lecture 11 - capm and factor modelsarmerin/fininsmathrwanda/lecture...capital asset pricing modelor...

Lecture 11CAPM and Factor models

Lecture 11 1 / 28

Introduction

So far we have only considered the portfolio optimization problem inisolation.

Now we are going to put the Markowitz model in an economy withindividuals.

In this market all individulas have the same belifs about the mean returnvector and covariance matrix of the rates of return.

They also want to maximise the expected return and minimise the variance(or standard deviation).

Lecture 11 2 / 28

Introduction

Lecture 11 2 / 28

Introduction

Lecture 11 2 / 28

Introduction

Lecture 11 2 / 28

The market portfolio (1)

The important quantity when connecting the Markowitz model with thereal-world economy is the market portfolio.

The market portfolio is the value weighted portfolio of all assets in theworld.

That is, we calculate each asset’s capitalisation and from this the weights.

Lecture 11 3 / 28

Example

In the world there are the following three assets:

Company No of shares Price per share CapitalisationA 10 000 4.00 40 000

B 20 000 5.00 100 000C 30 000 2.00 60 000

Total: 60 000 200 000

Hence, the market portfolio wM has weights

(40 000

200 000,

100 000

200 000,

60 000

200 000

)= (0.20, 0.50, 0.30).

Lecture 11 4 / 28

Example

Company No of shares Price per share CapitalisationA 10 000 4.00 40 000B 20 000 5.00 100 000

C 30 000 2.00 60 000

Total: 60 000 200 000

(40 000

200 000,

100 000

200 000,

60 000

200 000

)= (0.20, 0.50, 0.30).

Lecture 11 4 / 28

Example

Company No of shares Price per share CapitalisationA 10 000 4.00 40 000B 20 000 5.00 100 000C 30 000 2.00 60 000

Total: 60 000 200 000

(40 000

200 000,

100 000

200 000,

60 000

200 000

)= (0.20, 0.50, 0.30).

Lecture 11 4 / 28

Example

Total: 60 000 200 000

(40 000

200 000,

100 000

200 000,

60 000

200 000

)= (0.20, 0.50, 0.30).

Lecture 11 4 / 28

Example

Total: 60 000 200 000

(40 000

200 000,

100 000

200 000,

60 000

200 000

= (0.20, 0.50, 0.30).

Lecture 11 4 / 28

Example

Total: 60 000 200 000

(40 000

200 000,

100 000

200 000,

60 000

200 000

)= (0.20, 0.50, 0.30).

Lecture 11 4 / 28

The market portfolio is something we observe in the market.

It should consist of all traded assets in the world.

This is impossible in practice, so we usually approximate it using a subsetof traded assets.

Which assets to use depends on the modelling situation. If we model astock market, then we can take a broad stock index as the marketportfolio.

Lecture 11 5 / 28

Which assets to use depends on the modelling situation.

If we model astock market, then we can take a broad stock index as the marketportfolio.

Lecture 11 5 / 28

We assume from now on that the market model contains a risk-free asset.

The important connection between the Markowitz model and a financialmarket in equilibrium is that the tangent portfolio in the Markowitz modelis equal to the market portfolio.

This is true since the portfolio of only risky assets that investors hold inequilibrium is the market portfolio.

Lecture 11 6 / 28

The straight line representing the efficient frontier when we introduce themarket portfolio is called the Capital Market Line (CML).

The CML equation is

r̄ = rf +r̄M − rfσM

The quantityr̄M − rfσM

is sometimes referred to as the market price of risk.

Lecture 11 7 / 28

The CML equation is

Lecture 11 7 / 28

The CML equation is

Lecture 11 7 / 28

CAPM (1)

Even though we, in theory, only want to hold the market portfolio, therestill exists other financial assets that are parts of the market portfolio.

It turns out that the expected mean rate of return of any asset has asimple relation to the market portfolio. This relation is known as theCapital Asset Pricing Model or simply CAPM.

Lecture 11 8 / 28

CAPM (1)

Even though we, in theory, only want to hold the market portfolio, therestill exists other financial assets that are parts of the market portfolio.

It turns out that the expected mean rate of return of any asset has asimple relation to the market portfolio. This relation is known as theCapital Asset Pricing Model or simply CAPM.

Lecture 11 8 / 28

CAPM (2)

To derive the CAPM relation we take any asset’s rate of return ri and forma portfolio using it and the market portfolio with return rM .

If we let α denote the portfolio weight in asset i , then the expected returnof this portfolio is

r̄α = αr̄i + (1 − α)r̄M ,

and the standard deviation

σα =√α2σ2

i + 2α(1 − α)σiM + (1 − α)2σ2M .

