Presentation 2: ObjectivesTo introduce you with the Principles of

Investment Strategy and Modern Portfolio Theory

Topics to be coveredIndifference curve and utility functionDiversificationMean and Variance of a portfolioEfficient FrontierEffect of CorrelationSkills for the above

1

Asset Allocation Revisit

The process of selecting assets from a variety of different asset classes, designed to balance investors’ expected returns with their tolerance for risk.

A fundamental approach designed to reduce risk, preserve profits and improve total returns

Rational Investor Attitude to Risk

Blue: E, Red: B

Rational Investor Attitude to Risk

Risk Loving - high return for high riskRisk neutral - indifferent to riskRisk averse - low risk, low returnAttitudes to risk reflected by shape of

utility (indifference) curves

Indifference CurveIndifference curve captures the various

risk-return scenario providing the same utility level for investors.

The curve (Utility Function) of an individual investor allows us to measure the subjective value the individual would place on investment.

Indifference Curve

Indifference Curve Indifference curve and risk aversion

A B C

Utility Function

WhereU = utilityE ( r ) = expected return on the asset or

portfolioA = coefficient of risk aversions2 = variance of returns

21( )

2U E r A

Indifference Curve

Utility Function

An asset with expected rate of return 10% and standard deviation 28%.

What is the utility level of this specific asset to an risk Conservative(4) investors.

How about an risk Aggressive (2) investor?

Indifference Curve

Select the portfolio where the highest attainable indifference curve is tangential (just touching) to the efficient frontier.

Indifference Curve

Which is most accurate about indifference curve?A. For a risk averse person the indifference curve is flatterB. Investors expected utility may be different along the indifference curveC. Indifference Curve do not intersect

Indifference Curve

Asset Allocation

Diversification Investment Strategy Having eggs in many basket or having

eggs in the right baskets

Diversification

13Source: Sardonic Salad

Diversification

Diversification can help reduce risk only when you combine the right assets.

Right assets whose value movements are not in a perfect synchrony.

Risk averse investor will diversify to at least some extent

more risk-averse investors diversifying more completely than less risk-averse investors.

Diversification

15

CAR Magic CAR Dream Oil Extra2007 42% 18% -4%2008 60% 42% -10%2009 14% 14% 18%2010 -10% -4% 42%2011 -4% -10% 60%2012 18% 60% 14%

YearStock Returns

Diversification

16

CAR Magic CAR Dream Oil Extra2007 42% 18% -4%2008 60% 42% -10%2009 14% 14% 18%2010 -10% -4% 42%2011 -4% -10% 60%2012 18% 60% 14%

Stock ReturnsYear

Diversification

17

CAR Magic CAR Dream Oil Extra2007 42% 18% -4%2008 60% 42% -10%2009 14% 14% 18%2010 -10% -4% 42%2011 -4% -10% 60%2012 18% 60% 14%

Average Return 20% 20% 20%

Stock ReturnsYear

Diversification

18

CAR Magic CAR Dream Oil Extra2007 42% 18% -4%2008 60% 42% -10%2009 14% 14% 18%2010 -10% -4% 42%2011 -4% -10% 60%2012 18% 60% 14%

Average Return 20% 20% 20%Risk 26.8% 26.8% 26.8%

Stock ReturnsYear

Diversification

19

CAR Magic CAR Dream Oil Extra2007 42% 18% -4% 30.0% 7.0%2008 60% 42% -10% 51.0% 16.0%2009 14% 14% 18% 14.0% 16.0%2010 -10% -4% 42% -7.0% 19.0%2011 -4% -10% 60% -7.0% 25.0%2012 18% 60% 14% 39.0% 37.0%

Average Return 20% 20% 20% 20% 20%

Stock ReturnsYear Portfolio Returns

Diversification

20

CAR Magic CAR Dream Oil Extra2007 42% 18% -4% 30.0% 7.0%2008 60% 42% -10% 51.0% 16.0%2009 14% 14% 18% 14.0% 16.0%2010 -10% -4% 42% -7.0% 19.0%2011 -4% -10% 60% -7.0% 25.0%2012 18% 60% 14% 39.0% 37.0%

Average Return 20% 20% 20% 20% 20%Risk 26.8% 26.8% 26.8% 24.1% 10.2%

Stock ReturnsYear Portfolio Returns

Diversification

21

Diversification

22

Mean, Variance and Covariance of a Portfolio If we have a portfolio of assets we could compute

the expected return and the variance of the whole portfolio.

Suppose assets 1,2, …., n are held in the proportions X1, X2, …, Xn then the expected return on a portfolio is given by:-

where X1 + X2 + … + Xn = 1. Note that JP Morgan uses this as an

approximation (although the returns are calculated using log returns as opposed to percentage returns).

N

iiipp RXRER

1

23

Mean, Variance and Covariance of a Portfolio

When computing the variance we also have to concern ourselves with how the asset returns vary together - the covariance.

If returns tend to move in opposite directions then this reduces the overall variability of the portfolio.

But if returns tend to move in the same direction then the variability of the portfolio is increased.

In the analysis that follows s12 refers to the covariance between assets 1 and 2.

