summary of portfolio theory & investment analysis

8/10/2019 Summary of Portfolio Theory & Investment Analysis

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Kingdom of Saudi ArabiaKing Saud University

CBA - Finance Department

Summary of

Portfolio Theory & Investment Analysis

Dr. Chaker Aloui

Prepared by: Mohammed Alhassan

Copyright 2014, All Rights Reserved

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1)The Concept of Risk

... *.

ProbabilityEvents

0.80E1

0.20E2

0E3

1P i

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2) Stock Price Behavioral Risk

Almarai Salama Cooperative Insurance Co

Comparison between the fluctuation of the stock price of Almarai and Salama

Comments:

1. These graphs represent the evolution over time for two stocks listed on Tadawl.

2. Historical data was extracted from Tadawl and during the perios (2012-2014, daily

price).

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3. we notice a linear tendency for Almarai stock operating in the Retail sector.

4. Almarai stock is increasing during the whole period, however, Salama stock price

seems to be more volatile. because of its high fluctuation during the same period

so risk is higher for Salama

5. we expect a negative correlation (co-movement) between the two stocks.

6. having a negative correlation between the stocks will be useful for portfolio

management.

7. in fact, selecting these two stocks reduces the level of portfolio risk. Diversification

benefit in.

8. we expect negative correlation between the two stocks because we have two

firms operating in two different sectors.

3) Stock Return (HPR: holding period return)The stock return is defined as follows:

iR =P0

(P 1 P0)+D1 P0 = initial stock priceP1 = Ending Stock price (period 1)D1 = Dividends

eusethissymbol( ) topointthatthisvariableisrandom.w

i= stock, Ri= is the stock return during the holding period (t,t+1)

1 andD1 arerandomvariables(prices), sotheyarenotedandP 1 andD.P

.iwillbearandomvariable RiskonStockReturn R

contains two parts:i R

1. . profit or losses generated by the price fluctuation (change).P0

(P1 P0)

2. . profit generated by the distribution of dividends.P0D1

Example of the Total Stock Return Formula

the original price is $1000 and the ending price is $1020. The appreciation of the stock is then

$20. The $20 in price appreciation can then be added to dividends of $20 which would equal a

total return of $40. This can then be divided by the original price of $1000 which would equal

a percentage return of 4%.

i.e.: for Rajhi stock we have:

P0 = 74 SARP1= 78 SARD1= 2.55 SAR/share

= 0.0885 or 8.85%rajhiR =74

(78 74)+2.55

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4) The Expected Return and Risk4.1.1) The expected return: (with probability distribution)

(Ri) iRiE = n

i=1

P E(R i)= the expected return

Pi= the probability we must have in

i=1

=P = 1

i.e.: We assume D 1 = 0 , P0= 70 SAR todayprice

stock price

probabilities for next

week

Probability that this

price might happens

Stock Return

i R

Expected return

(Ri)E

75 0.2 7.14%70

75 70 = 0.01428

70 0.1 0%70

70 70 = 0

73 0.4 4.28%70

73 70 = 0.01712

78 0.2 11.42%70

78 70 = 0.02284

78 0.1

11.42%7078 70

=0.01142

total 1 0.06566 or 6.56%

4.1.1) The expected return: (with historical data)

(Ri) iE

=

n

i=1R

n

E(R i)= the expected return

n= number of prices

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i.e.: assume P0= 100 SAR todayprice

months closing price

last few months

Dividends Stock Return

i R

Expected return

(Ri)E

1 115 3 18%100

(115 100)+3=

2 110 3 13%100

(110 100)+3=

3 90 3 %100

(90 100)+3= 7

4 100 4 4%100

(100 100)+4=

5 105 5 10%100

(105 100)+5=

total 8%3 .6%5

38% = 7

*

%20= n1

(Ri) iRiE = n

i=1

P iP = n1

(Ri) i (Ri)E

=

n

i=1n

1

RE

=

n

i=1n

Ri

4.2 How To Measure Risk

in financial markets, risk is measured by the variance and the standard deviation.

