Drawdown
Drawdown
= max [ sum[r(t)] ] - r(t)
t = 0..T
Maximum Loss
Given a time series with N log returns r(t) over t = 1 to T, we can define...
= min [r(1), r(1)+r(2), r(1)+r(2)+r(3), ..., r(1)+...+r(N)]
= min [ sum [r(i)] ]
t=0..N i=1..t
In words: The minimum cumulated return from the beginning in a certain time period.
Maximum Drawdown
The maximum drawdown can be loosely defined as the largest drop from a peak to a bottom in a certain time period.
Maximum drawdown captures a path-dependant feature of a time series which is not represented in the histogram of the return time series.
= min [r(1), r(1)+r(2), r(1)+r(2)+r(3), ..., r(1)+...+r(N),
r(2), r(2)+r(3), r(2)+r(3)+r(4), ..., r(2)+...+r(N), ..., r(N)]
= min [ sum [r(j)] ]
i=1..t, t=1..N j=i..t
In words: The minimum cumulated return from any beginning points over a certain time period.
= max [ Drawdown(t) ]
t=0..T
...the last formula yields the end-point of the maximum
drawdown period. The starting point is found at the
last time point Drawdown(t) was equal to zero.
Maximum drawdown is always smaller than or equal to the difference between maximum loss and maximum gain.
Maximum loss & gain are the global extreme values, maximum drawdown is a concept base on the local minimum of a return time series.
Maximum drawdown is often used when not enough observations are available to calculate volatility measures (like for example standard deviation).
Maximum drawdown is highly dependent on the time interval chosen (annual, monthly, daily and so on) as well as the observation period.
For calculating Maximum Drawdown with the standard functionality of Microsoft Excel, see the spreadsheet in the download section.
As the calculation of Maximum Drawdown is a little bit cubersome (not difficult, just cubersome), I have written some Excel VBA functions. Note that these functions only work with log returns! The functions are available in the download section. There, you will also find a MATLAB function for calculating maximum drawdown and related statistics.
Many times, the period during which maximum drawdown has occured is as important as the maximum drawdown value itself. Also of interest is 'time under water', that is the period of time it takes for cumulative returns to recover to their value at the start of the maximum drawdown period.
The maximum drawdown statistics can be summarized in a chart similar to the one below...
It's also interesting to analyze the n-th drawdown respectively the full drawdown vector containing all sorted drawdowns.
Relationship between Drawdown and Value-At-Risk (VaR)
Drawdown can also be understood as the loss an investment may suffer with a certain confidence 1-a whatever the time period. In this formulation, it becomes clear that drawdown is closely related to the concept of 'Value-At-Risk', which describes the loss of an investment with a certain confidence over a certain time period. Therefore, drawdown is an "interim" loss, VaR on the other hand describes "end-of-period" loss.
If (log) returns are assumed to be independently and identically normal distributed, it is possible to express Drawdown in terms of expected return, volatility, confidence level, time horizon (and a constant threshold level, if applicable). This implies that it is only necessary to consider drawdown if returns are not IDD normal distributed. This is typically the case with certain hedge fund strategies, where returns are known to be serially correlated over time and having "fat tails".
Literature:
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M. M. L. de Prado / A. Peijan: "Measuring Loss Potential of Hedge Fund Strategies," Journal of Altnerative Investments, Summer 2004
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