Drawdown: from practice to theory and back again

Abstract

Maximum drawdown, the largest cumulative loss from peak to trough, is one of the most widely used indicators of risk in the fund management industry, but one of the least developed in the context of measures of risk. We formalize drawdown risk as Conditional Expected Drawdown (CED), which is the tail mean of maximum drawdown distributions. We show that CED is a degree one positive homogenous risk measure, so that it can be linearly attributed to factors; and convex, so that it can be used in quantitative optimization. We empirically explore the differences in risk attributions based on CED, Expected Shortfall (ES) and volatility. An important feature of CED is its sensitivity to serial correlation. In an empirical study that fits AR(1) models to US Equity and US Bonds, we find substantially higher correlation between the autoregressive parameter and CED than with ES or with volatility.

Appendices

Appendix A: Data and estimation methodologies

1.1 Data

The data were obtained from the Global Financial Data database. We took the daily time series for the S&P 500 Index and the USA 10-year Government Bond Total Return Index.

1.2 Portfolio construction

Rather than provide thorough realistic empirical analyses of portfolio risk and return, our goal behind the simulated portfolios is to illustrate this article’s theoretical development in relation to drawdown risk. For simplicity, we therefore do not account for transaction costs or market frictions in all hypothetical portfolios constructed throughout this study. Moreover, we assume that all portfolios are fully invested and long only.

Fixed-mix portfolios. In the fixed-mix portfolios, rebalancing to the fixed weights is done on a monthly basis. When comparing to other popular rebalancing schemes (quarterly, bi-annually and yearly), similar results were obtained.

Risk parity portfolios. In risk parity strategies, assets are weighted so their ex post risk contributions are equal. As mentioned in Sect. 5, parity portfolios are not restricted to volatility only, but can be constructed along other risk measures, such as Expected Shortfall and Conditional Expected Drawdown. Asset weights in the strategies depend on estimates of the underlying risk measures (see Sect. A.3), which are calculated using a 3-year rolling window of trailing returns. Varying the estimation methodology by changing the length of the rolling window or the weighting scheme applied to the returns within this window did not alter our results substantially. Similar to the fixed-mix portfolios, risk parity portfolios are rebalanced monthly, with other rebelancing schemes yielding similar results.

1.3 Risk estimation

Volatility. Portfolio volatility is calculated as the annualized standard deviation of the daily time series over the entire period under consideration. To obtain the volatility risk contributions for a n-asset portfolio $P = \sum _i w_i X_i$, note that the i-th total contribution $\mathrm{RC}_i^\sigma$ to portfolio volatility

$$\begin{aligned} \sigma (P) = \sum _i w_i^2 \sigma _i^2 + \sum _i \sum _{j \ne i} w_i w_j \sigma _{i,j} \end{aligned}$$

is

$$\begin{aligned} \mathrm{RC}_i^\sigma = w_i^2 \sigma _i^2 + \sum _{j \ne i} w_i w_j \sigma _{i,j}, \end{aligned}$$

where $\sigma _i^2$ is the variance of $X_i$ and $\sigma _{i,j}$ is the covariance of $X_i$ and $X_j$. Then, the i-th fractional contribution to volatility is given by

$$\begin{aligned} \mathrm {FRC}_i^\sigma (P) = \frac{w_i^2 \sigma _i^2 + \sum _{j \ne i} w_i w_j \sigma _{i,j}}{\sigma (P)} \quad . \end{aligned}$$

