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Asset expected return

Fundamentals of FinanceAsset expected return

Jukka Perttunen

University of Oulu - Department of Finance

Fall 2017

Jukka Perttunen Fundamentals of Finance


Expected value – the discrete case

The expected value of a random variable x is the sum of the probability-weighted values of x :

E(x) =n∑

i=1

xi p(xi ).

In the case of the discrete variable:

i xi p(xi ) xi p(xi )

1 10 0.10 1.00

2 20 0.25 5.00

3 30 0.35 10.50

4 40 0.20 8.00

5 50 0.10 5.00

1.00 29.50

E(x) =5∑

i=1

xi p(xi ) = 29.50



Variance – the discrete case

The variance of a random variable x is the expected value of the squared deviations of x from it’s expected value:

Var(x) = E[[

x − E(x)]2]

=n∑

i=1

[xi − E(x)

]2p(xi ).


i xi p(xi ) xi p(xi ) xi−29.50 (xi−29.50)2 (xi−29.50)2p(xi )

1 10 0.10 1.00 −19.50 380.25 38.0250

2 20 0.25 5.00 −9.50 90.25 22.5625

3 30 0.35 10.50 0.50 0.25 0.0875

4 40 0.20 8.00 10.50 110.25 22.0500

5 50 0.10 5.00 20.50 420.25 42.0250

1.00 29.50 124.7500

Var(x) =5∑

i=1

[x − E(x)

]2p(xi ) = 124.75

Std(x) =√

Var(x) =√

124.75 ≈ 11.17



Covariance – the discrete case

The covariance of two random variables, x and y , is the expected value of the product of their deviations from

their expected values:

Cov(x) = E[[

x − E(x)][y − E(y)

]]=

n∑i=1

[xi − E(x)

][yi − E(y)

]p(i).


i xi yi p(i) xi p(i) yi p(i) xi−29.50 yi−60.40 (xi−29.50)(yi−60.40) (xi−29.50)(yi−60.40)p(i)

1 10 44 0.10 1.00 4.40 −19.50 −16.40 319.80 31.98

2 20 52 0.25 5.00 13.00 −9.50 −8.40 79.80 19.95

3 30 60 0.35 10.50 21.00 0.50 −0.40 −0.20 −0.07

4 40 70 0.20 8.00 14.00 10.50 9.60 100.80 20.16

5 50 80 0.10 5.00 8.00 20.50 19.60 401.80 40.18

1.00 29.50 60.40 112.20

Cov(x, y) =5∑

i=1

[x − E(x)

][y − E(y)

]p(i) = 112.20



Expected value, variance and covariance

The expected value of a random variable x is the sum of the probability-weighted values of x :

E(x) =

∫ ∞−∞

x p(x) dx.

The variance of a random variable x is the expected value of the squared deviations of x from it’s expected value:

Var(x) = E[[

x − E(x)]2]

=

∫ ∞−∞

[x − E(x)

]2 p(x)dx.

The standard deviation of a random variable x is the square root of it’s variance:

Std(x) =

√E[[

x − E(x)]2].

The covariance of two random variables x and y is the expected value of the product of their deviations from

their expected values:

Cov(x) = E[[

x − E(x)][y − E(y)

]]=

∫ ∞−∞

∫ ∞−∞

[x − E(x)

][y − E(y)

]p(x, y)dxdy.



Portfolio of assets

In the Modern Portfolio Theory the characteristics of a portfolio of assets are analyzed in terms of the expected

return and the return variance (or standard deviation) of the portfolio.

The expected return µp of a portfolio is the weighted average of the expected returns µi of the individual assets:

µp =n∑

i=1

wiµi , wheren∑

i=1

wi = 1.

The return variance σ2p of a portfolio is determined by the variances σ2

i = σii of the individual assets,

and the return covariances σij , (i 6= j), of all the pairs of the assets in the portfolio:

σ2p =

n∑i=1

n∑j=1

wiwjσij .

Correspondingly, the return standard deviation σp of a portfolio appears as:

σp =

√√√√ n∑i=1

n∑j=1

wiwjσij .

The annual return standard deviation is called portfolio volatility.



Risky assets

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

A sample of risky assets

An asset with a higher expected return may has

a lower risk in terms of volatility, and vice versa.



