Under the MDE model, conditional volatility for each market state or regime is calculated as follows:
σ2 = Ʃw (xt-i)r2t-i
xt-i = the vector of relevant variables describing the market state or regime at time t – i
w(xt-i) = the weight used on observation t — i, as a function of the “distance” of the state xt-i from the current state xt.
The kernel function, w(xt-i), is used to measure the relative weight in terms of “near” or “distant” from the current state. The MDE model is very flexible in identifying dependence on state variables. Some examples of relevant state variables in an MDE model are interest rate volatility dependent on the level of interest rates or the term structure of interest rates, equity volatility dependent on implied volatility, and exchange rate volatility dependent on interest rate spreads.
To demonstrate mean reversion, consider a time series model with one lagged variable:
Xj = a + b x Xi-1
This type of regression, with a lag of its own variable, is known as an autoregressive (AR) model. In this case, since there is only one lag, it is referred to as an AR(1) model. The long- run mean of this model is evaluated as [a / (1 — b)].
The single-period conditional variance of the rate of change is σ2 and that the two-period variance is (1 + b2)σ2.
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.
A method for estimating future volatility is implied volatility. The Black-Scholes-Merton model is used to infer an implied volatility from equity option prices. Using the most liquid at-the-money put and call options, an average implied volatility is extrapolated using the Black-Scholes-Merton model.
Backtesting is the process of comparing losses predicted by the value at risk (VaR) model to those actually experienced over the sample testing period.
There are three desirable attributes of VaR estimates that can be evaluated when using a backtesting approach.
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