The variance is generated by:

4.2.1) (with probability distribution)

(Ri)2 = i(Ri (Ri)n

i=1

P E 2

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4.2.2) (with historical data)

(Ri)2 = n

i=1n

(Ri E(Ri) 2

Pi= the probabilities=1

n = the number of possible returns

Ri= stock return

E(Ri)= expected return = or(Ri) iRiE = n

i=1

P (Ri) iE = n

i=1

R n

the standard deviation = ,(Ri) = (Ri)2 2 , nd 0 a 0

Almarai Salama Cooperative Insurance Co

(R) (R) 0, because Ea>0 RFR and Oa>0[ aEa RF R]

4. is called the Risk Price >0, however (Ea-RFR): Risk Premium

[ a

Ea RF R

]

The risk premium is the compensation against risk when investors select risky assets.

The risk premium is positively correlated to the degree of risk aversion.

Risk Aversion Risk Premium 5. We can say that the CML is the NEW EFFICIENT FRONTIER representing efficient

portfolios containing both stocks and bonds. this new efficient frontier is linear

and positive

(Z) is an impossible portfolio when investing only in stocks because (Z) is above the

curve(Efficient frontier). However, using stocks and bonds, (Z) is a possible portfolio, but

inefficient. We can find another portfolio (Z1) better than (Z).

Z1>Z because .andEz1 zz1

= z >E

Therefore, all the Efficient portfolios of stocks and bonds are exactly located on the

straight line (CML).

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CML: for a portfolio A.p F R E =R + [ aEa RF R] p

we replace (A) by the Market:

The General Equation of the CMLp F R E =R + [ mEm RF R] p

7. According to the CML, the expected return is the sum of two parts:

Part1: RFR: Fixed return on T.Bills (Bonds)

Part2: Risk Price adjusted (multiplied) by the level of risk (measured by S.D).m

Em RF R

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7.1.1) Leverage effect in the CML:

(R1) F R E = R + [ mEm RF R] p

CML is the New Efficient frontier for portfolio containing stocks and bonds.

Portfolio (Z) is a possible and efficient because its located on the CML.

The main feature of the CML is to allow for the leverage effect. Its possible to

borrow money (short position on bonds) in order to invest more in the marketportfolio. Therefore, investor profit from borrowing money to have portfolios with

highest expected return and risk than the market.

Example: The initial investment is 1M. (Wm=1.8 and Wrfr=-0.8)=portfolio Z, We

invest 1.8M in the Market portfolio. We borrow 0.8M (we are shorting on 0.8M

bonds).

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Application: The market portfolio contains two stocks A and B.Ea = 12% Eb=15% and RFR= 4%

% %a = 3 b = 6 corr(A,B) =0.6

Wa=0.5, Wb=0.5

1. Write the Ep of CML

2. Draw the CML

p F R E =R + [ mEm RF R] p

we must find Em

Em = WaEa+WbEb= 13.5%

=0.13%WaWbCorr(A, )ab 2p = 2m =W a

2a2 + W b

2b

2+ 2 B

2p

= 2m

..

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7.2) The Security Market Line (SML): The SML represent a relationship between expected return and risk.

Risk in the SML is measured by the covariance between Ri and Rm

(Risk=Cov(Ri,Rm)).

The relationship between expected return and risk is positive because E(Rm)>RFR.

Eq. of the SML:

(Ri) F R R )E = R + [ 2mEm RF R] C ov(Ri, m

(Ri) F R R )E = R + [ 2mEm RF R] C ov(Ri, m

we also can write the SML differently:

< Beta (Ri) F RE = R + Em F R[ R ] 2mCov(R R )i, m

so.. The General Equation of the SML(Ri) F R E = R + Em F R[ R ]

, Beta is measuring the sensitivityof stock return to the market return. =2m

Cov(R R )i, m

(investopedia definition): Beta is a measure of the volatility, or systematic risk, of a

security or a portfolio in comparison to the market as a whole.

