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CAPM - How to measure risk

### Transcript of Lecture3 CAPM in Practise

• Lecture 3: CAPM in practice

Investments

FIN460-Papanikolaou CAPM in practice 1/ 59

• Overview

1. The Markowitz model and active portfolio management.

2. A Note on Estimating

3. Using the single-index model and the CAPM in the passiveportfolio problem

3.1 Using to get good covariance estimates3.2 Using to get good expected-return estimates

4. The Black and Litterman Method.

An Example: Global Asset Allocation

5. Conclusions

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• Using the Markowitz model

With a reasonable number of securities, the number ofparameters that must be estimated is huge:

For a portfolio of N (100) securities we need:is N 100E(ri)s N 100Cov(ri,r j)s 12N(N1) 4950Total 12N(N+3) 5150

About how much data will we need when we have 500 securities?1000 Securities?

Means and covariances are estimated with error.

Small errors in mean or covariance estimates often lead tounreasonable weights.

FIN460-Papanikolaou CAPM in practice 3/ 59

• Using the Markowitz model

A remedy for both of these problems is:

1. First, we useI the CAPM to determine what market believes expected returns

should be

I a single factor model to calculate asset covariances

2. then we can combine our views with the CAPM-derivedestimates to get portfolio weights.

The key input we will need for both of these is the set of asset s.So, first, we must consider the problem of estimating s.

FIN460-Papanikolaou CAPM in practice 4/ 59

• Estimating

1. Let ri,t , rm,t and r f ,t denote historical individual security, marketand risk free asset returns (respectively) over some periodt = 1,2, . . . ,T .

2. The standard way to estimate a beta is to use acharacteristic line regression:

ri,t r f ,t = i+i(rm,t r f ,t)+ i,t3. To estimate this, typically we would use monthly data4. Typically use 5 years (60 months) of data.

Why not use more data? (10 or 20 years)I Parameter instability

Can we use weekly, daily, or intraday data?I Non-synchronous pricesI Bid-ask bounce

FIN460-Papanikolaou CAPM in practice 5/ 59

• Estimating - ExampleHere is an example of a (12 month) characteristic-line regression forGM Common stock (from BKM, Ch. 10):

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• Estimating - Example

This figure illustrates this still better:

How do we read i, i, and i,t off of this scatterplot?

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• Estimating

There are a number of Institutions that supply s:1. Value-Line uses the past five years (with weekly data) with the

Value-Weighted NYSE as the market.

2. Merrill Lynch uses 5 years of monthly data with the S&P 500 asthe market

FIN460-Papanikolaou CAPM in practice 8/ 59

• Estimating

Merrill uses total rather than excess returns:

ri,t = ai+birm,t + ei,t

So, the ai here is equal to:

ai = i+(1i)r f

where r f is the average risk-free rate over the estimation period ai is not equal to zero if the CAPM is true. However, bi and i will be very close, as long as cov(r f ,t ,rm,t) 0

FIN460-Papanikolaou CAPM in practice 9/ 59

• Estimating

Note also that Merrill used adjusted s, which are equal to:

Ad ji 1/3+(2/3) i

Intuition?I Statistical Bias

I Why 1/3 and 2/3 ?

I What should these numbers be based on?

I Other more advanced shrinkage techniques.

FIN460-Papanikolaou CAPM in practice 10/ 59

• Estimating for new companies

In the absence of historical data, how would we estimate fornew companies?

Standard industry practice is to use comparables:

Find a similar company, that is traded on an exchange and use thebeta of that company.

The following approach is in the same spirit but is more powerful.

Often a comparable company cannot be found. Can yield a model-predicted beta from a sample of companies

with similar characteristics.

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• Estimating

Which characteristics to use?

Industry Firm Size Financial Leverage Operating Leverage Growth / Value

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• Estimating for new companies - Methodology

1. Select a sample of companies to estimate the model.

2. Estimate i,t for these companies using historical return data.

3. Regress estimated betas i,t on several characteristics that candrive betas. Example:

i,t = a0+j jINDUSTRYj,i+a1FLEVi,t+a2SIZEi,t+a3OLEVi,t+ui,t

INDUSTRYj,i is a dummy variable that take the value 1 if firm ibelongs in industry j and 0 otherwise.

