Muy Buena Treynor Black Sin Matlab

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THETHEORYOFACTIVEPORTFOLIOMANAGEMENT

Market efficiencyis based on the notion that investors are willing to depart from a passiveinvestment strategy to trade on any perceived mispricing(i.e., instances where the market price

of a security deviates from its intrinsic value).

There are economic and empirical arguments that suggest that it is possible for active portfolio

management strategies to beat passive strategies.

Economic logic: If it were impossible to beat a passive investment strategy, investors would

move their funds from active management strategies to less expensive passive strategies. As less

money is actively managed, prices would no longer reflect intrinsic values, and the consequent

mispricing and profit opportunities would lure back active managers who would then be able

to outperform passive strategies again.

Empirical evidence:

Some portfolio managers have consistently been able to earn abnormal returns. There is sufficient variability in returns realized by portfolio managers to statistically

preclude us from concluding that no money managers outperform passive investment

strategies.

Some anomalies have been persistent enough to suggest that managers who identifiedthem and traded on them would have been able to beat a passive investment strategy

over extended periods of time.

These arguments suggest that active portfolio managers can indeed enhance returns even if

markets are almost efficient.

Objectives of Active Portfolios

The mean-variance framework allows us to separate the product decision (how the optimal

risky asset portfoliois determined) and the consumption decision (how the investors funds are

allocated between the optimal risky asset portfolio and the risk-free asset). The separation theorem

tells us that all investors hold the same optimal risky asset portfolio regardless of their level of

risk aversion. They only differ with respect to the weights of the optimal risky asset portfolio

and the risk-free asset in their individual portfolios. These weights depend on the individual

investors risk-return preferences (utility function) that determine the shape of her indifference

curves. More (less) risk-averse investors hold a smaller (greater) proportion of their optimal

investor portfolios in the optimal risky asset portfolio and a greater (smaller) proportion in the

risk-free asset.

We have also learned that the optimal risky asset portfolio is the portfolio that maximizes the

slope of the capital allocation line(straight line drawn from the risk-free rate to a portfolio on

the efficient frontier of risky assets). Stated differently, the optimal risky asset portfolio is one

that maximizes the reward-to-variability ratio (Sharpe ratio).

LOS 55a: Justify active portfolio management when security markets are

nearly efficient.Vol 6, pg 457-463

THE THEORY OF ACTIVE PORTFOLIO MANAGEMEN

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The aim of an active portfolio manager is to tilt the weights (of asset classes and/or individual

securities within those asset classes) of the optimal risky asset portfolio such that the realized

Sharpe ratio (slope of the ex-post CAL) is greater (steeper) than that of the passive strategy of

simply holding the market portfolio of risky assets.

Further, the risk-free rate, expected returns on assets, variances of assets, covariances of assets,

and managers Sharpe ratios are all not constant over time. Therefore, we can make argumentsagainst a purely passive strategy (or in favor of active portfolio management) because active

portfolio theory is needed to:

Develop capital market forecasts (expected returns, variances and covariances) for allmajor asset classes.

Assign weights to different asset classes such that the Sharpe ratio of the optimal riskyasset portfolio is maximized.

Assign weights to the optimal risky asset portfolio and the risk-free asset in line withthe individual investors utility function (risk aversion).

Rebalance the portfolio in light of changing market expectations and investor circumstances.

The Value of Perfect Market Timing

Consider the following investment strategies:

Strategy A: Invest $1,000 in 30-day commercial paper and keep rolling over the proceeds into

30-day paper for the 52-year period.

Strategy B:Invest $1,000 in the NYSE Index and reinvest all dividends for the 52-year period.

The average monthly return on this investment was 10.70% and the standard deviation of returns

was 22.14%.

Strategy C:Invest withperfect market timing. What this assumes is that the investor is able topredict (with 100% accuracy) at the beginning of each month whether the return on the NYSE

Index for that particular month will be greater than that on 30-day paper. If the return on equities

is expected to be greater the investor shifts her funds into the NYSE index for that month;

otherwise, she leaves her funds invested in 30-day paper.

Over the period from January 1, 1927 to December 21, 1978:

The average annual return and standard deviation on Investment B was greater than thatof Investment A. This makes sense because equity investments entail greater risk, so

their expected return ought to be higher as well.

