Central Bank of Egypt Value at Risk Va
Central Bank of Egypt Value at Risk (Va. R) March-14 1
Central Bank of Egypt Index I. What is Va. R? II. Types of Va. R 1. 2. 3. Historical VAR Parametric VAR Monte Carlo VAR III. Back-testing Va. R IV. Stress Testing V. Scenario Analysis March-14 2
Central Bank of Egypt I. What is Va. R? • What is Va. R? – Estimate of the loss from a given position over a fixed time period that will be equaled or exceeded with a given probability • Va. R has two equivalent interpretations: – Worst Case Loss: over one day, there is a 95% probability that we will not lose more than $ yy – An unlikely event: on average, in one out of every 20 days, we should expect to incur a loss greater than or equal to a certain amount March-14 3
Central Bank of Egypt I. What is Va. R? Example: Suppose an investor placed $ 100, 000 in a certain type of investment with an expected annual return of 10% and 95% Va. R = -15%. This means that the investment is expected to generate a 10% return over 1 year but : Ø There is 5% probability that the investor can lose $ 15, 000 or more over the next year. Ø The investor is 95% confident that the maximum amount he is going to lose is not going to exceed $15, 000 Va. R = 100, 000 * -15% = -15, 000 March-14 4
Central Bank of Egypt I. What is Va. R? • Key aspects in Va. R: Ø Va. R measures the minimum potential loss at the stated probability; the actual loss that could be incurred could be higher. Ø Va. R is associated with a stated degree of probability. Lowering the probability (increasing the confidence interval) increases the Va. R. Ø Va. R measure is associated with a specific time period. Increasing the time interval will increase the Va. R. March-14 5
Central Bank of Egypt II. Types of Va. R • Different types of VAR measures: – Historical VAR – Analytic VAR (Variance Covariance, Parametric) – Monte Carlo VAR March-14 6
Central Bank of Egypt 1. Historical Va. R • Normality Assumption is not required. • Works on historical returns. March-14 7
Central Bank of Egypt 1. Historical Va. R Calculations • Step 1: Collect data on historical returns for asset/portfolio. These returns should be measured over a time interval = desired Va. R time period. • Step 2: from this info, construct a histogram of historical return data. • Step 3: Va. R is the return associated with the cumulative probability from the left tail of the histogram that equals March-14 8
Central Bank of Egypt 1. Historical Va. R • Advantages: Ø Because it is non-parametric, historical method does not require normality assumption Ø Easy to understand implement. Ø Based only on historical information. • Disadvantages: Ø History repeats itself (same return distribution) Ø A biased estimate of forward looking Va. R (taking historical returns of 10 years bonds misleading as duration was higher years ago) March-14 9
Central Bank of Egypt 2. Parametric Va. R • The most common measure of risk is standard deviation of the distribution of returns. • Higher volatility = higher risk = potential for higher losses. • Using standard deviation and some assumptions about returns, we can create a confidence interval, or a probable distribution of returns. • Accordingly, Va. R is the chance of losing a potential amount over the holding period with a certain confidence level. March-14 10
Central Bank of Egypt 2. Parametric Va. R Expected Return Value at Risk, p% Volatility Conditional Va. R March-14 11
Central Bank of Egypt 2. Parametric Va. R Assumptions: • Parametric Va. R assumes that asset returns are normally distributed with known mean and standard deviation over a specified time period • Covariances (correlations) among assets are known for the same time interval. March-14 12
Central Bank of Egypt 2. Parametric Va. R Assumption of Normality • Va. R is calculated assuming normal market conditions • What do we mean by “normal”? Ø Gaussian Ø Normal market circumstances –extreme market conditions are not included; they are examined separately • Therefore Va. R is meant to show what can happen to the portfolio on a day-to-day basis. March-14 13
Central Bank of Egypt 2. Parametric Va. R • Inputs into the Va. R calculation: Ø Market values of all securities in the portfolio Ø Their volatilities Ø The correlations between them • The assumption is that the movement of the components of the portfolio are random, and drawn from a normal distribution March-14 14