Lecture 11 9 / 28

CAPM (2)

r̄α = αr̄i + (1 − α)r̄M

σα =√α2σ2

i + 2α(1 − α)σiM + (1 − α)2σ2M .

Lecture 11 9 / 28

CAPM (2)

r̄α = αr̄i + (1 − α)r̄M ,

σα =√α2σ2

i + 2α(1 − α)σiM + (1 − α)2σ2M .

Lecture 11 9 / 28

CAPM (3)

The important observation here is that we must have

dr̄αdσα

∣∣∣∣α=0

=r̄M − rfσM

To calculate the left-hand side we use the fact that

dr̄αdσα

=dr̄αdαdσαdα

Nowdr̄αdα

= r̄i − r̄M

anddσαdα

=ασ2

i + (1 − 2α)σiM + (α− 1)σ2M

Lecture 11 10 / 28

CAPM (3)

dr̄αdσα

∣∣∣∣α=0

=r̄M − rfσM

dr̄αdσα

=dr̄αdαdσαdα

Nowdr̄αdα

= r̄i − r̄M

anddσαdα

=ασ2

i + (1 − 2α)σiM + (α− 1)σ2M

Lecture 11 10 / 28

CAPM (3)

dr̄αdσα

∣∣∣∣α=0

=r̄M − rfσM

dr̄αdσα

=dr̄αdαdσαdα

Nowdr̄αdα

= r̄i − r̄M

anddσαdα

=ασ2

i + (1 − 2α)σiM + (α− 1)σ2M

Lecture 11 10 / 28

CAPM (3)

dr̄αdσα

∣∣∣∣α=0

=r̄M − rfσM

dr̄αdσα

=dr̄αdαdσαdα

Nowdr̄αdα

= r̄i − r̄M

anddσαdα

=ασ2

i + (1 − 2α)σiM + (α− 1)σ2M

Lecture 11 10 / 28

CAPM (4)

It follows that

dr̄αdσα

= σαr̄i − r̄M

ασ2i + (1 − 2α)σiM + (α− 1)σ2

and from thisr̄M − rfσM

=dr̄αdσα

∣∣∣∣α=0

= σMr̄i − r̄MσiM − σ2

This is the CAPM relation.

Lecture 11 11 / 28

CAPM (4)

It follows that

dr̄αdσα

= σαr̄i − r̄M

ασ2i + (1 − 2α)σiM + (α− 1)σ2

=dr̄αdσα

∣∣∣∣α=0

Lecture 11 11 / 28

CAPM (4)

It follows that

dr̄αdσα

= σαr̄i − r̄M

ασ2i + (1 − 2α)σiM + (α− 1)σ2

=dr̄αdσα

∣∣∣∣α=0

Lecture 11 11 / 28

CAPM (5)

Usually one rewrites the CAPM equation, and formulates it as follows:

Theorem

(CAPM) If the market portfolio is efficient, then the expected return r̄i ofany asset i satisfies

r̄i = rf + βi (r̄M − rf ),

whereβi =

σiMσ2M

The parameter βi is referred to an assets beta-value.

Lecture 11 12 / 28

CAPM (5)

Theorem

r̄i = rf + βi (r̄M − rf ),

whereβi =

σiMσ2M

Lecture 11 12 / 28

CAPM (5)

Theorem

r̄i = rf + βi (r̄M − rf ),

whereβi =

σiMσ2M

Lecture 11 12 / 28

CAPM (6)

In the expressionr̄i = rf + βi (r̄M − rf )

rf is time compensation (the time value of money again).

βi (r̄M − rf ) is risk compensation.

For any rate of return r the expression

r̄ − rf

is called the risk premium. Hence CAPM can be rephrased

Risk premium of asset i = βi · The market’s risk premium.

Lecture 11 13 / 28

CAPM (6)

r̄ − rf

Lecture 11 13 / 28

CAPM (6)

r̄ − rf

Lecture 11 13 / 28

CAPM (6)

r̄ − rf

is called the risk premium.

Hence CAPM can be rephrased

Lecture 11 13 / 28

CAPM (6)

r̄ − rf

Lecture 11 13 / 28

CAPM (7)

If we have a portfolio with weights wi ,

r =n∑

wi ri ,

then the beta βP of the portfolio is given by

βP =Cov(r , rM)

=Cov (

∑ni=1 wi ri , rM)

wiCov(ri , rM)

wiβi .

