From Sharpe Ratio to Max-Drawdown, a numerical approach.

Sharpe ratio is the most widely accepted measure to quantify the quality and the performance of a systematic strategy. Most funds select PMs based on the Sharpe of their track record and performance-based ranking systems also rank participants based on their realized Sharpe ratio.

However, real trading does not know about statistic, and concretely what really matters when evaluating a live strategy is drawdown, not least because drawdowns can have a significant impact on an investor's emotional state and decision-making, which can affect the long-term success of their investment strategy.

It's crucial to understand the relationship between the Sharpe ratio and drawdown to accurately gauge the potential for risk in an investment strategy. The goal of this article is to establish this connection and provide a clearer understanding of what can be realistically expected in terms of maximum drawdown for a strategy that aims to deliver a given Sharpe ratio.

The results can assist in evaluating the appropriate moment to discontinue a strategy if the actual drawdown exceeds what is compatible with the assumed Sharpe ratio.

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We'll begin with a simulation of daily standard normal returns, represented by r_n. There are 250 trading days in a year, so we'll generate 250 of these random returns. The equity line is determined by taking the cumulative sum of the rn. To generate equity lines that correspond to a specified annual Sharpe ratio (S) and annual volatility (AV), we'll apply a linear transformation to the r_n

where:

It is easy to see that:

The cumulative returns of r’_n are thus equity lines of Sharpe S and annual vol AV as desired.

By generating N of these lines, it is possible study the statistical properties of a random sample with specific sharpe and annual vol

The following is the Python code for generating these paths:

import pandas as p
import numpy as np
import random

def generate_paths(s,STD,N):
??? returns = []
??? for _ in range(N):
??????? returno = np.random.normal(0, 1, 250)
??????? r_std = returno.std()
??????? returno = returno+(s*r_std/np.sqrt(250)-returno.mean())
??????? returno = (STD/(np.sqrt(250)*r_std))*returno
??????? returns.append(returno)
??? return pd.DataFrame(returns).Td        

By utilizing this technology, we can delve into the samples to explore the connection between Sharpe ratio and drawdowns. A drawdown refers to a fall in an investment's value from its peak to trough during a specific period.

To illustrate this relationship, we run simulations of 40 million paths, calculating the max drawdown for each generated equity line. The end result is a plotted distribution of max drawdowns, as shown in the charts for Sharpe ratios 1 and 2

The final step involves determining the expected and worst-case drawdowns over a specific period, based on the distribution of max drawdowns, for a given Sharpe ratio. The chart to the left displays the results. On the right, equity lines corresponding to the most severe drawdowns for various Sharpe ratios are plotted to give the reader a visual representation of how steep the drawdowns can be, even with a relatively high sharpe.?

The table below can assist in real-time trading by determining when to stop a strategy. If a strategy with a projected Sharpe of 1 incurs a loss exceeding 2.35 times the annual volatility, the assumption on the Sharpe ratio becomes invalid, and it may be advisable to discontinue it.

Thank you for your attention!

Francesco