How to calculate risk indicator for a portfolio as maximum drawdown, sortino ratio and ulcer index?

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Mattia Smanio 2023-4-22

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评论： LeoAiE 2023-5-7

Based on the following code, is it possible to have the correct formula to calculate the maximum drawdown, the sortino ratio and the ulcer index?

clear all;

clc;

NumPorts=2000;

% Set Up the Data

open ("MatlabBOGLE.xlsx");

t = readtable("MatlabBOGLE.xlsx");

symbol = t.Properties.VariableNames(2:end);

dailyreturn= tick2ret(t(:,2:end));

dailyreturn2 = table2array(dailyreturn);

% Create a Portfolio Object (with the risk-free rate)

RiskFreeRate=0.00/252;

p = Portfolio("AssetList",symbol,"RiskFreeRate",RiskFreeRate);

x0 =[0.1,0.1,0.3,0.5,0,0];

p = setInitPort(p,x0);

p = estimateAssetMoments(p,dailyreturn);

[initialrisk,initialreturn] = estimatePortMoments(p,p.InitPort);

display(initialrisk); %"Rischio" Iniziale del lazy Portfolio con le %di allocazioni inalterate

display(initialreturn); %Rendimento Iniziale del lazy Portfolio con le %di allocazioni inalterate

clf;

portfolioexamples_plot('Asset Risks and Returns', ...

{'scatter', initialrisk,initialreturn, {'StartPort'}}, ...

{'scatter', sqrt(diag(p.AssetCovar)), p.AssetMean, p.AssetList, '.r'});

% Set Up a Portfolio Optimization Problem

p=setDefaultConstraints(p);

pwgt = estimateFrontier(p,NumPorts);

[portrisk,portret] = estimatePortMoments (p,pwgt);

figure

portfolioexamples_plot('Efficient Frontier', ...

{'line', portrisk, portret}, ...

{'scatter', initialrisk, initialreturn, {'StartPort'}}, ...

{'scatter', sqrt(diag(p.AssetCovar)), p.AssetMean, p.AssetList, '.r'});

% Maximize the Sharpe Ratio

p = setInitPort(p, x0);

swgt = estimateMaxSharpeRatio(p);

[srsk,sret] = estimatePortMoments(p,swgt);

display(swgt);

display(srsk);

display(sret);

figure

portfolioexamples_plot('Efficient Frontier with Maximum Sharpe Ratio Portfolio', ...

{'line', portrisk, portret}, ...

{'scatter', srsk, sret, {'Sharpe'}}, ...

{'scatter', initialrisk,initialreturn, {'StartPort'}}, ...

{'scatter', sqrt(diag(p.AssetCovar)), p.AssetMean, p.AssetList, '.r'});

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Answer 1

LeoAiE 2023-4-23

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1951348-how-to-calculate-risk-indicator-for-a-portfolio-as-maximum-drawdown-sortino-ratio-and-ulcer-index#answer_1221228

在 MATLAB Online 中打开

In your current code, you've implemented a portfolio optimization strategy using the Sharpe ratio. To calculate the maximum drawdown, Sortino ratio, and ulcer index, you can

Calculate the portfolio returns using the optimal weights obtained from the portfolio optimization.
Calculate the maximum drawdown.
Calculate the Sortino ratio.
Calculate the ulcer index.

% Your existing code
% ...
% Step 1: Calculate portfolio returns using the optimal weights (swgt)
portfolio_returns = dailyreturn2 * swgt;
% Step 2: Calculate maximum drawdown
cumulative_returns = cumprod(1 + portfolio_returns) - 1;
cumulative_max = cummax(cumulative_returns);
drawdowns = (cumulative_max - cumulative_returns) ./ (cumulative_max + 1);
max_drawdown = max(drawdowns);
% Step 3: Calculate Sortino ratio
target_return = 0; % Define target return (usually 0 for Sortino ratio)
excess_returns = portfolio_returns - RiskFreeRate;
downside_returns = excess_returns;
downside_returns(downside_returns > target_return) = 0;
sortino_ratio = (mean(excess_returns) - target_return) / sqrt(mean(downside_returns .^ 2));
% Step 4: Calculate Ulcer index
ulcer_index = sqrt(mean(drawdowns .^ 2));
% Display the results
display(max_drawdown);
display(sortino_ratio);
display(ulcer_index);

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LeoAiE 2023-4-23

在 MATLAB Online 中打开

Glad the answer was helpful!

