Backtest a Profitable Trend-Following Strategy - Concretum Group (2024)

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% Download Data From Frenchurl = 'https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/48_Industry_Portfolios_daily_CSV.zip';zipFile = websave('temp.zip', url);pause(1);unzip(zipFile, 'temp_folder');pause(1);csvFile = dir(fullfile('temp_folder', '*.csv'));csvFilePath = fullfile('temp_folder', csvFile.name);movefile(csvFilePath, 'output.csv', 'f');delete(zipFile); pause(1);rmdir('temp_folder', 's');clear p csvData = readtable('output.csv');p.industry_names = csvData.Properties.VariableNames(2:end);rows_nan = find(isnan(table2array(csvData(:,1))));% MktCap ind_weightingfrom = 1; until = rows_nan(1)-1;data = table2array(csvData(from:until,1:end)); p.ret = data(:,2:end)/100;p.caldt = datenum(num2str(data(:,1)),'yyyymmdd'); % change in case daily datap.ret(find(p.ret<=-0.99))=NaN;%%clearvars -except pp.mkt_ret = market_french_reconciled(p.caldt); % download the time-series of market return and match the position with respect to daysp.tbill_ret = tbill_french_reconciled(p.caldt); % download the time-series of tbill daily ret and match the position with respect to dayssave('industry48.mat','p')

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function y = market_french_reconciled(caldt)url = 'https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/F-F_Research_Data_Factors_daily_CSV.zip';zipFile = websave('temp.zip', url);pause(1);unzip(zipFile, 'temp_folder');pause(1);csvFile = dir(fullfile('temp_folder', '*.csv'));csvFilePath = fullfile('temp_folder', csvFile.name);pause(1);movefile(csvFilePath, 'output.csv', 'f');delete(zipFile); pause(2);rmdir('temp_folder', 's');csvData = readtable('output.csv', 'ReadVariableNames', false);rows_nan = find(isnan(table2array(csvData(:,1))));% MktCap Weightingfrom = rows_nan(1); until = rows_nan(2)-1;data = table2array(csvData(from:until,1:end)); data(find(isnan(data(:,1))),:)=[];ret_mkt = data(:,2)/100+data(:,5)/100; % add back the risk freecaldt_mkt = datenum(num2str(data(:,1)),'yyyymmdd'); % change in case daily dataT = length(caldt);y = NaN(T,1);X_dates = datetime(caldt, 'ConvertFrom', 'datenum');[isMember, idx] = ismember(caldt_mkt, caldt);% Create a mapping of caldt_mkt to ret_mkt using idxmappedReturns = NaN(size(caldt));mappedReturns(idx(isMember)) = ret_mkt(isMember);% Assign the returns to y based on matching indicesy = mappedReturns;end

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function y = tbill_french_reconciled(caldt)url = 'https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/F-F_Research_Data_Factors_daily_CSV.zip';zipFile = websave('temp.zip', url);pause(1);unzip(zipFile, 'temp_folder');pause(1);csvFile = dir(fullfile('temp_folder', '*.csv'));csvFilePath = fullfile('temp_folder', csvFile.name);pause(1);movefile(csvFilePath, 'output.csv', 'f');delete(zipFile); pause(2);rmdir('temp_folder', 's');csvData = readtable('output.csv', 'ReadVariableNames', false);rows_nan = find(isnan(table2array(csvData(:,1))));% MktCap Weightingfrom = rows_nan(1); until = rows_nan(2)-1;data = table2array(csvData(from:until,1:end)); data(find(isnan(data(:,1))),:)=[];ret_mkt = data(:,5)/100; % caldt_mkt = datenum(num2str(data(:,1)),'yyyymmdd'); % change in case daily dataT = length(caldt);y = NaN(T,1);X_dates = datetime(caldt, 'ConvertFrom', 'datenum');[isMember, idx] = ismember(caldt_mkt, caldt);% Create a mapping of caldt_mkt to ret_mkt using idxmappedReturns = NaN(size(caldt));mappedReturns(idx(isMember)) = ret_mkt(isMember);% Assign the returns to y based on matching indicesy = mappedReturns;end

