Return Distributions And Market Regimes
Three common deviations from normality that are problematic in modeling risk result from asset returns that are fat-tailed, skewed, or unstable. Fat-tailed refers to a distribution with a higher probability of observations occurring in the tails relative to the normal distribution. A distribution is skewed when the distribution is not symmetrical and there is a higher probability of outliers. Parameters of the model that vary over time are said to be unstable.
The phenomenon of “fat tails” is most likely the result of the volatility and/or the mean of the distribution changing over time.
Conditional And Unconditional Distributions
If the mean and standard deviation are the same for asset returns for any given day, the distribution of returns is referred to as an unconditional distribution of asset returns. However, different market or economic conditions may cause the mean and variance of the return distribution to change over time. In such cases, the return distribution is referred to as a conditional distribution.
Market Regimes and Conditional Distributions
A regime-switching volatility model assumes different market regimes exist with high or low volatility. The probability of large deviations from normality (such as fat tails) occurring are much less likely under the regime-switching model because it captures the conditional normality.
Estimating Value At Risk
Historical-based approaches of measuring VaR typically fall into three sub-categories: parametric, nonparametric, and hybrid.
- The parametric approach typically assumes asset returns are normally or lognormally distributed with time-varying volatility (i.e., historical standard deviation or exponential smoothing).
- The nonparametric approach is less restrictive in that there are no underlying assumptions of the asset returns distribution (i.e., historical simulation).
- The hybrid approach combines techniques of both parametric and nonparametric methods to estimate volatility using historical data.
A major difference between the historical standard deviation approach and the two exponential smoothing approaches is with respect to the weights placed on historical returns. Exponential smoothing approaches give more weight to recent returns, and the historical standard deviation approach weights all returns equally.
The RiskMetrics® and GARCH approaches are both exponential smoothing weighting methods. RiskMetrics® is actually a special case of the GARCH approach. Exponential smoothing methods are similar to the historical standard deviation approach because they are parametric, attempt to estimate conditional volatility, use recent historical data, and apply a set of weights to past squared returns.
Return Aggregation
When a portfolio is comprised of more than one position using the RiskMetrics® or historical standard deviation approaches, a single VaR measurement can be estimated by assuming asset returns are all normally distributed. The historical simulation approach for calculating VaR for multiple portfolios aggregates each period’s historical returns weighted according to the relative size of each position. The weights are based on the market value of the portfolio positions today, regardless of the actual allocation of positions K days ago in the estimation window. A third approach to calculating VaR for portfolios with multiple positions estimates the volatility of the vector of aggregated returns and assumes normality based on the strong law of large numbers.
Implied Volatility
The implied-volatility-based approach for measuring VaR uses derivative pricing models such as the Black-Scholes-Merton option pricing model to estimate an implied volatility based on current market data rather than historical data.
Mean Reversion and Long Time Horizons
With an AR(1) model, long-run mean is computed as: [a / (1 − b)]. If b equals 1, the long-run mean is infinite (i.e., the process is a random walk). If b is less than 1, then the process is mean reverting.
Under the context of mean reversion, the single-period conditional variance of the rate of change is σ2, and the two-period variance is (1 + b2)σ2. Without mean reversion (i.e., b = 1), the two-period volatility is equal to the square root of 2σ2. With mean reversion (i.e., b < 1), the two-period volatility will be less than the volatility from no mean reversion.
If mean reversion exists, the long horizon risk (and resulting VaR calculation) will be smaller than square root volatility.
