4340 Risk Management

Arithmetic vs log returns

Value at Risk

• What loss is such that it will only be exceeded with a probability p over the given time horizon K?

• 

• Converting return VaR to dollar VaR:

  

• Downsides:

  • Ignores the magnitude of losses above VaR

  • Assumes that the portfolio is constant over the next K days, which is unrealistic when the horizon is larger than a week

  • Unclear how p and K should be chosen

Historical Simulation (HS) for VaR

• Have a portfolio with N_i units in each asset i at price P_i. get a time series of returns with P_i changing with t but N_i being constant

• VaR is the pth percentile of returns over the past m periods. Get a time series of VaR by moving backwards with these m-sized windows

• Pros:

  • Easy to implement

  • No parametric requirements

• Cons:

  • How is m chosen? Too low and VaR will not incorporate enough large losses. Too high and VaR will be constant over time

  • Does not adapt quickly enough to large losses

Weighted Historical Simulation (HS) for VaR

• Over the window of m periods, assign exponentially decreasing weights to past returns, so that they sum to 1 over the m periods

• Sort the returns from largest loss to largest gain, then the VaR is the return where the cumulative weights exceed p

• Pros:

  • Only required parameter in eta, moderating the rate of weight exponential decline

  • Weighting makes the choice of m less critical

  • WHS responds quickly to large losses, with immediately high VaR

• Cons:

  • How is eta chosen? Typically between 0.95 and 0.99, but more specifically?

  • Requires large amount of data for multi-day VaR

  • Since volatility is correlated with the magnitude of price movements, large price increases usually give high ensuing volatility and therefore VaR, but this is not captured by WHS, which only considers the largest magnitude losses

VaR transformation

• Eg one-day to m-day, multiply by sqrt(m)

• Works only where returns are IID, eg HS and RiskMetrics. Does not work for GARCH (due to autocorrelation) and WHS (due to weighting of more recent returns)

HS vs RiskMetrics trading

• Traders are allowed a certain amount of dollar VaR. therefore, the choice of VaR model influences the position size

• In a crash, RiskMetrics VaR increases quicker than HS, meaning lower positions in the crashes and lower losses

• After the crash, RiskMetrics VaR decreases quicker than HS, meaning higher positions and quicker recovery

Expected shortfall

• The magnitude of the mean loss higher than (positive) VaR

• 

• Relative difference between ES and VaR:

  

• Downside: Much higher estimation error than VaR, being in the tail, with fewer data points and higher modelling uncertainty

Autoregressive models

• Captures persistent patterns or trends

• 

• Predict future returns with lagged returns (in this case with one lag, AR(1)). 

• If then coefficient magnitude is < 1, then the unconditional mean (long-term average is)

  

  and the unconditional variance is

  

Autocorrelation function: the impact of a lagged variable on the regressand: for AR(1) models it is rho_t=phi_1^tau, ie, the coefficient to the power of the number of lags, since in each period the variable is multiplied by the same coefficient.

Moving average models

• Captures short-term noise, sudden corrections or overreactions

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• Predict future returns with lagged error terms (independent with zero expectation)

• 

  expected value is theta_0, but not conditional on the lagged residual

• ACF for MA(1):

  

  zero for larger time periods. In general, the ACF for MA(q) is non-zero for the first q lags and then 0

Issues with time series regression

• Spurious detection of mean reversion: erroneously finding that a variable mean-reverting when it is a truly random walk

• Spurious regression: erroneously finding that a variable x is significant when regressing y on x

• Spurious detection of causality: erroneously finding that the current value of x causes future values of y when in reality it cannot. More severe case of spurious regression

Simple variance forecasting

Over short time horizons, mean returns are zero, meaning variance is the sum of squared returns

RiskMetrics

• Exponential smoother:

  

  variance is a declining-weighted average of past squared returns

• 

• Advantages of RiskMetrics:

  • Only has one parameter, which is consistent (lambda=0.94) across asset classes

  • Captures autocorrelation, as large price changes give high immediate volatility

  • Only some 100 daily lags are required for the weight to converge to 1

• Disadvantages of RiskMetrics:

  • RiskMetrics is a special case of GARCH with alpha+beta=1, meaning shocks persist. 