Portfolios of risky assets

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

µp =n∑

i=1

wiµi

σp =

√√√√ n∑i=1

n∑j=1

wiwjσij

n∑i=1

wi = 1

wi ≥ 0

Shaded area represents all possible portfolios of

the risky assets, with all feasible combinations

of weights wi .

Diversification enables us to reach a lower level

of risk at the same level of the expected return.



Frontier of portfolios

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

We are able to identify the frontier of portfolios by minimizing the portfolio return variance

at a number of different levels of the expected return µ∗:

min e =n∑

i=1

n∑j=1

wiwjσij , s.t.n∑

i=1

wiµi = µ∗,

n∑i=1

wi = 1, wi ≥ 0.



Minimum variance portfolio

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

We are able to identify the minimum variance portfolio by minimizing the portfolio return variance

without the constraint of the required expected return:

min e =n∑

i=1

n∑j=1

wiwjσij ,n∑

i=1

wi = 1, wi ≥ 0.

Minumum variance portfolio o



Efficient portfolios

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

o

The efficient frontier represents the portfolios with the highest

expected return at a given level of volatility.

Minumum variance portfolio



Portfolio of the risk-free asset and a risky asset

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

o

Risk-free asset

r = 3%

po

An arbitrary risky asset i

µi = 12%, σi = 28%

A portfolio can be created as a combination of the risk-free asset

and an individual risky asset, or a portfolio of risky assets.

µp = wr r + wiµi

= 0.6× 3% + 0.4× 12%

= 6.6%

σp = wiσi

= 0.4× 28% = 11.2%



Tangent portfolio

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

o

Risk-free asset

r = 3%

oTangent portfolio t

The portfolios with the highest possible expected returns lie on the tangent line, which

consists of the combinations of the risk-free asset and the tangent portfolio t.

The tangent line now represents the set of efficient portfolios.



Tangent portfolio

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

o

Risk-free asset

r = 3%

oTangent portfolio t

We are able to identify the tangent portfolio by maximizing the slope coefficient of

a line, which goes through the risk-free asset and a portfolio of risky assets:

Slope coefficient:

µt − r

σt

σt

µt − r

max z =

n∑i=1

wiµi − r√√√√ n∑i=1

n∑j=1

wiwjσij

, s.t.n∑

i=1

wi = 1, wi ≥ 0.

µt = 15%, σt = 23%



Tangent portfolio and the risk-free asset

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

o

Risk-free asset

r = 3%

µp = wr r + wtµt

= 0.6× 3% + 0.4× 15%

= 7.8%

po

σp = wtσt

= 0.4× 23% = 9.2%

o

Tangent portfolio t

µt = 15%

σt = 23%

A portfolio may be created as a combination of the risk-free

asset and the tangent portfolio.



Tangent portfolio and the risk-free asset

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

o

Risk-free asset

r = 3%

o

Tangent portfolio t

µt = 15%

σt = 23%

µp = wr r + wtµt

= −1.0× 3% + 2.0× 15%

= 27.0%

po

σp = wtσt

= 2.0× 23%

= 46.0%

A portfolio may be leveraged by borrowing at the risk-free rate,

and investing both the initial capital and the borrowed capital

in the tangent portfolio.



Effect of diversification

In order to evaluate the effect of diversification on the portfolio return variance, we write the variance formula

in the matrix-form:

σ2p =

n∑i=1

n∑j=1

wiwjσij

= µ′Σµ

= [ w1 w2 w3 . . . wn ]

σ21 σ12 σ13 . . . σ1n

σ21 σ22 σ23 . . . σ2n

σ31 σ32 σ33 . . . σ2n

. . . . . . .

. . . . . . .

. . . . . . .

σn1 . . . . . σ2n

w1

w2

w3

.

.

.

wn

.︸︷︷︸

The row vector µ′ of asset weights.

The column vector µ of asset weights.︷︸︸︷

︸︷︷︸The variance/covariance matrix Σ of asset returns.

The variance/covariance matrix contains the variability of the assets in the portfolio.

There are n variance terms in the diagonal of the matrix, whereas the other n2 − n terms represent covariances.




In the case of two assets, the return variance appears as

σ2p = [ w1 w2 ]

σ21 σ12

σ21 σ22

w1

w2

= [ w1σ21 + w2σ21 w1σ12 + w2σ

22 ]

w1

w2

= w1w1σ21 + w1w2σ21 + w1w2σ12 + w2w2σ

22

= w21σ

21 + w2

2σ22 + 2w1w2σ21.

