Unit4 portfolio theory & CAPM

Unit 4

Portfolio Theory and Capital Asset Pricing Model

Diversification

Diversification is the financial equivalent of the cliche “Don’t put all your eggs in one basket”

Diversification results in reduced risk in the portfolio.

This is because of the averaging effect

Price

Time

Diversification

Diversification is the financial equivalent of the cliche “Don’t put all your eggs in one basket”

Diversification results in reduced risk in the portfolio.

This is because of the averaging effect

Price

Time

We think of “risk” as being linked to the volatility of returns – so averaging tends to smooth volatility

Diversification

Remember that the risk that is diversified away is the “specific” risk: the risk pertaining to the industries and businesses in that portfolio.

“Market” risk: the risk of investing in equities rather than , say, bonds, cannot be diversified away.

Diversification

Remember that the risk that is diversified away is the “specific” risk: the risk pertaining to the industries and businesses in that portfolio.

“Market” risk: the risk of investing in equities rather than , say, bonds, cannot be diversified away.

Hence:

Average risk

No shares in portfolio

Market risk

Portfolio Theory

This theory assumes that investors are “risk averse”

Note: “risk averse” does NOT means that investors will not take risks (“risk avoidance”)

Portfolio Theory

This theory assumes that investors are “risk averse”

Note: “risk averse” does NOT means that investors will not take risks

Instead it means that investors will take on risk if the returns on offer are sufficient

So the attractions of an investment are a combination of the expected returns and the risk, as measured by standard deviation

Portfolio Theory – Expected returns and standard deviation

Consider the following distribution of returns

Frequency

ReturnR

Portfolio Theory – 2 share portfolio

Let us assume that we have a portfolio comprising two shares

The expected return of this portfolio is given by:

E(Rp) = WxE(Rx) + WyE(Ry) (the weighted average of the two returns)





Proportion of the portfolio that is X or Y (for example, 0.6 & 0.4)

Expected returns on X & Y (eg 10% and 12%)

What is the expected returns here?





Proportion of the portfolio that is X or Y (for example, 0.6 & 0.4)

Expected returns on X & Y (eg 10% and 12%

Using these figures the expected return would be 0.6 x 10% + 0.4 x 12% = 10.8%





The risk of the portfolio is given by:

Sp = √(Wx2Sx

2 + Wy2Sy

2 + 2WxWySxSyCorrxy






Sp = √(Wx2Sx

2 + Wy2Sy

2 + 2WxWySxSyCorrxy

The standard deviations (ie risk) of the investment , say 30% and 40%

The correlation coefficient of X & Y – say 0.3

What is the risk (SD) of the portfolio? (recall that the weightings are 0.6 and 0.4)






Sp = √(Wx2Sx

2 + Wy2Sy

2 + 2WxWySxSyCorrxy

=√(0.6 x 0.6 x 30 x 30 + 0.4 x 0.4 x 40 x 40 + 2 x 0.6 x 0.4 x 30 x 40 x 0.3) = 27.4%


Combining these formulae one can start to plot a graph of return against risk for varying weights of portfolio

Return

Risk

This graph represents all the possible combinations of holding shares in X & Y (eg 10% X 90%Y, or 20% X, 80% Y etc)



Return

Risk

You would not pick this weighting. Why not?



Return

Risk

Because the portfolio at “A” has the same risk but a higher return

A

Portfolio Theory – many-share portfolio

If we had significantly more shares than 2 then the graph would be a follows

Return

Risk



Return

Risk

All of the possible combinations of the shares would be in the shaded area



Return

Risk

Any portfolios lying on this edge are the most efficient – they earn the highest returns at the lowest risk – hence it is called the “efficient frontier”



Return

Risk

Any portfolios lying on this edge are the most efficient – they earn the highest returns at the lowest risk – hence it is called the efficient frontier

You want the portfolio to be as far in the north west of this graph as possible

Portfolio Theory – Domination

Would we prefer A or B and C or D?

Return

Risk

A

C

B

D

Portfolio Theory – Domination

Would we prefer A or B and C or D?

Return

Risk

A

C

B

D

A has a higher return than B for the same risk and C has the same return as D for a lower risk. So we prefer A and C

Portfolio Theory – the Market Portfolio

Return

Risk

So the most efficient portfolio will be the one along the frontier that is most north-west. This is called the “market portfolio”

M

Capital Asset Pricing Model (CAPM)

Let us now combine our market portfolio with a risk free investment - what is meant by a risk free investment ?



In CAPM we assume that we can invest or borrow at the risk free rate

Rf

M

Here we are only investing in risk free




Rf

M


Here we are only investing in the market portfolio




Rf

M


Here we are investing in part risk free and part market





Rf

M


Here we have BORROWED at the risk free rate and invested in the market portfolio

Here we are investing in part risk free and part market


CAPM

As the risk-free – market portfolio relationship is a straight line it is straightforward to describe it mathematically.

Once we have that relationship we then have a formula for the return on a share compared to the risk of that share

CAPM

So we want an easy measure of risk. For this we use beta (β). This measures the relative volatility of the share against the market portfolio.

CAPM

So we want an easy measure of risk. For this we use beta (β). This measures the relative volatility of the share against the market portfolio.

Beta for the market is set as one.

Shares whose beta are greater than one are called “aggressive”; less than one “defensives”

CAPM

Let us build up the relationship. The y axis will be the expected return on a share and the x axis the risk associated with the share as measured by its beta

Risk = β

Return = Re

CAPM

A straight line is given as y = mx + c

Let us put in some points we know: the risk free and market .

What is c?

Rf

M

Risk = β

Rm

10

CAPM


What is c? It is the y-intercept = Rf

What is m?

Rf

M

Risk = β

Rm

10

CAPM


Y = the return on the share Re, X = risk, using beta

What is c? It is the y-intercept = Rf

What is m? It is the gradient

Rf

M

Risk = β

Rm

1

The gradient is given by:

Rm-Rf 1

0

CAPM

So the formula of the return on equity is:

Re = (Rm-Rf)β + Rf

Rf

M

Risk = β

Rm

1

Re

CAPM

A defensive will have a lower return than an aggressive

Re = (Rm-Rf)β + Rf

Rf

M

Risk = β

Rm

1

Defensivesβ< 1

Aggresivesβ > 1

Next Time

We shall look at project appraisals – discounted cash flow etc - from unit 5

Unit4 portfolio theory & CAPM

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Transcript of Unit4 portfolio theory & CAPM