Portfolio Management (Rohit)

8/9/2019 Portfolio Management (Rohit)

http://slidepdf.com/reader/full/portfolio-management-rohit 1/64

11K. Hartviksen

MPT ² Modern Portfolio Theory

Business 2039



22K. Hartviksen

Key Terms

Harry Markowitz

MPT

Expected return required return

portfolio

systematic risk

unsystematic risk

diversification

beta coefficient

security market line

market premium for risk

capital asset pricingmodel

cost of capital

mean, variance,standard deviation

correlation capital market line



Harry M

arkowitz

Modern portfolio theory was initiated byUniversity of Chicago gr aduate student,

Harry Markowitz in 1952. Markowitz showed how the risk of a portfolio

is NOT just the weighted aver age sum of therisks of the individual securities«but r ather,

also a function of the degree of comovementof the returns of those individual assets.



3

Risk and Return - MPT

Prior to the establishment of Modern PortfolioTheory, most people only focused upon investment

returns«they ignored risk.

With MPT, investors had a tool that they could useto dr amatically reduce the risk of the portfoliowithout a significant reduction in the expectedreturn of the portfolio.



10

Correlation

The degree to which the returns of twostocks co-move is measured by the

correlation coefficient. The correlation coefficient between the

returns on two securities will lie in ther ange of +1 through - 1.

+1 is perfect positive correlation.

-1 is perfect negative correlation.



11

Perfect Negatively Correlated Returnsover Time

Time1994 1995

1996

Returns on Stock A

10%

Returns on Stock B

A two-asset portfolio

made up of equal parts

of Stock A and B would

be riskless. There

would be no variability

of the portfolios returns

over time.

Returns on Portfolio

Returns

%



145K. Hartviksen

Ex Post Portfolio ReturnsSimply the Weighted Aver age of Past Returns

ii x

x

i

i

n

iii p

assetonreturnassetof weightrelative

:Where

1

!!

! §!



145K. Hartviksen

Ex Ante Portfolio ReturnsSimply the Weighted Aver age of Expected Returns

RelativeWeight

ExpectedReturn

WeightedReturn

Stoc X 4 8 %

Stoc Y %

Stoc Z % 6Expected Portfolio Return = 14.70%



Grouping Individual Assetsinto Portfolios

The riskiness of a portfolio that is made of differentrisky assets is a function of three different f actors:

the riskiness of the individual

assets th

at m

ake up theportfolio

the relative weights of the assets in the portfolio

the degree of comovement of returns of the assets makingup the portfolio

The standard deviation of a two-asset portfolio maybe measured using the Markowitz model:

B A B A B A B B A A p W W VW W W ,

2222 2!



Risk of

aThree-

asset Portfolio

C AC AC AC BC BC B B A B A B AC C B B A A p W W VW W VW W VW W W W ,,,

222222 222 !

The data requirements for a three-asset portfolio growsdr amatically if we are using Markowitz Portfolio selection formulae.

We need 3 (three) correlation coefficients between A and B; A andC; and B and C.

A

B C

a,b

b,c

a,c



R

isk of a

Four-asset Portfolio

The data requirements for a four-asset portfolio grows dr amaticallyif we are using Markowitz Portfolio selection formulae.

We need correlation coefficients between A and B; A and C; Aand D; B and C; C and D; and B and D.

A

C

B D

a,b a,d

b,c c,d

a,c

b,d



Diversification Potential

The potential of an asset to diversify a portfolio isdependent upon the degree of co-movement of

returns of thea

sset with those other a

ssets tha

tmake up the portfolio.

In a simple, two-asset case, if the returns of the twoassets are perfectly negatively correlated it ispossible (depending on the relative weighting) to

eliminate all portfolio risk. This is demonstr ated through the following chart.



