Question

(a) Explain the so called “pairs trading” rule in statistical arbitrage.

Suppose a stock A is currently valued at 80 and another stock B is valued at 110. How would you design a pairs trading rule that generates profits when the two prices converge?

(b) A portfolio has total variance 16%, and an average return of 9%. The market index has total variance 9% and an average return of 8%. The correlation between the return on the portfolio and the return on the market index is 0.6. The risk-free return is 4%.

Work out the Sharpe ratio and the Treynor ratio for the portfolio.

(c) Suppose you hold an equally weighted portfolio of 16 stocks. Each stock has a beta of 0.8 and total variance 16%. The market index has variance 9%. Assume that the unsystematic risk is independent across the stocks.

Calculate the total risk, market risk and unsystematic risk of the portfolio.

Answer 1

(A)     A pairs trade is a trading strategy that involves matching a long position with a short position in two stocks with a high correlation.

Statistical arbitrage consist of the number of strategies and these strategies are market neutral because they involve opening both a long and short position simultaneously to take advantage of inefficient pricing in correlated securities.

In the given case Stock A is valued at 80 and Stock B is valued at 110. To earn profits investors can go short in B and long in A.

(B) SHARPE RATIO = (Rp-Rf)/S.D.portfolio

      TREYNOR RATIO= (Rp-Rf)/Bp

       Standard deviation of portfolio = under root of varince

        standard deviation = 4%

       SHARPE RATIO = (9%-4%)/4%    = 1.25 TIMES

       BETA OF PORTFOLIO = (Covariance between security and market) / variance of market, Bp = 7.2/9 = 0.80

        CORRELATION between A&B = COVab / SDa*SDb

         COVab = 0.60*4*3 = 7.2

         treynor ratio = (9%-4%)/.80 = 6.25 times

(C)     SYSTEMATIC RISK of the portfolio is Beta of portfolio

          Bp = summation of ( Beta of individual security * weight of individual security)

          When B of individual security and weights are the same for all stocks Bp will be the same as B of individual security.

         Therefore Bp i.e SYSTEMATIC RISK i.e Market risk = 0.80

         UNSYSTEMATIC VARIANCE = (S.Dp)² - (Bp * S.D.m)²

                                                       16 - (0.80*3)²

                                                                           10.24