Active Portfolio Management Theory of Active Portfolio Management

Active Portfolio Management • – – Theory of Active Portfolio Management Market timing portfolio construction Portfolio Evaluation Conventional Theory of evaluation Performance measurement with changing return characteristics

Theory of Portfolio Management - Market Timing • Most managers will not beat the passive strategy (which means investing the market index) but exceptional (bright) managers can beat the average forecasts of the market • Some portfolio managers have produced abnornal returns that are beyond luck • Some statistically insignificant return (such 50 basis point) may be economically significant

• According the mean-variance asset pricing model, the objective of the portfolio is to maximize the excess return over its standard deviation(ie. , according to the Capital Allocation Line (CAL)) • buy and hold? Return CAL SD

Market Timing v. s Buy and Hold • Assume an investor puts $1, 000 in a 30 -day CP (riskless instrument) on Jan 1, 1927 and rolls it over and holds it until Dec 31, 1978 for 52 years, the ending value is $3, 600 $1, 000 $3, 600 52 yrs

• An investor buys $1, 000 stocks in in NYSE on Jan 1, 1978 and reinvests all its dividends in that portfolio. The the ending value of the portfolio on Dec 31, 1978 would be: $67, 500 $1, 000 1/1 1978 $67, 500 Dec 31, 1978 • Suppose the investor has perfect market timing in every month by investing either in CP or stocks , whichever yields the highest return, the ending value after 52 years is $5. 36 billion !

Treynor-Black Model • The Treynor-Black model assumes that the security markets are almost efficient • Active portfolio management is to select the mispriced securities which are then added to the passive market portfolio whose means and variances are estimated by the investment management firm unit • Only a subset of securities are analyzed in the active portfolio

Steps of Active Portfolio Management • Estimate the alpha, beta and residual risk of each analyzed security. (This can be done via the regression analysis. ) • Determine the expected return and abnormal return (i. e. , alpha) • Determine the optimal weights of the active portfolio according to the estimated alpha, beta and residual risk of each security • Determine the optimal weights of the entire risky portfolio (active portfolio + passive market portfolio)

Advantages of TB model • TB analysis can add value to portfolio management by selecting the mispriced assets • TB model is easy to implement • TB model is useful in decentralized organizations

TB Portfolio Selection • For each analyzed security, k, its rate of return can be written as: rk -rf = ak + bk(rm-rf) + ek ak = extra expected return (abnormal return) bk = beta ek = residual risk and its variance can be estimated as s 2(ek) • Group all securities with nonzero alpha into a portfolio called active portfolio. In this portfolio, a. A, b. A and s 2(e. A) are to be estimated.

Combining Active Portfolio with Market Portfolio (passive portfolio) Return New CAL p. A CML M Risk r. A=a. A + rf +b. A(rm-rf)

Given: rp = wr. A + (1 -w)rm The optimal weight in the active portfolio is: w = w 0/[1+(1 -b. A)w 0] where w 0= a. A/s 2(e. A) (rm-rf)/s 2 m The slope of the CAL (called the Sharpe index) for the optimal portfolio (consisting of active and passive portfolio) turns out to include two components, which are: [(rm-rf)/sm]2 + [a. A/s 2(e. A)]2

The optimal weights in the active portfolio for each individual security will be: wk = ak/s 2(ek) a 1/s 2(e 1)+. . . +an/s 2(en)

Illustration of TB Model • Stock a b s(e) 1 7% 1. 6 45% 2 -5 1. 0 32 3 3 0. 5 26 • rm-rf =0. 08; sm=0. 2 • Let us construct the optimal active portfolio implied by the TB model as: Stock a/s 2(e) Weight (wk) 1 0. 07/0. 452 = 0. 3457 2 -0. 05/0. 322 = -0. 4883 3 0. 03/0. 262 = 0. 4438 Total (T) 0. 3012 (1)/T = 1. 1417 (2)/T = -1. 6212 (3)/T = 1. 4735

Composition of active portfolio: a. A = w 1 a 1+w 2 a 2+w 3 a 3 =1. 1477(7%)-1. 6212(5%)+1. 4735(3%) =20. 56% b. A = w 1 b 1+w 2 b 2+w 3 b 3 = 1. 1477(1. 6)-1. 6212(1)+1. 4735(0. 5) = 0. 9519 s(e. A) = [w 21 s 21+w 22 s 22+w 23 s 23]0. 5 = [1. 14772(0. 452)+1. 62122(0. 322) +1. 47352(0. 262)]0. 5 = 0. 8262 Composition of the optimal portfolio: w 0 = (0. 2056/0. 82622) / (0. 08/0. 22) = 0. 1506 w = w 0 /[1+(1 -b. A) w 0 ] = 0. 1495

Composition of the optimal portfolio: Stock Final Position w (wk) 1 0. 1495(1. 1477)=0. 1716 2 0. 1495(-1. 6212)=-0. 2424 3 0. 1495(1. 1435)=0. 2202 Active portfolio 0. 1495 Passive portfolio 0. 8505 1. 0