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Friday, 24 January 2014

Drawdown Risk Budgeting: Contributions to Drawdown-At-Risk and the Drawdown Parity Portfolio

Similar to Value-At-Risk, Drawdown-At-Risk is defined as a point on the drawdown distribution defined by a probability interpreted as a "level of confidence". The well-known risk measure Maximum Drawdown is the 100% Drawdown-At-Risk, i.e. the drawdown which is not exceeded with certainty.

The table below shows Drawdown-At-Risk values for the constituents of a specific multi asset class universe (total returns, monthly figures, Jan 2001 to Oct 2011, base currency USD)...

In portfolio analytics, fully additive contributions to risk can be derived from Euler's homogeneous function theorem for linear homogeneous risk measures. Portfolio volatility and tracking error are examples of risk measure which are linear homogeneous in constituent weights.

Non-linear homogeneous risk measures can be approximated (e.g. Taylor series expansions, using the total differential as a linear approximation and so on). In the chart below, we show how the 95% DaR of an equal-weighted portfolio varies with variations in individual constituent weights (we make the assumptions that exposures are booked against a riskfree cash account with zero return)...

This chart is called "the Spaghetti chart" by certain people. In the case of the minimum 95% DaR portfolio, i.e. the fully invested long-only portfolio with minimum 95% Drawdown-At-Risk, all spaghettis must point downwards...

The full details of the risk decomposition for the equal-weighted portfolio...

...in comparsion with the minimum 95% DaR portfolio...

Additive contributions to portfolio Drawdown-At-Risk open up the door for drawdown risk budgeting. For example, the Drawdown Parity Portfolio can be calculated as the portfolio with equal constituent contributions to portfolio drawdown risk...

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Due to the residual, the DaR contributions are not perfectly equalized. Taking into account estimation risk and other implementation issues, this is acceptable for practical purposes.

Being able to calculate additive contributions to drawdown-at-risk is useful for descriptive ex post or ex ante risk budgeting purposes. The trade risk charts are useful indicators providing information on a) the risk drivers in the portfolio and b) the directions to trade.

Budgeting drawdown risk is really budgeting for future drawdowns ("ex ante drawdown"). This involves estimating future drawdowns. Whether future drawdowns can be estimated with the required precision is an empirical question. In order to assess what this task might involve, it is interesting reviewing certain findings in the theoretical literature related to the expected maximum drawdown for geometric Brownian motions (see for example "An Analysis of the Expected Maximum Drawdown Risk Measure" by Magdon-Ismail/Atyia. More recently, analytical results have been derived for return generating processes with time-varying volatility). In the long-run, the expected maximum drawdown for a geometric Brownian motion is...

\$MDD_{e} = (0.63519 + 0.5 \cdot ln(T) + ln(\frac{mu}{sigma})) \cdot \frac{sigma^2}{mu} \$

Expected maximum drawdown is function in investment horizon (+), volatility (+) and expected return (-).

While we have time series models with proven high predictive power to estimate volatility risk (e.g. GARCH), the estimation of maximum drawdown is a much more challenging task because it involves estimating expected returns, which is known to be subject to much higher estimation risk.

Monday, 6 January 2014

Resampling the Efficient Frontier - With How Many Observations?

Since optimizer inputs are stochastic variables, it follows that any efficient frontier must be a stochastic object. The efficient frontier we usually plot in mean/variance space is the expected efficient frontier. The realized efficient frontier will almost always deviate from the expected frontier and will lie within certain confidence bands.
Several attempts have been made to illustrate the stochastic nature of the efficient frontier, the most famous one probably being the so-called "Resampled Efficient Frontier" (tm) by Michaud/Michaud(1998).
Resampling involves setting the number of simulations as well as setting the number of observations to generate in each simulation. The importance of the latter decision is typically underestimated.
The chart below plots the resampled portfolios of 16 portfolios on a particular mean/variance efficient frontier...

The larger density of points at the bottom left end of the frontier is a result from the fact that there exist two very similar corner portfolios in this area of the curve.
The chart below plots the same frontier with the same number of simulations, but a much larger number of generated observations...

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As the confidence bands, average weights or any risk and return characteristics are largely determined by the choice of number of simulations and number of observations in each simulation, it is worth keeping an eye on these modelling decisions when using relying on a resampling approach for investment purposes.