General Approach Markowitz Model

MATH 11158: Optimization Methods in Finance

Portfolio Optimization1

Akshay Gupte

School of Mathematics, University of Edinburgh

Week 3 : 25 January, 2021

1Chapter 6 in the textbook

Email: [email protected]

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General Approach Markowitz Model

General Approach

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General Approach Markowitz Model

What is Portfolio Optimization ?

In arbitrage detection, we created a “good” portfolio with zero risk

But is it realistic to select a portfolio (i.e., make an investment) without

accounting for any risk in our decisions ? Financial markets are highly

nondeterministic

Question: What makes a good investment?

• High (expected) return

• Low risk

• Something else?

In fact we would simultaneously like high return and low risk. Improving

one often results in worse performance on the other2.

Select a portfolio that strikes a balance between these two objectives

=⇒ Multicriteria (or Pareto) optimization

2There ain’t no such thing as a free lunch

3 / 40

General Approach Markowitz Model

What is Portfolio Optimization ?

In arbitrage detection, we created a “good” portfolio with zero risk

But is it realistic to select a portfolio (i.e., make an investment) without

accounting for any risk in our decisions ? Financial markets are highly

nondeterministic

Question: What makes a good investment?

• High (expected) return

• Low risk

• Something else?

In fact we would simultaneously like high return and low risk. Improving

one often results in worse performance on the other2.

Select a portfolio that strikes a balance between these two objectives

=⇒ Multicriteria (or Pareto) optimization

2There ain’t no such thing as a free lunch

3 / 40

General Approach Markowitz Model

What is Portfolio Optimization ?

In arbitrage detection, we created a “good” portfolio with zero risk

But is it realistic to select a portfolio (i.e., make an investment) without

accounting for any risk in our decisions ? Financial markets are highly

nondeterministic

Question: What makes a good investment?

• High (expected) return

• Low risk

• Something else?

In fact we would simultaneously like high return and low risk. Improving

one often results in worse performance on the other2.

Select a portfolio that strikes a balance between these two objectives

=⇒ Multicriteria (or Pareto) optimization

2There ain’t no such thing as a free lunch

3 / 40

General Approach Markowitz Model

What is Portfolio Optimization ?

In arbitrage detection, we created a “good” portfolio with zero risk

But is it realistic to select a portfolio (i.e., make an investment) without

accounting for any risk in our decisions ? Financial markets are highly

nondeterministic

Question: What makes a good investment?

• High (expected) return

• Low risk

• Something else?

In fact we would simultaneously like high return and low risk. Improving

one often results in worse performance on the other2.

Select a portfolio that strikes a balance between these two objectives

=⇒ Multicriteria (or Pareto) optimization

2There ain’t no such thing as a free lunch

3 / 40

General Approach Markowitz Model

Assumptions on Market Conditions

• Rational decision-makers : investors want to maximise return while

reducing the risks associated with their investment

• No arbitrage : cannot make a costless, riskless profit

• Risky securities : S1, . . . , Sn for n ≥ 2, whose future returns are

uncertain. There is no risk-free asset S0 in the portfolio

• Equilibrium : supply equals demand for securities

• Liquidity : any # of units of a security can be bought and sold quickly

• Access to information : rapid availability of accurate information

• Price is efficient : Price of security adjusts immediately to new

information, and current price reflects past information and expected

further behaviour

• No transaction costs and taxes : transaction costs are assumed to be

negligible compared to value of trades and are ignored. No taxes

(capital-gains etc.) on transactions

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General Approach Markowitz Model

Notation

We have £1 to invest3 in n risky securities S1, . . . ,Sn

Goal is to select a portfolio x = (x1, . . . , xn) ∈ Rn, where

xi = amount invested in S

i

Intuitive to think of xi ≥ 0, however, doing so means we do not allow

short-selling on S i , and we generally allow short-selling of assets.

Hence, x ≥ 0 is not imposed as a constraint always.

