MS-E2114 Investment Science Lecture 5: Mean-variance ... · MS-E2114 Investment Science: Lecture 5,...

MS-E2114 Investment ScienceLecture 5: Mean-variance portfolio theoryA. Salo, T. SeeveSystems Analysis LaboratoryDepartment of System Analysis and MathematicsAalto University, School of Science

December 8, 2017

MS-E2114 Investment Science: Lecture 5, Mean-variance portfolio theoryDecember 8, 2017

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Overview

Random returns

Portfolio mean and variance

Markowitz model

Two-fund theorem

One-fund theorem


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This lecture

I In earlier lectures, we have considered practically certaincash flows

I Emphasis on mathematical analysis of fixed incomesecurities

I Credit risks have not been thoroughly addressedI Yet cash flows for most investments are uncertain

I Examples include stock prices, dividends, real property, ...I The period for which capital is tied can be uncertain, too

I We cover the Nobel prize winning framework of HarryMarkowitz for portfolio choice under uncertainty

I Markowitz H. (1952). Portfolio selection, Journal of Finance7, 77-91.

I Link to Markowitz on the Nobel Prize website

https://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1990/markowitz-bio.html


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Overview

Random returns


Markowitz model

Two-fund theorem

One-fund theorem


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Random returns

I Assume that you invest a fixed amount X0e now andreceive the random amount X1e one year later

I (Total) return R =X1X0

I (Rate of) return r =X1 − X0

X0I Thus R = 1 + r and X1 = (1 + r)X0

I X1 is random⇒ r is random, tooI If X1 < X0, then r will be negative


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Short sellingI Shorting (short selling) = Selling an asset that one does

not ownI To short an asset, one can borrow the asset from someone

who owns it (=has a long position) (e.g., brokerage firm)and sell it for, say, X0e

I At end of borrowing period, one has to buy the asset fromthe market for X1e to return it (plus possible dividends thestock may have paid during the period) to the original owner

I If the asset value declines to X1 < X0, the deal gives a profitX0 − X1 > 0

I If the asset value increases to X1 > X0, the cash flowX0 − X1 will be negative and the deal leads to a loss ofX1 − X0

I Because prices can increase arbitrarily, losses can becomevery large⇒ Shorting can be very risky and is thereforeprohibited by some institutions


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Short selling

I Total return for a short position: Receive −X0 and pay −X1

⇒ R = −X1−X0

=X1X0

= 1 + r

I This is same as for the long positionI Initial position X0 < 0 of asset⇒ profit rX0

I Allows one to bet on declining asset values (r < 0)I Short 100 stock and sell them for 1 000e . If price declines

by 10% and you buy the stock back for 900e , and youobtain a profit of 100e

r =−900− (−1 000)

−1 000= −0.1

rX0 = −0.1 · (−1 000) = 100


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Portfolio returnI Portfolio of n assets

I X0i = investment in the i-th asset (negative when shorting)I X0 =

∑ni=1 X0i = total investment

I Weight of the i-th asset i is wi = X0iX0 ⇒∑n

i=1 wi = 1I X1i = cash flow from investment in by the end of the periodI Return of i-th asset Ri = 1 + ri

I Portfolio return

R =∑n

i=1 X1iX0

=

∑ni=1 RiX0i

X0=

∑ni=1 RiwiX0

X0=

n∑i=1

wiRi

⇒ 1 + r =n∑

i=1

wi(1 + ri) = 1 +n∑

i=1

wi ri

⇒ r =n∑

i=1

wi ri


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Random variablesI Expected value E[x ] is the mean (average) outcome of a

random variableI For a finite number of possible realizations xi with

probabilities pi , i = 1,2, . . . ,n,

E[x ] =n∑

i=1

pixi = x̄

I Variance Var[x ] is the expected value of the squareddeviation from the mean x̄

σ2 = Var[x ] = E[(x − x̄)2]= E[x2 − 2xx̄ + x̄2]= E[x2]− 2E[x ]x̄ + x̄2

⇒ σ2 = Var[x ] = E[x2]− E[x ]2


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Random variablesI Standard deviation is the expected deviation from the

meanσ = Std[x ] =

√Var[x ]

I Covariance Cov[x1, x2] is the expected product ofdeviations from the respective means of two randomvariables x1, x2

