Fin management materials / 3 P4AFM-Session04_j08

SESSION 04 – PORTFOLIO THEORY AND CAPM

OVERVIEW

Objective

¾To apply portfolio theory to investment analysis and appreciate its link to the Capital Asset Pricing Model.

¾To apply CAPM in calculating project-specific discount rates.

¾To discuss the limitations of CAPM and the development of Arbitrage Pricing Theory.

PORTFOLIO

THEORY

CAPM

ARBITRAGE

PRICING THEORY

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SESSION 04 – PORTFOLIO THEORY AND CAPM

1WHAT IS RISK?

1.1 Definition

An investment is defined as having a degree of risk if its returns are uncertain or variable. The amount of risk an investment has will depend upon the variability of the returns of the investment around the average return.

1.2Measurement

¾Risk is measured by the standard deviation of the returns around the average return.

Example 1

Two investments have the following possible returns and associated probabilities:

What are the expected return and the standard deviation of these investments?

Solution

Investment 1

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SESSION 04 – PORTFOLIO THEORY AND CAPM

Investment 2

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1.3Investors and risk

¾Investors are assumed to be risk averse.

¾Investors in general will seek to maximise return in relation to risk - this is referred to as “mean –variance efficiency” and is the basis of portfolio theory.

Example 2

Which of the two investments in Example 1 is a risk averse investor likely to prefer?

Solution

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SESSION 04 – PORTFOLIO THEORY AND CAPM

2PORTFOLIO THEORY

2.1Risk reduction

¾Risk averse investors will accept more risk provided that they are compensated for it by an adequate return. However if they could reduce their risk without a reduction in return this would be even better.

¾This is possible by the process of diversification.

¾When an investment (a) is added to an existing portfolio (b) the risk of the new portfolio formed will usually be less than a simple weighted average of the individual risks of a and b alone.

This will almost always be the case

¾Furthermore, some investments can be added to an existing portfolio to give a new portfolio of lower risk than the existing portfolio.

This is because of the way that the investments’ returns interact with each other in different sets of economic conditions – “states of the world”.

¾Whether such investments are acceptable will depend on the effect that they have on the return of the existing portfolio.

¾Portfolio theory appraises investments by looking at the effect that they have on the risk/return characteristics of the total investments held. The approach involves

a comparison of the return of the existing portfolio to the return of the existing portfolio + the new investment

AND

a comparison of the risk of the existing portfolio to the risk of the existing portfolio + the new investment

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SESSION 04 – PORTFOLIO THEORY AND CAPM

¾Possible outcomes

¾It is theoretically possible to apply this approach to any number of combinations of investments. In practical terms the appraisal will look at two assets only. These two assets are the existing portfolio (which may of course be made up of lots of investments) and a new investment.

¾For an individual investor portfolio theory can be used to make decisions about including a new share in a portfolio of shares.

¾For a firm it can be used to make decisions about including a new project in the existing portfolio of projects.

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SESSION 04 – PORTFOLIO THEORY AND CAPM

2.2Two asset portfolios - mathematics

Risk is measured by standard deviation (s)

Weighting of investments should be by market values.

¾What will be the return and risk of the portfolio?

¾The return will be a simple weighted average of the return on each investment, but the risk will depend on how much the fluctuations in A and B counteract each other.

%Return

B

rb

ra

A

Time

¾The extent to which this happens will depend on the correlation between the two investments’ returns.

¾Correlation coefficients can have a value between –1 and +1:

−+1 means the two investments are perfectly positively correlated i.e. they move in the same direction and in the same proportion. In this (rare) situation the risk of the portfolio will be a simple weighted average of the risks of the two investments i.e. no risk reduction is available.

−–1 means the two investments are perfectly negatively correlated i.e. they move in the opposite direction but in the same proportion. Although this is also rare in practice it would be perfect for risk reduction.

−0 means that there is no correlation between the returns

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SESSION 04 – PORTFOLIO THEORY AND CAPM

Formulae for risk and return of a two asset portfolio:

Return = wara + wbrb

¾The final element of the two asset portfolio risk equation is the vital part:

¾It is the correlation coefficient which determines how much risk reduction is possible.

¾The formula for the risk of a two asset portfolio is published in the exam.

Illustration 1

Note that if r ab = +1 then the covariance becomes:

i.e. if r ab = +1 the risk of the portfolio is a simple weighted average of the individual risks .

In practice it is almost always the case that the correlation coefficient between two investments is less than +1. In this case the risk of the portfolio is less than a simple weighted average of the individual risks.

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SESSION 04 – PORTFOLIO THEORY AND CAPM

Example 3

The following information is given about securities A and B:

A portfolio is to be constructed with 60% of the available funds invested in A and 40% in B.

Determine the return and risk of the portfolio if the correlation coefficient between the returns of A and B is:

(a)0.9

(b)0.5

(c)−0.2

Solution

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SESSION 04 – PORTFOLIO THEORY AND CAPM

2.3The Markowitz efficient frontier

¾With many risky assets available, the risk and return combinations become an area:

Efficient frontier

Return

all possible risk and return combinations from all available risky assets: known as “OPPORTUNITY SET”

Risk

¾Only those portfolios lying along the “north west frontier” are mean-variance efficient.

¾A portfolio is mean-variance efficient if it offers

−maximum return for a given level of risk

OR

−minimum risk for a given level of return

¾The frontier is known as the Markowitz efficient frontier.

2.4The Capital Market Line

¾If an investor constructs a portfolio of shares then obviously these are all risky assets.

¾However an investor may also consider buying risk-free securities e.g. Treasury Bills

¾An individual investor may decide on one of the following courses of action:

to invest only in risky shares;

to withdraw some of his investment in shares and invest in risk-free securities;

to increase his investment in shares by borrowing at the risk-free rate (Rf) i.e. using leverage.

¾Mean-variance efficient portfolios now become a mix of risky shares and risk-free securities. This new efficient frontier is known as the Capital Market Line.

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SESSION 04 – PORTFOLIO THEORY AND CAPM

Return

Most efficient

portfolios Capital Market Line (CML)

M

Rf

Risk

¾Point M is where the frontier of equity portfolios touches the Capital Market Line.

¾The Capital Market Line represents various combinations of equity portfolio M and a risk free investment.

¾These most efficient portfolios lie along the line from Rf to M and beyond, i.e. these portfolios maximise return for a given level of risk.

¾Assuming that all investors are rational and risk averse and have access to the same information, they will all choose portfolios lying on the Capital Market Line.

¾M is therefore the only efficient portfolio of shares and is known as the Market Portfolio – an equity portfolio that contains every share on the market.

2.5Systematic vs. unsystematic risk

¾It is possible for an investor to diversify away a certain type of risk by investing in more shares. The type of risk that can be removed by diversification is known as unsystematic risk or unique risk.

¾A rational risk-averse investor will diversify away as much risk as is possible and will therefore hold the market portfolio of shares i.e. a portfolio containing every share on the market. This portfolio contains zero unsystematic risk.

¾The investor may then choose to combine the market portfolio with the risk-free asset by either borrowing or investing at the risk free interest rate.

¾However the market portfolio itself cannot be risk-free; its returns rise and fall due to factors such as macro-economic changes. Therefore some risk remains even in a perfectly diversified equity portfolio.

¾This is known as systematic risk or market risk. Systematic risk cannot be removed from equity investments.

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