• two time periods, now (time 0) and future (time 1)

• ri : Ω→ R is random variable for return of asset i (e.g., a 5% return

means r = 1.05). Probability distribution of ri usually unknown

µi = E[ri ], expected return of S i

• r := r(ω) = (r1(ω), . . . , rn(ω)) is vector of random returns

µ = (µ1, . . . , µn) ∈ Rn is vector of means

3In our analysis, investment of £1 can be scaled to £b for any b > 0

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General Approach Markowitz Model

Notation

We have £1 to invest3 in n risky securities S1, . . . ,Sn

Goal is to select a portfolio x = (x1, . . . , xn) ∈ Rn, where

xi = amount invested in S

i

Intuitive to think of xi ≥ 0, however, doing so means we do not allow

short-selling on S i , and we generally allow short-selling of assets.

Hence, x ≥ 0 is not imposed as a constraint always.

• two time periods, now (time 0) and future (time 1)

• ri : Ω→ R is random variable for return of asset i (e.g., a 5% return

means r = 1.05). Probability distribution of ri usually unknown

µi = E[ri ], expected return of S i

• r := r(ω) = (r1(ω), . . . , rn(ω)) is vector of random returns

µ = (µ1, . . . , µn) ∈ Rn is vector of means

3In our analysis, investment of £1 can be scaled to £b for any b > 0

5 / 40

General Approach Markowitz Model

Notation

We have £1 to invest3 in n risky securities S1, . . . ,Sn

Goal is to select a portfolio x = (x1, . . . , xn) ∈ Rn, where

xi = amount invested in S

i

Intuitive to think of xi ≥ 0, however, doing so means we do not allow

short-selling on S i , and we generally allow short-selling of assets.

Hence, x ≥ 0 is not imposed as a constraint always.

• two time periods, now (time 0) and future (time 1)

• ri : Ω→ R is random variable for return of asset i (e.g., a 5% return

means r = 1.05). Probability distribution of ri usually unknown

µi = E[ri ], expected return of S i

• r := r(ω) = (r1(ω), . . . , rn(ω)) is vector of random returns

µ = (µ1, . . . , µn) ∈ Rn is vector of means

3In our analysis, investment of £1 can be scaled to £b for any b > 0

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General Approach Markowitz Model

• Σ : covariance matrix for random vector r

Σij = Cov(ri , rj) for all i 6= j

Σij = E[(ri − µi )(rj − µj)]

Σii = Var[ri ] for all i = 1, . . . , n

Σi = E[(ri − µi )2] = E[r2i ]− (E[ri ])2 = E[r2i ]− µ2i

Fact

Covariance matrix Σ is symmetric and positive semidefinite.

x>Σx = x> E[(r − µ)(r − µ)>]︸ ︷︷ ︸

Σ

x

= E[x>(r − µ)(r − µ)>x ]

= E[(r>x − µ>x)2] = E[(r(x)− E[r(x)])2] = Var[r>x ] ≥ 0

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General Approach Markowitz Model

• Σ : covariance matrix for random vector r

Σij = Cov(ri , rj) for all i 6= j

Σij = E[(ri − µi )(rj − µj)]

Σii = Var[ri ] for all i = 1, . . . , n

Σi = E[(ri − µi )2] = E[r2i ]− (E[ri ])2 = E[r2i ]− µ2i

Fact

Covariance matrix Σ is symmetric and positive semidefinite.

x>Σx = x> E[(r − µ)(r − µ)>]︸ ︷︷ ︸

Σ

x

= E[x>(r − µ)(r − µ)>x ]

= E[(r>x − µ>x)2] = E[(r(x)− E[r(x)])2] = Var[r>x ] ≥ 0

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General Approach Markowitz Model

Admissible Portfolios

The set of feasible portfolios is denoted by the set X ⊂ Rn

Budget constraint is always included

n∑

i=1

xi = 1, or

n∑

i=1

xi = b if initial is $b

• No short selling : xi ≥ 0

• Short selling allowed : xi ≥ −`i

• Diversification : xi ∈ {0} ∪ [`i , ui ], such a semi-continuous variable

can be modeled using integer programming

Unless stated otherwise, we assume X = {x ∈ Rn : ∑ni=1 xi = 1}

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General Approach Markowitz Model