24

Mean, Variance and Covariance of a Portfolio

Variance of Portfolio with 2 assets:

Can we derive the above?

25

1221

2

2

2

2

2

1

2

1

2 2 XXXXp

Mean, Variance and Covariance of a Portfolio

Applying the derivation explained for 2 assets above now prove the following for 3 assets case:

26

23321331

122123

23

22

22

21

21

2

22

2

XXXX

XXXXXp

Mean, Variance and Covariance of a Portfolio

A measure of the association between two assets which is always in the range +1 to -1 is the correlation coefficient. This is defined as:-

27

21

1212

The covariance and

correlation coefficient always have the same sign

Mean, Variance, Covariance and Correlation of a Portfolio

Thus for a two variable portfolio:-

28

2112211221 2or 2 XXXX

1221

2

2

2

2

2

1

2

1

2 2 XXXXp

29

30

The opportunity set under risk: Efficient Portfolios

How can we use the previous analysis to construct portfolios?

31

The opportunity set under risk: Efficient Portfolios Suppose we consider a two asset (1 and 2)

portfolio with proportion X1 invested in asset 1 and X2 (1- X2) invested in asset 2.

32

The opportunity set under risk: Efficient Portfolios The expected return of such a portfolio is:-

The variance is then:-

33

21

_

1

_

1

_

1 RXRXR p

121122

21

21

21

2 )1(21 XXXXP

21121122

21

21

21

2 )1(21 or, XXXXP

The opportunity set under risk: Efficient Portfolios Let us focus on the correlation coefficient r12. If the correlation coefficient is negative between 1

and 2 then the risk of a single asset (1 or 2) portfolio will be reduced by combining them together.Since r12 < 0 then the returns on 1 and 2 tend

to move in opposite directions and will partially offset each other.

When the return on one is high the return on the other is low (and vice versa) and as a result portfolio returns are less variable.

37

The opportunity set under risk: Efficient Portfolios If r12 is +1 then the returns on assets will always

tend to move in the same direction and there is no risk reduction.

38

Negative correlationWhen correlation between two assets is “-

1” you have the ultimate in diversification benefits and a risk free portfolio.

The graph on the next slide shows such an outcome. Perfect negative correlation gives a mean combined return for two securities over time equal to the mean for each of them, so the returns for the portfolio show no variability.

39

40

Time series of returns for two assets with perfect negative correlation

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12Time

Re

turn

Returns from asset 1 over time

Returns from asset 2 over time

Mean Return from Portfolio of assets 1 and 2

Any returns above and below the mean for each of the assets are completely offset by the return for the other asset, so there is no variability in total returns, that is, no risk, for the portfolio.

Negative correlation

Perfect Positive Correlation (r = +1)

41

Risk and Return

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 0.01 0.02 0.03 0.04 0.05 0.06

Risk (weekly standard deviation)

Retu

rn (

weekly

)

All money in Glaxo

All money in Lloyds

Perfect Positive Correlation (r = -1)

42

Risk and Return

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 0.01 0.02 0.03 0.04 0.05 0.06

Risk (weekly standard deviation)

Retu

rn (

weekly

)

All money in Glaxo

All money in Lloyds

No Correlation (r = 0)

43

Risk and Return

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 0.01 0.02 0.03 0.04 0.05 0.06

Risk (weekly standard deviation)

Retu

rn (

weekly

)

All money in Glaxo

All money in Lloyds

Actual Correlation (r )

44

Risk and Return

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 0.01 0.02 0.03 0.04 0.05 0.06

Risk (weekly standard deviation)

Retu

rn (

weekly

)

All money in Glaxo

All money in Lloyds

45

Single Index Model “The mean variance approach” to portfolio

analysis involves estimating the mean and variance of alternative portfolios and then selecting the portfolio that offers the best mean-variance combination.

What do we need to do this ? Mean and variance of each asset. Correlation between each asset. Mean and variance of differing combinations of assets.

This is fine when the portfolio consists of only two assets. What if there are 20 ?

46

If there are 20 assets we need the same info as for 2:-Mean and variance of each asset

40 parametersCorrelation between each asset

each one of the assets can be correlated with each of the 19 other assets = 19 x 20 = 380.

But correlation matrices are symmetric = 190

So for each combination we need 230 parameters, or:-2N + N(N-1)/2 parameters for an N asset portfolio.

E.g. N = 200, parameters = 400+200(199)/2 = 20,300 parameters.

Single Index Model

47

Thank you

48

Variance (X) = Cov (X, X)Variance (Rp) = Cov (Rp, Rp)When R1, R2 are the returns from X1 and X2 respectivelyVariance = Cov (x1R1 + X2R2, X1R1+ X2R2)Applying Cov(A+B,C+D) = Cov(A,C)+Cov(A,D)+Cov(B,C)+Cov(B,D)Variance (Rp) =X2

1 Var(R1) +X22Var(R2)+ 2X1,X2Cov(R1,R2)

Replacing Covariance by Correlation:X2

1 Var (R1) + +X22Square Var (R2)+ 2X1,X2 Corr(R1 R2)

SD(R1) SD(R2)

Note to Formula Derivation