MATLAB

url = 'https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/48_Industry_Portfolios_daily_CSV.zip';zipFile = websave('temp.zip', url);pause(1);unzip(zipFile, 'temp_folder');pause(1);

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csvFile = dir(fullfile('temp_folder', '*.csv'));csvFilePath = fullfile('temp_folder', csvFile.name);movefile(csvFilePath, 'output.csv', 'f');

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delete(zipFile); pause(1);rmdir('temp_folder', 's');

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clear pcsvData = readtable('output.csv');p.industry_names = csvData.Properties.VariableNames(2:end);rows_nan = find(isnan(table2array(csvData(:,1))));

MATLAB

from = 1; until = rows_nan(1)-1;data = table2array(csvData(from:until,1:end)); p.ret = data(:,2:end)/100;p.caldt = datenum(num2str(data(:,1)),'yyyymmdd');p.ret(find(p.ret<=-0.99))=NaN;

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p.mkt_ret = market_french_reconciled(p.caldt);p.tbill_ret = tbill_french_reconciled(p.caldt);

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save('industry48.mat','p')

MATLAB

clear allload('industry48.mat')% INDICATORS PARAMETERSup_day = 20;down_day = 40;kelt_mult = 2;adr_vol_adj = 1.4; % we noticed that ATR is usually 1.4xVol(close2close)kelt_mult = kelt_mult*adr_vol_adj;% FROM DAILY TOTAL RETURNS, CREATE THE HYPOTHETICAL PRICE TIME-SERIES FOR% EACH INDUSTRYp.price = cumprod(1+p.ret,'omitmissing');% Rolling Volatility of Daily Returnsp.vol = Rolling_Vol(p.ret,up_day,'omtinan');% Technical Indicatorsp.ema_down = Rolling_Ema(p.price,down_day,'omitnan');p.ema_up = Rolling_Ema(p.price,up_day,'omitnan') ;% Donchain Channels (Eq.5 and Eq.6)p.donc_up = Rolling_Max(p.price,up_day);p.donc_down = Rolling_Min(p.price,down_day);% Keltner Bands (Eq. 3 and 4)p.kelt_up = p.ema_up + kelt_mult*Rolling_Mean(abs(price2change(p.price,1)),up_day,'omitnan');p.kelt_down = p.ema_down - kelt_mult*Rolling_Mean(abs(price2change(p.price,1)),down_day,'omitnan');% Model Bands p.long_band = min(p.donc_up,p.kelt_up); % Eq.7p.short_band = max(p.donc_down,p.kelt_down); % Eq.11% Model Long Signalp.long_signal = p.price>=lag_TS(p.long_band,1) & lag_TS(p.long_band,1)>lag_TS(p.short_band,1);

MATLAB

function y=Rolling_Vol(price,window,nanflag)T = size(price,1);N = size(price,2);y = NaN(T,N);if prod(double(nanflag=='omitnan'))==1  for i=window:1:T y(i,:)=std(price(i-window+1:i,:),1,'omitmissing'); endelse for i=window:1:T y(i,:)=std(price(i-window+1:i,:),1,'includenan'); endend end

MATLAB

function ema = rolling_ema(price, window, nanflag)% Size of the price matrixT = size(price, 1);N = size(price, 2);% Initialize output matrixema = NaN(T, N);% Smoothing factoralpha = 2 / (window + 1);% Loop through each element and compute EMAfor n = 1:N for t = 2:T if isnan(price(t, n)) if strcmp(nanflag, 'omitnan') % If omitnan flag is used, skip NaN values continue; end else % If the previous EMA is NaN (start of the series), initialize it as the first data point if isnan(ema(t-1, n)) ema(t-1, n) = price(t-1, n); end % Compute the EMA ema(t, n) = alpha * price(t, n) + (1 - alpha) * ema(t-1, n); end endendend

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function y=Rolling_Max(price,window)y = movmax(price,[window-1 0],1,"omitmissing");if size(y,1) >= windowy(1:window-1,:) = NaN;endend

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function y=Rolling_Min(price, window)y = movmin(price,[window-1 0],1,"omitmissing");if size(y,1) >= windowy(1:window-1,:) = NaN;endend