In the case of two assets, there are two variance terms and two covariance terms driving the return variance of the

portfolio.




In the case of three assets, the return variance appears as

σ2p = [ w1 w2 w3 ]

σ2

1 σ12 σ13

σ21 σ22 σ23

σ31 σ32 σ23

w1

w2

w3

.

Now there are three variance terms and six covariance terms driving the return variance of the portfolio.

In general, in the return variance formula of an n-asset portfolio, there are n variance terms, representing the risk in

the individual assets’ returns, and n2 − 1 covariance terms, representing the correlation between the asset returns.

As the number of assets in the portfolio grows, the proportion of variance terms gets lower, and the proportion of

covariance terms gets higher:

Number of assets Variance terms Covariance terms

2 2 (50%) 2 (50%)

3 3 (33%) 6 (67%)

4 4 (25%) 12 (75%)

10 10 (10%) 90 (90%)



Risk-contribution of an asset

We may rewrite the portfolio return variance formula in terms of the sum of the risk-contributions of the individual

assets in the riskiness of the portfolio:

σ2p =

n∑i=1

n∑j=1

wiwjσij

= w1

n∑j=1

wjσ1j + w2

n∑j=1

wjσ2j + . . . + wi

n∑j=1

wjσij + . . . + wn

n∑j=1

wjσnj .

Risk-contribution of an asset i︷︸︸︷

With an alternative notation, we may rewrite the risk-contribution of an individual asset i as

wi

n∑j=1

wjσij = wi

[w1Cov(Ri , R1) + w2Cov(Ri , R2) + . . . + wnCov(Ri , Rn)

]

= wi

[Cov(Ri ,w1R1) + Cov(Ri ,w2R2) + . . . + Cov(Ri ,wnRn)

]

= wi

[Cov(Ri ,w1R1 + w2R2 + · · · + wnRn)

]

= wi

[Cov(Ri , Rp)

]

= wiσip .



Risk-contribution of an asset in the market portfolio

In the Capital Asset Pricing Model we consider the risk-contribution of an individual asset from the point of view

of the market portfolio m, which contains all risky assets i with their relative market weights wi .

The expected return µm and the return variance σ2m of the market portfolio appear as

µm =n∑

i=1

wiµi , σ2m =

n∑i=1

n∑j=1

wiwjσij .

The return-contribution of an individual asset i in the market portfolio is

wiµi .

The risk-contribution of an individual asset i in the market portfolio is

wi

n∑j=1

wjσij = wiσim.

In the model, the return contribution wiµi of an individual asset is assumed to be reward to the risk-contribution

wiσim of the asset.

Such being the case, we should be able to express the expected return µi of an individual asset as a function of

the risk-contribution σim of the asset:

µi = f (σim).



Model of expected return

In any model of expected return, we want to build on the risk-free rate r , and add an asset-specific risk-premium pi :

µi = r + pi .

Furthermore, we want to express the risk-premium pi in terms of a product of the unit-price um of the market-related

risk, and an asset-specific amount xi of the market-related risk:

µi = r + umxi .

The amount of the market-related risk of the market portfolio m itself may be defined to be equal to one, and thus

the expected return of the market portfolio appears as

µm = r + um.

From the above result, we are able to solve the unit-price of the market-related risk:

um = µm − r.

The model of the expected return appears now as

µi = r + (µm − r)xi .

To complete the model, we want to replace the variable xi of the amount of the market-related risk with the risk-

contribution σim of the asset.



Capital Asset Pricing Model

When it comes to the market portfolio itself, the amount of the market-related risk was defined to be equal to one:

xm = 1.

To have the same identity to hold true in the case of the the risk-contribution σmm = σ2m of the market portfolio,

we need to divide it by the return variance σ2m :

σmm

σ2m

= 1.

With this adjusted measurement of risk-contribution, the expected return of an asset i appears now as

µi = r +(µm − r

)βi , where βi =

σim

σ2m

.

It can be shown, that for any portfolio of assets, the beta is the weighted average of the individual assets’ betas:

βp =n∑

i=1

wiβi .



Capital Market Line

The set of all risky assets/portfolios

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

Risk-free asset

r = 3%

On the market level, all investors’ risky holdings, together, form the market portfolio m.

oMarket portfolio m

The market portfolio is believed to be an efficient portfolio.

Individual investors, on average, create their portfolios p ascombinations of the market portfolio and the risk-free asset.

po

Efficient portfolios lie on the Capital Market Line.