σ12 = Cov[x1, x2] = E[(x1 − x̄1)(x2 − x̄2)]= E[x1x2 − x1x̄2 − x̄1x2 + x̄1x̄2]= E[x1x2]− E[x1]x̄2 − x̄1 E[x2] + x̄1x̄2

⇒ σ12 = Cov[x1, x2] = E[x1x2]− E[x1]E[x2]

I Covariance and variance closely related

σ21 = Var[x1] = Cov[x1, x1] = σ11


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Random variables

I (Pearson) correlation Corr[x1, x2] measures the strength ofthe linear relationship of two random variables

ρ12 = Corr[x1, x2] =σ12σ1σ2

Cov[x1, x2]√Var[x1]

√Var[x2]

I No correlation⇔ ρ12 = 0⇔ σ12 = 0I Positive correlation⇔ ρ12 > 0I Negative correlation⇔ ρ12 < 0I Perfect correlation⇔ ρ12 = ±1

I We have|ρ12| ≤ 1⇔ |σ12| ≤ σ1σ2


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Random variables

I Variance of a linear combination of two random variables

σ2a1x1+a2x2 = Var[a1x1 + a2x2]

= a21 Var[x1] + a22 Var[x2] + 2a1a2 Cov[x1, x2]

I More generally, the variance of a linear combination ofrandom variables x1, x2, . . . , xn is

σ2∑ni=1 ai xi

= Var

[n∑

i=1

aixi

]

=n∑

i=1

n∑j=1

aiaj Cov[xi , xj ] =n∑

i=1

n∑j=1

aiajσij


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Overview

Random returns


Markowitz model

Two-fund theorem

One-fund theorem


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Random returnsI Consider n assets with random returns ri , i = 1,2, . . . ,n,

such that E[ri ] = r̄iI Expected return of the portfolio

r =n∑

i=1

wi ri

⇒ E[r ] =n∑

i=1

wi E[ri ] =n∑

i=1

wi r̄i

I Portfolio variance

σ2 = Var

[n∑

i=1

wi ri

]

⇒ σ2 =n∑

i=1

n∑j=1

wiwj Cov[ri , rj ] =n∑

i=1

n∑j=1

wiwjσij


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DiversificationI Investing in several assets leads to lower variance

I Variability in the realized returns of different assets tend toaverage out

I Consider investing an equal amount in n assetsI Expected return of each asset is m and variance σ2I Uncorrelated returns (σij = 0, i 6= j)

r̄ =n∑

i=1

wi r̄i =n∑

i=1

1n

m = m

Var[r ] =n∑

i=1

n∑j=1

wiwjσij =n∑

i=1

w2i σii =n∑

i=1

1n2σ2 =

1nσ2

⇒ limn→∞

Var[r ] = limn→∞

1nσ2 = 0

I No variation, yet the expected return is the same!


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Diversification

I Uncorrelated assets are ideal for diversificationI If the returns are correlated, say, σij = 0.3σ2, i 6= j , we have

Var[r ] =n∑

i=1

n∑j=1

wiwjσij =n∑

i=1

1n2σii +

n∑i=1

n∑j=1j 6=i

1n2σij

=1nσ2 + n(n − 1) 1

n20.3σ2 =

(0.7

1n

+ 0.3)σ2

⇒ limn→∞

Var[r ] = limn→∞

(0.7

1n

+ 0.3)σ2 = 0.3σ2

I Hence, variance cannot be reduced to zero


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Mean-standard deviation diagram

I Variance (or standard deviation) of returns can be used asa measure of risk

I If two portfolios have equal expected return, then the onewith smaller variance is preferred

I Consider 3 assetsAsset i 1 2 3

ρ =

1 0.2 0.20.2 1 0.30.2 0.3 1

E[ri ] 10% 12% 14%σi 8% 10% 12%

ρ12 =σ12σ1σ2

⇒ Σ =

σ11 σ12 σ13σ21 σ22 σ23σ31 σ32 σ33

=0.64% 0.16% 0.19%0.16% 1% 0.36%

0.19% 0.36% 1.44%


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Mean-standard deviation diagramI What are the possible combinations of returns and

variances?I Pairs (σ,E[r ]) such that{

E[r ] =∑n

i=1 wi E[ri ]σ2 =

∑ni=1∑n

j=1 wiwjσij

I Portfolio A with w1 = w2 = w3 = 1/3 hasI Expected return

r̄A =3∑

i=1

13E[ri ] = 0.12

I Standard deviation

σA =

√√√√ n∑i=1

n∑j=1

132σij = 7.07%


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Mean-standard deviation diagram