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function y=Rolling_Mean(price,window,nanflag)T = size(price,1);N = size(price,2);y = NaN(T,N); for i=window:1:T y(i,:)=mean(price(i-window+1:i,:),1,nanflag);endy(find(isnan(price)==1))=NaN; % if input variable is missing, consider NaN end

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function y=price2change(price,n) % This function converts prices into returns T=size(price,1); N=size(price,2); y=NaN(T,N);   y(n+1:end,:)=price(n+1:T,:)-price(1:T-n,:); y(isnan(y))=NaN;  end

MATLAB

function y=lag_TS(data,n) % This function lag the data matrix to n-times T=size(data,1); N=size(data,2); y=NaN(T,N);  if n>0 y(n+1:T,:) = data(1:T-n,:); elseif n<0 n = -n; y(1:T-n,:) = data(1+n:T,:);  else  y = data; end y(isnan(y))=NaN;  end

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clear allload('industry48.mat')

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up_day = 20;down_day = 40;kelt_mult = 2;adr_vol_adj = 1.4; % we noticed that ATR is usually 1.4xVol(close2close)kelt_mult = kelt_mult*adr_vol_adj;

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p.price = cumprod(1+p.ret,'omitmissing');

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p.vol = Rolling_Vol(p.ret, up_day, 'omitnan');

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p.ema_down = Rolling_Ema(p.price, down_day, 'omitnan');p.ema_up = Rolling_Ema(p.price, up_day, 'omitnan');p.donc_up = Rolling_Max(p.price, up_day);p.donc_down = Rolling_Min(p.price, down_day);p.kelt_up = p.ema_up + kelt_mult * Rolling_Mean(abs(price2change(p.price, 1)), up_day, 'omitnan');p.kelt_down = p.ema_down - kelt_mult * Rolling_Mean(abs(price2change(p.price, 1)), down_day, 'omitnan');

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p.long_band = min(p.donc_up, p.kelt_up); p.short_band = max(p.donc_down, p.kelt_down); p.long_signal = p.price >= lag_TS(p.long_band, 1) & lag_TS(p.long_band, 1) > lag_TS(p.short_band, 1);

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AUM_0 = 1;invest_cash = "YES"; % invest cash in short-term bonds, and pay for cash borrowedtarget_vol = 0.02; % the risk budget per industry is target_vol/Nmax_leverage = 2; % max leverage at portfolio level.max_not_trade = 0.20; % Limit the max equity exposure per industry% initiliaze portfolio variablesp.exposure = zeros(size(p.price)); % matrix that take 1 or 0 based on if we hold a long positionp.ind_weight = zeros(size(p.price)); % optimal weight at industry levelp.trail_stop_long = NaN(size(p.price)); % trailing stop lossN_ind = length(p.industry_names); % how many industries in database% for each industry, compute the optimal weight at daily frequency, and% update the level of trailing stop in case the stop has not been hitT = length(p.caldt);for j = 1:length(p.industry_names) for t = 1:T  if ~isnan(p.ret(t,j)) && ~isnan(p.long_band(t,j)) % start the backtest when key-variable are not NaN % EXPOSURE CALCULATION  % Open a new long position if yesterday exposure was 0 and % today we have a long_signal at the closure if p.exposure(t-1,j) <= 0 && p.long_signal(t,j) == 1 % New long signal  p.exposure(t,j) = 1; p.trail_stop_long(t,j) = p.short_band(t,j); % set the value for the trailing stop  % leverage required to reach the target vol  lev_vol = target_vol/p.vol(t,j);  % optimal weight at industry level p.ind_weight(t,j) = p.exposure(t,j)*lev_vol;  % Yesterday we were long. Today the price didn't cross the trailing stop. We confirm the long epxosure and we update the trailing stop  elseif p.exposure(t-1,j) == 1 && p.price(t,j) > max(p.trail_stop_long(t-1,j),p.short_band(t,j))  p.exposure(t,j) = 1; % the long exposure is confirmed p.trail_stop_long(t,j) = max(p.trail_stop_long(t-1,j),p.short_band(t,j)); % update trailing stop as in Eq.12 % leverage required to reach the target vol  lev_vol = target_vol/p.vol(t,j); % optimal weight at industry level %p.ind_weight(t,j) = min(p.ind_weight(t-1,j),lev_vol); % in the paper i used this but it's a small typo, it deflated %the results in terms of IRR p.ind_weight(t,j) = p.exposure(t,j)*lev_vol;  % yesterday we were long but today the closing price is below the trailing stop  elseif p.exposure(t-1,j) == 1 && p.price(t,j) <= max(p.trail_stop_long(t-1,j),p.short_band(t,j))   % set exposure and weight to 0 p.exposure(t,j) = 0;  p.ind_weight(t,j) = 0; end  end endend% Initialize a strucutre str where we store the results at aggregate% portfolio levelstr.caldt = p.caldt;str.available = sum(~isnan(p.ret),2,'omitmissing'); % how many industries were available each day (some industries may became available later in the database)str.port_weight = p.ind_weight./str.available; % optimal weight at portfolio level. See Eq.8% Limit the exposure of each industry at "max_not_trade"idx_above_max_not = find(str.port_weight > max_not_trade);str.port_weight(idx_above_max_not) = max_not_trade;% In case the portfolio exposure exceed the maximum leverage, rescaled each% position proportionallystr.sum_exposure = sum(str.port_weight,2,'omitmissing'); % implied daily leverageidx_above_max_lev = find(str.sum_exposure > max_leverage); % find days when the maximum leverage was exceededstr.port_weight(idx_above_max_lev,:) = str.port_weight(idx_above_max_lev,:)./str.sum_exposure(idx_above_max_lev)*max_leverage; % Eq10str.sum_exposure = sum(str.port_weight,2,'omitmissing');% Compute portfolio daily returnsstr.ret_long = sum(lag_TS(str.port_weight,1).*p.ret,2,'omitmissing');str.ret_tbill = (1-sum(lag_TS(str.port_weight,1),2,'omitmissing')).*p.tbill_ret; % return of cash investment/borrowedif strcmp(invest_cash,"YES") % include cash returns only when selected str.ret_long = str.ret_long + str.ret_tbill;end% Compute the timeseries of the AUMstr.AUM = AUM_0*cumprod(1 + str.ret_long, 'omitnan');str.AUM_SPX = AUM_0*cumprod(1 + p.mkt_ret, 'omitnan'); % Calculating the adjusted AUM for SPX