Example of Portfolio

Combinations

and Correl

ation

sset

Expected

Ret rn

tandard

eviation

Correlation

Coefficient

A

B

ei t of ei t of

Expected

Ret rn

tandard

eviation

7

7 7 7

7

7

7

Portfolio Components Portfolio C ar acteristics

PerfectPositive

Correlation ±no

diversification




Combina

tionsa

nd Correla

tionsset

Expected

Ret rn

tandard

eviation

Correlation

Coefficient

A

B

ei t of ei t of

Expected

Ret rn

tandard

eviation

7

7 7 7

7

7


PositiveCorrelation ±

weakdiversification

potential




Combina

tionsa

nd Correla

tionsset

Expected

Ret rn

tandard

eviation

Correlation

Coefficient

A

B

ei t of ei t of

Expected

Ret rn

tandard

eviation

7 7 7

7

7


NoCorrelation ±

somediversification

potential

Lower risk t anasset




Combina

tionsa

nd Correla

tionsset

Expected

Ret rn

tandard

eviation

Correlation

Coefficient

A -

B

ei t of ei t of

Expected

Ret rn

tandard

eviation

7 7 7

7

7


Ne ativeCorrelation ±

reater diversification

potential




Combina

tionsa

nd Correla

tionsset

Expected

Ret rn

tandard

eviation

Correlation

Coefficient

A -

B

ei t of ei t of

Expected

Ret rn

tandard

eviation

7 7 7

7

7


PerfectNe ative

Correlation ±reatest

diversificationpotential

Risk of t eportfolio isalmost

eliminated

at7 asset



The Effect of Correlation on Portfolio Risk:The Two-Asset Case

Expected Return

Standard Deviation

0%

0% 10%

4%

8%

20% 30% 40%

12%

B

VAB= +1

A

VAB = 0

VAB= -0.5

VAB = -1

Diversification of a TwoAsset Portfolio Demonstrated Graphically



An Exercise using T-bills, Stocksa

nd BondsBase ata: Sto -b Bond

Ex ected eturn

Standard e iation

orrelation oe icient atrix:

Sto 1 - 16 0.048

-b -0.216 1.000 0.380

Bond 0.048 0.380 1.000

Port olio o binations:

Co bination Stocks T-bills Bonds

Ex ected

eturn Variance

Standard

e iation

1 100.0% 0.0% 0.0% 12.7 0.0283 16.8%

2 90.0% 10.0% 0.0% 12.1 0.0226 15.0%

3 80.0% 20.0% 0.0% 11.4 0.0177 13.3%

4 70.0% 30.0% 0.0% 10.8 0.0134 11.6%

5 60.0% 40.0% 0.0% 10.1 0.0097 9.9%

6 50.0% 50.0% 0.0% 9.4 0.0067 8.2%

7 40.0% 60.0% 0.0% 8.8 0.0044 6.6%

8 30.0% 70.0% 0.0% 8.1 0.0028 5.3%

9 20.0% 80.0% 0.0% 7.5 0.0018 4.2%

10 10.0% 90.0% 0.0% 6.8 0.0014 3.8%

11 0.0% 100.0% 0.0% 6.2 0.0017 4.2%

Weig ts Port olio



Results Using only Three Asset Classes

Attainable Portfolio Combinationsand Efficient Set of Portfolio Combinations

Standard Deviation of the Portfolio (%)

P o r t f o l i o E x p e c t e d R e t u r n ( % ) i ient et

ini u

arian e

ort olio



PlottingAchievable Portfolio Combinations

Expected Return onthe Portfolio

Standard Deviation of the Portfolio

0%

0% 10%

4%

8%

20% 30% 40%

12%



The Efficient Frontier



0%

0% 10%

4%

8%

20% 30% 40%

12%



The Capital Market Line



0%

0% 10%

4%

8%

20% 30% 40%

12%

Risk-free

r ate

Capital

Market Line



The Capital Market Line and Iso Utility Curves



0%

0% 10%

4%

8%

20% 30% 40%

12%

Risk-free

r ate

Capital

Market Line

HighlyRisk

AverseInvestor

A risk-taker



The Capital Market Line and Iso Utility Curves



0%

0% 10%

4%

8%

20% 30% 40%

12%

Risk-free

r ate

Capital

Market Line

A risk-taker¶sutility curve

The risk-taker¶soptimalportfolio

combination



CML versus SML

Please notice that the CML is used toillustr ate all of the efficient portfolio

combinations available to investors.

It differs significantly from the SML thatis used to predict the required return

that investors should demand giventhe riskiness (beta) of the investment.



