程序代写案例-MATH 11158
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
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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
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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
• 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.
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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.
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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
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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