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AUM_0 = 1;invest_cash = "YES";target_vol = 0.02;max_leverage = 2;max_not_trade = 0.20;p.exposure = zeros(size(p.price));p.ind_weight = zeros(size(p.price));p.trail_stop_long = NaN(size(p.price));N_ind = length(p.industry_names);T = length(p.caldt);

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for j = 1:length(p.industry_names) for t = 1:T if ~isnan(p.ret(t,j)) && ~isnan(p.long_band(t,j)) if p.exposure(t-1,j) <= 0 && p.long_signal(t,j) == 1 p.exposure(t,j) = 1; p.trail_stop_long(t,j) = p.short_band(t,j); lev_vol = target_vol/p.vol(t,j); p.ind_weight(t,j) = p.exposure(t,j) * lev_vol; elseif p.exposure(t-1,j) == 1 && p.price(t,j) > max(p.trail_stop_long(t-1,j), p.short_band(t,j)) p.exposure(t,j) = 1; p.trail_stop_long(t,j) = max(p.trail_stop_long(t-1,j), p.short_band(t,j)); lev_vol = target_vol/p.vol(t,j); p.ind_weight(t,j) = p.exposure(t,j) * lev_vol; elseif p.exposure(t-1,j) == 1 && p.price(t,j) <= max(p.trail_stop_long(t-1,j), p.short_band(t,j)) p.exposure(t,j) = 0; p.ind_weight(t,j) = 0; end end end}

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str.caldt = p.caldt;str.available = sum(~isnan(p.ret), 2, 'omitmissing');str.port_weight = p.ind_weight./str.available;idx_above_max_not = find(str.port_weight > max_not_trade);str.port_weight(idx_above_max_not) = max_not_trade;str.sum_exposure = sum(str.port_weight, 2, 'omitmissing');idx_above_max_lev = find(str.sum_exposure > max_leverage);str.port_weight(idx_above_max_lev, :) = str.port_weight(idx_above_max_lev, :) ./ str.sum_exposure(idx_above_max_lev) * max_leverage;