CML



Capital Market Line


-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

Risk-free asset

r = 3%

oMarket portfolio m

po

CML

The reward-to-risk ratio of an efficient portfolio p is always

µp − r

σp.

σp

µp − r



Capital Market Line


-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

Risk-free asset

r = 3%

oMarket portfolio m

po

CML

The reward-to-risk ratio of an efficient portfolio p is always

µp − r

σp.

The reward-to-risk ratio of the market portfolio m is

µm − r

σm.

σm

µm − r



Capital Market Line


-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

Risk-free asset

r = 3%

oMarket portfolio m

po

CML

For any efficient portfolio p, the reward-to-risk ratio is always

µp − r

σp=µm − r

σm.

σm

µm − r



Capital market line and individual risky assets

-

6

10% 20% 30% 40% 50% σ

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

a

d

b

f

c

e

h

g

i

j

o

Risk-free asset

r

oMarket portfolio m

CML

Individual risky assets are located below the Capital Market Line.

An asset with a higher expected return may has a lower riskin terms of volatility, and vice versa.



Security market line

-

6

0.5 1.0 1.5 2.0 β

10%

20%

30%

µ

o

o

o

o

o

o

o

o

o

o

a

d

b

f

c

e

h

g

i

j

o

Risk-free asset

r

oMarket portfolio m

SML

Individual risky assets are located at the Security Market Line.

An asset with a higher expected return always has a higher riskin terms of beta, and vice versa.



Capital Asset Pricing Model – an alternative approach

It is assumed that in the market equilibrium, each asset i is hold in the market portfolio m, with the weight of wi ,

and the excess demand or supply ewi of any asset i is equal to zero.

Let us, however, consider a portfolio p, where the weight of the asset i is increased by ewi , and the weight of the

market portfolio m is decreased by the same amount.

The expected return and the return variance of this excess demand portfolio p are

µp = ewiµi + (1− ewi )µm, σp = ew2i σ

2i + (1− ewi )

2σ

2m + ew2

i (1− ewi )σim.

The sensitivities of the expected return and the return variance to the changes in excess demand are

dµp

dewi

= µi − µm,dσp

dewi

=ewiσ

2i + σim − 2 ewiσim − σ2

m + ewiσ2m√

ew2i σ

2i + 2ewi (1− ewi )σim + (1− ewi )

2σ2m

.

The ratio of the results above provides the reward-to-risk ratio of the excess demand portfolio:

dµp

dσp=

(µi − µm)√

ew2i σ

2i + 2ewi (1− ewi )σim + (1− ewi )

2σ2m

ewiσ2i + σim − 2 ewiσim − σ2

m + ewiσ2m

.

In the market equilibrium, however, the excess demand ewi is always equal to zero, and for all assets i , we have

dµp

dσp

∣∣∣∣ewi = 0

=(µi − µm)σm

σim − σ2m



Capital Asset Pricing Model – an alternative approach

The Capital Market Line showed us that for an efficient portfolio the reward-to-risk ratio is always

µp − r

σp=µm − r

σm⇒ µp =

µm − r

σmσp − r.

From the above equation we find that

dµp

dσp=µm − r

σm.

The analysis of the excess demand in the market equilibrium showed that

dµp

dσp=

(µi − µm)σm

σim − σ2m

.

The combining of the reward-to-risk ratios of an efficient portfolio and the excess demand portfolio results:

(µi − µm)σm

σim − σ2m

=µm − r

σm

⇒ (µi − µm)σ2m = (µm − r)(σim − σ

2m)

⇒ µi = r +(µm − r

)σimσ2m

.



Variance and covariance estimates with daily data

Historical n-day time-series of logarithmic asset returns: Rit = ln Sit − ln Si,t−1-

Ri Ri Ri . . .

︸︷︷︸Annualized return variance estimate:

σ̂2i = 250×

1

n − 1

n∑t=1

(Rit − R̄i )2 R̄i =

1

n

n∑t=1

Rit

We assume daily market data with 250 observations per year, on average.

Term volatility refers to the annualized return standard deviation.

Historical n-day time-series of logarithmic returns of asset i : Rit = ln Sit − ln Si,t−1-

RiRiRi. . .

Historical n-day time-series of logarithmic returns of asset j : Rjt = ln Sjt − ln Sj,t−1-R

jRjRj. . .