The average annual return and standard deviation on Investment C was greater than thatof Investment B (and A). However note that:

o The standard deviation of returns for Investment C is a misleading measure ofrisk. This is because the return on Strategy C is at least as much as that of Strategy

B, if not greater (in months when the 30-day paper return exceeds the NYSE

index return). In other words, Strategy C only provides positive surprises, so no

matter how significant its standard deviation, you would always prefer Strategy

C to Strategy B.

For our purposes we

define apurelypassive strategy asone that uses onlyindex funds andapplies fixed weightsthat do not vary inresponse to perceivedmarket conditions.

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o Even if the net return on Strategy C (after accounting for fair managementfees) were lower than that on Strategy B, Strategy C would still be preferred

(despite its higher standard deviation of returns). This is because (as mentioned

earlier) standard deviation is not an appropriate measure of risk for Strategy C.

o The skewness of returns on Strategy B is very low as over a large sample size,equity returns are approximately normally distributed. However, the skewness

of returns on Strategy C is quite high (the distribution is skewed to the right).This is because this strategy effectively eliminates the negative tail of the

distribution (returns below the 30-day paper return).

The bottom line is that mean-variance analysis is not adequate for valuing market timing. Another

approach to valuing perfect market timing ability is to view it as a call option where the return

equals the greater of (1) the 30-day rate and (2) the return on the NYSE Index.

The Value of Imperfect Forecasting

The appropriate measure of market forecasting ability is not the overall proportion of perfect

forecasts. If the market rises for three days out of four and the manager always predicts a bullish

market, the 75% success ratio is not the correct measure of forecasting ability. The correct

approach would be to examine the proportion of bull markets correctly forecasted andthe

proportion of bear markets correctly forecasted. Under this approach, the true measure of our

managers forecasting ability would be calculated as:

P1+ P2 1 = 1.0 + 0 1= 0

P1= Proportion of bull markets correctly forecasted

P2= Proportion of bear markets correctly forecasted

The value of imperfect market timing will be calculated by multiplying the call option value of

a perfect market timer (C) by P1+ P2 1.

LOS 55b: Describe the steps and the approach of the Treynor-Black Model

for security selection.Vol 6, pg 464-472

Overview of the Treynor-Black Model

Recall from Level I, that a mispriced security is one that offers a positive risk-adjusted return

or alpha. From time to time, an analyst may identify securities that appear to be underpriced and

offer positive anticipated alphas to the investor. However, increasing the weight of the particular

security in her portfolio would mean losing some of the benefits of full-diversification. Therefore,

active portfolio managers must strike a balance between (1) aggressively pursuing alpha bytaking positions on mispriced securities and (2) ensuring that their portfolios remain diversified

by not allowing a few stocks to dominate.

The Treynor-Black modelis an optimization model for portfolio managers who use security

analysis. The model assumes that markets are nearly efficient. The bullets that follow provide

a basic overview of the model.


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Creating the Active Portfolio

If all securities are fairly priced, the rate of return on security i is given by:

ri=r

f

i(r

M

rf) e

i

ei represents firm-specific disturbance (also known as the nonsystematic component of return),

which (1) has an expected value of 0, and (2) is independent across securities. This expression

can be converted into the following form, which is known as the single index model:

ri

rf

=i

i(r

M

rf) e

iR

i=

i

iR

M e

i

Ri= r

i

rf

: The excess return on security i

RM

=rM

rf

: The excess return on the market

The single index model asserts that the excess return on security iis composed of:

The systematic (non-diversifiable) component of return, iR

M. This component of excess

return is related to the return on the market.

o irepresents the sensitivity of the excess return on security ito the excess return

on the market.

The nonsystematic (diversifiable) component of return, eiis also known as the residualreturn. We have already described this component earlier.

The excess risk-adjusted return or alpha, i.

In order to determine the weight of each security in the active portfolio (using the formulas

provided below) we first need to determine i

,

i

and 2ei

(variance of the residual return). We

can run a regression of the excess returns of security i on the excess returns of the market to

come up with estimates of i,iand 2

ei.

iis the intercept term of the regression.

iis the slope coefficient.

eiis the standard error of regression (presented in the regression statistics). In terms of

the single index model, it represents the standard deviation of the residual return.

