i

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...

﻿
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.