︸︷︷︸Annualized return covariance estimate:

σ̂ij

= 250×1

n − 1

n∑t=1

(Rit − R̄i )(Rjt − R̄j )



Variance and covariance estimates with monthly data

Historical n-month time-series of logarithmic asset returns: Rit = ln Sit − ln Si,t−1-

Ri Ri Ri . . .

︸︷︷︸Annualized return variance estimate:

σ̂2i = 12×

1

n − 1

n∑t=1

(Rit − R̄i )2 R̄i =

1

n

n∑t=1

Rit

We assume monthly market data with 12 observations per year.

Term volatility refers to the annualized return standard deviation.

Historical n-month time-series of logarithmic returns of asset i : Rit = ln Sit − ln Si,t−1-

RiRiRi. . .

Historical n-month time-series of logarithmic returns of asset j : Rjt = ln Sjt − ln Sj,t−1-R

jRjRj. . .

︸︷︷︸Annualized return covariance estimate:

σ̂ij

= 12×1

n − 1

n∑t=1

(Rit − R̄i )(Rjt − R̄j )



Beta estimate with daily/monthly total returns

Historical n-day/month time-series of logarithmic asset returns: Rit = ln Sit − ln Si,t−1-

RiRiRi. . .

Historical n-day/month time-series of logarithmic market returns: Rmt = ln Smt − ln Sm,t−1-Rm Rm Rm. . .

︸︷︷︸Estimation of the market model:

Rit = α̂i + β̂iRmt + εit

The market model regresses the asset returns against the market returns.

The estimate of the regression coefficient corresponds to the beta of the Capital Asset Pricing Model:

β̂i =σ̂im

σ̂2m

.

The frequency of the data does not systematically affect to the value of the beta estimate.

Daily data provides more observations with a higher measurement error in asset returns.

Monthly data provides less observations with a lower measurement error in asset returns.



Market model of Olvi Plc

-

6

−20% 0% 20% Rm

−20%

0%

20%

Ri

α = 0.01203

(0, 0)

β =∆Ri

∆Rm= 0.86957

Ri = 0.01203 + 0.86957Rm

Olvi Plc

Monthly data 2001–2010

∆Ri

∆Rm

..

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Market model of Konecranes Plc

-

6

−20% 0% 20% Rm

−20%

0%

20%

Ri

}α = 0.00463

(0, 0)

β =∆Ri

∆Rm= 1.23477

Ri = 0.00463 + 1.23477Rm

Konecranes Plc

Monthly data 2001–2010

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∆Rm

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Beta estimate with monthly excess returns

Historical n-month time-series of logarithmic asset excess returns: eRit = Rit − Rft-

eRieR

ieR

i. . .

Historical n-month time-series of logarithmic market excess returns: eRmt = Rmt − Rft-

eRmeRm

eRm. . .

︸︷︷︸Estimation of the market model:

Rit − Rft = α̂i + β̂i (Rmt − Rft ) + εit

The market model regresses the asset excess returns against the market excess returns.

The estimate of the regression coefficient corresponds to the beta of the Capital Asset Pricing Model:

β̂i =eσ̂imeσ̂2

m

.

The Capital Asset Pricing Model is a single-period model, where the risk-free rate remains constant.

The alpha of the market model may be interpreted as the average abnormal return over the estimation period.

The version of the model requires the availability of the risk-free return.



Applying the Capital Asset Pricing Model

µi = r +(µm − r

)βi

Historical n-day/month time-series of logarithmic asset returns: Rit = ln Sit − ln Si,t−1-

RiRiRi. . .

Historical n-day/month time-series of logarithmic market returns: Rmt = ln Smt − ln Sm,t−1-Rm Rm Rm. . .

︸︷︷︸Rit = α̂i + β̂iRmt + εitThe market model provides the

estimate of the beta.

?The current long-term government

bond yield serves as an estimate of

the future-period risk-free rate.

?

Every year, over a long period (say 30 years) of yearly historical data, calculate:

– the yearly logarithmic return, Rm , of the stock market,

– the yearly logarithmic return, Rf , of a portfolio of government-issued risk-free bonds,

– the yearly excess return, Re = Rm − Rf , of the stock market over the return of the risk-free bond portfolio.

The time-series average, R̄e , of the excess returns serves as an estimate of the expected price of the market risk in

terms of an annual percentage rate.

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fundamentals of finance - university of oulucc.oulu.fi/~jope/fof/c09/printable/asset expected...

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