Applied Portfolio VaR Decomposition. (2) Impact vs Moving Elements.

Calculations of daily Value-at-Risk (VaR) for any $N$-asset portfolio, as we have studied it already in Part 1, heavily depend on the covariance matrix we need to estimate. This estimation requires historical return time-series. Often negligible but superbly important question one should ask here is: How long?! How long those return-series ought to be in order to provide us with fair and sufficient information on the daily return distribution of each asset held in the portfolio?

It is intuitive that taking data covering solely past 250 trading days and, for example, past 2500 days (moving element) would have a substantial impact on the calculation of current portfolio VaR. In this short post, we will examine this relation.

Case Study (Supplementary Research)

Using the example of 7-asset portfolio from Part 1, we modify our Matlab code that allows us to download the history of stock (adjusted) close prices covering past 10 calendar years (ca. 3600 days),

% Applied Portfolio VaR Decomposition. (2) Impact vs Moving Elements.
% (c) 2015 QuantAtRisk.com, by Pawel Lachowicz


clear all; close all; clc;
format short

% Read the list of the portfolio components
fileID = fopen('portfolio.lst');
tmp = textscan(fileID, '%s');
fclose(fileID);
pc=tmp{1};  % a list as a cell array

% fetch stock data for last 10 years since:
t0=735976; % 12-Jan-2015
date2=datestr(t0,'yyyy-mm-dd');        % from
date1=datestr(t0-10*365,'yyyy-mm-dd');  % to

%{
% create an empty array for storing stock data
stockd={};
% scan the tickers and fetch the data from Quandl.com
for i=1:length(pc)
    quandlc=['WIKI/',pc{i}];
    fprintf('%4.0f %s\n',i,quandlc);
    % fetch the data of the stock from Quandl
    % using recommanded Quandl's command and
    % saving them directly into Matlab's FTS object (fts)
    fts=0;
    [fts,headers]=Quandl.get(quandlc,'type','fints', ...
                  'authcode','YourQuandlCode',...
                  'start_date',date1,'end_date',date2);
    stockd{i}=fts; % entire FTS object in an array's cell
end
save data2
%}
load data2

Assuming the same initial holdings (position sizes) for each asset:

% an initial dollar expose per position
d=[   55621.00; ...
     101017.00; ...
      23409.00; ...
    1320814.00; ...
     131145.00; ...
     321124.00; ...
    1046867.00];

we aim at calculation of the portfolio VaR,
$$
\mbox{VaR}_P = 1.65 \sigma_P C = 1.65\left(\boldsymbol{d}’ \boldsymbol{M}_2 \boldsymbol{d} \right)^{1/2}
$$ in a function of time or, alternatively speaking, in a function of varying length of return time-series:

% examine VaR_P as a function of historical data
% taken into account
e=[];
for td=(10*250):-1:60   % loop over trading days
    rs=[];
    for i=1:length(pc)
        cp=fts2mat(stockd{i}.Adj_Close,0);
        rv=cp(2:end,1)./cp(1:end-1,1)-1;
        rs=[rs rv(end-td:end)];
    end
    rs(isnan(rs))=0.0;
    % covariance matrix
    M2=cov(rs);
    % the portfolio volatility
    vol_P=sqrt(d'*M2*d);
    % diversified portfolio VaR
    VaR_P=1.65*vol_P;
    e=[e; td VaR_P];
end

The meaning of the loop is straightforward: How does $\mbox{VaR}_P$ change if we take only last 60 trading days into account, and what is its value when we include more and more data going back in time as far as 10 years?

The best way to illustrate the results is to compare running portfolio VaR with the stock market itself, here reflected by ETF of SPY,

% download close price time-series for SPY (ETF)
% tracking S&P500 Index
[fts,headers]=Quandl.get('GOOG/NYSE_SPY','type','fints', ...
                  'authcode','YourQuandlCode',...
                  'start_date',date1,'end_date',date2);
SPY_cp=fts2mat(fts.Close,0);
    
% plot results
subplot(2,1,1)
xtmp=(2501:-1:1)./250;
plot(xtmp,SPY_cp(end-10*250:end,1));
xlim([min(xtmp) max(xtmp)]);
set(gca,'xdir','reverse');
ylabel('SPY Close Price [$]')

subplot(2,1,2)
plot(e(:,1)./250,e(:,2),'r');
set(gca,'xdir','reverse');
ylabel('VaR_P [$]')
xlabel('Time since 12-Jan-2015 [trading years]')

uncovering the big picture as follows:

And now all becomes clear! Firstly, $\mbox{VaR}_P$ of our 7-asset portfolio is a function of data. Secondly, $\mbox{VaR}_P$ changes in the way that we might expect as contrasted with the market behaviour. In this point, note an evident increase in $\mbox{VaR}_P$ value if we include not 6 but 7 years of data. Why? The market went down in 2008 and the impact of it had been imprinted in the daily losses of single stocks. Therefore this contribution enriched the left tail of their return distributions. An inclusion of this information in our covariance matrix in a natural way increases the level of $\mbox{VaR}_P$.

So, how long? How many data should we include to get fair estimation of $\mbox{VaR}_P$? It’s not so easy to address. I would go with a lower bound of at least 1 year. As for the upper limit, it depends. If you are optimistic that in 2015 stocks will keep their upward momentum, look back up to 5 years. If you are a cautious investor, include 7 to 10 years of data.

Footnote

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