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Investment Management

Active Portfolio Management

Road Map

The Efficient Markets Hypothesis (EMH) and ‘beating the market’Active portfolio management

− Market timing

− Security selection

Security selection: Treynor&Black model

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The problem

Is it possible to beat the performance of a passiveportfolio?

− If markets are fully efficient, the answer is NO

− But… what if markets are only nearly efficient

Active portfolio management seeks to exploit perceived market inefficiencies

Two forms of active portfolio management

− market timing

− security selection

Market inefficiency

Actively-managed portfolios

− are not fully diversified

− may include mispriced securities

Competition amongst active managers ensures that securities trade close to their fair values

− on a risk-adjusted basis, most portfolio managers will notbeat passive strategies

− managers with exceptional skills and/or privileged information can beat passive strategies

Some active managers must earn abnormal profits

− otherwise …° there would be no active portfolio managers

° security prices would stray from fair values, inducing portfolio managers to adopt active strategies

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Beating the market portfolio…

Market timing

Two techniques for improving performance:

− adjust the bond/stock mix of the portfolio in anticipation of market changes

° modify portfolio in favour of equities (bonds) if investor is “bullish” (“bearish”) about the stock market

− adjust the equity beta of the portfolio

° invest in high-beta (low-beta) stocks if investor is “bullish”(“bearish”) about the stock market

Both techniques impact the average beta of the overall portfolio therefore we can use beta to measure how successful market timing has been

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Market timing: Adjusting bond-stock mix

Little evidence of market timing ability

Direct comparisonwith market return

Evaluate market timing by comparing the return on the portfolio with the return on the market

No market timing

− average beta of portfolio (fairly) constant

− portfolio return is a constant fraction of return on market (assuming no specific risk)

Market timing

− high β in rising market, low β in falling market

− portfolio return > market return

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Evidence of market timing (returns comparison)

Market timing

Example: Merton’s exercise− holding period 1 January 1927 - 31 December 1978

− investment strategies:1. invest $1000 in 30-day T-bills, rolled over every 30 days

2. invest $1000 in NYSE index, and re-invest dividends

3. invest $1000 in either 30-day T-bills or NYSE index, switching from one to the other every 30 days on the basis of perfect foresight (certainty)

How do the three strategies compare (December 1978)?

$5,360,000,000Stretegy 3. Perfect foresight

$67,500Strategy 2. All-equity

$3,600Strategy 1. All-safe asset

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Market timing

Why does perfect market timing result in such huge gains?

− compounding over a large number of years

− privileged information

Monthly returns on NYSE index and switching

portfolio (σTBill = 2.10%, σequities = 22.14%) :

Security selection

Aims at identifying mispriced securities which offer opportunity to achieve returns higher than pre-specified benchmarks (or the market portfolio)

Problems:− adjusting composition of portfolio in favor of

mispriced securities moves away from being full diversified

− incur costs of carrying specific (or diversifiable) risk

Trade-off between− achieving abnormal returns

− reducing risk via diversification

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Treynor-Black model: preamble

The Treynor-Black model is a model embedding the use of security analysis

Analysts look at the market and investigate in depth only few securities (the others are assumed to be fairly priced). They form active portfolios as follows:

− Estimate for each security betas and residual risk

− Given a certain equilibrium model (say the CAPM) the identify the magnitude of mispricing (alpha)

− They also estimate the impact of holding a less than fully diversified portfolio looking at the variance of stock residuals(=residual risk)

− Given the estimates of betas, alphas and residual risks they compute the optimal weights of each security in the active

portfolio

Treynor-Black model: assumptions

Assume:

− there is a single common source of risk

− the market is nearly efficient (i.e. the CAPM/single index model nearly holds)

Asset returns:

where αk is the abnormal return (=Jensen’s alpha) of the mispriced assets k.

Expected return on active portfolio (comprising mispricedsecurities):

With variance:

( )k f k M f k kr r r r eβ α= + − + +

( ) ( )A i f A M fE r r E r rα β ⎡ ⎤= + + −⎣ ⎦

( ) ( )A A M Ar e2 2 2 2σ β σ σ= +

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Treynor-Black model: intuition

Treynor-Black model: solution

The optimal active portfolio is a combination of the passive market portfolio and the active portfolio of mispriced securities

How do we judge the success of the strategy? The mathematics of the efficient frontier reveals that

( ) ( )

NA i

P M MiA iSharpe ratio of the Market

portfolio (squared)Information ratio of theactive portfolio (squared)

S S Se e

2 2

2 2 2

1

α ασ σ=

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= + = +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑

Only for the optimal active portfolio (P), the Sharpe ratio (squared) is given by the sum of

− the sharpe ratio of the (passive) market portfolio (squared)

− the standardised degree of mispricing, appraisal ratio orinformation ratio, (squared)

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Example

Equity analyst in HK selects the following mispriced stocks:

40%8%0.20Cathay Pac Air

45%14%0.50Esprit Holdgs

50%17%0.60MTR Corp

60%20%0.60BOC HK

σ(ei)E(ri)βi

Other information: E(rM) = 10%, σRM = 25%, RF = 4%

Example (cont’d)Step 1: calculate expected abnormal returns and information ratios

2.80%

7.00%

9.40%

12.40%

αi

0.07

0.16

0.19

0.21

αi/ σ(ei)