4. Suppose you are asked to find the beta for new company XYZ inthe tech industry, given XYZs characteristics. Your estimate willbe

xyz,t = a0+ tech+a1FLEVxyz,t +a2SIZExyz,t +a3OLEVxyz,t

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• Estimating

Betas may change over time, thus we use short windows of data(5-years)

Possible reasons for this are

1. changes in the firms leverage

2. changes in the type of a firms operations

3. the firm acquires targets in other industries

We can use rolling window regressions to estimate a time seriesof s:

at month t estimate using months t60 through t1 at month t+1 estimate using months t59 through t.

Alternatively, use more sophisticated statistical models that allowfor time-variation in beta.

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• Estimating

s can be quite volatile.Example: AT&T

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• Estimating

s of industries also changes over time.Example: Oil Industry

0

0.5

1

1.5

2

1 96 2

1 96 4

1 96 6

1 96 8

1 97 0

1 97 2

1 97 4

1 97 6

1 97 8

1 98 0

1 98 2

1 98 4

1 98 6

1 98 8

1 99 0

1 99 2

1 99 4

1 99 6

1 99 8

2 00 0

2 00 2

2 00 4

0

5

10

15

20

25

30

35

40

45

50Oil industry market beta

Real Oil Price (1980=100)

FIN460-Papanikolaou CAPM in practice 16/ 59

• The CAPM and Active Portfolio Management

To create an optimal portfolio we need to estimate the minimumvariance/efficient frontier, MVE portfolio, and the best capitalallocation line (CAL).

Where are the inputs coming from?

1. One way to do this is to ignore market opinion, and to estimate theE(ri)s and i, j s for all assets.

I For example, one could use past empirical data.

I we have already seen that there are problems with this approach

2. An alternative approach is to accept market opinion and simplyhold the market portfolio.

I This approach doesnt tell you what to do if you have informationthat you believe has not yet been incorporated into market prices.

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• The CAPM and Active Portfolio Management

The approach we will instead take is to1. Calculate s for the securities we plan to hold using the methods

discussed earlier .2. Using these s, calculate i, j s and E(ri)s, assuming the CAPM

holds exactly.3. Then, incorporate our information by perturbing the values away

from the CAPM-calculated values.4. Finally, using these estimates and the Markowitz portfolio

optimization tools (from Lecture 2), determine our optimal portfolioweights.

Note that if(i) We start with all of the assets in market portfolio(ii) We use the unmodified i, j s and E(ri)s we get from step 2

Then the weights we calculate in step D.4 will be exactly those ofthe market portfolio. Why??

FIN460-Papanikolaou CAPM in practice 18/ 59

• A single index model

An alternative to estimating i, j s is to assume that a single indexmodel describes returns:

1. ri,t r f ,t = i+i(rm,t r f )+ i,t2. for two different securities i and j,

cov(i,t , j,t) = 0.I The CAPM does not imply this.I Is this a reasonable simplification?

To get covariances/correlations, use:

i, j = cov(ri,t ,r j,t) = i j2m i 6= j

and

i, j =i j2mi j

For variances:i,i = 2i =

2i 2m+2,i

FIN460-Papanikolaou CAPM in practice 19/ 59

• The single index model vs the CAPM

1. The single index model is a statistical model.

It specifies that all common movements between stocks can becaptured by a single index.

It is also often called a single factor model. Can be generalized to multiple common factors. It is a statistical technique designed to estimate large covariance

matrices.

2. The CAPM is an economic model of expected returns.

It specifies that the market portfolio captures systematic risk. However, it does allow securities i and j to be correlated on top of

what their covariance with the market portfolio implies, forinstance if they are in the same industry. The only requirement isthat these correlations wash out as we add more stocks.

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• The CAPM and Active Portfolio Management

To get the markets beliefs about expected returns, use theCAPM:

E(ri) r f = i (E(rm) r f )Remember, this is estimated by the regression

ri,t r f ,t = i+i(rm,t r f )+ i,t

Given our estimates i , i and E(rm,t r f ), if we calculate

i+iE(rm,t r f )

it will be the same as estimating E(ri,t r f ,t) using historicalaverages. Why?

The CAPM estimate of E(ri,t r f ,t) is obtained by imposingi = 0.

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• The CAPM and Active Portfolio Management

Then, to construct the efficient frontier based on the Single IndexModel (for 100 securities) we need estimates of the following:

r f 1 1E(rm) 1 12m 1 1i N 100i N 1002,i N 100Total 3N+3 303

This is considerably smaller than the 5150 we had before.

Where are the E(ri)?

