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?

CALReturn

SD

Market Timing v.s Buy and Hold

• Assume an investor puts $1,000 in a 30-day CP (riskless instrument) on Jan 1, 1927and 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 $67,500

1/1 1978 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 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 s2(ek)

• Group all securities with nonzero alpha into a portfolio called active portfolio. In this portfolio, aA, bA and s2(eA) are to be estimated.

Combining Active Portfolio withMarket Portfolio (passive portfolio)

A.

M

p

CML

New CALReturn

Risk

rA=aA + rf +bA(rm-rf)

Given:rp = wrA + (1-w)rm

The optimal weight in the active portfolio is:w = w0/[1+(1-bA)w0]

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 + [aA/s2(eA)]2

aA/s2(eA)(rm-rf)/s2

m

where w0=

The optimal weights in the activeportfolio for each individual securitywill be:

ak/s2(ek)a1/s2(e1)+...+an/s2(en)

wk =

Illustration of TB Model• Stock a b s(e)

1 7% 1.6 45%2 -5 1.0 323 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/s2(e) Weight (wk)1 0.07/0.452 = 0.3457 (1)/T = 1.14172 -0.05/0.322 = -0.4883 (2)/T = -1.62123 0.03/0.262 = 0.4438 (3)/T = 1.4735Total (T) 0.3012

Composition of active portfolio:aA = w1a1+w2a2+w3a3

=1.1477(7%)-1.6212(5%)+1.4735(3%) =20.56%bA = w1b1+w2b2+w3b3

= 1.1477(1.6)-1.6212(1)+1.4735(0.5) = 0.9519s(eA) = [w2

1s21+w2

2s22+w2

3s23]0.5

= [1.14772(0.452)+1.62122(0.322) +1.47352(0.262)]0.5

= 0.8262

Composition of the optimal portfolio:w0 = (0.2056/0.82622) / (0.08/0.22)

= 0.1506w = w0 /[1+(1-bA) w0 ] = 0.1495

Composition of the optimal portfolio:

Stock Final Positionw (wk)

1 0.1495(1.1477)=0.17162 0.1495(-1.6212)=-0.24243 0.1495(1.1435)=0.2202Active portfolio 0.1495Passive portfolio 0.8505

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