2eirepresents unsystematic risk.

Only a limited number of securities are analyzed in detail to determine whether they aremispriced. The securities that are not analyzed are assumed to be fairly priced.

The model treats the market portfolio as the passive portfolio. The market portfolio isthe baseline portfolio that is efficiently diversified.

The investment manager knows the expected rate of return on the passive (market)portfolio,E(rM), and the variance of the market portfolio,

2.

The aim here is to form an active portfolio that consists of a limited number of stocksthat are expected to generate alpha. The (1) amount of perceived mispricing and (2) the

specific risk of the security determine the weight of an individual security in the active

portfolio.

The next step is to create the active portfolio. This process is described in detail in thenext section.

The final step is to create the optimal combination of the active portfolio and the passive(market) portfolio. This process is described in the section after the next one.

M

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Example 1: Determining Weights of Individual Securities in the Active Portfolio

The macroeconomic forecasting unit of Drex Portfolio Inc. (DPF) issues the following forecasts:

Expected return on the market:E(rM

) = 15%

Risk-free rate: rf= 7% Excess return on the market:RM=E(rM) rf= 8% Standard error of this forecast: M= 20%

Security analysts at the company come up with the following forecasts for three securities that

they believe are mispriced:

Stock

1

2

3

7%

5%3%

1.6

1

0.5

e45%

32%

26%

Determine the following:

1. The weight of each stock in the active portfolio. 2. The anticipated alpha of the active portfolio. 3. The beta of the active portfolio. 4. The unsystematic risk of the active portfolio.

Solution:

We first need to determine the anticipated alpha-unsystematic risk ratios (/2e) for each of

the three stocks. This is done by first computing unsystematic risk (2e) for each of the stocks

and then dividing the anticipated alpha () by unsystematic risk (2e).

wk =

n

i=1i /

ei

k

/ek

This expression should make sense to you:

k/2

ekrepresents anticipated alpha for the security per unit of unsystematic risk entailed

in the security.

The greater the anticipated alpha per unit of unsystematic risk for a security, the greaterthe reward-to-risk ratio, and the greater the weight of the security in the active portfolio.

unsystematic risk for the active portfolio as a whole. It equals the sum of the anticipated

alpha-unsystematic risk ratios for all the securities in the active portfolio. It is basically

a scale factor to ensure that the portfolio weights sum to one.

The expression n

i=1i /

ei

( ) represents the total anticipated alpha per unit of

The weight of security kin the active portfolio (Portfolio A) is determined using the followingexpression:


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1. Therefore: wStock 1= 1.1477

wStock 2= 1.6212 wStock 3= 1.4735

2. The anticipated alpha of a portfolio equals the weighted average of theanticipated alphas of the individual securities comprising the portfolio.

A= 1.1477 0.07 + (6212) (0.05) + 1.4735 0.03 = 0.2056 = 20.56%

3. The beta of a portfolio equals the weighted average of the betas of the individualsecurities comprising the portfolio.

A

= 1.14771.6 + (1.6212)1.0 + 1.4735 0.5 = 0.9519

4. The unsystematic variance of the active portfolio is calculated using the formula forthe variance of a 3-asset portfolio. However, we only use the unsystematic variances

of the three stocks here. This is because the unsystematic risks of the stocks are not

correlated so all terms in the variance formula that include the covariance equal 0.

2eA= 1.14772 0.45

2 + (1.6212)2 0.322+ 1.47352 0.262= 0.6826

Stock

1

2

3

7%

5%3%

1.6

1

0.5

e

45%

32%

26%

2e

0.452= 0.2025

0.322= 0.1024

0.262= 0.0676

Total

Stock

1

2

3

7%

5%3%

1.6

1

0.5

e

45%

32%

26%

2e

0.2025

0.1024

0.0676

Total

(/2e) / 0.3012

0.3457/0.3012 = 1.1477

0.4883/0.3012 = 1.62120.4438/0.3012 = 1.4735

1.0000

Creating the Optimal Risky Portfolio

Now that we have learned how to determine the weights of individual securities in the active

portfolio, we turn our attention to determining the weights of the active portfolio and the passive

(market) portfolio in the overall optimal risky portfolio.

