Unit-2_Mean-Variance.pdf

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Portfolio Optimization

Unit 3: Mean-Variance Portfolio Selection

Duan LI & Xiangyu Cui

India Institute of Technology KharagpurMay 26 - 30, 2014


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Return

Asset : An investment instrument that can be bought and sold

Single (investment) period : An investor invests at the beginning of the period and holds it until the end of the period

Total return of investing on an asset (for a single period):

R = total return = amount received (later)amount invested (initially)

= X 1X 0

Rate of return : r = X 1 X 0X 0 Prot : p = X 1 X 0 = rX 0

R = 1 + r

X 1 = (1 + r )X 0


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Short Sales

Short selling or shorting : the process of borrowing an asset, selling

it, and returning the asset at a later date Short selling is regarded very risky: the potential for loss is unlimited.

Suppose we receive X 0 initially and pay X 1 later. The total return of

the shorting is

R = total return = amount received (later)amount invested (initially)

= X 1

X 0 = X 1

X 0


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The rate of return is

r = rate of return = X 1 ( X 0 )

X 0=

X 1 X 0X 0

R = 1 + r Prot p = X 0 X 1 = rX 0

In practice, the short selling is supplemented by certain restrictions

and safeguards: To short a stock, you are required to deposit anamount equal to the initial price X 0 . At the end of the time period(with stock price changing to X 1 ), you recover your original position(liquidate your position) and receive your prot from shorting equal

to X 0 X 1 .


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Example: John Goes Short

John short sold 100 shares of stock ABC at the price $10/share. Thestock dropped to $9/share after one year.

Question : Evaluate the return of this investment

Solution : X 0 = 1000, X 1 = 900

R = 9001000 = 0 .9

r = R 1 = 0.1

p = rX 0 = 0 .1 1000 = 100


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Portfolio Return

A portfolio or a master asset : Allocation of the initial amount of X 0to available n different assets

= ( X 01 , X 02 , , X 0 n ),

where X 0 i is the amount invested in the ith asset, with

n

i =1X 0 i = X 0

Weight of asset i in the portfolio:

wi = X 0 iX 0

n

i=1

wi = 1 .


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Portfolio Return (Contd)

Let Ri and r i be the total return and rate of return of asset i. Then

The total return of the portfolio is

R =ni =1 R i X 0 i

X 0=

n

i =1

wi R i

The rate of return of the portfolio is

r = R 1 =

n

i =1wi r i


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Portfolio Mean and Variance

Consider n assets with random rate of return r1 , r 2 , , r n , and a port-folio using the weights wi , i = 1 , 2, , n . Let w = ( w1 , , wn ) .

Let r i be the expected return of r i and ij be the covariance betweenr i and r j . Let r = (r 1 , , r n ) and = ( ij )n n .

The mean rate of return of the portfolio is

r =n

i =1wi r i = w

r

The variance of the return of the portfolio is

2 = var (r ) =n

i,j =1wi wj ij = w

w


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Example: A Two-Stock World

Consider a market consisting of two stocks, with r 1 = 0 , 12, r 2 = 0 .15,1 = 0 .2, 2 = 0 .18 and 12 = 0 .01. Bill holds a portfolio with w1 =

0.25, w2 = 0 .75 What are the mean return and variance of Bills portfolio?Solution :

r = 2i =1 wi r i = 0 .1425

2 = 2i,j =1 wi wj ij = 0 .024475 or = 0 .1564

If 12 is instead 0.1. Then 2 = 0.095.

One seminal work of Prof. Harry Markowitzs is to use the variance of the nal wealth as a risk measure for investment.

Suitably constructing a portfolio can reduce investment risk.


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Diversication

Diversication : A process of including additional assets in the port-

folio to reduce the variance of its return. Consider n assets that are mutually uncorrelated. The rate of return

of each asset has mean m and variance 2 . Form a portfolio withwi = 1n . Then

The mean rate of return of the portfolio is E (r ) = m

The variance of the portfolio return is var (r ) = 2

n

The variance decreases as n increases while the mean return rateremains the same.

In the case when some assets are correlated, there may be a lowerlimit of variance that may be achieved by diversication.


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Diagram of Portfolios

Mean-standard deviation diagram or r diagram : A two-dimensionaldiagram, representing the return of assets or portfolios. The horizon-tal axis is for the standard deviation , and the vertical axis for themean rate of return r .

Assume that assets 1 and 2 are characterized by (r 1 , 1 ) and (r 2 , 2 ),

respectively, and their correlation coefficient is . Let the decisionvariable be which is the percentage of wealth you invest in asset 2 You invest (1 ) of your wealth in asset 1.

The random return of the portfolio, (1 )r 1 + r 2 , has the followingmean and standard deviation,

r ( ) = (1 )r 1 + r 2

() =

(1 )2 21 + 2 (1 )1 2 + 2 22


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If = 1, then

() = (1 )2 21 + 2 (1 )1 2 + 2 22= [(1 )1 + 2 ]2 = (1 )1 + 2 .

If = 1, then

() = (1 )2 21 2(1 )1 2 + 2 22= [(1 )1 2 ]

2 = | (1 )1 2 |

= (1 )1 2 if 1 1 + 2 2 (1 )1 if 1 1 + 2

Portfolio diagram lemma. The curve in an r diagram denedby portfolios made from two assets 1 and 2 lies within the triangularregion dened by the two original assets and the point on the verticalaxis of height A = r 1 2 +r 2 1 1 + 2 .