Asset 1

Asset 2

Standard deviation 𝜎

Exp

ecte

d r

etu

rn 𝐸

𝑟

Portfolio A

Asset 3

I Yellow area = Set of all possible (σ,E[r ]) that can beobtained from portfolios such that wi ≥ 0,

∑ni=1 wi = 1

I Blue area = As above but with shorting allowed(wi ’s, i = 1,2,3, can be negative as well)


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Efficient frontier


Exp

ecte

d r

etu

rn 𝐸

𝑟

Efficient points

I Green curve = Minimum variance set (minimum varianceattainable for a given return)

I Green point = Minimum variance point (minimum varianceattainable using assets 1,2 and 3)

I Efficient frontier = Curve above (and including) the point


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Overview

Random returns


Markowitz model

Two-fund theorem

One-fund theorem


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Markowitz model

I Portfolios of the efficient frontier r̄ can be found by solving

minw

12

n∑i=1

n∑j=1

wiwjσij

s.t.n∑

i=1

wi r̄i = r̄

n∑i=1

wi = 1


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Markowitz modelI Set up the Lagrangian

L =12

n∑i=1

n∑j=1

wiwjσij − λ

(n∑

i=1

wi r̄i − r̄

)− µ

(n∑

i=1

wi − 1

)I Equations of the efficient set are solved by setting the

partial derivatives of L to zero

∂

∂wiL =

n∑j=1

wjσij − λr̄i − µ = 0, ∀i = 1,2, . . . ,n

∂

∂λL =

n∑i=1

wi r̄i − r̄ = 0

∂

∂µL =

n∑i=1

wi − 1 = 0


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Example: Solving minimum variance portfolioAsset i 1 2 3

ρ = Σ =

1 0 00 1 00 0 1

E[ri ] 1 2 3σi 1 1 1

L =12

n∑i=1

n∑j=1

wiwjσij − λ

(n∑

i=1

wi r̄i − r̄

)− µ

(n∑

i=1

wi − 1

)

⇒

∂

∂w1L = w1σ21 − λr̄1 − µ = w1 − λ− µ = 0

∂

∂w2L = w2σ22 − λr̄2 − µ = w2 − 2λ− µ = 0

∂

∂w3L = w3σ23 − λr̄3 − µ = w3 − 3λ− µ = 0

∂

∂λL =

n∑i=1

wi r̄i − r̄ = w1 + 2w2 + 3w3 − r̄ = 0

∂

∂µL =

n∑i=1

wi − 1 = w1 + w2 + w3 = 0

(1a)

(1b)

(1c)

(1d)

(1e)


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Example: Solving minimum variance portfolio

I Equations (1a)-(1c) yield

w1 = λ+ µ, w2 = 2λ+ µ, w3 = 3λ+ µ

I Substituting these into (1d) and (1e) yields{14λ+ 6µ = r̄6λ+ 3µ = 1

⇒

{λ = 12 r̄ − 1µ = 23 − r̄

,

and thus

w1 =43− 1

2r̄ , w2 =

13, w3 =

12

r̄ − 23


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I Substituting the optimal weights w1,w2,w3 into theobjective of minimizing the portfolio variance yields

minwσ2 = min

w

3∑i=1

w2i =12

r̄2 − 2r̄ + 73

(2)

I The expected return r̄ for which variance is minimized canbe found by differentiating (2) with respect to r̄

⇒r̄ − 2 = 0⇒ r̄ = 2

⇒σ2min =13, σmin =

1√3≈ 0.577


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Asset 3

Asset 2

Asset 1

(𝜎𝑚𝑖𝑛, ҧ𝑟)


Exp

ecte

d r

etu

rn ҧ𝑟

I Blue area = The set of all possible pairs (σ,E[r ]) that aportfolio can obtain for some w1,w2,w3 ≥ 0,

∑ni=1 wi = 1


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Overview

Random returns


Markowitz model

Two-fund theorem

One-fund theorem


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Two-fund theorem

Theorem(Two-fund theorem) There are two efficient funds (portfolios)such that any efficient portfolio can be duplicated, in terms ofmean and variance, as a combination of these two. In otherwords, all investors seeking efficient portfolios need invest incombinations of these two funds only.Proof : Let w1 and w2 be efficient portfolios with correspondingLagrange multipliers be λ1, µ1 and λ2, µ2. Form the portfoliowα = αw1 + (1− α)w2, α ∈ R.

