Folio Lab/Documentation

Drawdown at Risk (DaR)

Drawdown at Risk applies the quantile logic of Value at Risk to the drawdown function of a single portfolio path. It reads that path across time and returns the threshold that separates the worst drawdowns from the rest. FolioLab reports it at the 95% level as a positive magnitude.

Overview

While Maximum Drawdown captures the single worst peak-to-trough decline in a given sample, it is a point estimate that provides no probabilistic context. Drawdown at Risk (DaR) addresses this limitation by framing drawdown risk in terms of a confidence level, analogous to how VaR frames return risk.

DaR was developed as part of a broader framework for drawdown-based risk management by Chekhlov, Uryasev, and Zabarankin (2005), and further refined by Goldberg and Mahmoud (2017). Definition 3.1 of that paper sets the drawdown as a function of time on one sample path, not as a distribution of maximum drawdowns across many paths. DaR is a quantile of that function. It therefore describes how much of one path lies deep underwater, and it needs only that one path.

This distinction matters for a historical run. A single backtest gives one path, so a cross-path distribution of maximum drawdowns does not exist for it. The quantile of the drawdown function does exist, and that is what FolioLab reports.

In portfolio optimization, DaR serves as a constraint or objective function that controls the tail risk of the drawdown distribution. This is particularly useful for strategies where maintaining the high-water mark is critical, such as hedge funds with performance fee structures.

Mathematical Formulation

Drawdown Process

Let denote the portfolio value at time . The running maximum (high-water mark) is:

The drawdown at time is the fractional decline from the running maximum:

Note that , with when the portfolio is at its all-time high. The maximum drawdown over the period is .

Drawdown at Risk

DaR at confidence level is the -quantile of the drawdown function over the observed period:

Here the probability runs over time on one path. Each date contributes one drawdown observation. DaR is the smallest level that at least of the dates stay at or above.

For example, a 95% DaR of 15% means that the portfolio sat more than 15% below its running peak on 5% of the observed dates. This is a statement about the observed path. It is not a probability statement about a future drawdown.

How FolioLab computes it

FolioLab takes the 5% quantile of the signed drawdown series and reports its absolute value, so the result is a positive magnitude:

The signed series is at or below zero at every date, so its 5% quantile is the deep tail, and Proposition 3.1 of Chekhlov, Uryasev and Zabarankin gives the nonnegative magnitude that FolioLab prints.

One difference from the paper is worth stating. Definition 3.1 builds the drawdown series on the uncompounded cumulative rate of return, and the paper notes that a relative drawdown does not carry the same properties. FolioLab computes on a compound equity curve. This is the common library form, and it is not literally Definition 3.1.

Analogy with Value at Risk

DaR mirrors the structure of VaR but operates on drawdowns rather than returns:

Value at Risk

  • Quantile of the return (loss) distribution
  • Measures single-period loss risk
  • Not path-dependent

Drawdown at Risk

  • Quantile of the drawdown function across time
  • Measures cumulative peak-to-trough risk
  • Path-dependent

Empirical Estimation

FolioLab estimates DaR directly from the one observed path. No resampling and no simulation take part:

  • Step 1: Build the compound equity curve from the daily portfolio returns.
  • Step 2: Form the signed drawdown series against the running peak , one value per date.
  • Step 3: Take the 5% quantile of that series and report its absolute value as .

A resampling estimate of the maximum drawdown distribution is a different measure. FolioLab does not compute one, and the number on the results page is not that quantity.

Advantages & Limitations

Advantages

  • Whole-path view: Unlike MDD, which reports one episode, DaR summarises the whole drawdown function and states how much of the period sat deep underwater.
  • Path-dependent: Captures the cumulative nature of losses, which is more relevant for long-term investors than single-period measures.
  • Intuitive interpretation: Easy to communicate as "the portfolio was deeper than this on 5% of the days in the period."
  • Optimization-compatible: Can be used as a constraint in portfolio optimization to control drawdown risk directly.

Limitations

  • Backward looking: The quantile describes the one observed path. It is not a forecast, and a different sample period can move it a long way.
  • Not subadditive: Like VaR, DaR inherits the non-coherence problem -- the DaR of a combined portfolio may exceed the sum of individual DaRs.
  • Data intensive: The drawdown observations are serially correlated, so a short series gives few independent episodes and an unstable quantile.
  • No tail information: DaR, like VaR, does not describe the severity of drawdowns beyond the threshold.

References

  1. Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). "Drawdown Measure in Portfolio Optimization." International Journal of Theoretical and Applied Finance, 8(1), 13-58.
  2. Goldberg, L. R., & Mahmoud, O. (2017). "Drawdown: From Practice to Theory and Back Again." Mathematics and Financial Economics, 11(3), 275-297.
  3. Zabarankin, M., Pavlikov, K., & Uryasev, S. (2014). "Capital Asset Pricing Model (CAPM) with Drawdown Measure." European Journal of Operational Research, 234(2), 508-517.
  4. Alexander, G. J., & Baptista, A. M. (2006). "Portfolio Selection with a Drawdown Constraint." Journal of Banking & Finance, 30(11), 3171-3189.

Not investment advice. Past performance is not indicative of future results.