MS-E2114 Investment Science Lecture 5: Mean-variance ... · MS-E2114 Investment Science: Lecture 5,...
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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
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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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MS-E2114 Investment Science: Lecture 5, Mean-variance portfolio theoryDecember 8, 2017
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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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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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MS-E2114 Investment Science: Lecture 5, Mean-variance portfolio theoryDecember 8, 2017
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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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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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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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MS-E2114 Investment Science: Lecture 5, Mean-variance portfolio theoryDecember 8, 2017
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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
Standard deviation 𝜎
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
Portfolio mean and variance
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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Example: Solving minimum variance portfolio
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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Example: Solving minimum variance portfolio
Asset 3
Asset 2
Asset 1
(𝜎𝑚𝑖𝑛, ҧ𝑟)
Standard deviation 𝜎
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
Portfolio mean and variance
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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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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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
Standard deviation 𝜎
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
Portfolio mean and variance
Markowitz model
Two-fund theorem
One-fund theorem
Random returnsPortfolio mean and varianceMarkowitz modelTwo-fund theoremOne-fund theorem