fm - modern portfolio theory

7/27/2019 FM - Modern Portfolio Theory

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o ern ort o o eory- ar ow z c en ron er

nanc a o e ng

Prof. Doug Blackburn


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Some Assumptions

There are many types of portfolios that meet alarge number of investor needs.

The optimal portfolio is directly related to

.

The portfolios we will be discussing are based

on the following assumptions: Investors want return

Investors hate risk


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What is risk?

This seems like an easy question yet it isdifficult to answer.

Individuals dislike uncertainty

This implies that individuals like variances to besmall so that actual returns are relatively close to

expected returns.


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Measuring return and variance

...

N

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Assume on y r s y assets no r s - ree secur ty

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Using Linear Algebra

.s,covarianceandvariancesofmatrixsymmetricNNanDefine

.weightsportfolioofvector1NanDefine

.

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Using Linear Algebra

12

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1 1,

i j


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Setting up the problem

We need to find the optimal portfolio weightsbut we first need to identify the objective.

We have two choices:

.

Minimize variance for a given level of return.

Mathematically, it is more convenient to

minimize the uadratic variance function.


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Changing to matrix notation

It is much more convenient to write theproblem using matrix notation.

matrixcovariance-variancetheisVV11Min 2 wwT

.ts

weightsportfolioofvectortheis11

returnsexpectedofvectortheis

ww

rrwr

T

p


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Objective and first order conditions

sMultiplierLagrangeofmethodUse

11V2

21 p wrrwwwL

01V)1( 21

rw

w

0)2(1

pT rrw

011)3(2

TwL


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Solving the problem

01V(1) .bymultiplyleftthenandVbymultiplyleft(1),Using 21

-1

T

rwr

1VV)4(

1-

2

1-

1

rw

1VV 1-21-1

TTT rrrwr

.(2),fromand,thatalgebralinearfromRecall

pTTT rrwrwwr

1VV)5(1-

2

1-

1

TT

p rrrr


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Solving the problem

.11(3)fromand,1by(4)multiplyLeft

TT

w

1VV)4( 1-21-

1

rw

11V1V11)6( 1-21-

1

TTT rw


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Solving the problem

This gives us two equations with two unknowns the two Lagrange Multipliers

1VV)5( -12-1

1

TTp rrrr

11V1V11)6( 1-21-1

TTT rw

)5( 21 rAB p Simplify

1)6( 21 CA

A,B, and C.


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Solving the problem

r

CA

B p

2

1

1

rABp

1

1

rC 1

BAABC 22 1

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BrA

D

ArC pp

21


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The Optimal Portfolio Weights

rw

-12

-1

1 1VV:(4)equationFrom

pp rAB

r

ArC

w

1-1-* 1VV

:su t p er agrangeor t enu st tute

:gRearrangin

prArCDrABDw

1-1-1-1-* 1VV1V1V1

prhgw

*


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Minimum variance portfolios for given returns

Return Two portfolios with the

same return.

Efficient Frontier

E r1

o a n ar ance or o o

Standard deviation

1

2


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Global Minimum Portfolio

1VV1

andV1V1

whereV

:port o ovar ancem n mume

1-1-1-1-T2 ArChrABgrhgrhg

.algebra......some

12

2

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Ar

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C pp

minimum.aexistsThereequation.quadraticconvexaisThis

.1

andThus,02 22

C

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Ar

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pppp

p


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Global Minimum Portfolio

1r

1

mv

rVTReturn of the MV portfolio

mv ar ance o e por o o

11

11

V

VwTmv

Weights of the MV portfolio


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Extending the model

Assume that the investor has access to a riskfree asset.

n f rm h m r l m n in h hrisk-free asset does not affect the objective

function.

Risk-free asset has zero variance and zero

correlation with all other assets.


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Extending the model

However, including the risk free asset doesaffect the two linear constraints

The expected portfolio return

The sum of the portfolio weights

11

1

N

iip

N

i rwrw

Combining the two conditions yieldsN

i

fiifp rrwrr

1


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Extending the model

The new problem has the Lagrangian:

fpfTT rrrrwwwL

1V

2

This can be solved in a similar fashion as the

prev ous pro em. I will let you work out the details or see

er on .


