This document discusses portfolio theory and the Capital Asset Pricing Model (CAPM). It explains that diversification reduces risk through averaging effects. While specific risk can be diversified away, market risk cannot. Portfolio theory assumes investors are risk averse and seek the highest returns for a given level of risk. The CAPM uses beta to measure a stock's risk relative to the market and defines the security market line, where expected return is equal to the risk-free rate plus a risk premium based on beta. Stocks with beta greater than one are more volatile than the market.

1. Unit 4
Portfolio Theory and Capital Asset Pricing
Model

2. 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

3. 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

4. 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.

5. 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

6. Portfolio Theory
This theory assumes that investors are “risk averse”
Note: “risk averse” does NOT means that investors will not take risks (“risk
avoidance”)

7. 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

8. Portfolio Theory – Expected returns and
standard deviation
Consider the following distribution of returns
Frequency
ReturnR

9. Portfolio Theory – Expected returns and
standard deviation
Consider the following distribution of returns
Frequency
ReturnR

10. 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)

11. 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?

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

13. 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)
The risk of the portfolio is given by:
Sp = √(Wx
2Sx
2 + Wy
2Sy
2 + 2WxWySxSyCorrxy

14. 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)
The risk of the portfolio is given by:
Sp = √(Wx
2Sx
2 + Wy
2Sy
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)

15. 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)
The risk of the portfolio is given by:
Sp = √(Wx
2Sx
2 + Wy
2Sy
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%

16. Portfolio Theory – 2 share portfolio
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)

17. Portfolio Theory – 2 share portfolio
Combining these formulae one can start to plot a graph of return against risk for
varying weights of portfolio
Return
Risk
You would not pick this
weighting. Why not?

18. Portfolio Theory – 2 share portfolio
Combining these formulae one can start to plot a graph of return against risk for
varying weights of portfolio
Return
Risk
Because the portfolio at
“A” has the same risk
but a higher return
A

19. Portfolio Theory – many-share portfolio
If we had significantly more shares than 2 then the graph would be a follows
Return
Risk

20. Portfolio Theory – many-share portfolio
If we had significantly more shares than 2 then the graph would be a follows
Return
Risk
All of the possible
combinations of the shares
would be in the shaded area

21. Portfolio Theory – many-share portfolio
If we had significantly more shares than 2 then the graph would be a follows
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”

22. Portfolio Theory – many-share portfolio
If we had significantly more shares than 2 then the graph would be a follows
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

23. Portfolio Theory – Domination
Would we prefer A or B and C or D?
Return
Risk
A
C
B
D

24. 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

25. 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

26. 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 ?

27. 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

28. 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
Here we are
only investing in
the market
portfolio

29. 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
Here we are
investing in part risk
free and part market
Here we are
only investing in
the market
portfolio

30. 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
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
Here we are
only investing in
the market
portfolio

31. 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

32. 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.

33. 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”

34. 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

35. 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

36. CAPM
A straight line is given as y = mx + c
What is c? It is the y-intercept = Rf
What is m?
Rf
M
Risk = β
Rm
10

37. CAPM
A straight line is given as y = mx + c
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

38. CAPM
So the formula of the return on equity is:
Re = (Rm-Rf)β + Rf
Rf
M
Risk = β
Rm
1
Re

39. CAPM
A defensive will have a lower return than an aggressive
Re = (Rm-Rf)β + Rf
Rf
M
Risk = β
Rm
1
Defensives
β< 1
Aggresives
β > 1

40. Next Time
We shall look at project appraisals – discounted cash
flow etc - from unit 5