scales and returns VR: Critique Sub-optimal decisions: Can we be ensured that the use of VaR leads to more reasoned and sensible decisions by managers and investors? Overexposure to risk: VaR does not measure how bad things can become once we are beyond the threshold Agency problems: Managers might try to game VaR Diversification: There are cases in which VaR does not take into account diversifi-cation benefits [Within the class of normal distributions, diversification is always reflected in the VaR.This problem only becomes prevalent in a non-normal world.] Researchers and practitioners have come up with extensions that allow to circumvent some of these limitations Tail value at risk: Definition VaR can encourage managers to hide risks underneath the rug (i.e., the upper tail of the loss distribution). Statisticians recommend Tail VaR or Expected Shortfall: Perspective:Among those losses that only happen in 1 %of the cases, what is the average loss? (tail average) For Normal distribution: standard normal density function (=NORM.DIST(x,0,1,0)) Φ inverse of standard normal CDF (=NORM.INV(x,0,1,1)) 16
Tail value at risk: Advantages and drawbacks TVaR takes diversification into account in any case. It is harder to game than VaR and uses more information of the probability distribution. (+) It can be just as wrong as VaR. Furthermore, its focus is somewhat broader than that of VaR but the upside potential is still ignored. (+/-) It is more complex and therefore harder to calculate. It might be very tricky to get the tail right. How much do we know about extreme events that only happen very rarely? (-) For VaR we only need to know where they start. For TVaR we need to know what exactly they look like (how they are distributed). Higher standard errors in the estimation leave more room for managerial discretion. 17