Return of a Portfolio

Return of a portfolio x is a random variable that is a linear function of x

Random return on x = sum of returns on each asset

R(x) =

n∑

i=1

rixi = r

>x

Expected return on x

E[R(x)] =

n∑

i=1

E[rixi ] =

n∑

i=1

E[ri ]xi =

n∑

i=1

µixi = µ

>x

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General Approach Markowitz Model

Risk of a Portfolio

Risk of a portfolio is given by a risk function (risk measure)

Risk : x ∈ X → R, where Risk(x) = Risk(R(x))

Commonly used risk measure is variance Risk(x) = Var[R(x)]

More generally, we want risk measures to be convex and a few other

properties

Convexity implies that diversification reduces risk

Risk

(

1

2

x +

1

2

x ′

)

≤ 1

2

Risk(x) +

1

2

Risk(x ′)

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General Approach Markowitz Model

Optimizing Risk-Return Tradeoff

Want to maximize expected return while minimizing risk

There are three kinds of problems we could solve

max

x∈X

E[R(x)] s.t. Risk(x) ≤ “risk budget”

min

x∈X

Risk(x) s.t. E[R(x)] ≥ “target return”

max

x∈X

E[R(x)]− δRisk(x) “risk vs return trade-off”

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General Approach Markowitz Model

Estimating Data

How to calculate mean, covariance, etc. of asset return rates ri for future

time T + 1?

Sampling : Use historical returns r(t) for t ∈ [t1, t2] with

0 ≤ t1 < t2 ≤ T as a sample and take sample mean and variance.

Linear Factor Models : Linear regression using some underlying factors

that affect return rates (e.g., Fama-French three factor model)

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General Approach Markowitz Model

Efficient Portfolios

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General Approach Markowitz Model

Efficiency

Definition

Portfolio x is efficient if is satisfies one of the following conditions:

(1) it has the maximum expected return among all admissible portfolios

of the same (or smaller) risk:

max

x

µ>x

s.t. Risk(x) ≤ σ2 (EP1)

x ∈ X

(2) it has the minimum variance among all admissible portfolios with the

same (or larger) expected return:

min

x

Risk(x)

s.t. µ>x ≥ R (EP2)

x ∈ X

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General Approach Markowitz Model

Visualization via Efficient Frontier

We can plot the efficient portfolios x on a “return-vs-risk” diagram. This

is called the Efficient Frontier

It shows the highest return achievable for a given level of risk, or the

lowest risk achievable for a given (target) return.

y-axis : Expected return of portfolio

x-axis : Some risk measure of return of portfolio

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General Approach Markowitz Model

Expected Return (µ>x) vs Risk (Risk(x) = variance of x) for 10000

random portofolios

ERet(x)

Risk(x)

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General Approach Markowitz Model

→Efficient Frontier

Risk(x)

ERet(x)

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General Approach Markowitz Model

3rd version of Efficient Portfolio: Risk-Adjusted Return

max

x

µ>x − δRisk(x)︸ ︷︷ ︸

risk-adjusted return

s.t. x ∈ X (EP3)

• Parameter δ > 0 sets the relative importance of return and risk

• Choose a small value of δ if you have small sensitivity to risk

• Choose a large value of δ if you have large sensitivity to risk

• Problem: neither return nor risk are controlled directly

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General Approach Markowitz Model

Equivalence of the 3 Models

Theorem

For any convex risk measure, the three characterisations (EP1), (EP2)

and (EP3) of Efficient Portfolios are equivalent, i.e.,

• A portfolio that is efficient under one characterisation is also

efficient under the other two,

• they lead to the same efficient frontier.

Let us see why EP1 ⇐⇒ EP3. Let x∗ be an efficient portfolio under

(EP1), that is, there is σ > 0, s.t. x∗ is the (unique) solution to

max

x∈X

µ>x s.t. Risk(x) ≤ σ2

(EP1) is convex, therefore we can dualise the constraint, i.e. ∃λ∗ > 0

such that x∗ is also the solution to

max

x∈X

µ>x − λ∗

(

Risk(x)− σ2

)

= σ2λ∗ + max

x∈X

µ>x − λ∗Risk(x)

that is x∗ is also a solution to (EP3) for δ = λ.