Data Limitations

Because of the need for so much data,MPT was a theoretical idea for many

years.

Later, a student of Markowitz, namedWilliam Sharpe worked out a way

around that«creating the Beta Coefficient as a measure of volatilityand then later developing the CAPM.



CAPM

The Capital Asset Pricing Model was

the work of Willia

m Sharpe,

astudentof Harry Markowitz at the University of

Chicago.

CAPM is an hypothesis «



6

Capital Asset Pricing Model

R eturn

%

Beta Coe icient

R f

Security Market

Line

B M =1.0

k m

Market

Premium

for risk

Real Return Premium for expected inflation

R equired return = R f + Fs [k M - R f ]



CAPM

This model is an equilibrium based model.

It is called a single-f actor model because the slope of the SMLis caused by a single measure of risk « the beta.

Although this model is a simplification of reality«it is robust (itexplains much of what we see happening out there) and itenjoys widespread use in a great variety of applications.

Although it is called a µpricing model¶ there are not prices onthat gr aph«.only risk and return.

It is called a pricing model because it can be used to help usdetermine appropriate prices for securities in the market.



Risk

Risk is the chance of harm or loss;danger.

We know that various asset classeshave yielded very different returns inthe past:



Historical Returns and Standard Deviations19 - 9 1

Aver age Return Standard Deviation

Canadian common stock 12. 3% 1 . 1%

U.S. common stock (Cdn $) 1 .09 1 . 0Long term bonds .01 10.20

Small cap stocks 15. 2 . 0

Inflation .52 3.5

Treasury bills .15 .1

___________________ 1The Alexander Group



Risk and Return

The foregoing data point out that thoseasset classes that have offered the highest

r ates of return, have also offered the highestrisk levels as measured by the standarddeviation of returns.

The CAPM suggests that investors demand

compensation for risks that they areexposed to«and these returns are built intothe decision-making process to invest or not.



6

Capital Asset Pricing Model

R eturn

%

Beta Coeff icient

R f

Security Market

Line

B M =1.0

k m

Market

Premium

for risk

Real Return Premium for expected inflation

R equired return = R f + Fs [k M - R f ]



CAPM

The foregoing gr aph shows that investors: demand compensation for expected inflation

demand a real r ate of return over and above expected inflation

demand compensation over and above the risk-free r ate of return for any additional risk undertaken.

We will make the case that investors don¶tneed compensation for all of the risk of an

investment because some of that risk canbe diversified away.

Investors require compensation for risk theycan¶t diversify away!



7

Beta Coefficient

The beta is a measure of systematic risk of an investment.

Systematic risk is the only relevant risk to a diversified investor according to the CAPM since all other risk may be diversifiedaway.

Total risk of an investment is measured by the securities¶standard deviation of returns.

According to the CAPM total risk may be broken into twoparts«systematic (non-diversifiable) and unsystematic(diversifiable)

TOTAL RISK S STEMATIC RISK + UNS STEMATIC RISK

The beta can be determined by regressing the holding periodreturns (HPRs) of the security over 30 periods against the

returns on the over all market.



Measuring Risk of theIndividual Security

Risk is the possibility that the actual return that willbe realized, will turn out to be different than whatwe expect (or have forecast).

This can be measured using standard statisticalmeasures of dispersion for probability distributions.They include: variance

standard deviation coefficient of variation



Standard Deviation

The formula for the standard deviationwhen analyzing population data

(realized returns) is:

1

)(1

2

!§!

n

k k n

i

ii

W



Standard Deviation

The formula for the standard deviation whenanalyzing forecast data (ex ante returns) is:

it is the square root of the sum of thesquared deviations away from the expectedvalue.

§!

!n

i

iii P k k 1

2)(W



Using Forecasts to EstimateBeta

The formula for the beta coefficient for a stock µs¶ is:

Obviously, the calculate a beta for a stock, you mustfirst calculate the variance of the returns on themarket portfolio as well as the covariance of the

returns on the stock with the returns on the market.

)(

)(

M

M s

s k Variance

k k Cov

B !



