VaR Pro and Contra Pro: Easy to calculate and to understand. It is a common language of communication within the organizations as well as outside (e.g. regulators, auditors, shareholders). It is not really complicated, but messy and time consuming. Contra: It is not a coherent measure, more specifically it does not satisfy superposition. It fails to recognize the concentration of risks. Most parametric approaches neglect the heavy tails and the skewness of the return series.
Calculating VaR There are several approaches for calculating the Value at Risk figure. The most popular are the Variance covariance approach Historical simulation Parametric VaR Monte Carlo simulation
Definition of Value-at-Risk (VaR) Definition of VaR (confidence level 99% and horizon 1 day): VaR is the answer to the question: What is the most I can - with a 99% level of confidence - expect to lose tomorrow 1% probability 0.4 0.3 0.2 For a portfolio with benchmark we calculate the VaR as the active portfolio (i.e.. a long position in the fond and a short position in the benchmark). 0.1-3 -2-1 1 2 3 P&L (msek) VaR 99% 99% probability
Absolute & Relative VaR Speaking in mathematical terms, Absolute VaR is simply the (1 q) quantile of the distribution of the d day change of value for a given portfolio P: VaR q,d (P) = F 1 P d (1 q) PV(P where P d is the change of value for a given portfolio over d days, F P d is the distribution function of P d and PV(P) the present value of the portfolio P. The Relative VaR is defined as 1 VaR qd, ( P) E[ P] F (1 q) PV ( P) P d If we write the VaR in percent of a potential loss, just remove the term PV(P) above and replace E[P] by the mean of the underlying distribution F.
Relation between Relative VaR 99 / MtM (%) TE (%) For linear portfolios: ex ante Tracking Error Relativ VaR For portfolios including options, VaR is a much better risk measure compared with Tracking Error. ex ante means before the event. In finance expected Relativ VaR Tracking Error 0.05% 0.34% 0.10% 0.68% 0.20% 1.36% 0.30% 2.04% 0.40% 2.71% 0.50% 3.39% 0.60% 4.07% 0.70% 4.75% 0.80% 5.43% 0.90% 6.11% 1.00% 6.79%
VaR Value-at-Risk - Components The time horizon to be analysed may relate to the time period to holding a portfolio, or to the time required to liquidate assets. Typical periods using VaR are 1 day, 10 days, or 1 year. The confidence level is the interval estimate in which the VaR would not be expected to exceed the maximum loss. Commonly used confidence levels are 99% and 95%. The loss amount (or loss percentage).
VaR The answer What is the most I can - with a 95% or 99% level of confidence - expect to lose in dollars over the next month? What is the maximum percentage I can - with 95% or 99% confidence - expect to lose over the next year?
Standard Normal Density and the 99% Quantile
VaR The historical method VaR MV d 2.3263 h Where d is the number of trading days, MV the market value of the portfolio and h the historical volatility. The value 2.3263 is a given value used to calculate the level of certainty of 99%. It can be calculated solving: 2 1 1 z /2 N0,1( x) e dz 0.99 2 x Using Excel, use NORM.S.INV(99%) = 2.3263, NORM.S.INV(95%) = 1.6449
VaR
VaR - Variance-Covariance Method The idea behind the variance-covariance is similar to the ideas behind the historical method - except that we use the familiar curve instead of actual data. The advantage of the normal curve is that we automatically know where the worst 5% and 1% lie on the curve. N p i i i1 p T ω Σω VaR99% MV p 2.3263 p ω i = V i /V p is the return on asset i in the portfolio. Σ the covariance matrix of the N assets. µ p the expected value i.e., the mean.
VaR - Variance-Covariance Method The blue curve above is based on the actual daily standard deviation of the index, which is 2.64%. The average daily return happened to be fairly close to zero Confidence # of Calculation Equals 95% (high) 1.65 x 1.65 x 2.64 % 4.36 % 99% (very high) 2.33 x 2.33 x 2.64 % 6.16 %
Parametric VaR (analytic) Parametric VaR assume a probability distribution, such as normal distribution, for a portfolio s revenue. VaR can then be calculated using a closed analytic formula. The main advantage: the calculation can be performed very quickly. A significant disadvantage: the model don't capture nonlinear risk, such as optionality unless relatively complicated adjustments are made. If the portfolio s relative return is assumed to have a univariate normal distribution, it is called a Delta-Normal or Variance-Covariance model.
Parametric VaR Since the changes of instrument prices in a portfolio is approximated by linear functions of the first order sensitivities w.r.t. changes in the risk factors, the prices can be derived as the first order Taylor expansion in the risk factors: dp j P j F 1 df 1 + P j F 2 df 2 + + P j F k df k The derivatives above are similar to Delta, Vega, Rho... The change in the value of holding j, X j, is the change in the price, P j, times the nominal value Y j. The change dx j can be interpreted as the P/L of instrument holding j: k dx j = dp j Y j P j F 1 df 1 + P j F 2 df 2 +... + P j F k df k Y j = i=1 P j F i Y j df i
Parametric VaR Since we assume log-normal distribution we use logarithmic return, then k k Pj Pj dx j Yj d ln Fi Fi Yj d ln Fi i1 ln F i i1 F i The terms in the curly brackets are called delta equivalent values of the position j. Risk positions are then aggregated across portfolio holdings to give the portfolio P/L.