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General Approach Markowitz Model

Risk vs Return Curve

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General Approach Markowitz Model

Minimum Risk as a Function of Target Return

Consider this family of optimization problems parameterized by R:

σ(R) = min

x

Risk(x)

(†) s.t. µ>x ≥ R

x ∈ X

• σ(R) is the smallest risk of an admissible portfolio whose expected

return is at least R

• σ(R) is the functional form of the Efficient Frontier

• Define X (R) = {x ∈ Rn : µ>x ≥ R, x ∈ X}

This set is nonempty if and only if there exists an admissible

portfolio (x ∈ X ) yielding return at least R

Let Rmax > 0 be the largest R for which X (R) is nonempty.

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General Approach Markowitz Model

Convexity of σ(R)

Theorem

The function σ(R) : (−∞,Rmax ]→ R is convex when Risk is a convex

risk measure.

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General Approach Markowitz Model

Proof of Convexity of σ(R)

We proceed directly from definition of convexity. Fix arbitrary 0 < α < 1

and R1, R2 ≤ Rmax . We need to show that

σ(αR1 + (1− α)R2︸ ︷︷ ︸

R3

) ≤ ασ(R1) + (1− α)σ(R2). (1)

Let f (x) = Risk(x) and let xi for i = 1, 2, 3 be the optimal solution in (†)

for R = Ri . Clearly, f (xi ) = σ(Ri ), and hence (1) becomes

f (x3) ≤ αf (x1) + (1− α)f (x2).

We will show that the convex combination of portfolios x1 and x2:

y = αx1 + (1− α)x2

is feasible for R3 (so that y ∈ X (R3)) but Risk(y) ≥ Risk(x3)

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General Approach Markowitz Model

Since xi is feasible for (†) with right hand side R = Ri , we have

µT xi ≥ Ri , i = 1, 2. (2)

xi ∈ X , i = 1, 2. (3)

By adding an α multiple of the first inequality in (2) to the (1− α)

multiple of the second inequality, we get

µ>y = µ>(αx1 + (1− α)x2) ≥ αR1 + (1− α)R2 = R3. (4)

Moreover, since X is a convex set, from (3) we get

y = αx1 + (1− α)x2 ∈ X . (5)

(4) and (5) combined say that y ∈ X (R3). Since x3 is the optimal point

for (†) with R = R3, we have f (x3) ≤ f (y). Finally, since f is convex,

f (x3) ≤ f (y) = f (αx1 + (1− α)x2) ≤ αf (x1) + (1− α)f (x2).

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General Approach Markowitz Model

Markowitz Model

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General Approach Markowitz Model

Portfolio Optimization as a Convex QP

Portfolio Theory was pioneered by Harry Markowitz in 1950’s through his

seminal paper4 which won a Nobel Prize in Economics in 1990

It uses Risk(x) = Var[R(x)] to find efficient portfolios

min

x

x>Σx

s.t. µ>x ≥ R (EP2)

x ∈ X

Similarly for EP1 and EP3

4Markowtiz, “Portfolio selection”, J. Finance, 1952. https://doi.org/10.2307/2975974

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General Approach Markowitz Model

Variance vs Standard Deviation

Standard deviation (

√

x>Σx) instead of the variance x>Σx).

Note:

√

x>Σx is strictly convex! (try proving it!))

max

x

µ>x

s.t. x>Σx ≤ σ2 (EP1)

x ∈ X

max

x

µ>x

s.t.

√

x>Σx ≤ σ (EP1′)

x ∈ X

(EP1) and (EP1’) are clearly equivalent

min

x

x>Σx

s.t. µ>x ≥ R (EP2)

x ∈ X

min

x

√

x>Σx

s.t. µ>x ≥ R (EP2′)

x ∈ X

(EP2) and (EP2’) are clearly equivalent (lead to the same x∗ for given R)

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General Approach Markowitz Model

max

x

µ>x − δx>Σx (EP3)

s.t. x ∈ X

max

x

µ>x − δ′

√

x>Σx (EP3′)

s.t. x ∈ X

• (EP3) and (EP3’) are equivalent : lead to the same efficient frontier.