Lets begin with an overview of what we are trying to do here. Go back to LOS 54b in Reading

54 where we came up with the minimum variance frontier between Stock A and Stock B (2 risky

assets). The minimum variance frontier was a function of:

The expected returns on Stock A and Stock B. The standard deviation of returns for Stock A and Stock B. The covariance of returns between Stock A and Stock B.

We then determine the weight of each security in the active portfolio by dividing the anticipated

alpha-unsystematic risk ratio (/2e) for the security by the sum of the anticipated alpha-

unsystematic risk ratios for all three securities (0.3012).

/2e

0.3457

0.48830.4438

0.3012

/2e

0.07/0.452= 0.3457

0.05/0.322= 0.48830.03/0.262= 0.4438

0.3012

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Note that theseformulas for varianceand covariance weregiven in an earlierreading under themarket model.

Figure 1: The Optimization Process with Active and Passive Portfolios

E(r)

CAL(P)

P

CML

A

M

XrM

RFR

M

EF0

EF1

M2

The initial efficient frontier (EF0)consists of all

risky assets.

The point of tangency betweenEF0and the CML

defines the market portfolio (Portfolio M).

The active portfolio (Portfolio A) is expected to

generate positive alpha. Therefore it plots above

the CML.

The new efficient frontier (EF1)combines

Portfolios M and A with varying weights.

The (new) optimal risky portfolio (Portfolio P)

is the point of tangency between a straight line

drawn from RFR and EF1.The CAL with the

highest Sharpe ratio is now CAL(P).

The point of tangency between an investors

indifference curveand CAL(P), Point X

determines the weight of Portfolio P in theoptimal investor Portfolio.

A= eA2 2 2

Once we have determined the minimum variance frontier, it is easy to determine the efficient

frontier(portion of minimum variance frontier above and to the right of the global minimum

variance portfolio). Once the risk-free asset is added to the mix, we then determine the optimal

risky asset portfolio as the tangency portfoliowhere a line drawn from the risk-free rate is tangent

to the efficient frontier. This tangency portfolio offers the highest excess return per unit of risk.

We are going to try to accomplish the same thing here, except that instead of Stock A and StockB, we will be working with Portfolio M (the passive, market portfolio) and Portfolio A (the active

portfolio). The minimum variance frontier that considers only these two risky portfolios can be

drawn up based on the following parameters:

The expected return on the market: rM The expected return on the active portfolio: E(rA) = A+ A RM+ rf where

RM=E(rM) rf The variance of the market: 2M The variance of the active portfolio: Systematic risk + Unsystematic risk = 2

A2

M+ 2

eA


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The covariance of the active portfolio with the market portfolio: CovA,M

= A

2M

Once we have drawn up the minimum variance frontier (and the efficient frontier) the (new)optimal risky portfolio (Portfolio P) is determined through an optimizationprocess. Portfolio P

consists of only Portfolios M and A. It occurs at the point of tangency between a straight line

drawn from the risk-free rate and the (new) efficient frontier, EF1.

E(rA) =

AR

f+

i(R

M)


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Assuming (for simplicity) that the beta of Portfolio A equals 1, the optimal weight, w0, of Portfolio

A in Portfolio P is calculated as:

w0=

A

RM

2(eA)

2M

A

/2(eA)

RM /2

M

=

According to this equation (see part between the = signs), if Portfolio A has average systematic

risk (A= 1), then the optimal weight of Portfolio A in Portfolio P equals the relative advantageof Portfolio A (the ratio of Portfolio As alpha to market excess return) divided by the disadvantage

of Portfolio A (the ratio of nonsystematic risk of Portfolio A to market risk).

If the beta of Portfolio A does not equal 1, we can use the following equation to determine the

optimal weight, w*, of Portfolio A in Portfolio P.

w*=w0

(1A

)w0

Notice that w* increases with greater A(systematic risk). The greater the systematic risk ofPortfolio A, the smaller the benefit of diversifying it with the market, and the more beneficial

it is to take advantage of mispriced securities.

