Relative VaR or Earnings-at-Risk (EaR)

Incorporating Earnings in the picture of risk

As described in the absolute VaR section, Mean Zero Absolute VaR assumes expected returns are Normal, which implies a mean of zero. Stated in other words, this  assumption considers that our portfolio will not yield any returns.


This approach makes complete sense when dealing with highly liquid trading books that are revalued daily.


In this framework p&l movements as well a fees are the main driving force behind the trades. Returns stemming from dividends and coupons are minimal when prices are sampled daily, especially when costs of carrying the positions are taken into account. 


There are many rationales behind the mean-zero framework. From a theoretical standpoint, the mean is computed by summing the returns (i.e. the difference in prices).
Since the returns are the difference in prices and these returns are summed as they are sampled, they all cancel out,  except for the first and last price which is then divided by the number of samples minus one. From this, It is quite obvious that in most cases dealing with daily sampling will result in a rather small mean, compared to the volatility. (to get a feel for this have a look at the volatility analyzer module by requesting advanced statistics)  From a modeling standpoint, this has different consequences.

The Zero Mean assumption affects many parts of the Parametric VaR model:

  • The computation of sigma. (Volatility and Correlations) that are fed into the model are mean zero volatilities and correlations. (See unitized - standardized returns with mean zero 0 and variance one).
  • The simulation process. (A relative diffusion, but with zero (0) drift).
  • The final VaR which is a de-facto one-day absolute VaR does not take into account the portfolio mean returns (see Daily Earnings at Risk or DEaR) which assumes volatility is stationary (=no drift).
  • Predictive components, such as conditional correlation, used mainly for what-if or stress testing purposes. 

This produces a very neat and consistent model for analyzing risks over very short horizons.

However, If your investment horizon goes beyond ten days (2 weeks). you should probably consider Relative VaR (EaR: Earnings -at-Risk) and other byproducts such as Benchmark VaR.

 

Relative Var

Value at Risk (VaR) is computed as the value at sa given percentile p where q is the p-th quantile

Relative VaR or Earnings at Risk, as it is commonly known, is designed around one simple concept: 

Instead of assuming the position's mean is zero,

Absolute VaR=( 0  - Portfolio Volatility)*Confidence.

we compute the average mean of the portfolio which we incorporate into our volatility / Value-at-Risk computation.
     
Relative VaR=( Portfolio Mean Expected Return  - Portfolio Volatility)*Confidence.

So, instead of assuming risk as pure volatility, we incorporate expected returns.

When discussing relative VaR, most practitioners tend to emphasize the mean expected return of the portfolio from which scaled volatility will be deducted in order to obtain Value-at-Risk.

As mentioned above, the mean zero assumption affects different parts of the VaR computation. As such,  a consistent framework must accommodate these same points with a mean expected return:

 The Expected Mean assumption affects the same parts of the VaR model:

  • The computation of sigma and rho. (Volatility and Correlations) includes an expected mean.
  • The simulation process. A relative diffusion, with drift:
    Foreign Exchange with Covered Interest Parity (simulation of interest rate differential).
    Equity includes systematic and specific returns
    Interest Rates: short term and long term drifts specifically  mean reversion.
    Commodity. drift as the net convenience yields.
  • Predictive components used mainly for what-if or stress testing purposes.
  • Earnings at Risk is computed with a drift.
  • The final VaR result must cover the specific horizon sought via multi-stepped simulation with reinvestments, ageing, etc.

Needless to say, the term expectation leaves a lot of room for interpretation.

How do we estimate these returns ?

From a theoretical standpoint, the most obvious choice is to use the average mean of projected returns, but we could just as easily incorporate estimated or forecasted returns from economic factors, historical returns, returns from budgets or analysts forecasts.

As Absolute VaR makes complete sense for traders who mark-to-market positions daily, Relative VaR is ideally suited for individual investors, portfolio managers and corporations who rebalance positions weekly, monthly or quarterly.
There is indeed a very close relationship between your risk horizon, the frequency at which portfolios are rebalanced, publication of results and the sampling of the data that feeds the model! 

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