• Let x∗ = x∗(δ) be the optimal portfolio in (EP3) for a given δ > 0.

Then there exists a δ′ > 0 such that x∗ is the solution to (EP3’).

• (EP3’) is dual to (EP1’)/(EP2’), as with variance

• In some sense, the standard deviation is the correct risk measure to

use. But variance leads to nicer (easier) optimization problems.

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General Approach Markowitz Model

max

x

µ>x − δx>Σx (EP3)

s.t. x ∈ X

max

x

µ>x − δ′

√

x>Σx (EP3′)

s.t. x ∈ X

• (EP3) and (EP3’) are equivalent : lead to the same efficient frontier.

• Let x∗ = x∗(δ) be the optimal portfolio in (EP3) for a given δ > 0.

Then there exists a δ′ > 0 such that x∗ is the solution to (EP3’).

• (EP3’) is dual to (EP1’)/(EP2’), as with variance

• In some sense, the standard deviation is the correct risk measure to

use. But variance leads to nicer (easier) optimization problems.

27 / 40

General Approach Markowitz Model

max

x

µ>x − δx>Σx (EP3)

s.t. x ∈ X

max

x

µ>x − δ′

√

x>Σx (EP3′)

s.t. x ∈ X

• (EP3) and (EP3’) are equivalent : lead to the same efficient frontier.

• Let x∗ = x∗(δ) be the optimal portfolio in (EP3) for a given δ > 0.

Then there exists a δ′ > 0 such that x∗ is the solution to (EP3’).

• (EP3’) is dual to (EP1’)/(EP2’), as with variance

• In some sense, the standard deviation is the correct risk measure to

use. But variance leads to nicer (easier) optimization problems.

27 / 40

General Approach Markowitz Model

max

x

µ>x − δx>Σx (EP3)

s.t. x ∈ X

max

x

µ>x − δ′

√

x>Σx (EP3′)

s.t. x ∈ X

• (EP3) and (EP3’) are equivalent : lead to the same efficient frontier.

• Let x∗ = x∗(δ) be the optimal portfolio in (EP3) for a given δ > 0.

Then there exists a δ′ > 0 such that x∗ is the solution to (EP3’).

• (EP3’) is dual to (EP1’)/(EP2’), as with variance

• In some sense, the standard deviation is the correct risk measure to

use. But variance leads to nicer (easier) optimization problems.

27 / 40

General Approach Markowitz Model

Positive Definiteness Condition

Covariance matrix Σ is always psd. What if we require it to be positive

definite ? What does it mean for the Markowitz model ?

Positive definite means that x>Σx 6= 0 for all x 6= 0. If x is a portfolio

then its variance x>Σx = 0 means that that there are some

“redundancies” in the model ? returns of some assets depend

deterministically on the returns of others.

We can keep removing these assets from the model until we get positive

definiteness.

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General Approach Markowitz Model

Positive Definiteness Condition

Covariance matrix Σ is always psd. What if we require it to be positive

definite ? What does it mean for the Markowitz model ?

Positive definite means that x>Σx 6= 0 for all x 6= 0. If x is a portfolio

then its variance x>Σx = 0 means that that there are some

“redundancies” in the model ? returns of some assets depend

deterministically on the returns of others.

We can keep removing these assets from the model until we get positive

definiteness.

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General Approach Markowitz Model

Illustrative Example

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General Approach Markowitz Model

Example

Consider investing into an index fund of stocks (S&P 500), bonds (10y

US Treasury Bond) and money market CDs (1-day Federal Funds Rate).

Step 1. Get Historical Data

Ii,t = price of asset i = 1, . . . , n at time t = 0, 1, . . . ,T

S (asset 1) B (asset 2) MM (asset 3)

1960 (t = 0) 20.26 262.94 100.00

1961 (t = 1) 25.69 268.73 102.33

1962 (t = 2) 23.43 284.09 105.33

1963 (t = 3) 28.75 289.16 108.89

...

...

...

...

