Value at risk (VaR) is a probabilistic method of measuring the potential loss in portfolio value over a given time period and for a given distribution of historical returns. There is an X percent probability that the loss in portfolio value will be equal to or greater than the VaR measure.
Calculating delta-normal VaR requires assuming that asset returns conform to a standard normal distribution. A standard normal distribution is defined by two parameters, its mean (|i = 0) and standard deviation (a = 1), and is perfectly symmetric with 50% of the distribution lying to the right of the mean and 50% lying to the left of the mean.
VaR (X%) = zx% σ
VaR (X%) = the X% probability value at risk
zx% = the critical z-value based on the normal distribution and the selected X% probability.
σ = the standard deviation of daily returns on a percentage basis.
Risk managers may be interested in measuring risk over longer time periods, such as a month, quarter, or year. VaR can be converted from a 1-day basis to a longer basis by multiplying the daily VaR by the square root of the number of days (J) in the longer time period (called the square root rule).
For example, to convert to a weekly VaR, multiply the daily VaR by the square root of 5 (i.e., five business days in a week). We can generalize the conversion method as follows:
VaR(X%)J-days = VaR(X%)1-day J
LINEAR METHODS: Linear methods replace portfolio positions with linear exposures on the appropriate risk factor.
v0 = V(S0)
With this expression, we can describe the relationship between the change in portfolio value and the change in the risk factor as:
dV = Δ0 x dS
Here, Δ0 is the sensitivity of the portfolio to changes in the risk factor, S. As with any linear relationship, the biggest change in the value of the portfolio will accompany the biggest change in the risk factor. The VaR at a given level of significance, z, can be written as:
VaR = |Δ0| x (zσS0)
The delta-normal method for estimating VaR requires the assumption of a normal distribution. This is because the method utilizes the expected return and standard deviation of returns.
Advantages of the delta-normal VaR method
Disadvantages of the delta-normal method
Advantages of the historical simulation method
Disadvantages of the historical simulation method
Advantages of the Monte Carlo method
Disadvantages of the Monte Carlo method
Three common deviations from normality that are problematic in modelling risk result from asset returns that are fat-tailed, skewed, or unstable.
The second possible explanation for “fat tails” is that the second moment or volatility is time-varying. This explanation is much more likely given observed changes in interest rate volatility. Increased market uncertainty following significant political or economic events results in increased volatility of return distribution.
This is a time-series model used by analysts to predict time-varying volatility. Volatility is measured with a general GARCH(p,q) model using the following formula:
σ2t =a + b1r2t-1,t + b2r2t-2,t-1 + …+bpr2t-p, r-p+1+c1σ2t-1+c2σ2t-2+…+cqσ2t-q
Where: parameters a,b1 through bp, and c1 through cq = parameters estimated using historical data with p lagged terms on historical returns squared & q lagged terms on historical volatility
HISTORICAL SIMULATION METHOD
Historical simulation is a procedure for predicting the value at risk by simulating or constructing the cumulative distribution function of assets return overtime.
Under the historical simulation, all returns are weighted equally based on the number of observations in the estimation window (1/K).
The hybrid approach uses historical simulation to estimate the percentiles of the return and weights that decline exponentially (similar to GARCH or RiskMetrics.
Advantages of nonparametric methods compared to parametric methods:
The Taylor Series of order two is represented mathematically as:
f (x) = f (x0) + f ‘(x0) (x – x0 ) + 1/2 f “(x0) (x – x0)2
The first derivative tells us the delta, or slope of the line. The second derivative tells us the rate of change. The last term including the second derivative captures the convexity or curvature. This approximation is only useful for “well-behaved” quadratic functions of order two.
The full revaluation approach calculates the VaR of the derivative by valuing the derivative based on the underlying value of the index after the decline corresponding to an x% VaR of the index.
The delta-normal approach calculates the risk using the delta approximation, which is linear or the delta-gamma approximation, f(x) = f (x0) + f'(x0)(x – x0) + 1/2 f “(x0)(x – x0)2 ,which adjusts for the curvature of the underlying relationship.
The worst case scenario (WCS) assumes that an unfavourable event will occur with certainty. The focus is on the distribution of worst possible outcomes given an unfavourable event. An expected loss is then determined from this worst case distribution analysis. Thus, the WCS information extends the VaR analysis by estimating the extent of the loss given an unfavourable event occurs.
The structured Monte Carlo (SMC) approach simulates thousands of valuation outcomes for the underlying assets based on the assumption of normality. The VaR for the portfolio of derivatives is then calculated from the simulated outcomes. The general equation assumes the underlying asset has normally distributed returns with a mean of μ, and a standard deviation of σ. Advantage: It is able to address multiple risk factors by assuming an underlying distribution and modelling the correlations among the risk factors.
Disadvantage: In some cases it may not produce an accurate forecast of future volatility and increasing the number of simulations will not improve the forecast.
CORRELATIONS DURING CRISES
In times of crisis, correlations increase (some substantially) and strategies that rely on low correlations fall apart in those times. Certain economic or crisis events can cause diversification benefits to deteriorate in times when the benefits are most needed. A contagion effect occurs with a rise in volatility and correlation causing a different return generating process.
A simulation using the SMC approach is not capable of predicting scenarios during times of crisis if the covariance matrix was estimated during normal times. Unfortunately, increasing the number of simulations does not improve predictability in any way.
Stressing the correlation is a method used to model the contagion effect that could occur in a crisis event.
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