Hypothetical Returns

[Jaskarn]

@David Harper CFA FRM
In Backtesting VAR chapter, one idea is shared wherein its been told that we can't use actual returns for VAR backtesting as they are volatile so we either use hypothetical returns or cleaned returns. Can you give one example as to how we get hypothetical returns from actual returns?

I really didn't understand this line," The hypothetical returns represents a frozen portfolio, obtained from fixed position applied to the actual returns on all the securities, measured from close to close.

 

[Matthew Graves]

Actual portfolio NAV changes are noisy i.e. will contain the effects of trading, inflows, outflows etc. I think what they're getting at is that the backtest should not contain these effects. In addition, usually with VaR you're interested in the current exposure profile of the fund rather than the historic profile which will be time-varying.

You have a distribution of returns for each security and the current portfolio weights. You can combine the two (e.g. weighted average return of the portfolio at each time point) to give a distribution of portfolio returns based on the current exposures which you can then use to assess you VaR.

 

[David Harper CFA FRM]

Thank you @Matthew Graves
@Jaskarn Below I copied (i) the relevant section from Jorion and (ii) the explanation by Carol Alexander:

Jorion, VaR 3rd Edition:

6.1.2. Which Return? Before we even start addressing the statistical issue, a serious data problem needs to be recognized. VAR measures assume that the current portfolio is "frozen" over the horizon. In practice, the trading portfolio evolves dynamically during the day. Thus the actual portfolio is "contaminated" by changes in its composition. The actual return corresponds to the actual P&L, taking into account intraday trades and other profit items such as fees, commissions, spreads, and net interest income.

This contamination will be minimized if the horizon is relatively short, which explains why backtesting usually is conducted on daily returns. Even so, intraday trading generally will increase the volatility of revenues because positions tend to be cut down toward the end of the trading day. Counterbalancing this is the effect of fee income, which generates steady profits that may not enter the VAR measure.

For verification to be meaningful, the risk manager should track both the actual portfolio return R(t) and the hypothetical return R*(t) that most closely matches the VAR forecast. The hypothetical return R*(t) represents a frozen portfolio, obtained from fixed positions applied to the actual returns on all securities, measured from close to close.

Sometimes an approximation is obtained by using a cleaned return, which is the actual return minus all non-mark-to-market items, such as fees, commissions, and net interest income. Under the latest update to the market-risk amendment, supervisors will have the choice to use either hypothetical or cleaned returns. [2 See BCBS (2005b)].

Since the VAR forecast really pertains to R*, backtesting ideally should be done with these hypothetical returns. Actual returns do matter, though, because they entail real profits and losses and are scrutinized by bank regulators. They also reflect the true ex post volatility of trading returns, which is also informative. Ideally, both actual and hypothetical returns should be used for backtesting because both sets of numbers yield informative comparisons. If, for instance, the model passes backtesting testing with hypothetical but not actual returns, then the problem lies with intraday trading. In contrast, if the model does not pass backtesting with hypothetical returns, then the modeling methodology should be reexamined." --- Philippe Jorion. Value at Risk, 3rd Ed.: The New Benchmark for Managing Financial Risk (p. 142). Kindle Edition.

Carol Alexander, MRA, Vol IV:

"IV.6.4.2 Guidelines for Backtesting from Banking Regulators ... VaR estimates are based on one of two theoretical assumptions about trading on the portfolio. Either the portfolio is assumed to be rebalanced over the risk horizon to keep its asset weights or risk factor sensitivities constant, or it is assumed that the portfolio is held static so that no trading takes place and the holdings are constant. The assumption made here influences the VaR estimate for option portfolios over risk horizons longer than 1 day. But both assumptions lack realism. In practice, portfolios are actively managed at the trader’s discretion, and the actual or realized P&L on the portfolio is not equal to the hypothetical, unrealized P&L, i.e. the P&L on which the VaR estimate is based. In accountancy terminology the unrealized P&L is the mark-to-market P&L, whereas the realized P&L includes all the P&L from intraday trading and is based on prices that are actually traded. Realized P&L may also include fee income, any use of the bank’s reserves and funding costs. With these additional items we call it actual P&L and without these it is called cleaned P&L. To avoid confusion, we shall call the hypothetical, unrealized P&L the theoretical P&L.

Many banking regulators (for instance, in the UK) require two types of backtests, both of which are based on the simple methodology described above. Their tests must be based on both realized (actual or cleaned) P&L and on theoretical P&L. Backtests based on theoretical P&L are testing the VaR model assumptions. However, those based on realized 1-day P&L are not testing how the model will perform in practice, as a means of estimating regulatory capital, unless the scaling of 1-day VaR to 10-day VaR is accurate."