2003 (t = 43) 1622.94 5588.19 1366.73

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General Approach Markowitz Model

Step 2. Transform into Historical (yearly) Return Rates

ri,t =

Ii,t

Ii,t−1

S (asset 1) B (asset 2) MM (asset 3)

1960 (t = 0) - - -

1961 (t = 1) 1.2681 1.0220 1.0233

1962 (t = 2) 0.9122 1.0572 1.0293

1963 (t = 3) 1.2269 1.0179 1.0338

...

...

...

...

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General Approach Markowitz Model

Step 3. Estimate Mean Return Rates

r¯i =

1

T

>∑

t=1

ri,t︸ ︷︷ ︸

arithmetic mean

µi =

>∏

t=1

ri,t

1/T

︸ ︷︷ ︸

geometric mean

S (asset 1) B (asset 2) MM (asset 3)

r¯i 1.1206% 1.0785% 1.0632%

µi 1.1073% 1.0737% 1.0627%

µ = (1.1073, 1.0737, 1.0627)>

Note: Stocks have highest expected return

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General Approach Markowitz Model

Step 4. Estimate Covariance Matrix

Σij =

1

T

>∑

i=1

(ri,t − r¯i )(rj,t − r¯j), i , j ∈ {1, 2, . . . , n}

Σ =

0.02778 0.00387 0.000210.00387 0.01112 −0.00020

0.00021 −0.00020 0.00115

Note: MM has lowest variance (risk)

Step 5. Find Efficient Portfolio without Short-Selling

min

x

x>Σx

(EP2) s.t. µ>x ≥ R

x ∈ X := {x ∈ R3 : x ≥ 0, x1 + x2 + x3 = 1}

for R ∈ [6.5%, 10.5%].

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General Approach Markowitz Model

2 4 6 8 10 12 14 16

0.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

0.105

0.11

Efficient Frontier

Standard deviation (%)

7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Composition of efficient portfolios

Expected return of efficient portfolios (%)

Pe

rc

en

t i

nv

es

te

d

in

d

iff

er

en

t a

ss

et

c

la

ss

es

Stocks

Bonds

Money Market

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General Approach Markowitz Model

Analysis of Efficient Portfolios

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General Approach Markowitz Model

Allowing Short Selling

Take the minimum variance portfolio problem

min

x

f (x) =

1

2

x>Σx s.t. e>x = 1, µ>x = R

assuming some positive definite covariance matrix Σ

Since we have equality constraints5, Method of Lagrange Multipliers

can be used to derive analytical form of optimal portfolio x

Lagrange function is L(x , λ) = f (x) + λ1(1− e>x) + λ2(R − µ>x)

Lagrange Multiplier Theorem and positive definiteness of Σ (which is the

Hessian matrix of L(x , λ) w.r.t. x) says that we need to solve first-order

gradient conditions

∂L

∂x

= 0 =⇒ Σx − λ1e − λ2µ = 0

∂L

∂λ

= 0 =⇒ e>x = 1, µ>x = R

5If we had µ>x ≥ R, we would need to use Karush-Kuhn-Tucker (KKT) optimality conditions

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General Approach Markowitz Model

Allowing Short Selling

Take the minimum variance portfolio problem

min

x

f (x) =

1

2

x>Σx s.t. e>x = 1, µ>x = R

assuming some positive definite covariance matrix Σ

Since we have equality constraints5, Method of Lagrange Multipliers

can be used to derive analytical form of optimal portfolio x

Lagrange function is L(x , λ) = f (x) + λ1(1− e>x) + λ2(R − µ>x)

Lagrange Multiplier Theorem and positive definiteness of Σ (which is the

Hessian matrix of L(x , λ) w.r.t. x) says that we need to solve first-order

gradient conditions

∂L

∂x

= 0 =⇒ Σx − λ1e − λ2µ = 0

∂L

∂λ

= 0 =⇒ e>x = 1, µ>x = R

5If we had µ>x ≥ R, we would need to use Karush-Kuhn-Tucker (KKT) optimality conditions

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General Approach Markowitz Model