Central Bank of Egypt 2. Parametric Va. R α = μ - Z α * σ A more conservative approach: Va. R α = Z α * σ • A zero expected return is assumed when performing a Va. R calculation using the parametric method. This is generally an acceptable assumption for daily Va. R calculations because daily returns usually are very small; it is less acceptable for Va. R calculations with longer time horizons because returns tend to deviate substantially from zero as the time horizon increases. • The effect of assuming a zero expected return is to produce a more conservative (larger) projected loss. March-14 15
Central Bank of Egypt 2. Parametric Va. R • Here is the standardized normal distribution table March-14 16
Central Bank of Egypt 2. Parametric Va. R a) Measuring Risk (Individual level) Va. R α = Market Value * Z α * σ • Example: Compute the 5% annual Va. R of a $100, 000 portfolio with an annual expected return of 8% and standard deviation of 12%. Va. R 0. 05, 1 yr = μ - Z α * σ = 8% - 1. 645 (12%) = -11. 74% Va. R 0. 05, 1 yr = -11. 74% * $ 100, 000 = -$11, 740 Or Va. R 0. 05, 1 yr = Z α * σ = - 1. 645 (12%) = -19. 74% Va. R 0. 05, 1 yr = -19. 74% * $ 100, 000 = -$19, 740 March-14 17
Central Bank of Egypt 2. Parametric Va. R a) Measuring Risk (Individual level) • Va. R for periods longer than 1 day can be determined by multiplying the daily Va. R by the square root of the number of trading days that compose the longer Va. R period. • Example: Compute the 5% daily Va. R of a $100, 000 portfolio with an annual expected return of 8% and standard deviation of 12%. μdaily = μ annual / 250 = 8%/250 = 0. 032% σ daily = σ annual / √ 250 = 12% / √ 250 = 0. 76% Va. R 0. 05, daily = μdaily - Z αdaily * σ = 0. 032% - 1. 645 (0. 76%) = -1. 216% Va. R 0. 05, daily = -1. 216% * $ 100, 000 = $1, 216 March-14 18
Central Bank of Egypt 2. Parametric Va. R a) Measuring Risk (Individual level) • Compute weekly and monthly Va. R for the same exercise Va. R 0. 05, weekly = $1, 216 * √ 5 = $2, 720 Va. R 0. 05, monthly = $1, 216 * √ 22 = $ 5, 706 • Compute 10 days Va. R from an annual Va. R = $ 2, 000 March-14 19
Central Bank of Egypt 2. Parametric Va. R b) Measuring Risk (Aggregate level) • Portfolio Va. R is the sum of individual Va. Rs only if instruments are perfectly correlated. • In all other cases, portfolio Va. R is less than the sum of individual Va. Rs. • We have to account for correlation in calculating portfolio Va. R. March-14 20
Central Bank of Egypt 2. Parametric Va. R • Example: A portfolio of $ 100, 000 is composed of 2 assets: - A stock whose expected return is 10% with a standard deviation of 20% - A bond whose expected annual return is 5% with a standard deviation of 12%. If an investor puts 60% in the stock and 40% in bonds. what is the expected annual return, standard deviation and 95% Va. R assuming a correlation of 0. 30. March-14 21
Central Bank of Egypt 2. Parametric Va. R Interest rate Risk: • We need to make a few assumptions about how yields move through time • In particular, we generally assume that–Changes in yields are normally distributed with a mean of zero and a given standard deviation • The change in bond prices, for a given change in yields, can be described by modified duration March-14 22
Central Bank of Egypt 2. Parametric Va. R Interest rate Risk: • Modified duration and DV 01 tell us, reasonably accurately, the change in the value of our portfolio for a given change in yields. • There is, nevertheless, a drawback Ø They don’t tell us how much yields are likely to move Ø What we need is some idea of the probability of a given change in bond yields, and the likely effect of this change on our portfolio March-14 23
Central Bank of Egypt 2. Parametric Va. R • Advantages: Ø Easy to compute Ø Take into account correlations • Disadvantages: Ø Requires that individual asset returns be normally distributed Ø The method assumes expected asset returns, standard deviation and correlations are constant. Ø Not easily measured for portfolio with options (because return distribution of such portfolios are highly skewed, non-linear payoff) March-14 24
Central Bank of Egypt 3. Monte Carlo Va. R • Analytic method: used when distribution of returns is normal and the parameters of the model can be estimated with normal accuracy. • Historical method: used when historical return distribution can be reasonably expected to reflect future return historical distribution whether or not the distribution is normal. March-14 25
Central Bank of Egypt 3. Monte Carlo Va. R • However, there are circumstances when it is desired to determine the Va. R of an item for which: Ø There are insufficient historical return data to construct an accurate picture of its return distribution (ex newly issued securities or new financial instruments) Ø Future return distribution will be different from the past return distribution. Ø Parameters that define the return distribution are unknown but they can modeled. Use Monte Carlo simulation method to estimate Va. R. March-14 26