Systematic Risk

The returns on most assets in our economy are influenced bythe health of the µsystem¶

Some companies are more sensitive to systematic changes in

the economy. For example dur able goods manuf acturers. Some companies do better when the economy is doing poorly

(bill collection agencies).

The beta coefficient measures the systematic risk that thesecurity possesses.

Since non-systematic risk can be diversified away, it isirrelevant to the diversified investor.



Systematic Risk

We know that the economy goesthrough economic cycles of expansion

and contr action as indicated in thefollowing:



Canada¶s Business cycles from 1873-1992

Trough to ExpansionPeak to Contraction

(months f rom trough to peak)(months f rom peak to trough)

Nov 1873 66

May 1879 38 July 1882 32

Mar 1885 23 Fe b 1887 12

Fe b 1888 29 July 1890 9

Mar 1891 23 A pr 1893 13

Mar 1894 17 Aug 1895 12

Aug 1896 44 A pr 1900 10

Fe b 1901 22 Dec 1902 18

June 1904 30 Dec 1906 19July 1908 20 Mar 1910 16

July 1911 16 Nov 1912 25

Jan 1915 36(WWI) Jan 1918 15

A pr 1919 14 June 1920 15

Sep 1921 21 June 1923 14

Aug 1924 56 A pr 1929 47 (Depression)

Mar 1933 52 July 1937 15 (Depression)

Oct 1938 80(WWII) June 1945 8Fe b 1946 33 Oct 1948 11

Sep 1949 44(Korean War) May 1953 14²uly 195431 Fe b 1957 12

Fe b 1958 26 A pr 1960 10

Fe b 1961 160 June 1974 10

A pr 1975 58 Fe b 1980 6

July 1980 12 July 1981 6

Nov 1982 89 A pr 1990 22

Fe b 1992



Companies and Industries

Some industries (and by implication the companiesthat make up the industry) move in concert with theexpansion and contr action of the economy.

Some lead the over all economy. (stock market)

Some lag the over all economy. (ie. automotiveindustry)



Amount of Systematic Risk

Some industries may find that their fortunes arepositively correlated with the ebb and flow of theover all economy«but that this relationship is veryinsignificant.

An example might be Imperial Tobacco. This firmdoes have a positive beta coefficient, but very littleof the returns of this company can be explained by

the beta. Instead, most of the variability of returnson this stock is from diversifiable sources.

A Char acteristic line for Imperial Tobacco wouldshow a very wide dispersion of points around the

line. TheR2

would be very low (.05 5%

or lower).



Char acteristic Line for ImperialTobacco

R eturns on the

Market %

(TSE 300)

R eturns on

Imperial

To bacco %

Characteristic

Line f or Imperial

To bacco



High R2

An R2 that approaches 1.00 (or 100%) indicatesthat the char acteristic (regression) line explainsvirtually all of the variability in the dependentvariable.

This means that virtually of the risk of the security isµsystematic¶.

This also means that the regression model has a

strong predictive ability. « if you can predict whatthe market will do«then you can predict the returnson the stock itself with a great deal of accur acy.



Char acteristic Line Gener alMotors

R eturns on the

Market %

(TSE 300)

R eturns on

General

Motors %

Characteristic

Line f or GM

(high R 2)



Diversifiable Risk(non-systematic risk)

Examples of this type of risk include: a single company strike

a spectacular innovation discovered through thecompany¶sR&D progr am

equipment f ailure for that one company

management competence or management incompetencefor that particular firm

ajet c

arrying the senior m

an

agement te

am of the firmcr ashes

the patented formula for a new drug discovered by thefirm.

Obviously, diversifiable risk is that unique f actor that

influences only the one firm.



Partitioning Risk under theCAPM

Remember that the CAPM assumes that total risk (variability of a security¶s returns) can be separ ated into two distinct components:

Total risk system

atic risk + unsystem

atic risk

100% 0% + 0% (GM)

or

100% 5% + 95% (Imperial Tobacco)

Obviously, if you were toa

dd Imperia

l Toba

cco to your portfolio, youcould diversify away much of the risk of your portfolio. (Not tomention the f act that Imperial has realized some very high r ates of return in addition to possessing little systematic risk!)



