This document discusses active portfolio management. It covers the theory of market timing and portfolio construction. It also discusses evaluating portfolios using conventional and performance measurement methods. The document then provides details on the Treynor-Black model for active portfolio management, including estimating security alphas and betas, determining optimal active portfolio weights, and combining the active portfolio with a passive market portfolio. Worked examples are provided to illustrate applying the Treynor-Black model to determine optimal portfolio weights.

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

2. 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

3. • 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?
CAL
Return
SD

4. 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

5. • 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 !

6. 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

7. 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)

8. 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

9. 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.

10. Combining Active Portfolio with
Market Portfolio (passive portfolio)
A
.
M
p
CML
New CAL
Return
Risk
rA=aA + rf +bA(rm-rf)

11. 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=

12. The optimal weights in the active
portfolio for each individual security
will be:
ak/s2(ek)
a1/s2(e1)+...+an/s2(en)
wk =

13. 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/s2(e) Weight (wk)
1 0.07/0.452 = 0.3457 (1)/T = 1.1417
2 -0.05/0.322 = -0.4883 (2)/T = -1.6212
3 0.03/0.262 = 0.4438 (3)/T = 1.4735
Total (T) 0.3012

14. 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)
= 0.9519
s(eA) = [w2
1s2
1+w2
2s2
2+w2
3s2
3]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)
= 0.1506
w = w0 /[1+(1-bA) w0 ]
= 0.1495

15. 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
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