Multiplying first gradient condition by Σ−1, we get

x∗ = λ1Σ−1e + λ2Σ−1µ

Substituting this x into the two linear constraints and solving for λ yields

λ1 =

C − RB

AC − B2 , λ2 =

RA− B

AC − B2

where A = e>Σ−1e, B = µ>Σ−1e and C = µ>Σ−1µ. Hence,

x∗ := x∗R =

C − RB

AC − B2 Σ

−1e +

RA− B

AC − B2 Σ

−1µ

Markowitz Efficient Frontier is produced by the portfolios{

x∗R : R ≥

B

A

}

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General Approach Markowitz Model

Global and Diversified Portfolios

• Global Minimum Variance Portfolio is obtained by setting the second

Lagrange multiplier to zero

λ2 = 0 =⇒ R = B

A

=⇒ xG = Σ

−1e

A

• Diversified Portfolio is obtained by setting the first Lagrange

multiplier to zero

λ1 = 0 =⇒ R = C

B

=⇒ xD = Σ

−1µ

B

38 / 40

General Approach Markowitz Model

Mutual Fund Theorem

Theorem

Any minimum variance portfolio x∗ can be written as a convex

combination of two distinct minimum variance portfolios x ′ and x ′′ where

x ′ 6= x ′′,

x∗ = αx ′ + (1− α)x ′′, some α ∈ [0, 1].

In particular, we can take x ′ = xG and x ′′ = xD .

39 / 40

General Approach Markowitz Model

No Short Selling

X = {x ≥ 0 : ∑ni=1 xi = 1} is a simplex in Rn. Do not need Σ to be pd.

Exercise

Show that the set of optimal portfolios is a compact convex set.

Theorem

For every minimum variance portfolio x , there exist n + 1 “mutual funds”

w1, . . . ,wn+1 such that x is a convex combination of {w1, . . . ,wn+1}

x =

n+1∑

i=1

αiw

i , some α ≥ 0,

n+1∑

i=1

αi = 1

Here, compactness of X implies set of optimal solutions X ∗ is bounded,

and then the theorem is a consequence of Krein-Milman theorem and

Caratheodory theorem for convex sets

40 / 40

General Approach Markowitz Model

No Short Selling

X = {x ≥ 0 : ∑ni=1 xi = 1} is a simplex in Rn. Do not need Σ to be pd.

Exercise

Show that the set of optimal portfolios is a compact convex set.

Theorem

For every minimum variance portfolio x , there exist n + 1 “mutual funds”

w1, . . . ,wn+1 such that x is a convex combination of {w1, . . . ,wn+1}

x =

n+1∑

i=1

αiw

i , some α ≥ 0,

n+1∑

i=1

αi = 1

Here, compactness of X implies set of optimal solutions X ∗ is bounded,

and then the theorem is a consequence of Krein-Milman theorem and

Caratheodory theorem for convex sets

40 / 40

General Approach Markowitz Model

No Short Selling

X = {x ≥ 0 : ∑ni=1 xi = 1} is a simplex in Rn. Do not need Σ to be pd.

Exercise

Show that the set of optimal portfolios is a compact convex set.

Theorem

For every minimum variance portfolio x , there exist n + 1 “mutual funds”

w1, . . . ,wn+1 such that x is a convex combination of {w1, . . . ,wn+1}

x =

n+1∑

i=1

αiw

i , some α ≥ 0,

n+1∑

i=1

αi = 1

Here, compactness of X implies set of optimal solutions X ∗ is bounded,

and then the theorem is a consequence of Krein-Milman theorem and

Caratheodory theorem for convex sets

40 / 40

General Approach Markowitz Model

No Short Selling

X = {x ≥ 0 : ∑ni=1 xi = 1} is a simplex in Rn. Do not need Σ to be pd.

Exercise

Show that the set of optimal portfolios is a compact convex set.

Theorem

For every minimum variance portfolio x , there exist n + 1 “mutual funds”

w1, . . . ,wn+1 such that x is a convex combination of {w1, . . . ,wn+1}

x =

n+1∑

i=1

αiw

i , some α ≥ 0,

n+1∑

i=1

αi = 1

Here, compactness of X implies set of optimal solutions X ∗ is bounded,

and then the theorem is a consequence of Krein-Milman theorem and

Caratheodory theorem for convex sets

40 / 40

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