Lecture 11 14 / 28

CAPM (7)

r =n∑

wi ri ,

βP =Cov(r , rM)

=Cov (

∑ni=1 wi ri , rM)

wiCov(ri , rM)

wiβi .

Lecture 11 14 / 28

CAPM (7)

r =n∑

wi ri ,

βP =Cov(r , rM)

=Cov (

∑ni=1 wi ri , rM)

wiCov(ri , rM)

wiβi .

Lecture 11 14 / 28

CAPM (7)

r =n∑

wi ri ,

βP =Cov(r , rM)

=Cov (

∑ni=1 wi ri , rM)

wiCov(ri , rM)

wiβi .

Lecture 11 14 / 28

CAPM (7)

r =n∑

wi ri ,

βP =Cov(r , rM)

=Cov (

∑ni=1 wi ri , rM)

wiCov(ri , rM)

wiβi .

Lecture 11 14 / 28

CAPM (8)

When the expected rate of return of assets are plotted against thebeta-value (i.e. we have r̄ on the y -axis and β on the x-axis) we get whatis known as the Security Market Line (SML).

Using real-life data we often get results that does not perfectly fit thetheory. In order to determine these deviations two methods are

For SML: Jensen’s index (or alpha)

J = ˆ̄r − rf − β̂(ˆ̄rM − rf )

For CML: the Sharpe ratio:

S =ˆ̄r − rfσ̂

Here a ‘hat’ˆmeans an estimated value.

Lecture 11 15 / 28

CAPM (8)

Using real-life data we often get results that does not perfectly fit thetheory.

In order to determine these deviations two methods are

J = ˆ̄r − rf − β̂(ˆ̄rM − rf )

S =ˆ̄r − rfσ̂

Lecture 11 15 / 28

CAPM (8)

J = ˆ̄r − rf − β̂(ˆ̄rM − rf )

S =ˆ̄r − rfσ̂

Lecture 11 15 / 28

CAPM (8)

J = ˆ̄r − rf − β̂(ˆ̄rM − rf )

S =ˆ̄r − rfσ̂

Here a ‘hat’ˆmeans an estimated value.Lecture 11 15 / 28

CAPM (9)

If we letεi = ri − rf − βi (rM − rf )

then we can writeri = rf + βi (rM − rf ) + εi .

It follows from CAPM that E (εi ) = 0. Furthermore εi and rM areuncorrelated:

Cov(εi , rM) = Cov(ri − rf − βi (rM − rf ), rM)

= Cov(ri , rM) − βiCov(rM , rM)

= σiM − σiMσ2M

Lecture 11 16 / 28

CAPM (9)

If we letεi = ri − rf − βi (rM − rf ),

= σiM − σiMσ2M

Lecture 11 16 / 28

CAPM (9)

It follows from CAPM that E (εi ) = 0.

Furthermore εi and rM areuncorrelated:

= σiM − σiMσ2M

Lecture 11 16 / 28

CAPM (9)

= σiM − σiMσ2M

Lecture 11 16 / 28

CAPM (9)

= σiM − σiMσ2M

Lecture 11 16 / 28

CAPM (9)

= σiM − σiMσ2M

Lecture 11 16 / 28

CAPM (9)

= σiM − σiMσ2M

Lecture 11 16 / 28

CAPM (10)

Let us return tori = rf + βi (rM − rf ) + εi .

We haveσ2i = β2

i σ2M + σ2

since rM and εi are uncorrelated.

β2i σ

2M is called the systematic risk, and

σ2εi

the nonsystematic, idiosyncratic or specific risk.

Lecture 11 17 / 28

CAPM (10)

We haveσ2i = β2

i σ2M + σ2

β2i σ

2M is called the systematic risk

σ2εi

Lecture 11 17 / 28

CAPM (10)

We haveσ2i = β2

i σ2M + σ2

β2i σ

2M is called the systematic risk, and

σ2εi

Lecture 11 17 / 28

CAPM (11)

Assume that we have an asset with price P today and which is later soldfor Q. The rate of return is

r =Q − P

Using CAPM yields

r̄ =Q̄ − P

P= rf + β(r̄M − rf ).

We can write this as

P =Q̄

1 + rf + β(r̄M − rf ).

Lecture 11 18 / 28

CAPM (11)

r =Q − P

Using CAPM yields

r̄ =Q̄ − P

P= rf + β(r̄M − rf ).

P =Q̄

1 + rf + β(r̄M − rf ).

Lecture 11 18 / 28

CAPM (11)

r =Q − P

Using CAPM yields

r̄ =Q̄ − P

P= rf + β(r̄M − rf ).