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str.ret_long = sum(lag_TS(str.port_weight, 1) .* p.ret, 2, 'omitmissing');str.ret_tbill = (1 - sum(lag_TS(str.port_weight, 1), 2, 'omitmissing')) * p.tbill_ret;if strcmp(invest_cash, "YES") str.ret_long = str.ret_long + str.ret_tbill;endifstr.AUM = AUM_0 * cumprod(1 + str.ret_long, 'omitnan');str.AUM_SPX = AUM_0 * cumprod(1 + p.mkt_ret, 'omitnan');

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freq = 21;fig = figure(); plot(str.caldt([1:freq:end end]), str.AUM([1:freq:end end]), 'LineWidth', 2,'Color','k'); hold onplot(str.caldt([1:freq:end end]), str.AUM_SPX([1:freq:end end]), 'LineWidth', 1, 'Color','r'); hold offgrid off; set(gca,'GridLineStyle',':');set(gca, 'GridLineStyle', ':');set(gca, 'XTick');xlim([str.caldt(1) str.caldt(end)]);datetick('x', 'yyyy', 'keepticks'); % Keep the ticks and format themxtickangle(90); ytickformat('$%,.0f');ax=gca; ax.YAxis.Exponent = 0;set(gca, 'FontSize', 8); % Adjust font sizelegend('Timing Ind.','Market','Location','northwest');title(['Timing ' num2str(N_ind) ' Industries Strategy'], 'FontWeight', 'bold','FontSize',12);y_tick(1) = AUM_0;for j = 2:1:100, y_tick(j) = y_tick(j-1)*4; endyticks([y_tick]); set(gca, 'YScale', 'log');table_statistics = table_of_stats([str.AUM str.AUM_SPX],[size(p.ret,2) 1],{"Timing Ind.","Market"},str.caldt,p.mkt_ret,p.tbill_ret);display(table_statistics)

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function table_statistics = table_of_stats(AUM,N_assets,names,caldt,mkt_ret,tbill_ret) table_statistics = struct(); for s = 1:size(AUM,2)  ret = price2return(AUM(:,s),1); table_statistics(s).strategy = names{s};  table_statistics(s).assets = N_assets(s); table_statistics(s).irr = round((prod(1+ret,'omitmissing')^(252/length(ret))-1)*100,1); table_statistics(s).vol = round(std(ret,'omitmissing')*sqrt(252)*100,1); table_statistics(s).sr = round(mean(ret,'omitmissing')/std(ret,'omitmissing')*sqrt(252),2); table_statistics(s).sortino = round(SortinoRatio(ret)*sqrt(252),2);  table_statistics(s).ir_d = computeHitRatio(ret,caldt,"daily"); table_statistics(s).ir_m = computeHitRatio(ret,caldt,"monthly"); table_statistics(s).ir_y = computeHitRatio(ret,caldt,"yearly"); table_statistics(s).skew_d = round(skewness(ret),2); table_statistics(s).skew_m = round(computeSkewness(ret,caldt,"monthly"),2); table_statistics(s).skew_y = round(computeSkewness(ret,caldt,"yearly"),2); table_statistics(s).mdd = round(maxdrawdown(AUM(:,s))*100,0); XX = [mkt_ret-tbill_ret]; YY = ret-tbill_ret;  regr = fitlm(XX,YY); coef = regr.Coefficients.Estimate; tstas = regr.Coefficients.tStat; table_statistics(s).alpha = round(coef(1)*100*252,2); table_statistics(s).alpha_t = tstas(1); table_statistics(s).beta = round(coef(2),2); table_statistics(s).beta_t = tstas(2); table_statistics(s).worst_ret= round(min(ret)*100,2); table_statistics(s).worst_day= datestr(caldt(find(ret == min(ret)))); table_statistics(s).best_ret = round(max(ret)*100,2); table_statistics(s).best_day = datestr(caldt(find(ret == max(ret))));  end  originalTable = struct2table(table_statistics); % Step 1: Extract the column names colNames = originalTable.Properties.VariableNames; % Step 2: Transpose the data transposedData = table2array(originalTable)'; % Step 3: Create the new table newTable = array2table(transposedData);%, 'VariableNames', colNames); % Step 4: Add the original column names as a new column newTable = addvars(newTable, colNames', 'Before', 1);  % Step 5: Name the column of the new Table newTable.Properties.VariableNames = [" " names];  % Display the result table_statistics = newTable; end