Example 2: Determining Weights of the Active and Passive Portfolios in the Optimal

Portfolio

Using the information provided in Example 1, determine the weights of the active and market

portfolios in the optimal risky portfolio (Portfolio P) for DPF. Also determine the final weights

of Stocks 1, 2 and 3 in the optimal risky portfolio.

Solution:

In Example 1, we obtained the following:

RM= 8% M= 20% A= 20.56%

A=0.9519

2eA

=0.6826

The expression for the optimal weight, w*, of the active portfolio (Portfolio A) in the optimal

risky portfolio (Portfolio P) is given as:

w*=

A

A

(1A

) + RM

2(eA)

2M

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Evaluation of Performance

Sharpe Ratio

The square of the sharpe ratio of the optimal risky portfolio (Portfolio P) can be separated into

contributions from the square of the market sharpe ratio and the square of the active portfolios

infromation ratio:

SP=

2SM

2+

2(eA)

2A

=R

M

M + (eA)

A 22

Notice from this equation that Portfolio P improves upon the square of the sharpe ratio for the

market portfolio, S2Mby the amount:

(eA)

A 2

A /2(eA)

=RM /

2

M

=w0=0.2056/0.6826

0.08/(0.2)20.1506

And:

w*=w0

(1A)w0=

0.1506

1 + (10.9519)(0.1506)0.1495=

Therefore:

Notice the following:

The alpha for Portfolio A is impressively high (20.56%) but its weight in the optimalrisky portfolio (before adjustment for beta) is quite low (15.06%). This can be attributed

to the fact that it has a high nonsystematic standard deviation ( = 82.62%). The beta for Portfolio A is close to 1.0 (0.9519) which is why the adjustment for beta

is small (from 15.06% to 14.95%).

Since the beta of Portfolio A is lower than 1.0, its adjusted weight (w* = 14.95%) is

lower than its original weight (w0= 15.06%). This makes sense because the activeportfolio has lower-than-average risk that is correlated with the market (systematic

risk).In this case, more of the market portfolio must be added (and the weight of the

active portfolio reduced) to gain full diversification benefits.

The weights of the mispriced securities, the active portfolio and the market portfolio in the

optimal risky portfolio, Portfolio P, are calculated below:

Weight in Portfolio A Calculation of Final Weight Final Weight

Stock 1

Stock 2

Stock 3Active Portfolio

Market Portfolio

wStock 1

wStock 2

wStock 3

w*

1 w*

1.1477

1.6212

1.4735

0.1495 1.1477

0.1495 (1.6212)

0.1495 1.4735

0.1716

0.2424

0.22030.1495

0.8505

1.0000


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(eA)

A 2

(ei)

i 2

n

i=1=

Example 3: Determining the Impact of the Active Portfolio on the Sharpe Ratio

Using the information in Example 1, determine the Sharpe ratio of (1) the market and (2) the

optimal risky portfolio.

Solution:

1. SM=RM

M=

8

20= 0.40

2. The table below computes the information ratios for each of the three mispriced stocks:

Stock e

/e

1

2

3

7%

-5%

3%

1.6

1

0.5

45%

32%

26%

0.1556

-0.1563

0.1154

SP=

2SM

2+

2(eA

)2A = RM

M + (eA)

A

22

SP= 0.42+ (0.1556

2+ (0.1563)2+ 0.11542)Sp= 0.4711

The Sharpe ratio of the optimal risky portfolio is higher than that of the market portfolio by

0.0711. This increase represents the improvement in the risk-return tradeoffs available to the

investor by adding the active portfolio to the market portfolio.

M2measures the difference in expected returns between:

1. The market portfolio (that has a standard deviation of M). 2. A portfolio (Portfolio P*) that lies on the new CAL, CAL(P), and has the same standard

deviation as the market portfolio (M).

The ratio of the degree of mispricing to the non-systematic standard deviation (i/ei)measuresthe performance of the active component of the optimal risky portfolio. It is referred to as the

information ratio. The information ratio for each individual security measures the contribution

of that security to the performance of the active portfolio. Further, the sum of the squared

information ratios of the individual securities equals the squared information ratio of the active

portfolio.