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Maximizing the Sharpe Ratio

Finding a closed-form solution is always nice,but sometimes it is not possible.

n fin n im l r f liwith the additional constraints on the portfolio

wei hts.

No short selling (weights are non-negative)

asset (weights must be less than x%).


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Maximizing the Sharpe Ratio

We must use numerical methods for obtainingthe portfolio weights.

In hi n n i r n ifunction that looks like

MaximizeP

fp rr

(The Sharpe Ratio)

s.constra ntsu ect to


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The Sharpe Ratio and the CAPM

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fmifi rRErRE

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!theory!financeofheartat theisRatio)Sharpe(theanalysisVariance-Mean


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An alternative / flexible approach

To solve this problem, we can simply useExcels solver functionality to find the

optimal portfolios weights.

Notice that this is a well-behaved objective

function as a function of wei hts

Portfolio return is linear


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Inputs to the model

To use the model, all we need is historicalreturn data forNstocks.

Expected Returns for each stock can be estimated from

the historic mean.

Variance/Covariance matrix can be estimated by taking

the historic variance from the returns of each stock and

the historic covariance from all pairs of stocks.

Risk-free rate can be estimated using the yield on a 3-

month T-bill.


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Where to get data?

Prices / Returns

Yahoo!, Bloomberg, or other data source.

- St. Louis Federal Reserve FRED database.


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Expected returns

The classic approach is to use the historic

mean as a proxy for the expected return.

n l n ili ri m m l f returns (e.g. CAPM) as a proxy.

We will investigate this approach next time.


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Some things to think about

We have derived the optimal portfolio of a

mean-variance optimizer.

There are many other objective functions.

or o os can e orme o:

Consider skewness

Maximize dividend yield Preserve capital

And so on


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Portfolios and Higher Moments

Modern portfolio theory considers only mean

returns and variances/covariances of returns.

Quadratic utility function

Normal distributions

Returns, however, are not normal

ega ve s ewness ncreases e pro a y oextreme bad events.

extreme events (good or bad)


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Which security is preferred?

For a cost of $1 you can by either of the two

following gambles:

1. $40 with =0.1 or-$1 with pL=0.9

2. $1,000,000 with pW=0.1

-$1 with =0.9


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Which security is preferred?

Suppose we maximize Sharpe Ratio:

Shar e Ratio for choice 1 = 0.013

Sharpe Ratio for choice 2 = 0.000

The increase in volatility is in the positive

direction The probability of the same loss is equal across

the two choices


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Skewness Constraints

It is simple to minimize the portfolio variance

subject to both a constraint on the mean and

on skewness.

We can then compare the Sharpe ratios across

the various ortfolios


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Skewness Constraints

10

Skew= -0.2

6

8

p

eRatio

Skew= 0.1

Skew= 0.3

2

4Sha

Skew= 0.5

0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07


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Which Portfolio is Best?

Portfolio 1 Portfolio 2 Portfolio 3

Mean 1.50% 1.50% 1.50%

Skewness -0.49 0.00 0.00

Kurtosis1.15 0.69 0.00

Sharpe Ratio 8.48 5.88 4.72

Variance 0.18% 0.25% 0.32%

Min Ret -14.97% -12.87% -12.77%Max Ret 11.34% 17.98% 18.17%

uar e 4.64% 4.68% 5.53%


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Minimum Variance Portfolios

All three cases, variance was minimized.

Portfolio 1:

= .

Portfolio 2:

ean = . , ew = Portfolio 3:

Mean = 1.5%, Skew = 0, Kurtosis = 0


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Minimum Variance Portfolio

All Variance is not equal.

Large upward movements in prices is good.

bad particularly during down markets.

Perhaps maximum Sharpe Ratio is not the

es r s re urn s a s c.


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Semi-Variance

Some have considered an alternative measure

for risk that captures downside volatility

T

21

i

iT

1

,


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A New Measure

Perhaps it is time for a new measure for

portfolio risk.

A HALLEN ECan you develop a theory such that investors

, ,

skewness, and some reasonable amount of

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