Using the CAPM to PriceStock

The CAPM is a µfundamental¶ analyst¶s tool toestimate the µintrinsic¶ value of a stock.

The analyst needs to measure the beta risk of thefirm by using either historical or forecast risk andreturns.

The analyst will then need a forecast for the risk-free r ate as well as the expected return on the

market. These three estimates will allow the analyst to

calculate the required return that µr ational¶ investorsshould expect on such an investment given the

other benchmark returns available in the economy.



Required Return

The return that a r ational investor should demand istherefore based on market r ates and the beta risk of the investment.

To find this, you solve for the required return in theCAPM:

This is a formula for the str aight line that is the SML.

][)( f s f k k ! F



Security Market Line

This line can easily be plotted.

Dr aw Cartesian coordinates.

Plot the yield on 91-day Government of Canada Treasury Bills

as the risk-free r ate of return on the vertical axis. On the horizontal axis set a scale that includes Beta=1 (this is

the beta of the market)

Plot the point in risk-return space that represents your expected return on the market portfolio at beta =1

Dr aw a str aight line to connect the two points. Plot the required and expected returns for the stock at it¶s

beta.



Plot the Risk-Free Rate

Beta Coeff icient1.0

R eturn

%

R f



Plot Expected Return on theMarket Portfolio


R eturn

%

R f = 4%

k m=12%



Dr aw the Security Market Line


R eturn

%

R f = 4%

k m=12%

SML



Plot Required Return(Determined by the formula = Rf + Fs[kM - Rf ]


R eturn

%

R f = 4%

k m=12%

SML

1.2

R (k) = 4% + 1.2[8%] = 13.6%

R (k) = 13.6%



Plot Expected ReturnE(k) = weighted aver age of possible returns


R eturn

%

R f = 4%

k m=12%

SML

1.2

R (k) = 4% + 1.2[8%] = 13.6%R (k) = 13.6%

E(k)



If Expected = Required ReturnThe stock is properly (f airly) priced in the market. It is inEQUILIBRIUM.


R eturn

%

R f = 4%

k m=12%

SML

1.2

R (k) = 4% + 1.2[8%] = 13.6%R (k) = 13.6%

E(k)



If E(k) < R(k)The stock is over-priced. The analyst would issue a sell recommendation in anticipationof the market becoming µefficient¶ to this f act. Investors may µshort¶ the stock to takeadvantage of the anticipated price decline.


R eturn

%

R f = 4%

k m=12%

SML

1.2

R (k) = 4% + 1.2[8%] = 13.6%R (k) = 13.6%

E(k)E(k) = 9%



Let¶s Look at the PricingImplications

In this example:

E(k) = 9%

R(k) = 13. %

If the market expects the company to pay a dividend of $1.00 next

year, and the stock is currently offering an expected return of 9%, thenit should be priced at:

But, given the other r ates in the economy and our judgement aboutthe riskiness of this investment we think that this stock should beworth:

11.11$09.

00.1$

)(

0

10

!!

!

P

k E

d P

s

35.7$136.

00.1$0 !! P



Pr actical Use of the CAPM

Regulated utilities justify r ate increases using the model to

demonstr ate that their shareholders require an appropriate return ontheir investment.

Used to price initial public offerings (IPOs)

Used to identify over and under value securities

Used to measure the riskiness of securities/companies

Used to measure the company¶s cost of capital. (The cost of capital isthen used to evaluate capital expansion proposals).

The model helps us understand the variables that can affect stockprices«and this guides managerial decisions.



Rf rises

R eturn

%

Beta Coeff icient

R f1

SML1

B s=1.2

k s1

R f2

SML2

k s2

R ising interest rates ill

cause all required rates of

return to increase and this

ill f or ce do n stock and

bond prices.



The Slope of The SML rises(indicates growing pessimism about the future of the economy)

R eturn

%

Beta Coeff icient

R f1

SML1

B s=1.2

k s1

SML2

k s2 Gro ing pessimism

ill cause investors todemand greater

compensation f or

taking on risk«this

ill mean prices on

high beta stocks ill

f all more than lo beta stocks.

Documents

Portfolio Management (Rohit)