Where are the i, j =Cov(ri, r j)?What will the efficient portfolio weights be if we assume all i arezero? (and include all assets in our analysis)

FIN460-Papanikolaou CAPM in practice 22/ 59

• Example (cont)

Monthly data for GE, IBM, Exxon (XON), and GM:

Excess Returnsmean std alpha beta std R2ad j

IBM 3.22% 8.44% 1.31% 1.14 7.13% 28.5%XON 1.41% 4.03% 0.42% 0.59 3.28% 33.7%GM 0.64% 7.34% -1.06% 1.02 6.14% 30.0%GE 2.26% 5.86% 0.53% 1.04 4.15% 49.9%

VW-Rf 1.67% 4.02%Rf 0.36% 0.05%

VW is the Value-Weighted index of all NYSE, AMEX, andNASDAQ common stocks.

Rf is the (nominal) 1-month T-Bill yield, 4.394%/year(0.359%/month)

FIN460-Papanikolaou CAPM in practice 23/ 59

• Example (cont)

For expected returns, lets not impose the CAPM just yet.

To get the correlation structure, we use:

i, j =i j2mi j

i 6= j

IBM XON GM GEIBM 1 0.32 0.30 0.39XON 0.32 1 0.33 0.42GM 0.30 0.33 1 0.40GE 0.39 0.42 0.40 1

FIN460-Papanikolaou CAPM in practice 24/ 59

• Example (cont)

Plugging the (1) expected returns, (2) return standarddeviations, and (3) correlation matrix into the Excel spreadsheetwe get the following weights for the tangency portfolio:

Weight

IBM 29.6 %XON 49.7 %GM -21.4 %GE 42.4 %

It seems unreasonable that we should hold such extremeportfolio positions. The equilibrium arguments we used in developing the CAPM

suggest that the market knows something we dont about futureexpected returns!

FIN460-Papanikolaou CAPM in practice 25/ 59

• Example (cont)

Lets use the CAPM as a way of getting around this problem.

This is equivalent to setting i = 0 for all securities, or using theregression equation:

E(ri) = r f +i[E(rm) r f ]

and the past (average) return on the market to get equilibriumestimates of the expected returns.

Stock CAPM E(rei )IBM 1.91 %XON 0.99 %GM 1.70 %GE 1.73 %

FIN460-Papanikolaou CAPM in practice 26/ 59

• Example (cont)

With this equilibrium set of expected returns, we now get theportfolio weights:

weights

IBM 13.6%XON 33.3%GM 16.4%GE 36.6%

Why are the weights different? Why are these not the market weights? When would these be the

actual market weights? Is this the portfolio you would want to hold, given that you were

constrained to hold these four assets?

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• Example (cont)

However, there may be times when we think that the market is alittle wrong along one or more dimensions (a very dangerousassumption!)

How can we combine our views with what the market expects?

1. First, suppose that I think that the market has underestimatedthe earnings that IBM will announce in the next month, and thatIBMs expected return is 2% higher than the market expects.

2. Also, I have no information on the other three securities that wouldlead me to think that they are mispriced,

3. and I believe that the past betas, and residual std devs are goodindicators of the relative future values.

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• Example (cont)

In this case, we would use the same variance and covarianceinputs, but would change the expected returns. The new portfolioweights would be:

Stock E(rei ) weightsIBM 3.91% 54.1%XON 0.99% 17.7%GM 1.70% 8.7%GE 1.73% 19.5%

compared to the old allocation

E(rei ) weightsIBM 1.91% 13.6%XON 0.99% 33.3%GM 1.70% 16.4%GE 1.73% 36.6%

FIN460-Papanikolaou CAPM in practice 29/ 59

• Example (cont)

Alternatively, suppose that I think the risk () of Exxon isincreasing.

1. I guess that Exxons will rise from 0.59 to 0.8.

2. I also expect that Exxons idiosyncratic risk will not change.

First, I should recalculate almost everything using the equations:

E(ri) = r f +i[E(rm) r f ]2i =

2i 2m+2,i

i, j =i j2mi j

FIN460-Papanikolaou CAPM in practice 30/ 59

• Example (cont)

The new correlations we come up with are:

IBM XON GM GEIBM 1.00 0.44 0.30 0.39XON 0.44 1.00 0.45 0.57GM 0.30 0.45 1.00 0.40GE 0.39 0.57 0.40 1.00

as opposed to the old correlation matrix of:

IBM XON GM GEIBM 1 0.32 0.30 0.39XON 0.32 1 0.33 0.42GM 0.30 0.33 1 0.40GE 0.39 0.42 0.40 1

FIN460-Papanikolaou CAPM in practice 31/ 59

• Example (cont)