P =Q̄

1 + rf + β(r̄M − rf ).

Lecture 11 18 / 28

CAPM (12)

Since r = Q/P − 1 we have

β =Cov(Q/P − 1, rM)

=Cov(Q, rM)

Using this relation we can write the pricing equation as

1 + rf

[Q̄ − Cov(Q, rM)(r̄M − rf )

This is the certainty equivalent version of the pricing formula.

Lecture 11 19 / 28

CAPM (12)

=Cov(Q, rM)

1 + rf

Lecture 11 19 / 28

CAPM (12)

=Cov(Q, rM)

1 + rf

Lecture 11 19 / 28

Factor models (1)

In a factor model, a set of factors is used to model the random behavior ofasset returns.

A factor could be ‘anything’ from the return on specific assets or thereturn on a share index to economic variables such as inflation and GDP.

We can use factors in order to reduce the number of parameters in amodel.

Lecture 11 20 / 28

Factor models (1)

Lecture 11 20 / 28

Factor models (1)

Lecture 11 20 / 28

Factor models (2)

If we have n number of assets in a mean-variance model, then the numberof parameters is equal to

No of mean rate of returns + No of variances + No of covariances

= n + n +n(n − 1)

n2 + 3n

If n = 100, then the number of parameters is 5150, and if n = 1 000 thenumber of parameters is 501,500 – more than 500 times the number ofassets.

Lecture 11 21 / 28

Factor models (2)

If we have n number of assets in a mean-variance model, then the numberof parameters is equal to

No of mean rate of returns + No of variances + No of covariances

= n + n +n(n − 1)

n2 + 3n

If n = 100, then the number of parameters is 5150, and if n = 1 000 thenumber of parameters is 501,500 – more than 500 times the number ofassets.

Lecture 11 21 / 28

Single-factor models (1)

We start by considering single-factor models in which we use one factor.

The model isri = ai + bi f + ei , i = 1, 2, . . . , n.

f = the factor

ai = the intercept

bi = the factor loading

ei = the error

Without loss of generality we can assume that E (ei ) = 0, which we will do.

We will also assume that the error terms are uncorrelated with the factorand with each other:

Cov(f , ei ) = 0 and Cov(ei , ej) = 0.

Lecture 11 22 / 28

f = the factor

ai = the intercept

ei = the error

Lecture 11 22 / 28

f = the factor

ai = the intercept

ei = the error

Lecture 11 22 / 28

f = the factor

ai = the intercept

ei = the error

Lecture 11 22 / 28

f = the factor

ai = the intercept

ei = the error

Lecture 11 22 / 28

f = the factor

ai = the intercept

ei = the error

Lecture 11 22 / 28

f = the factor

ai = the intercept

ei = the error

Lecture 11 22 / 28

f = the factor

ai = the intercept

ei = the error

Cov(f , ei ) = 0

and Cov(ei , ej) = 0.

Lecture 11 22 / 28

f = the factor

ai = the intercept

ei = the error

Lecture 11 22 / 28

In the single factor model the factor and the error terms are random.

The intercepts and the factor loadings in the model are constants.

How should the factors be chosen?

The theory does not guide us here – we must use common sense.Thisgives us flexibility, but has the drawback that we have no rules to follow.

Lecture 11 23 / 28

The theory does not guide us here – we must use common sense.

Thisgives us flexibility, but has the drawback that we have no rules to follow.

Lecture 11 23 / 28

If we takef = rM − rf ,

the return of the market portfolio minus the risk-free rates, then we are(almost) back to CAPM.

This is called the market model.

Lecture 11 24 / 28

If we takef = rM − rf ,

the return of the market portfolio minus the risk-free rates, then we are(almost) back to CAPM.

This is called the market model.

Lecture 11 24 / 28

Recall thatri = ai + bi f + ei .

In this model we have

r̄i = ai + bi f̄

σ2i = b2

i σ2f + 2bi Cov(f , ei )︸︷︷︸

+σ2ei

= b21σ

2f + σ2

σij = Cov(ri , rj) = Cov(ai + bi f + ei , aj + bj f + ej)

= bibjCov(f , f ) + bi Cov(f , ej)︸︷︷︸=0

+bj Cov(ei , f )︸︷︷︸=0

+ Cov(ei , ej)︸︷︷︸=0

= bibjσ2f

Hence, the means and covariances of the rates of return depend on

ai , bi , σei , f̄ and σf .