I Weights in wα sum to 1I The return of wα is r = αr1 + (1− α)r2

I If r1 6= r2, then any r can be obtained by choosing asuitable α (this α may be negative)


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Two-fund theoremI Is wα efficient? Check optimality

∂

∂wiL =

n∑j=1

wjσij − λr̄i − µ = 0, ∀i = 1,2, . . . ,n

∂

∂λL =

n∑i=1

wi r̄i − r̄ = 0

∂

∂µL =

n∑i=1

wi − 1 = 0

I (w i , λi , µi), i = 1,2 satisfies theseI But so does the point α(w1, λ1, µ1) + (1− α)(w2, λ2, µ2)

I Can be verified by substitutionI Hence wα is optimal, which completes the proof.


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Overview

Random returns


Markowitz model

Two-fund theorem

One-fund theorem


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Risk-free asset

I Thus, two funds suffice if:1. Only expected return and variance matter to the investor2. Everyone has the same estimates about expectations and

variances3. There is a single investment period

I What if one asset is risk-free?I Return rf and variance σ2f = 0

I Assume that unlimited lending and borrowing are possibleat the risk-free rate rf


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Risk-free asset

I Let us invest the share 1− α in a portfolio of risky assets Awith expected return r̄A and variance σ2A, and the share αto the risk-free asset

I The expected return is

rα = αrf + (1− α)r̄A

I Standard deviation is

σα =√

(1− α)2σ2A = (1− α)σA


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One-fund theorem

Theorem(One-fund theorem) There is a single fund F of risky assetssuch that any efficient portfolio can be constructed as acombination of the fund F and the risk-free asset.

Proof.I Unlimited lending and borrowing at rate rf ⇒ (σα, rα) forms

a lineI (σα, rα) should be selected so that the line is as steep as

possible

⇒ maxw

tan θ =

∑ni=1 wi(r̄i − rf )√∑ni=1∑n

j=1 wiwjσij


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One-fund theorem

Exp

ecte

d r

etu

rn ҧ𝑟

𝜃

𝑟𝑓

F


I Green point = The portfolio that maximizes the slope k ofline r̄ = rf + kσ from (σα, rα) through the feasible set


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One-fund theoremI How to determine this portfolio F?

0 =∂

∂wktan θ

0 =r̄k − rf√∑n

i=1∑n

j=1 wiwjσij− 1

2

∑ni=1 wi(r̄i − rf )(√∑n

i=1∑n

j=1 wiwjσij)3 2 n∑

i=1

wiσik

⇒r̄k − rf =∑n

i=1 wi(r̄i − rf )∑ni=1∑n

j=1 wiwjσij

n∑i=1

wiσik = λ(w)n∑

i=1

wiσik

I Change of variable vi = λ(w)wi ⇒ r̄k − rf =n∑

i=1

viσik

I Solve for vi and retrieve the weightsvk∑ni=1 vi

=λ(w)wk

λ(w)∑n

i=1 wi=

wk∑ni=1 wi

= wk , k = 1,2, . . . ,n


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Example: One-fund theoremAsset i 1 2 3 Risk-free

Σ =

1 0 00 1 00 0 1

E[ri ] 1 2 3 1/2σi 1 1 1 0

I Optimality conditions

r̄k − rf =n∑

i=1

viσik ⇒

v1 = 1− 1/2 = 1/2v2 = 2− 1/2 = 3/2v3 = 3− 1/2 = 5/2

I Normalization of weights

wk =vk∑ni=1 vi

,

3∑i=1

vi = 9/2⇒

w1 = (1/2)/(9/2) = 1/9w2 = (3/2)/(9/2) = 3/9w3 = (5/2)/(9/2) = 5/9


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Overview

Random returns


Markowitz model

Two-fund theorem

One-fund theorem

Random returnsPortfolio mean and varianceMarkowitz modelTwo-fund theoremOne-fund theorem

MS-E2114 Investment Science Lecture 5: Mean-variance ... · MS-E2114 Investment Science: Lecture 5,...

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