portfolio analysis

Portfolio analysisManagement 4430.02

Class notes by A.P. Palasvirta

Geometric average return

Annualized return if you know the purchase price and the sale price of an asset

04/20/234430.02 Portfolio analysis Notes: A.P. Palasvirta, Ph.D.

2

R t =PT

P0

⎛

⎝⎜

⎞

⎠⎟

1T

−1

Forecasting risk & return

Expected return

Standard deviation

Covariance

E = Ri ⋅Prii =1

n

∑

SD = Ri −Exyz( )2⋅Pri

i =1

n

∑⎡

⎣⎢

⎤

⎦⎥

0.5

Covx, y = Ri, x −Ex( )⋅ Ri, y −Ey( ) ⋅Prii =1

n

∑

3


Historical risk & returnMean return

Standard deviation

Covariance


4

R =1T

Rtt =1

T

∑

SD =1

T −1Rt −R( )

t =1

T

∑2⎡

⎣⎢⎢

⎤

⎦⎥⎥

0.5

Covx, y =1

T −1Rt, x −Rx( )⋅ Rt, y −Ry( )

t =1

T

∑

Contrasting methods

Forecasting Forward looking Must determine possible outcomes

Probabilities of those outcomes Completely speculative

Historical Backward looking Using time series data Completely definitive

Statistics derived mean exactly the same thing


5

Covariance/variance matrix

Covariance Matrix

1 2 3 4 5 M

1 0.1240 0.0676 0.0650 0.0396 0.0104 0.0283 2 0.0792 0.0642 0.0341 0.0120 0.0242 3 0.0628 0.0309 0.0097 0.0216 4 0.0343 0.0117 0.0196 5 0.0071 0.0083

M 0.0135

Standard Deviation for Assets 0.3521 0.2814 0.2506 0.1852 0.0843 0.1162

Expected Return for Assets 0.1910 0.1750 0.1590 0.1520 0.0920 0.1220

6


Correlation

Correlation normalizes covariance

value of +1 means perfect positive correlation value of 0 means independence between data

series value of -1 means perfect negative correlation

−1 ≤ ρ ≤ 1

ρ a, b =Cov a, b

SD a ⋅SD b

7


DiversificationCorrelation is a measure of how two

assets react together to economic market conditions• High positive correlation

Two assets are affected similarly by economic events

• High negative correlationTwo assets are affected in completely

opposite ways by economic events• Independence -zero correlation

Two assets are act independent of each other relative to economic events

8


Portfolio Statistics

Expected return to the portfolio of multiple assets

Standard deviation to a portfolio of multiple assets

E x E s t xPort i ii

n

ii

n

1 1

1. .

SD x x COVporti

n

i j i jj

n

1 1

0 5

,

.

9


Two-asset Portfolio

Ep

SDP

Efficient Set

Risk preferences

Preferred Portfolio

Opportunity Set

10


Implications of Correlation StatisticsThe lower the pair-wise correlation of two

assets, the greater the diversification benefit of adding those assets to your portfolio

Adding assets which have low pair-wise correlation with each other to your portfolio reduces overall portfolio risk

11


Affects of Correlation

Ep

SDP

HighCorrelation

LowCorrelation

12


Three-asset Portfolio

Ep

SDP

13


Calculating Portfolio values - three asset portfolio

E x E x E x E x EP i ii

1

3

1 1 2 2 3 3

SD x x Cov

x x Cov x x Cov x x Cov

x x Cov x x Cov x x Cov

x SD x SD x SD x x Cov

P i j i jji

,

.

, , ,

, , ,

.

,

1

3

1

30 5

1 1 1 1 2 2 2 2 3 3 3 3

1 2 1 2 1 3 1 3 2 3 2 3

0 5

12

12

22

22

32

32

1 2 1

2 2 2

2 2 1 3 1 3 2 3 2 3

0 5

2 2 x x Cov x x Cov, ,

.

14


Multiple-asset Portfolio

Ep

SDP

15


Systematic Risk

SDp

N

SDM

16


Systematic Risk

As you diversify your portfolio by adding assets, your portfolio standard deviation decreases Optimal approx 20 assets = 21.68% 20*100*$20 = $40,000 buying individual stocks Buy funds which are already diversified

When you are fully diversified, your risk is the risk of the market portfolio

By changing portfolio proportions you can modify risk to suit your preferences

17


Optimal Portfolio

EM

SDM

Less risk averse

Moreriskaverse

CapitalMarketLine

U1

U2

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Capital market line


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Any efficient portfolio will be found on the capital market line

E RP( ) = R f +E RM( ) −R f

SD RM( )⋅SD RP( )

Capital Asset Pricing Model

Prices the risk of asset relative to its systematic risk

gives the required rate of return relative to its systematic risk Risk-free rate of return Beta Risk premium

R i = R f + β i ⋅ RM −R f( )

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The Risk-free Asset

The risk-free asset does not exist except as a theoretical concept

assets used t-bills or t-bonds Low default risk - government backing

For calculation of the company β, companies often use the t-bond rate that has a similar term

21


Beta

β measures only the systematic risk of an assets

β is also a covariance that is normalized by something, the variance of the market

β measures the risk of holding that firms stock in a fully diversified portfolio

βXYZ =Cova, market

Covm, m

=Cova, m

Varm=

Cova, m

SDm2

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Risk premium

Return to the market portfolio less the return to the risk-free rate of return Return to the market portfolio

Uses the 90 day return to the a broad based stock index Toronto stock exchange index Dow index S&P 500 index

T-bond rate of appropriate term

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Statistics

Correlation Matrix

1 2 3 4 5 Beta

1 1.0000 0.6821 0.7366 0.6072 0.3505 2.0963

2 1.0000 0.9103 0.6543 0.5060 1.7926

3 1.0000 0.6658 0.4594 1.6000

4 1.0000 0.7690 1.7926

5 1.0000 0.0083

M 1.0000

Required Rate of Return on Assets

0.2009 0.1791 0.1652 0.1791 0.0506 0.1220

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Security Market Line

β = 1

RM

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Security Market Line - Change in rate of inflation

β = 1

RM

Inflation adjustment

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RM

Security Market Line -Change in Risk Premium

β = 1

RM Slope changereflecting increased Systematic Risk

RM

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