If, I believe that these are the new correlations, but that themarket still believes that the past correlations represent the future(and will not discover this information over the next severalmonths) then I would use the old expected returns, giving newportfolio weights of:

old newE(rei ) weight weight

IBM 1.91% 13.6% 17.0%XON 0.99% 33.3% 16.7%GM 1.70% 16.4% 20.5%GE 1.73% 36.6% 45.7%

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• Example (cont)

If, I believe that market knows that the of Exxon is higher, andthe expected return on Exxon is higher now to compensate forthe increased risk:

old newE(rei ) weight E(r

e) weight

IBM 1.91% 13.6% 1.91% 9.2%XON 0.99% 33.3% 1.34% 54.8%GM 1.70% 16.4% 1.70% 11.1%GE 1.73% 36.6% 1.73% 24.8%

One other alternative is that I dont believe that the market yetknows that the risk of Exxon is higher, but will discover this in thenext few months.

1. What will happen as the market finds out?2. What should I do in this case?

FIN460-Papanikolaou CAPM in practice 33/ 59

• Forming views

Where is our information coming from?

1. It could come from public sources of information that may or maynot have been impounded in prices yet.

2. It could be private, that is it could come from our own analysis offundamentals.

Nevertheless, we may not be 100% confident that our informationis accurate.

For instance, in the previous example, if I am only somewhatconfident of my belief that the market will not adjust the price ofExxon properly, I might want to adjust the portfolio weights onlypart-way.

FIN460-Papanikolaou CAPM in practice 34/ 59

• Forming views - Example

Consider the Morningstar report on AIG:

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• Forming views - Example

1. Market price: P0 = 58.952. Your view:

2.1 V0 = (1+a)P0 = 65.00 a= 10.26%2.2 rCAPM = [E(P1)V0]/V0

3. You think the actual return will be:

r = [E(P1)P0]/P0= [E(P1)V0+V0P0]/P0= rCAPM(V0/P0)+ [V0P0]/P0= rCAPM+a+ rCAPMa rCAPM+a

FIN460-Papanikolaou CAPM in practice 36/ 59

• Forming views - Example

If you were 100% confident in your view, then you would alsothink that AIGs expected return will be higher than the CAPMimplied return by an amount a= 10.26%.

This is a very large number and will likely lead to extremeportfolio allocations.

Suppose that you are only 10% confident in your view, then youmight use a= 0.110.26%= 1.26%.NEXT: The Black and Litterman model gives a systematicframework that enables you to incorporate your views with whatthe markets views in forming the optimal portfolio.

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• Black-Litterman model - Global Asset Allocation

Focus on asset allocation between country indices.

Use an international version of the CAPM.

Calculate the equilibrium expected returns based on estimatedvariances and covariances. This is appropriate, sincecovariances can be estimated accurately, especially using dailydata, for a small number of assets.The portfolio manager can specify any number of market views"in the form of expected returns, and a variance (measure ofuncertainty) for each of the views. If the manager holds no views, she will hold the market portfolio. If her views are high variance (low certainty), she will hold close to

the equilibrium portfolio. When her views are low variance, she will move considerably

away from the market portfolio.

FIN460-Papanikolaou CAPM in practice 38/ 59

• Black-Litterman model - Global Asset Allocation

First, lets look at the sort of portfolio allocation we get if we usehistorical returns and volatilities as inputs:

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• Black-Litterman model - Global Asset Allocation

also, we use historical correlations:

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• Black-Litterman model - Global Asset Allocation

If you use our procedures from Lecture 2 and calculate and optimalportfolio, with p = 10.7%, you will get portfolio weights of:

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• Black-Litterman model - Global Asset Allocation

CAPM Based Estimates

assuming market risk premium=7.15%

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• Black-Litterman model - Global Asset Allocation

There are N assets in the market, returns are normallydistributed:

R N(,) is an unknown quantity. The CAPM estimate says

N(,)What this means is that the investor is uncertain about what theexpected returns are. He will form his best guess or in Bayesianterminology his posterior beliefs, by combining information fromtwo sources: The first, his prior , will be based on the CAPM. The variance of

his prior, , denotes how much confidence the investor has onthe CAPM. High values of means that he attaches less weight tothe CAPM.

The second will be his own views, as we will see next.

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• Black-Litterman model - Global Asset Allocation

Suppose that we believe that Germany will have an expectedreturn of 4%

Black-Litterman allows you to express this view as

ger = q+ , N(0,)

Here, q= 4% is your belief about Germanys expected return.

