Minimizing Risk in Cash Equity Trades using an Algorithmic Model
Team Latte
June 22, 2006

The recent turmoil in the stock markets, especially the Asian stock markets and the resultant movement in the index and stock volatilities have left quite a few cash equity traders scarred. It is not just the equity derivatives and index linked traders who have been taken on a roller coaster ride by the markets; the cash equity traders in banks and fund management houses have also had a very rough time.

There is a special class of quantitative trading strategy that many cash equity traders, especially the ones who do algorithmic trading, use which is called the Risk Minimization strategy (or the MCR optimization strategy). Essentially, a cash equity trader calculates what should be the size of an order (to buy or sell a stock or a list of stocks) that should be traded in a highly volatile market that will either minimize the total (market) risk of the trade list or at least minimize the residual risk in the trade list. This risk minimization strategy was overlooked by many a cash traders this time, which caused significant increase in the risk of trading portfolios.

Here is a simple example, which is a bit text bookish but is nevertheless adapted and simplified to fit a recent live example of a trader's book that we recently came across. Say a trader is trading SP500 stocks and there are three names (stocks) in his list, GoodStock, BadStock and OKStock with shares of 135, -135 and -40 respectively. This means he needs to buy 135 shares of GoodStock, sell 135 shares of BadStock and sell 40 shares of OKStock. He can do it in one go and execute the order or he can do it in a piecemeal basis over a period of time depending on the execution price (bid-ask spread), etc. However, all this might increase the risk of the trade list, i.e the residual risk trajectory might be such that his MCR (marginal contribution to risk) might be positive, which in turn means that all his trades would contribute to his overall risk. What he needs to do is to find out what is the exact order size in the in the given trade list that he must execute which will minimize his initial risk.

For this he needs to his variance-covariance matrix , C, in terms of Dollar per share of the trade list (for the names GoodStock, BadStock and OKStock). Most traders have this readily available with them (either done by themselves or provided to them by their research or quantitative analysis teams). The variance-covariance matrix in terms of Dollar per share of stocks is calculated from the current prices of the stocks and their return volatilities and correlations. This matrix C is highly dynamic in nature and varies with the underlying volatilities and correlations all the time. And this time around this matrix C caused a lot of trouble for the traders given the rising volatilities and dynamic correlations of the stocks.

Let us say that this matrix C is given in this case and is:

The initial risk of the trade list for the trader is calculated as:

The above is expressed in matrix notation which makes the calculations far simpler if we have many stocks in a trade list (which is how it happens in most real life cases).

If we calculate the above we get $16.81 as the initial risk in Dollars for the trade list:

To minimize this Risk (initial risk of the trade list) we need to find out the first mathematical derivative of the above and then work out the optimum value of the order size.

If we solve the above matrix equation then we will find the resultant value of the column vector of the stocks (i.e. the number of shares that needs to be traded in each order) that needs to be traded if the residual risk has to be minimized. The solution is pretty simple and we leave it to the readers to do it. The result is:

Therefore, the number of shares that needs to be traded in each order to minimize the residual risk is: 40 shares of GoodStock needs to be bought, 37 shares of BadStock needs to be bought and 13 shares of OKStock needs to be sold. However, we face a constraint here. The constraint is in BadStock. The trader has an overall short position and hence cannot buy the underlying stock. Therefore, the optimum order size for the three stocks would be:

Given the above variance-covariance matrix and the trader's trade list the above is the optimum order size and if he executes his trades according to the above order size he will have minimized his risk.

Reference: A very good book for Algorithmic trading strategies is Optimal Trading Strategies by Robert Kissell and Morton Glantz (American Management Association) where many of the above strategies are discussed in great detail .

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