Risk and Return – Part 3

For 9.220, Term 1, 2002/0302_Lecture14.pptStudent Version

Outline

1. Introduction2. The Markowitz Efficient Frontier3. The Capital Market Line (CML)4. The Capital Asset Pricing Model

(CAPM)5. Summary and Conclusions

Introduction We have seen that holding portfolios of

more than one asset gives the potential for diversification.

We will now look at what might be an optimal strategy for portfolio construction – being well diversified.

We extend the results from this into a model of Risk and Return called the Capital Asset Pricing Model (CAPM) that theoretically holds for individual securities and for portfolios.

The Opportunity Set and The Efficient Set

Expected Return and Standard Deviation for Portfolios of Two Assets Plotted for Different Portfolio Weights

0%

5%

10%

15%

20%

25%

30%

0% 5% 10% 15% 20% 25%

Portfolio Standard Deviation

Po

rtfo

lio E

xpec

ted

Ret

urn

100% Stock 1

100% Stock 2

The portfolios in this area are all dominated.

The Opportunity Set when considering all risky securities

Consider all the risky assets in the world; we can still identify the Opportunity Set of risk-return combinations of various portfolios.

E[R]

Individual Assets

The Efficient Set when considering all risky securities

The section of the frontier above the minimum variance portfolio is the efficient set. It is named the Markowitz Efficient Frontier after researcher Harry Markowitz (Nobel Prize in Economics, 1990) who first discussed it in 1959.

E[R]

minimum variance portfolio

efficient frontier

Individual Assets

Optimal Risky Portfolio with a Risk-Free Asset

In addition to risky assets, consider a world that also has risk-free securities like T-bills.

We can now consider portfolios that are combinations of the risk-free security, denoted with the subscript f and risky portfolios along the Efficient Frontier.

E[R]

The riskfree asset: riskless lending and borrowing

Consider combinations of the risk-free asset with a portfolio, A, on the Efficient Frontier.

With a risk-free asset available, taking a long f position (positive portfolio weight in f) gives us risk-free lending combined with A.

Taking a short f position (negative portfolio weight in f) gives us risk-free borrowing combined with A.

P

E[R]

Rf

Portfolio A

The riskfree asset: riskless lending and borrowing

Which combination of f and portfolios on the Efficient Frontier are best?

P

E[R]

Rf

What is the optimal strategy for every investor?

M: The Market Portfolio

The combination of f and portfolios on the Efficient Frontier that are best are…

All investors choose a point along the line…

In a world with homogeneous expectations, M is the same for all investors.

P

E[R]

Rf

CML stands for the Capital Market Line

M

CML

A new separation theorem

This separation theorem states that the market portfolio, M, is the same for all investors. They can separate their level of risk aversion from their choice of the risky component of their total portfolio.

All investors should have the same risky component, M!

P

E[R]

Rf

M

CML

Given Separation, what does an investor choose?

While all investors will choose M for the risky part of their portfoio, the point on the CML chosen depends on their level of risk aversion.

P

E[R]

Rf

M

CML

The Capital Market Line (CML) Equation

The CML equation can be written as follows:

Where EPi = efficient portfolio i (a portfolio on the CML composed of

the risk-free asset, f, and M) E[ ] is the expectation operator R = return σ = standard deviation of return f denotes the risk-free asset M denotes the market portfolio

M

fMEPfEP

RRERRE

ii

Note: the CML is our first formal relationship between risk and expected return. Unfortunately it is limited in its use as it only works for perfectly efficient portfolios: composed of f and M.

The Capital Asset Pricing Model (CAPM) If investors hold the market portfolio, M, then the risk of

any asset, j, that is important is not its total risk, but the risk that it contributes to M.

We can divide asset j’s risk into two components: the risk that can be diversified away, and the risk that remains even after maximum diversification.

The division is found by examining ρjM, the correlation between the returns of asset j and the returns of M.

Asset j’s total risk is defined by σj

The part of σj that can be diversified away is (1-ρjM)● σj

The part of σj that remains is ρjM● σj

Non-diversifiable risk and the relation to expected return.

We can extend the CML to a single asset by substituting in the asset’s non-diversifiable risk for σEPi:

fMjfj

M

jjM

fMM

jjMfj

EPjjM

M

fMEPfEP

RREβRRESML

σρ

RREσ

σρRRESML

σσρ

RRERRECML

i

ii

:

Let

:

for in sub

:

j

SML stands for Security Market Line. It relates expected return to β and is the fundamental relationship specified by the CAPM.

The Security’s Beta The important measure of the risk of a security in a large

portfolio is thus the beta ()of the security. Beta measures the non-diversifiable risk of a security –

i.e., the risk related to movements in the market portfolio.

22

, )(

M

MiiM

M

Mi

M

iiMi

RRCov

Estimating with regression

Sec

uri

ty R

etu

rns

Sec

uri

ty R

etu

rns

Return on Return on marketmarket

Characteris

tic Line

Characteris

tic Line

Know your betas! The possible range for β is -∞ to +∞ The value of βM is… The value of βf is… For a portfolio, if you know the individual

securities’ β’s, then the portfolio β is…

where the xi values are the security weights.

nn

n

iiip xxxx ...2211

1

Estimates of for Selected Stocks

Stock Beta

C-MAC Industries 1.85

Nortel Networks 1.61

Bank of Nova Scotia 0.83

Bombardier 0.71

Investors Group. 1.22

Maple Leaf Foods 0.83

Roger Communications 1.26

Canadian Utilities 0.50

TransCanada Pipeline 0.24

Examples What would be your portfolio beta, βp, if you had

weights in the first four stocks of 0.2, 0.15, 0.25, and 0.4 respectively.

What would be E[Rp]? Calculate it two ways. Suppose σM=0.3 and this portfolio had ρpM=0.4, what

is the value of σp? Is this the best portfolio for obtaining this expected

return? What would be the total risk of a portfolio composed

of f and M that gives you the same β as the above portfolio?

How high an expected return could you achieve while exposing yourself to the same amount of total risk as the above portfolio composed of the four stocks. What is the best way to achieve it?

Summary and Conclusions The CAPM is a theory that provides a relation between

expected return and an asset’s risk. It is based on investors being well-diversified and

choosing non-dominated portfolios that consist of combinations of f and M.

While the CAPM is useful for considering the risk/return tradeoff, and it is still used by many practitioners, it is but one of many theories relating return to risk (and other factors) so it should not be regarded as a universal truth.