Cathay Pac Air

Esprit Holdgs

MTR Corp

BOC HK

( ) ( )i i f i M fE r r E r rα β ⎡ ⎤= − − −⎣ ⎦

This implies a Sharpe ratio of the active portfolio of

( )

( ) ( ) ( ) ( ) ( )

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2

1

2 2 2 2 20.24 0.21 0.19 0.16 0.07 0.41

iP M

i i

S Se

ασ=

⎧ ⎫⎪ ⎪⎡ ⎤⎪ ⎪⎪ ⎪⎢ ⎥= + =⎨ ⎬⎢ ⎥⎪ ⎪⎪ ⎪⎣ ⎦⎪ ⎪⎩ ⎭

⎡ ⎤= + + + + =⎢ ⎥⎣ ⎦

∑

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Example (cont’d)

0.175/1.241 = 0.140.028/(0.402) = 0.175Cathay Pac Air

1.241

0.070/(0.452) = 0.346

0.094/(0.502) = 0.376

0.124/(0.602) = 0.344

1.00

0.346/1.241 = 0.28

0.376/1.241 = 0.30

0.344/1.241 = 0.28

total

Esprit Holdgs

MTR Corp

BOC HK

( )2/i ieα σ

( )

( )

2

42

1

/

/

i i

i

i i

i

ew

e

α σ

α σ=

=

∑

Step 2: construct the active portfolio implied by the security analyst input list (algebra is not required)

( )( )

2

2

//

i i

i ii

ee

α σα σ∑

Example (cont’d)

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1

1

2 2 2 22 2

1

12 2 2 2 2

0.28 0.124 0.30 0.094

0.28 0.070 0.14 0.028 0.086 (8.6%)

0.28 0.6 0.30 0.6

0.28 0.5 0.14 0.2 0.516

0.28 0.60 0.30 0.50

0.28 0.45 0.14 0.40 0.26

n

A i i

i

n

A i ii

n

A i i

i

w

w

e w e

α α

β β

σ σ

=

=

=

= = × + × +

× + × =

= = × + × +

× + × =

⎡= = × + × +⎢⎣

⎤× + × =⎥⎦

∑

∑

∑

4 (26.4%)

The formed active portfolio exhibits the following estimates

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Example (cont’d)

( ) ( )

[ ] ( )

( ) ( ) ( ) ( )

( )

( ) ( )

( )( )

2 2 22 2 2

22

0.086 0.04

0.516 0.10 0.04 0.16 16%

0.516 0.25 0.264

0.29 29%

cov , 0.516 0.25 0.032

cov , 0.032, 0.44

0.29 0.25

A A f A M f

A A M

A M A M

A M

A M

A M

E r r E r r

e

r r

r rr r

α β

σ β σ σ

β σ

ρσ σ

⎡ ⎤= + + − = + +⎣ ⎦+ − =

⎡ ⎤= + = + =⎢ ⎥⎣ ⎦=

= = × =

= = =×

The expected return and standard deviation of the active portfolio are equal to:

Example (cont’d)Step 3: determine composition of the overall risky portfolio (active portfolio + market portfolio)

where w* denotes the share of the active portfolio within the overall risky portfolio. The weight attached to the market portfolio within the overall risky portfolio is given by 1 – w*

In our example:

( )( )

A A

M M A

e ww w

R w

2

00 2

0

/*

/ 1 1

α σσ β

= → =+ −

( )( ) ( )

( )

2

0 2

0.086/ 0.2641.29

0.10 0.04 / 0.25

1.29* 0.80

1 1 0.519 1.29

w

w

= =−

= =+ −

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Example (cont’d)

0.14 × 0.80 = 0.11

0.28 × 0.80 = 0.22

0.30 × 0.80 = 0.24

0.28 × 0.80 = 0.22

wi

0.80Active portfolio

0.20Market portfolio

w*

Cathay Pac Air

Esprit Holdgs

MTR Corp

BOC HK

0.41 0.24A MS S= =

Example (cont’d)

CML (SR=0.24)

CAL (SR= 0.41)

0.04

0.09

0.14

0.19

0.24

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

total risk (sigma)

expe

cted

ret

urn

E(r)

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Treynor-Black model: summing up

Optimizing model for portfolio managers who use security analysis under the assumption that markets are nearly efficient

− security analysis can assess in depth only a small number of securities (securities not assessed are assumed to be fairly priced)

− market index portfolio is the passive portfolio

− perceived mispricing guides the composition of the active portfolio

Treynor-Black model: summing up

− analysts follow several steps to make up the portfolio and evaluate its expected performance

1. Estimate beta and residual risk for each analysedsecurity; from these, determine the required return

2. Given the degree of mispricing, determine the expected return for each security (abnormal return)

3. The nonsystematic risk component of the mispricedstock is the cost of not fully diversifying by specialisingin underpriced securities

4. Determine the optimal weight of each security in the active portfolio

5. determine the optimal risky portfolio, which is a combination of passive and active portfolios

6. Compare CAL w.r.t CML

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Readings

BKM

− Section 24.4, Chapter 27

Other readings (optional)

− Treynor, J.J. and Black F. (1973), “How to use security analysis to improve portfolio selection,” Journal of Business, 46, 66-86

− Black, F. and Litterman, R. (1992), “Global portfolio optimization”, Financial Analysts Journal, September-October 1992