Central Bank of Egypt 3. Monte Carlo Va. R • In Monte Carlo Va. R, the analyst specifies a model consisting of at least one random variable that is used to determine the return on the asset or portfolio/distribution of the possible values of every random variable in the model. • Based on these inputs, the computer generates the return distribution for this item and its Va. R. March-14 27
Central Bank of Egypt 3. Monte Carlo Va. R Example: Black Scholes Merton Model • A portfolio manager wants to determine the weekly Va. R at the 1% probability level for a call option on a highly volatile stock. Ø Ø Price of the underlying stock. Risk free interest rate Option strike price Dividend declared on the underlying stock during the time until the option expires. Ø Volatility of the underlying stock Ø Time until the option expires. March-14 28
Central Bank of Egypt 3. Monte Carlo Va. R • All other parameters are random variables that can take any value over the forecast period. • The manager can specify the probability distribution that every parameter based on experience. • When the process is complete, the computer sorts the possible returns and selects the 1 st percentile return and multiply it by the amount invested in the option to determine the 1% weekly Va. R. March-14 29
Central Bank of Egypt 3. Monte Carlo Va. R • Those distributions are the inputs to the Monte Carlo Simulation. • The computer then randomly selects a value for each random variable in the option pricing model from a random number generator that is conforming with the distributions described by the manager. • This process is repeated 100, 000 times with each iteration generating a new weekly return. March-14 30
Central Bank of Egypt 3. Monte Carlo Va. R • Advantages: Ø Flexibility: ability to determine the type of the distribution (normal, Poisson, exponential etc. . ) and the numerical parameter values of the distribution (mean, standard deviation etc. . ) Ø It can be used to analyze nonlinear (non-normal return distribution) as well as linear risks (normally distributed returns). Ø More likely to generate outlier possibilities than would historical analysis ---- which are referred to as the disaster scenarios. March-14 31
Central Bank of Egypt 3. Monte Carlo Va. R • Disadvantages: Ø It requires the risk analyst to develop appropriate valuation models for the assets in a portfolio and to specify realistic values for the parameters of the random variables contained in the models. Otherwise, “garbage in, garbage out”. Ø It requires more computer time and power and more analyst judgment than other methods. Ø Because it is based on random number generations, different runs of the process on the same parameters can produce different Va. Rs. March-14 32
Central Bank of Egypt III. Backtesting Va. R • Backtesting: is the process of comparing losses predicted by the Va. R model to those actually experienced over the sample testing period. • If a model were completely accurate, we would expect Va. R to be exceeded with the same frequency predicted by the confidence level used in the Va. R model. • 3 desirable attributes of Va. R estimates that can be evaluated when using a backtesting approach: » Unbiased » Adaptable » Robust March-14 33
Central Bank of Egypt IV. Stress Testing • During time of crisis, a contagion effect often occurs where correlations and volatility both increase and thus reduce any diversification benefits. • Stressing the correlation is a method used to model the contagion effect that could occur in a crisis event. • One approach for stress testing is to examine historical crisis events such as the Asian crisis. • Advantage: no assumptions of underlying asset returns or normality are needed. • Disadvantage: it is limited to only evaluating events that have actually occurred. March-14 34
Central Bank of Egypt V. Scenario Analysis • Analyze different pre-determined stress scenarios such as 200 bps increase in short term rates, an extreme inversion of the yield curve or an increase in volatility. • Advantage: It is not limited to the evaluation of risks that have occurred historically. • Disadvantage: the risk measure can be deceptive. • The worst case scenario (WCS): assumes that an unfavorable event will occur with certainty. The focus is on the distribution of worst possible outcomes given an unfavorable event. March-14 35
Central Bank of Egypt Thank you March-14 36
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