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function sortinoRatio = SortinoRatio(returns) % Compute the mean return meanReturn = mean(returns,'omitmissing'); % Compute the downside deviation downsideReturns = returns(returns < 0); downsideDeviation = std(downsideReturns,'omitmissing'); % Compute the Sortino Ratio sortinoRatio = (meanReturn) / downsideDeviation;end

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function hitRatio = computeHitRatio(ret,caldt, timeframe) % Ensure the timeframe is valid validTimeframes = {'daily', 'monthly', 'yearly'}; if ~ismember(timeframe, validTimeframes) error('Invalid timeframe. Choose "daily", "monthly", or "yearly".'); end % Convert datenum to datetime for easier manipulation dates = datetime(caldt, 'ConvertFrom', 'datenum'); returns = ret; % Calculate the Hit Ratio based on the timeframe switch timeframe case 'daily' hitRatio = round(sum(returns > 0) / sum(abs(returns) > 0) * 100, 0); case 'monthly' % Group by year and month [yearMonth, ~, idx] = unique(dates.Year*100 + dates.Month); monthlyReturns = accumarray(idx, returns, [], @(x) prod(1 + x,'omitmissing') - 1); hitRatio = round(sum(monthlyReturns > 0) / sum(abs(monthlyReturns) > 0) * 100, 0); case 'yearly' % Group by year [years, ~, idx] = unique(dates.Year); yearlyReturns = accumarray(idx, returns, [], @(x) prod(1 + x,'omitmissing') - 1); hitRatio = round(sum(yearlyReturns > 0) / sum(abs(yearlyReturns) > 0) * 100, 0); otherwise error('Unexpected timeframe.'); endend

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function skw = computeSkewness(ret,caldt, timeframe) % Ensure the timeframe is valid validTimeframes = {'daily', 'monthly', 'yearly'}; if ~ismember(timeframe, validTimeframes) error('Invalid timeframe. Choose "daily", "monthly", or "yearly".'); end % Convert datenum to datetime for easier manipulation dates = datetime(caldt, 'ConvertFrom', 'datenum'); returns = ret; % Calculate the Hit Ratio based on the timeframe switch timeframe case 'daily' skw = skewness(returns); case 'monthly' % Group by year and month [yearMonth, ~, idx] = unique(dates.Year*100 + dates.Month); monthlyReturns = accumarray(idx, returns, [], @(x) prod(1 + x,'omitmissing') - 1); skw = skewness(monthlyReturns); case 'yearly' % Group by year [years, ~, idx] = unique(dates.Year); yearlyReturns = accumarray(idx, returns, [], @(x) prod(1 + x,'omitmissing') - 1); skw = skewness(yearlyReturns);; otherwise error('Unexpected timeframe.'); endend

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freq = 21;fig = figure();plot(str.caldt([1:freq:end end]), str.AUM([1:freq:end end]), 'LineWidth', 2, 'Color', 'k'); hold onplot(str.caldt([1:freq:end end]), str.AUM_SPX([1:freq:end end]), 'LineWidth', 1, 'Color', 'r'); hold offgrid off; set(gca, 'GridLineStyle', ':');set(gca, 'XTick');xlim([str.caldt(1) str.caldt(end)]);datetick('x', 'yyyy', 'keepticks');xtickangle(90); ytickformat('$%,.0f');ax = gca; ax.YAxis.Exponent = 0;set(gca, 'FontSize', 8);legend('Timing Ind.', 'Market', 'Location', 'northwest');title(['Timing ' num2str(N_ind) ' Industries Strategy'], 'FontWeight', 'bold', 'FontSize', 12);y_tick(1) = AUM_0;for j = 2:1:100, y_tick(j) = y_tick(j-1)*4; endyticks([y_tick]);set(gca, 'YScale', 'log');

MATLAB

table_statistics = table_of_stats([str.AUM str.AUM_SPX], [size(p.ret,2) 1], {"Timing Ind.", "Market"}, str.caldt, p.mkt_ret, p.tbill_ret);display(table_statistics)