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orr(Ri, m) ov(Ri, m) orr(Ri, m)( )( )C R = m

i

Cov(R R )i, m C R =C R

m

i

= m

m

Corr(Ri,Rm)(m

= m

Corr(Ri,Rm)( )i

How we comment on Beta?:

eknowthatCov(X.X)W =n

(x xbar)(x xbar)

=n

(x xbar)

2

= x2

oifRp ms =R = 2mCov(R R )p, m =

2m

2m = 1

Differences between CML & SML:

CML SML

Form Linear Linear

Relationship E(Rp) & Risk E(Rp) & Risk

Sign of the slope + +

Slopem

Em RF R > 0 2m

Em RF R > 0

Risk p Cov(Rp,Rm)

Allows for Leverage effect No leverage effect

Portfolio Stocks+Bonds Stocks+Bonds

Graph

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7.3) Total risk, systematic risk and unsystematic risk:7.3.1) The Market Model:is linear relationship between stock return and market return

x ty =a +b +

i (Rm) iR =a + +

.0036 .34(Rm)y = 0 1

The estimated coefficient of anda

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Beta is the slope coefficient of the Market Model.

So, The Market Model is written as follows:

(R )Ri=

i+

i m +

i

is the stock returnRi

is a constanti is Beta coefficientB

i or =

( )m

2

Cov(R ,R)m im

Corr(R ,R)( )m i i

is the market returnRm

is an error term (Epsilon)i

The Market Model is linear. The direction is depending on the sign of the slope

coefficient Beta.

The Market Model is called the one-factor Model, because we have only one

variable (Rm) to equation. The stock return.

All the other variables are included in the error term ( ). These variables are, fori

example:(inflation, economic growth, interest rate, political risk, oil prices, gold

prices...etc).

7.3.2) Derivation of systematic and unsystematic risk using The

Market Model (PROOF 5 stars) :

(R )Ri=

i+

i m +

i

(R) ( (R ) )2i = 2

i+

i m +

i

( ) ( (R )) ( ) Cov( , (R )) C ov( , ) C ov( (R ), )= 2i + 2

i m + 2

i + 2

i

i m + 2

i

i + 2

i m

i

because its a constant.( )2i = 0

constantov( , (R ))Ci

i m = 0

constantov( , )Ci

i = 0

, Because is independent from the marketov( (R ), )Ci m

i = 0

i

(R) ( (R )) ( ) . ( ) 2

i = 2

i m + 2

i i2 2

m+ 2

i

So, we find that:

Total Risk = Systematic Risk + Unsystematic Risk (Residual, Specific Risk)

(R) . ( )2 i = i2 2m+ 2 i

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Systematic Risk: is the risk related to the market.

Unsystematic Risk: risk that is specific to the firm.

,PartofSystematicRiks% = Syst.Risk

TotalRisk = i2

i

2 2m

PartofUnsystematicRisk% =T otalrisk

Unsyst.risk=

i2

( )2 i

% syst.Risk + % unsyst.risk = 100% = 1

In fact, we make diversification to reduce only one part of risk (Unsystematic

Risk) diversification. In other word, the market risk can not be reducedby

diversification.

Reducing unsystematic Risk implies that total risk will be reduced (Systematic is

constant and depends only on the Beta).