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To summarize, the TB model exploits any perceived security mispricing through active portfolio

management (Portfolio A), but at the same time maintains a well-diversified portfolio (Portfolio

M). Once the optimal risky asset portfolio (Portfolio P) has been identified, the optimal investor

portfolio occurs at the point of tangency between CAL (P) and the investors indifference curve.

Multifactor Models and Active Portfolio Management

So far, our entire analysis of active portfolio management has been based on the single (single-

factor) index model. The Treynor-Black model can also be applied to a multifactor framework.

With multifactor models:

TB analysis continues to focus on the residuals of the model. The active portfolio would still be formed by calculating each securitys alpha relative

to its expected return given its beta (sensitivity) to all factors incorporated in the model.

However, the determination of variance and covariance estimates is a lot more complicatedthan when applying the TB model to a single-factor framework.

Example 4: Calculating and InterpretingM2

Continuing with the DPF example that we have been working with in this reading, calculate

theM2.

Solution:

The risk premium on Portfolio P* can be calculated by multiplying the slope of CAL(P) i.e.

the Sharpe ratio of the optimal risky portfolio (which we have already calculated in Example

3) by the standard deviation of the market portfolio (given in Example 1 as 20%).

RP*=E(rP*) rf= SP M= 0.4711 0.20 = 0.0942 or 9.42%

TheM 2is then calculated as the difference betweenRP*

andRM

.RM

was given as 8% in

Example 1.

M2=RP*RM= 9.42 8 = 1.42%

For the given level of risk (M) adding the optimal amount of the active portfolio (PortfolioA) to the market portfolio (Portfolio M) improves the investors expected return by 1.42%.


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We can evaluate the quality of the analysts forecasts by calculating the coefficient of determination

of the regression described above. TheR2can be calculated as the proportion of the total variation

in the analysts forecasts of alpha that is explained by variation in actual alpha:

R

f

+

This estimate of R2is used as a shrinking factorto adjust the analysts forecasts of alpha. The

adjusted forecast equals R2multiplied by forecasted alpha (f ). If the analysts has a record of predicting alpha with 100% accuracy (i.e., R2= 100%)

then we should take the forecast as it is to implement the Treynor-Black model.

If the analysts has a record of predicting alpha with 0% accuracy (i.e., R2= 0%) thenwe should ignore the analysts forecasts of alpha.

f

+

LOS 55c: Explain how an analyst's accuracy in forecasting alphas can be

measured and how estimates of forecasting can be incorporated into the

Treynor-Black approach. Vol 6, pg 472-473

Imperfect Forecasts of Alpha Values

An analysts past record in predicting alpha accurately should be considered by a portfolio

manager in determining the weight of the active portfolio (Portfolio A) in the overall risky

portfolio (Portfolio P) going forward. Based on the index model and stock betas, we can determine

actual realized alphas on the analysts previous recommendations using average excess realized

returns on the security (R) and the market (RM):

Actual (realized) alpha: R RM

To measure the forecasting accuracy of the analyst, we can regress alpha forecasts (f ) on realizedalpha ().

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f a a

For simplicity, we assume that aand aequal 0 and 1 respectively. Given that forecast errors

() are uncorrelated with true alpha () i.e., Cov,equals 0, the variance of the forecast is givenas:


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Example 5: Adjusting Forecast of Alpha for Imperfect Forecasts

Continuing with the DPF portfolio example, now assume that historically, variation in actual

alphas has explained 20% of the variation in alphas forecasted by DPFs security analysis

team. Determine the weight of the active portfolio in the overall risky portfolio if the analysts

forecast of alpha for Stocks 1, 2 and 3 are adjusted for imperfect forecasts.

Solution:

We will once again use the formulas for w0and then w* to determine the weight of the active

portfolio in the overall risky portfolio. However, instead of using an alpha of 20.56% as we

did in Example 2, we will now use an alpha of:

A /2(eA)

=RM /

2

M

=w0=

0.04112/0.6826

0.08/0.040.0312

Adjusted Alpha = Shrinking factor Alpha forecast

0.0301

1 + (10.9519)(0.0301)0.03008=w*=

Note that the purpose of this exercise is not to make up for deficiencies in the TB model. We

are just adjusting our expectations for the quality of forecasts.

= 0.20 20.56

= R2 A

= 4.112%


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