Lecture 11 25 / 28

r̄i = ai + bi f̄

σ2i = b2

i σ2f + 2bi Cov(f , ei )︸︷︷︸

+σ2ei

= b21σ

2f + σ2

+bj Cov(ei , f )︸︷︷︸=0

+ Cov(ei , ej)︸︷︷︸=0

= bibjσ2f

Lecture 11 25 / 28

r̄i = ai + bi f̄

σ2i = b2

i σ2f + 2bi Cov(f , ei )︸︷︷︸

+σ2ei

= b21σ

2f + σ2

+bj Cov(ei , f )︸︷︷︸=0

+ Cov(ei , ej)︸︷︷︸=0

= bibjσ2f

Lecture 11 25 / 28

r̄i = ai + bi f̄

σ2i = b2

i σ2f + 2bi Cov(f , ei )︸︷︷︸

+σ2ei

= b21σ

2f + σ2

σij = Cov(ri , rj)

= Cov(ai + bi f + ei , aj + bj f + ej)

+bj Cov(ei , f )︸︷︷︸=0

+ Cov(ei , ej)︸︷︷︸=0

= bibjσ2f

Lecture 11 25 / 28

r̄i = ai + bi f̄

σ2i = b2

i σ2f + 2bi Cov(f , ei )︸︷︷︸

+σ2ei

= b21σ

2f + σ2

+bj Cov(ei , f )︸︷︷︸=0

+ Cov(ei , ej)︸︷︷︸=0

= bibjσ2f

Lecture 11 25 / 28

r̄i = ai + bi f̄

σ2i = b2

i σ2f + 2bi Cov(f , ei )︸︷︷︸

+σ2ei

= b21σ

2f + σ2

+bj Cov(ei , f )︸︷︷︸=0

+ Cov(ei , ej)︸︷︷︸=0

= bibjσ2f

Lecture 11 25 / 28

r̄i = ai + bi f̄

σ2i = b2

i σ2f + 2bi Cov(f , ei )︸︷︷︸

+σ2ei

= b21σ

2f + σ2

+bj Cov(ei , f )︸︷︷︸=0

+ Cov(ei , ej)︸︷︷︸=0

= bibjσ2f

Lecture 11 25 / 28

r̄i = ai + bi f̄

σ2i = b2

i σ2f + 2bi Cov(f , ei )︸︷︷︸

+σ2ei

= b21σ

2f + σ2

+bj Cov(ei , f )︸︷︷︸=0

+ Cov(ei , ej)︸︷︷︸=0

= bibjσ2f

Lecture 11 25 / 28

If we have n assets in a one-factor model, then the number of parametersis equal to

n + n + n + 1 + 1 = 3n + 2.

If n = 100, then the number of parameters is 302, and if n = 1 000 thenthe number of parameters is 3 002.

This is a drastic reduction in the number of parameters.

Single-factor models are closely connected to simple linear regressions.

Lecture 11 26 / 28

n + n + n + 1 + 1 = 3n + 2.

Lecture 11 26 / 28

n + n + n + 1 + 1 = 3n + 2.

Lecture 11 26 / 28

n + n + n + 1 + 1 = 3n + 2.

Lecture 11 26 / 28

Multifactor models (1)

We can generalise and use m factors instead of just one.

In this case we have

ri = ai +m∑j=1

bij fj + ei .

The interpretation of ai is the same as before, but now each return has afactor loading bij with respect to each factor fj .

We again assume

E (ei ) = 0

So the expected value is

r̄i = ai +m∑j=1

bij f̄j .

Lecture 11 27 / 28

ri = ai +m∑j=1

bij fj + ei .

We again assume

E (ei ) = 0

r̄i = ai +m∑j=1

bij f̄j .

Lecture 11 27 / 28

ri = ai +m∑j=1

bij fj + ei .

We again assume

E (ei ) = 0

r̄i = ai +m∑j=1

bij f̄j .

Lecture 11 27 / 28

ri = ai +m∑j=1

bij fj + ei .

We again assume

E (ei ) = 0

r̄i = ai +m∑j=1

bij f̄j .

Lecture 11 27 / 28

ri = ai +m∑j=1

bij fj + ei .

We again assume

E (ei ) = 0

r̄i = ai +m∑j=1

bij f̄j .

Lecture 11 27 / 28

We assumeCov(fj , ei ) = 0

but allow the factors to have non-zero covariance between each other.

This model can also be written on vector form:

r = a + Bf + e.

Lecture 11 28 / 28

r = a + Bf + e.

Lecture 11 28 / 28

r = a + Bf + e.

Lecture 11 28 / 28

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