Expected shortfall. For confidence level $\alpha \in (0,1)$, an estimate for the Expected Shortfall of a portfolio is calculated by ordering the daily return time series over the whole period according to the magnitude of the returns, then averaging over the worst $(1-\alpha )$ percent outcomes, more specifically:

$$\begin{aligned} {\widehat{\mathrm{ES}}}_\alpha = \frac{1}{K} \sum _{i=1}^{K} r_{(i)} , \end{aligned}$$

where T is the length of the daily time series, $K = \lfloor T(1-\alpha ) \rfloor$, and $r_{(i)}$ is the i-th return of the magnitude-ordered time series. To obtain the contributions to shortfall risk, recall that under a continuity assumption, the Expected Shortfall of an asset $X \in {\mathcal {M}}$ can be expressed as $\mathrm{ES}_\alpha (X) = {\mathbb {E}}\left( X \mid X \ge \mathrm{VaR}_\alpha (X)\right)$, or the expected loss in the event that its Value-at-Risk at $\alpha$ is exceeded.¹⁴ As usual, let $P = \sum _i w_i X_i$ be the portfolio in consideration. Assuming differentiability of the risk measure VaR, the marginal contribution of $X_i$ to portfolio shortfall $\mathrm{ES}_\alpha (P)$ is given by

$$\begin{aligned} \mathrm {MRC}_i^{\mathrm{ES}_\alpha }(P) =\frac{\partial \mathrm{ES}_\alpha (P)}{\partial w_i} = {\mathbb {E}}(X_i \mid P \ge \mathrm{VaR}_\alpha (P)) \quad . \end{aligned}$$

An estimate for the i-th marginal contribution to shortfall risk is then obtained by averaging over all the returns of asset $X_i$ that coincide with portfolio returns exceeding the portfolio’s Value-at-Risk at threshold $\alpha$.

Conditional expected drawdown. The first step in calculating an estimate for Conditional Expected Drawdown is to obtain the empirical maximum drawdown distribution. From the historical time series of returns, we generate return paths of fixed length n using a one-day rolling window. This means that consecutive paths overlap. The advantage is that for a return time series of length T, we obtain a maximum drawdown series of length $T-n$, which for large T and small n is fairly large, too. From these $T-n$ return paths we calculate the maximum drawdown as defined in Sect. 2. An estimate for the Conditional Expected Drawdown at confidence level $\alpha \in (0,1)$ is then calculated as the average of the largest $(1-\alpha )$ percent maximum drawdowns. To obtain an estimate for the i-th contribution to drawdown risk CED, we take the average over all the drawdowns of the i-th asset in the path $[t_{j*},t_{k*}]$ that coincide with the overall portfolio’s maximum drawdowns that exceed the portfolio’s drawdown threshold $\text {DT}_\alpha$ at confidence level $\alpha$ (Recall that $j^*<k^*\le n$ are such that $\varvec{\mu }(P_{T_n}) = P_{t_{k*}} - P_{t_{j*}}.$)

Appendix B: Drawdown risk decomposition along a balanced portfolio of US Equity and US treasury bonds

See Figs. 9 and  10.

Fig. 9

Daily 6-month rolling fractional risk contributions (FRC) along 90 % conditional expected drawdown (CED) of US Equity and US Treasury Bonds to the balanced 60/40 portfolio over the period 1982–2013. Also displayed is the daily VIX series over the same period, with the right-hand axis indicating its level

Fig. 10

Decomposition of drawdown risk contributions $\mathrm{RC}^\mathrm {CED}_i(P) = w_i \mathrm {CED}(X_i) \mathrm Corr^\mathrm {CED}_i$ for the 60/40 allocation to US Equity and US Treasury Bonds over the period 1982–2013. The top two panels show the daily 6-month rolling standalone 90 % Conditional Expected Drawdown (CED) of the two assets, while the bottom two panels show the 6-month rolling generalized correlations of the individual assets along CED

Appendix C: Risk decomposition along expected shortfall

See Fig. 11.

Fig. 11

Decomposition of contributions $\mathrm{RC}^\mathrm{ES}_i(P) = w_i \mathrm{ES}(X_i) \mathrm Corr^\mathrm{ES}_i$ to 90 % expected shortfall (ES) for the 60/40 allocation to (A) US Equity and US Government Bonds, and (B) US Equity and US Treasury Bonds over the time period 1982–2013