The variable allows for a margin of error in your view.

Your confidence in your view is represented by , i.e. thevariance in your margin of error.

The higher is, the less confident we are.

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• Black-Litterman model - Global Asset Allocation

The Black-Litterman model also allows you to form relative views.

Suppose that we believe that Germany will outperform France by5%

Black-Litterman allows you to express this view as

P= Q+ , N(0,)

Here, P is a matrix, for instance if there are only two countriesand you only form a relative view, it will equal

P=(

1 10 0

)As before, is the variance of your views.

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• Black-Litterman model - Global Asset Allocation

Given our views and the CAPM prior, our best guess about whatexpected returns are is is going to be a weighted average of theCAPM expected returns, and our views, Q.

= [()1+P1P]1[()1+P1Q]

In the case of one asset with prior pi and one view, q, it simplifiesto

= pi1/(2)

1/+1/(2)+q

1/1/+1/(2)

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• Forming posterior beliefs - Example

Suppose that you would like to for your best guess about GEsexpected excess return going forward.

GE has a beta of 1.2 and a standard deviation of 0.33. Assumingthe equity premium is 6%, this gives you (in excess of the risk-freerate)

capmGE = 1.26%= 7.2% and var(capmGE )= 0.332= 0.1

You estimated the historical average return of GE to be 8.3% overthe last 10 years. This gives you another estimate or view:

histGE = 8.3% and var(histGE ) =

(0.3310

)2 0.01

FIN460-Papanikolaou CAPM in practice 47/ 59

• Forming posterior beliefs - Example

Your best guess will be:

GE =1

var(capmGE )capmGE + 1var(histGE )

histGE

1var(capmGE )

+ 1var(histGE )

=1

0.1 7.2%+ 10.01 8.3%1

0.1 +1

0.01

It is a weighted average of your prior (the CAPM) and your view(the historical average return).

The relative weights depend on the precision (the inverse of thevariance) of the signals.

Your confidence in the CAPM determines .

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• Forming posterior beliefs - Example

Your posterior belief as a function of

0 0.5 1 1.5 2 2.5 30.072

0.074

0.076

0.078

0.08

0.082

0.084

GE

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• Black-Litterman model - Global Asset Allocation

5 4 3 2 1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

prior (CAPM)best guessview

bar

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• Black-Litterman model - Global Asset Allocation

Finally, the optimal portfolio weights will be:

w = wMKT +P

where P are the investors view portfolios, and is acomplicated set of weights. These weights have the followingproperties:

1. The higher the expected return on a view, q, the higher the weightattached to that view, .

2. The higher the variance of a view, , the lower the absolute valueof the weight attached to the view.

If you hold no view on an asset, then your optimal allocation willbe the market weight.

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• Black-Litterman model - Global Asset Allocation

Suppose that we believe that Germany will outperform Franceand UK.

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• Black-Litterman model - Global Asset Allocation

The optimal allocations according to Markowitz is:

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• Black-Litterman model - Global Asset Allocation

Our view is that Germany will outperform France and UK.

However, our best guess, , will be shifted for all countries.

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• Black-Litterman model - Global Asset Allocation

The Black-Litterman model internalizes the fact that assets arecorrelated. Your views about Germany vs France and UKtranslate into implicit views about other countries.

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• Black-Litterman model - Global Asset Allocation

Lets add a view: Canada will outperform US by 3%:

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• Black-Litterman model - Global Asset Allocation

weights depend

How bullish is the view, i.e. (Q)

How confident is the investor in the view ().

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• Black-Litterman model vs Markowitz

Black Litterman1. The optimal portfolio equals the (CAPM) market portfolio plus a

weighted sum of the portfolios about which the investor has views.

2. An unconstrained investor will invest first in the market portfolio,then in the portfolios about which views are expressed.

3. The investor will never deviate from the market weights on assetsabout which no views are held.

4. This means that an investor does not need to hold views abouteach and every asset!

Markowitz1. Need to estimate vector of expected returns for all assets.

2. Estimation error often leads to unrealistic portfolio positions

3. If we change the expected return for one asset, this will changethe weights on all assets.

FIN460-Papanikolaou CAPM in practice 58/ 59

• Summary

In order to use the CAPM in our mean-variance analysis, weneed estimates of . Regression analysis yields that.

The CAPM assumes that all investors hold the same views so allof them hold the market-portfolio.

The Black-Litterman model is one way to bring the informationfrom the CAPM in our asset allocation decision.

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