Value at Risk (VaR)

Index What is VaR? Types of VaR Back-testing VaR Stress Testing Historical VAR Parametric VAR Monte Carlo VAR Back-testing VaR Stress Testing Scenario Analysis

I. What is VaR? What is VaR? Estimate of the loss from a given position over a fixed time period that will be equaled or exceeded with a given probability VaR 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

I. What is VaR? Example: Suppose an investor placed $ 100,000 in a certain type of investment with an expected annual return of 10% and 95% VaR = -15%. This means that the investment is expected to generate a 10% return over 1 year but : There is 5% probability that the investor can lose $ 15,000 or more over the next year. The investor is 95% confident that the maximum amount he is going to lose is not going to exceed $15,000 VaR = 100,000 * -15% = -15,000

I. What is VaR? Key aspects in VaR: VaR measures the minimum potential loss at the stated probability; the actual loss that could be incurred could be higher. VaR is associated with a stated degree of probability. Lowering the probability (increasing the confidence interval) increases the VaR. VaR measure is associated with a specific time period. Increasing the time interval will increase the VaR.

II. Types of VaR Different types of VAR measures: Historical VAR Analytic VAR (Variance Covariance, Parametric) Monte Carlo VAR

1. Historical VaR Normality Assumption is not required. Works on historical returns.

1. Historical VaR Calculations Step 1: Collect data on historical returns for asset/portfolio. These returns should be measured over a time interval = desired VaR time period. Step 2: from this info, construct a histogram of historical return data. Step 3: VaR is the return associated with the cumulative probability from the left tail of the histogram that equals

1. Historical VaR Advantages: Because it is non-parametric, historical method does not require normality assumption Easy to understand and implement. Based only on historical information. Disadvantages: History repeats itself (same return distribution) A biased estimate of forward looking VaR (taking historical returns of 10 years bonds misleading as duration was higher years ago)

2. Parametric VaR 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, VaR is the chance of losing a potential amount over the holding period with a certain confidence level.

2. Parametric VaR Expected Return Value at Risk, p% Volatility Conditional VaR

2. Parametric VaR Assumptions: Parametric VaR 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.

2. Parametric VaR Assumption of Normality VaR is calculated assuming normal market conditions What do we mean by “normal”? Gaussian Normal market circumstances –extreme market conditions are not included; they are examined separately Therefore VaR is meant to show what can happen to the portfolio on a day-to-day basis.

2. Parametric VaR Inputs into the VaR calculation: Market values of all securities in the portfolio Their volatilities The correlations between them The assumption is that the movement of the components of the portfolio are random, and drawn from a normal distribution

2. Parametric VaR VaR α = μ - Z α * σ A more conservative approach: A zero expected return is assumed when performing a VaR calculation using the parametric method. This is generally an acceptable assumption for daily VaR calculations because daily returns usually are very small; it is less acceptable for VaR 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.

2. Parametric VaR Here is the standardized normal distribution table

2. Parametric VaR Measuring Risk (Individual level) VaR α = Market Value * Z α * σ Example: Compute the 5% annual VaR of a $100,000 portfolio with an annual expected return of 8% and standard deviation of 12%. VaR0.05,1yr = μ - Z α * σ = 8% - 1.645 (12%) = -11.74% VaR0.05,1yr = -11.74% * $ 100,000 = -$11,740 Or VaR0.05,1yr = Z α * σ = - 1.645 (12%) = -19.74% VaR0.05,1yr = -19.74% * $ 100,000 = -$19,740

2. Parametric VaR Measuring Risk (Individual level) VaR for periods longer than 1 day can be determined by multiplying the daily VaR by the square root of the number of trading days that compose the longer VaR period. Example: Compute the 5% daily VaR 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% VaR0.05,daily = μdaily - Z αdaily * σ = 0.032% - 1.645 (0.76%) = -1.216% VaR0.05,daily = -1.216% * $ 100,000 = $1,216

2. Parametric VaR Measuring Risk (Individual level) Compute weekly and monthly VaR for the same exercise VaR0.05,weekly = $1,216 * √5 = $2,720 VaR0.05,monthly = $1,216 * √22 = $ 5,706 Compute 10 days VaR from an annual VaR = $ 2,000

2. Parametric VaR b) Measuring Risk (Aggregate level) Portfolio VaR is the sum of individual VaRs only if instruments are perfectly correlated. In all other cases, portfolio VaR is less than the sum of individual VaRs. We have to account for correlation in calculating portfolio VaR.

2. Parametric VaR 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% VaR assuming a correlation of 0.30.

2. Parametric VaR 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

2. Parametric VaR Interest rate Risk: Modified duration and DV01 tell us, reasonably accurately, the change in the value of our portfolio for a given change in yields. There is, nevertheless, a drawback They don’t tell us how much yields are likely to move 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

2. Parametric VaR Advantages: Easy to compute Take into account correlations Disadvantages: Requires that individual asset returns be normally distributed The method assumes expected asset returns, standard deviation and correlations are constant. Not easily measured for portfolio with options (because return distribution of such portfolios are highly skewed, non-linear payoff)

3. Monte Carlo VaR 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.

3. Monte Carlo VaR However, there are circumstances when it is desired to determine the VaR of an item for which: There are insufficient historical return data to construct an accurate picture of its return distribution (ex newly issued securities or new financial instruments) Future return distribution will be different from the past return distribution. Parameters that define the return distribution are unknown but they can modeled. Use Monte Carlo simulation method to estimate VaR.

3. Monte Carlo VaR In Monte Carlo VaR, 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 VaR.

3. Monte Carlo VaR Example: Black Scholes Merton Model A portfolio manager wants to determine the weekly VaR at the 1% probability level for a call option on a highly volatile stock. 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. Volatility of the underlying stock Time until the option expires.

3. Monte Carlo VaR 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 1st percentile return and multiply it by the amount invested in the option to determine the 1% weekly VaR.

3. Monte Carlo VaR 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.

3. Monte Carlo VaR Advantages: 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..) It can be used to analyze nonlinear (non-normal return distribution) as well as linear risks (normally distributed returns). More likely to generate outlier possibilities than would historical analysis ---- which are referred to as the disaster scenarios.

3. Monte Carlo VaR Disadvantages: 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”. It requires more computer time and power and more analyst judgment than other methods. Because it is based on random number generations, different runs of the process on the same parameters can produce different VaRs.

III. Backtesting VaR Backtesting: is the process of comparing losses predicted by the VaR model to those actually experienced over the sample testing period. If a model were completely accurate, we would expect VaR to be exceeded with the same frequency predicted by the confidence level used in the VaR model. 3 desirable attributes of VaR estimates that can be evaluated when using a backtesting approach: Unbiased Adaptable Robust

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.

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.

Thank you Deceptive: for ex, a shift in the domestic yield curve could cause estimation errors by overstating the risk for a long and short position and understating the risk for a long-only position. Asset class specific risk is another disadvantage. WCS: The tail of the original return distribution is more thoroughly examined wth another distribution that includes only probably extreme events.