Analysis > Market Risks > Relative VaR > Ex-Ante Tracking Error

Benchmark VaR and Ex-Ante Tracking Error

 

To Compute Benchmark VaR or Benchmark Earnings-at-Risk, we create a Combined portfolio. This Combined-Portfolio is designed to have zero Net Present Value.

Note:
  • We often refer to Combined Returns when evoking net returns of the Combined Portfolio.
  • The Net Present Value must take special care of products that are valued at Par at Inception if the Inception date coincides with the Present Value Date. This is espeiclaly true when dealing with Financial Futures and Interest Rate Swaps.


The trick is to scale the portfolio so that the Net Present Value is Equal to the Benchmark Net Present Value.

  • + Long the Benchmark
  •  - Short the Portfolio * Scaling Factor
  • =Combined Portfolio
 
 

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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