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Feasible Set

Suppose there are n basic assets.

Feasible set or feasible region : The set of points that correspondingto all possible portfolios forming from the n basic assets

The feasible region must be convex to the left.

Minimum-variance set : The left boundary of a feasible set Minimum-variance point (MVP) : The point on the minimum-variance

set that has minimum variance

Risk-averse investor : An investor who, under the same rate of return,prefers the portfolio with the smallest standard deviation


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Graphical Illustration of Mean-Variance of TwoRisky Assets


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Feasible Set in the Mean-Variance Space


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Mean-Variance Formulation in PortfolioSelection


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Feasible Set (Contd)

Risk-seeking or risk-preferring investor : An investor who, under the

same rate of return, chooses the portfolio other than the one of min-imum standard deviation

Non-satiation investor : An investor who, under the same level of standard deviation, selects the portfolio with the largest mean rate of return

Efficient frontier : The upper portion of the minimum-variance set

Efficient frontier provides the best trade off between return and risk

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Formulation of Markowitzs Model

Suppose there are n basic assets with mean rates of return r i (i =1, 2, , n ) and covariances ij (i, j = 1 , 2, , n ). Given a value r , the

objective of a Markowitzs mean-variance portfolio selection problem isMinimize 12 w

wsubject to w r = r

1 w = 1 ,

where 1 = (1 1) .

This classical Markowitzs Model is a convex quadratic optimization

problem.

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Solution to Markowitzs Mean-Variance Model

Introduce the Lagrangian

L = 12w

w (w

r r ) (1

w 1)

where and are two Lagrangian multipliers.Equations for Efficient Set. The portfolio weights wi (i = 1 , 2, , n )

and the two Lagrange multipliers and for an efficient portfolio (withshorting allowed) having mean rate of return r satisfy

w r 1 = 0r w = r1 w = 1 ,

where 0 = (0 0) .

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Linearity of Efficient Equations

Let {w 1 , 1 , 1 }, and {w 2 , 2 , 2 } be two efficient portfolios corre-sponding to r 1 and r 2 , respectively.

w j r 1 = 0r w j = r j1 w j = 1 , j = 1 , 2

Then {w = w 1 + (1 )w 2 , 1 + (1 ) 2 , 1 + (1 )2 }satises

w r 1 = 0

r

w

= r1

+ (1 )r2

1 w = 1

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Two-Fund Theorem

Two efficient funds (portfolios) can be established so that any efficientportfolio can be duplicated, in terms of mean and variance, as a combination

of these two. In other words, all investors seeking efficient portfolios needonly invest in combinations of these two funds.Implications:

Two mutual funds could provide a complete investment service.

Individuals do not need to purchase individual stocks separately.

Note the underlying assumptions:

Mean-variance framework Everyone has the same assessment of means, variances and co-

variances. Single period

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Computational Implication

Find two particular solutions!

Take (a) = 0 and (b) = 0

The constraint 1 w = 1 may be violated, so one needs to normalize.

The solution to (a) is the minimum-variance point.

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Example

n = 2; r = 0.1510.125 ; = 0.023 0.00930.0093 0.014

Let = 0 . 0.023 0.00930.0093 0.014 v = 11

v = 19.956258.1716 Normalization = 1

78 .1278 and w = 0.25540.7446

Let = 0 . 0.023 0.00930.0093 0.014 v = 0.1510.125

v = 4.046.24 Normalization = 110 .28 and w =

0.3930.607

All efficient solutions can be expressed as

0.3930.607 + (1 ) 0.25540.7446 , 0

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M-V Selection: No Shorting

A modied Markowitzs mean-variance selection problem with shortingprohibited :

Minimize 12 w w

subject to w r = r1 w = 1wi 0, i = 1 , 2, , n.

This problem is still a convex quadratic optimization problem and canbe solved by quadratic programming algorithms.

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Inclusion of Riskless Asset

Riskless or risk-free asset : An asset that has a deterministic return

When riskless borrowing and lending are available, the efficient setbecomes a single straight line.

This line is tangent to the original feasible set of risky assets from the

riskless point. Tangent portfolio : The portfolio that corresponds to the point F in

the original feasible set that is on the line segment dening the efficientset.

The One-Fund Theorem. There is a single fund F of risky assetssuch that any efficient portfolio can be constructed as a combinationof the fund F and the risk-free asset.

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Calculation of Tangent Fund

Let the tangent point be ( p , r p). To identify the tangent point isequivalent to solving

max tan = r p r f

p=

w r r f (w w )1 / 2

Fractional programming

From the theory of fractional programming, the above problem can be solvedby considering the following auxiliary problem:

max w (r 1 r f ) w w

where is a parameter to be determined.

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Tangent Portfolio and One Fund Theorem

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Calculation of Tangent Fund (Cont)

Algorithm to nd tangent portfolio w F :

1. Solve

v = r r f 1

2. Normalize

w F = v1 v

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Example

n = 2; r = 0.1510.125 ; = 0.023 0.00930.0093 0.014 ; rf = 0.08.

0.023 0.00930.0093 0.014 v = 0.1510.125

0.080.08

v = 2.444

1.591 w = 2.444

1.591/ (2.444 + 1.591) = 0.6057

0.3943

All efficient solutions can be expressed by

(1 ) 0.6057

0.3943with 1.

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