Monte Carlo VaR In a Monte Carlo simulation, a probability distribution is assumed for the market changes of a number of risk factors, and the distribution s parameters are estimated using historical data. By revaluing a portfolio s instruments for a large number of random outcomes (simulations) sampled from the distribution, VaR can then be calculated. One advantage is that it correctly captures nonlinear risk in financial instruments. A disadvantage is that it is calculation-intensive. Model-based measurement such as volatilities and correlations are extracted from historical data. Such data may be irrelevant when financial markets are under stress.
Monte-Carlo Simulations The stock price is simulated by a stochastic process dst rstdt Stdzt For simplicity, study the natural logarithm of the stock price: x t = ln(s t ) which gives: 1 2 dxt dt dzt r 2 x x t z z tt t tt t z t t z t t xt x i t t t i1 S exp t x i ti
Monte-Carlo Simulations
Monte-Carlo Simulations (10 000)
Monte-Carlo Simulations For each scenario, we then calculate the profit of the call options as: max(s T X, 0). To find the theoretical option value we calculate the mean value of the discounted pay-off: N 1 C exp( rt ) max S X,0 0 Ti, N i 1 where X is the strike price of the option. The standard deviation (SD) and the standard error (SE) of the simulations is given by: (Remember: the annualized volatility σ is the standard deviation of the instrument's yearly logarithmic returns.) N N 1 1 SD C C rt N N SE SD N 2 T, i T, i exp 2 1 i1 N i1 2
Monte-Carlo and Risk factors Market risk refers to the exposure of a particular portfolio to potential losses due to movements in the market: share prices volatilities interest rates exchange rates Such variables are called risk factors. Ideally, we have quoted price on instruments (mark-to-market). If not we use theoretical prices (mark-tomodel). Historical values are used to determine the scenarios that change current values in the simulation. Using a spectral decomposition, we obtain transformation. between the correlated risk factors. All instruments in indices- and benchmark instruments are used
Risk factors cont. In order to statistically manipulate time series and use them in scenarios, standardization is required. Bootstrapping is used to convert of swap rates to zero coupon rates. Interpolation is used for dates between the quoted maturities. We call tis risk factors standardized risk factors. These are used to calculate covariance's and generate the scenarios. The nodes in all yield curves is standardized (19 nodes): 1d, 2d, 3d, 7d, 30d, 90d, 180d, 270d, 1y, 2y, 3y, 4y, 5y, 7y, 9y, 10y, 15y, 20y, 30y
Risk factors cont. Type of risk factor IR (zero coupon rate) FX (exchange rate) EQ (share price) EQ vol (implied volatility) Risk factor sensitivity(sek) Delta (P&L at node shift +1 basis point) NPV Delta (P&L at share price +0.1%) Vega (P&L at volatility +1 percentage point) Historical volatility (vol) Absolute daily vol (expressed in basis points) Relative daily vol (expressed as a decimal number) Relative daily vol (expressed in 0.1%) Absolute daily vol (expressed as percentage) points)
Risk factors cont. A simple measurement of correlation sensitivity is obtained by observing how VaR changes when all correlations approach 0 (fully diversified or uncorrelated VaR) or 1 (fully correlated VaR). These VaR calculations are as follows: Fully diversified VaR: VaR ρ=0 = σ 1 x 1 2 + σ 2 x 2 2 +... + σ n x n 2 Fully correlated VaR: VaR ρ=1 = σ 1 x 1 + σ 2 x 2 +... σ n x n where j is volatility for the j th risk factor.
Expected shortfall An alternative to Value-at-Risk. The "expected shortfall at q% level" is the expected return on the portfolio in the worst q% of the cases. Expected shortfall is also called Conditional Value at Risk (CVaR) and Expected Tail Loss (ETL). As we have seen, VaR is defined as the level of loss that will not exceeded with a confidence level during a certain period of time. For example, if a bank's one-day 99% VaR is $3 million, there is considered to be only a 1% chance that losses will exceed $3 million a day. ES ( X ) E X X VaR ( X ) CVaR is a measure that produces better incentives for traders than VaR itself. Where VaR asks the question 'how bad can things get?', CVaR asks 'if things do get bad, what is our expected loss?'
Expected shortfall
Stress Testing Stress testing involves estimating how the portfolio would perform under some of the most extreme market moves seen in the past. However, it should be estimated how the portfolio would perform under some made up worst case scenario as well. The aim of stress testing is to understand (or at least to get an idea of) the risk exposure of the portfolio. Examples are historical extreme movement such as October 19, 1987 when the S&P 500 moved by 22.3 standard deviations. or worst case scenario such as a sudden increase/decrease of volatility of ±20 percent. a sudden increase (or devaluation) of a currency which is important for the portfolio. a default of a major customer.
Backtesting Backtesting is a way to estimate the model risk. compares the d day VaR estimation with the actually observed profit/loss over the next d days. If the actually observed profit/loss exceeds the VaR estimation too often, the model is not appropriate. For example, the 99 percent VaR quantile estimation should be exceeded by the actually observed profit/loss on average 2.5 times given 250 observations.
Approach of the Banking Industry The banking industry uses the VaR approach to measure the market risk in normal times. uses stress testing for estimating the impact of crash times to their portfolio. takes the liquidity risk into consideration by calculating the 10 day VaR. evaluates the model risk by doing backtesting.