  • If RiskMetrics & GARCH have same variance today, lower than the unconditional variance, RiskMetrics will give lower variance forecast than GARCH, meaning lower VaR & more comfortable risk managers

GARCH

• Generalised autoregressive conditional heteroskedasticity

• 

  (persistence)

• Unconditional variance:

  

  (long-term average)

• Using this expression for the unconditional variance, can substitute in to find that

  

  ie, future variance depends on shocks away from the long-term variance in today’s returns and today’s variance

• Forecasting variance in k days’ time:

  

  that is, the persistence of a shock in variance depends on the size of alpha + beta. 

Leverage & nonlinear GARCH

• A negative return increases variance by more than a same-magnitude positive return

• 

  , persistence =

  

• Where the bigger the spread between the z-score and theta, the bigger the volatility. Since we want this to be higher for negative returns, theta is positive

QLIKE

• When training a regression model for variance, we want to penalise forecasted volatilities beneath the volatility more than forecasted volatilities above 

• 

  The target is R_{t+1}^2, so if sigma < R, then R^2/sigma^2 < 1 and ln(R^2/sigma^2) << 0

Realised variance

• 

  daily variance estimated as the sum of m squared minute-returns

• Stylized facts:

  • RVs are more precise indicators of daily variance than daily squared returns

  • Higher autocorrelation in RVs than daily squared returns: greater forecastability

  • Log of RVs is approximately normally distributed

  • Daily returns divided by square root of RV, is approximately standard normal

Heterogeneous Autoregression (HAR)

• OLS forecast of tomorrow’s RV based on lagged RV variables

• 

  Where RV_D is today’s RV, RV_W is the mean RV over the past trading week, and RV_M is the mean RV over the past trading month (5 & 21 days, respectively)

• Considering the stylised fact that log RVs are normally distributed, we can regress log(RV) tomorrow on the log RV variables

• Benefits of HAR:

  • Captures long-memory, with 21 lags from monthly RV

  • Parsimonious, with only three coefficients (plus intercept)

  • Uses RVs, which are more precise than squared daily returns

Quantile-Quantile plots

• Quantile: groups of points, eg, percentiles are 100 groups

• Process:

  • Sort returns

  • Standardised returns on the vertical, z_i

  • The horizontal quantile is the inverse density of (i-0.5)/T

  • 

Filtered historical simulation

• Generate variance estimates, eg from GARCH & take the sequence of past returns. Calculate the series of standardised returns from these

• Use these z-scores to calculate VaR & ES, rather than make distributional assumptions

• Can generate large losses in the forecast period even without having observed a large loss in the recorded past returns, if a return was much lower than volatility would predict

Cornish-Fisher approximation

where CF is an adjustment to the inverse normal cumulative density function, using observed skewness and excess kurtosis

Standardised student’s t distribution

• t(d) for general t distribution, t-tilde (d) for standardised

• T-distribution generally is centred at 0 and has one parameter, the degrees of freedom, d, giving

  

• The standardised t distribution has the same mean, variance and skewness as the standardised normal. The only difference is excess kurtosis, which decreases in d. Gives method of calculating d, from measuring excess kurtosis: 

  

Extreme value theory (EVT)

• Beyond a threshold u, observations converge to Generalised Pareto Distribution

• Using the approximation for this distribution of

  

  allows us to get simple estimators for c and xi

  

• Choosing u: T_u is a good rule of thumb. Otherwise where Q-Q plot deviates

What EVT does is it gives us F(y), the density of tail returns. Can use this to estimate VaR & ES, in conjunction with our forecast of volatility, with 

Variance reduction techniques

• Stratification:

  • Wanting to draw n data points, create n strata between 0 and 1, distributing one point uniformly within each stratum

  • Then compute the inverse cumulative distribution of each point to get their distribution on the interpretable axis

• Antithetic variates:

  • In one path of the MC, we have the set of z-scores. Simply also take the set of negative z-scores

  • The output of these two paths will negatively covary, giving decreased variance

• Control variates:

  • Want to estimate some quantity using Monte Carlo. You simulate an unbiased but noisy estimator, as well as another quantity for which you have an exact value.

  • The estimate for the true quantity is then the simulated value plus an adjustment based on the difference between the other quantity and its true value

• Moment matching:

  • Simulate a vector of some quantity, for which we know a moment, eg true mean. 

  • If the sample mean does not match, adjust all observations

Models for variance

• Simple variance forecasting

• RiskMetrics

• GARCH

• Leverage models (GARCH)

• Heterogenous Autoregression (Realised Variance)

Models for VaR/ES

• Any variance model + assumption about innovation generation

• HS

• WHS

• FHS

• Cornish-Fisher

• Extreme Value Theory