To draw the drawdown graph, you can use the following code:

figure;
plot(cumulative_returns, 'b-', 'LineWidth', 1.5);
hold on;
plot(cumulative_max, 'r--', 'LineWidth', 1.5);
hold off;
legend('Cumulative Returns', 'Cumulative Maximum');
xlabel('Time');
ylabel('Cumulative Returns');
title('Drawdown Graph');
grid on;

To calculate other risk-adjusted performance metrics, maybe this code will help but not sure what exactly you are looking for

% Calmar Ratio
calmar_ratio = mean(portfolio_returns) / max_drawdown;
% Treynor Ratio
beta = cov(portfolio_returns, p.AssetMean) / var(p.AssetMean);
treynor_ratio = (mean(portfolio_returns) - RiskFreeRate) / beta;
% Information Ratio
benchmark_returns = mean(p.AssetMean);
tracking_error = std(portfolio_returns - benchmark_returns);
information_ratio = (mean(portfolio_returns) - benchmark_returns) / tracking_error;
% Cumulative Annual Growth Rate (CAGR)
years = length(portfolio_returns) / 252; % Assuming 252 trading days per year
cagr = (1 + cumulative_returns(end))^(1/years) - 1;
% Display the results
display(calmar_ratio);
display(treynor_ratio);
display(information_ratio);
display(cagr);

and finally I used this book before and was helpful

MATLAB Guide

Desmond J. Higham and Nicholas J. Higham, MATLAB Guide, Third Edition, SIAM, 2017. xxvi+476 pages, hardcover, ISBN 978-1-61197-465-2.

ther are many other good options out there so do your research to find something suitable for your application.

Hope this helps!

Mattia Smanio 2023-4-25

Thank you for your kind cooperation (I will buy for sure the book that you have recomended), at least i would ask two more questions:

Is there a summary to calculate the other indicators like Volatility, Upside Potential Ratio, Sterling Ratio, Omega Ratio, and MAR?
In the following picture, is it possible to change x axis with the first column of my file excel named "Date" (connected with the previous part of the code)?

LeoAiE 2023-5-7

在 MATLAB Online 中打开

Volatility (annualized): Since you already have daily returns, you can calculate the annualized volatility using the standard deviation of daily returns.

volatility = std(portfolio_returns) * sqrt(252);

Upside Potential Ratio: It measures the expected value of returns above a target return.

upside_returns = excess_returns;
upside_returns(upside_returns < target_return) = 0;
upside_potential_ratio = mean(upside_returns) / sqrt(mean(downside_returns .^ 2));

Sterling Ratio: It measures the reward-to-pain ratio.

sterling_ratio = (mean(portfolio_returns) - RiskFreeRate) / (max_drawdown + 0.1); % Adding a constant to the denominator

Omega Ratio: It measures the ratio of the probability weighted gains to losses.

gain_returns = excess_returns;
gain_returns(gain_returns < target_return) = 0;
loss_returns = excess_returns;
loss_returns(loss_returns >= target_return) = 0;
omega_ratio = sum(gain_returns) / abs(sum(loss_returns));

MAR (Managed Account Reports) Ratio: It is similar to the Calmar ratio.

mar_ratio = (mean(portfolio_returns) - RiskFreeRate) / max_drawdown;

To plot the graph with the x-axis as dates, you can use the following code:

dates = t.Date(2:end); % Assuming you have dates in the first column, skipping the header
figure;
plot(dates, cumulative_returns, 'b-', 'LineWidth', 1.5);
hold on;
plot(dates, cumulative_max, 'r--', 'LineWidth', 1.5);
hold off;
legend('Cumulative Returns', 'Cumulative Maximum');
xlabel('Time');
ylabel('Cumulative Returns');
title('Drawdown Graph');
grid on;
datetick('x', 'yyyy-mm-dd', 'keepticks'); % Format the x-axis dates