Proof: Diversification and Reduction of unsystematic risk:

, Lets consider a portfolio containing equally weighted stocks(R )Ri = i + i m + i (W1=W2=....Wn)(n Stocks)

p R1 R2 .. RnR =W1 +W2 + . +Wn

W .. ) W Rm Rm .. Rm)= ( 1 1 +W2 2 + . +Wn n + ( 1 1 +W2 2 + . +Wn n

W .. )+ ( 1 1 +W2 2 + . +Wn n

W .. )( 1 1 +W2 2 + . +Wn n = p W .. ) ( 1 1 +W2 2 + . +Wn n = p

(W .. )Rp = p +Rm 1 1 +W2 2 + . +Wn n + p

unsystematic risk = ( ) (W .. )2 p = 2

1 1 +W2 2 + . +Wn n

= ( ) ( ) .. ( ) Cov( ) Cov( )...W212

1 +W22

22 + . +W

2n

2n + 2 1, 2 + 2 1, n

all the Covs equal 0

= )(n1 2 ( ) ( ) .. ( )[ 2 1 + 2 2 + . 2 n ] = ) ( )( n1 2

n

i=12 i

, loserto0, ( ) orecloserto0n n1 c

n1 L m ( )

2p 0

We show that for fully diversified portfolio the unsystematic risk will be reduced to Zero.

as its shown in the graphs. (Reducing specific risk for fully diversified portfolio)

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The Capital Asset Pricing Model (CAPM): The CAPM is describing a linear relationship between return and risk.

Risk in the CAPM is measured by Beta.

The CAPM is written as follows:

(Ri) F R E = R + E F R[ m R ] i According to the CAPM, The expected return is the sum of two parts:

Part 1: Risk free rate = interest rate on T.Bills

Part 2: is the risk premium adjusted (multiplied) by beta.

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8) Valuation of Portfolios Performance:

8.1) Over-valuation and under-valuation stocks:We compare the expected return and risk given by the SML or CAPM and the observed

return (return on the market).

*Observed Return > The expected return our investment decision is that the Stock is

over-valued, we should take a short position (Sell).

*Observed Return < The expected return our investment decision is that the Stock is

under-valued, we should take a long position (buy).

8.2) Treynors composite performance measure: Treynor's objective was to find a performance measure that could apply to all

investors, regardless of their personal risk preferences.

Treynors measure is based on the capital market theory.

Treynor showed that rational investors always prefer portfolios possibility line

with large slope. in other word Treynor introduced the concept of the SML, which

defines the relationship between portfolio returns and market rates of returns,

whereby the slope of the line measures the relative volatility between the

portfolio and the market (beta). The beta coefficient is simply the volatility

measure of a stock portfolio to the market itself. The greater the line's slope, the

better the risk-return tradeoff.

Treynors Measure:

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T= i

Ri RF R

Ri: The average rate of return for a portfolio using a given holding period.

RFR: The average of the RFR using the same period.

Bi: Portfolio i's beta

iskP remium Tm= =1mRm RF R

R

TS > TM

Application: E(Rm)= 14% =Rm , RFR =8% , Holding period= 1 Year

Managers RpAverage Annual Return Beta

Manager W 12% 0.9

Manager X 16% 1.05

Manager Y 18% 1.20

we compute T for each portfolio:

always start with Tm(market)= .061.00

0.14 0.08 = 0

Tw= .0440.900.12 0.08 = 0 Tx= .076

1.050.16 0.08 = 0

Ty= .0831.20

0.18 0.08 = 0

- Lowest performance (Manager W)

- Highest Performance (Manager Y)

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8.3) Sharpes measure:The Sharpe measure of performance is based on

- The CAPM

- The CML

Si=

i

Ri RF R Ri: The average rate of return for a portfolio (i) using a given holding period.

RFR: The average of the RFR using the same period.

: the standard deviation of the rate of return using the same period.i

Si: Risk Premium return earned per unit of total risk

*T is using only systematic risk assuming that ( )2

i= 0

*S is using total risk ( )i2

. ( )

i

2 = i

2 2m+

2

i

Application: RFR=7%

PF Return Beta i Sp Tp Rank S Rank T

P 0.15 1 0.05 1.6 0.08 2 3

Q 0.20 1.5 0.10 1.3 0.0866 4 2

R 0.10 0.6 0.03 1 0.05 5 5

S 0.17 1.1 0.06 1.666 0.09 1 1

Market 0.13 1 0.04 1.5 0.06 3 4

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