CAPITAL ASSET PRICING MODEL

AN INTRODUCTION

by Richard Diamond

September 2011

PRESENTATION INTRODUCTION

• What is CAPM? Background to model derivation, purposes and issues that CAPM aims to address

• Statistical workings and pre-requisites of in-depth understanding of CAPM (Modern Portfolio Theory)

• CAPM weaknesses and applications in financial management

• Issues of presentation to the UG cohort. Pedagogic advice stemming from the threshold concepts paradigm

CAPM INVENTION

• Capital Asset Pricing Model was developed as an attempt to predict capital markets’ behaviour under the condition of risk

• CAPM is based on equilibrium assumption and used to estimate a risk-adjusted return on asset (‘cost of equity’)

• An academic paper that presented CAPM is Sharpe (1964). Capital Asset Prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (3), 425-442

72

The CAPM

• The Capital Asset Pricing Model (CAPM) – Is a linear factor model, in which the factor is the market return;– Is derived directly from the mean-variance analysis (see “Fundamentals of

Optimization and Application to Portfolio Selection” for more details);;– Is an equilibrium model: it can be used to predict asset prices;– Can be applied to any security or portfolio;– Is expressed in terms of expectations.

• For an investment I, It takes the form

or, alternatively

RrERrE MII

RrERrE MII

CAPM FORMULATION USINGEXPECTATIONS ALGEBRA

72

The CAPM

• The Capital Asset Pricing Model (CAPM) – Is a linear factor model, in which the factor is the market return;– Is derived directly from the mean-variance analysis (see “Fundamentals of

Optimization and Application to Portfolio Selection” for more details);;– Is an equilibrium model: it can be used to predict asset prices;– Can be applied to any security or portfolio;– Is expressed in terms of expectations.

• For an investment I, It takes the form

or, alternatively

RrERrE MII

RrERrE MII

• This reads as “expected excess return on Individual asset equals beta times expected excess return on Market”

• Excess return is historical return r minus risk free rate R. Excess return on market is regarded as ‘the market price of risk’

• Most of textbooks present a re-arranged expression as follows:

THE STRUCTURE OF CAPM This kind of explanation is typical for a US classroom. Being

difficult to comprehend, it shows a complex structure

WHAT IS BETA OF AN ASSET?

• Beta is a parameter that gives information about individual stock and its specific (idiosyncratic) risk

• Beta is calculated as the slope of the simple linear regression (You can remember ‘beta’ from regression formulae before!)

• What is an independent variable in the regression? (Excess return on the market or market price of risk)

BETA CALCULATIONStep-by-step Excel demonstration together with re-explanation of the linear regression model and Q&A. Offer Youtube links

Month S&P 500 MSFT1 1,147.39 30.162 1,076.92 26.133 1,067.14 25.464 989.82 27.355 911.62 23.996 916.07 24.547 815.28 21.878 885.76 26.749 936.31 28.84

10 879.82 25.85

Monthly Stock Prices

S&P 500 MSFT-6.14% -13.35%-0.91% -2.58%-7.25% 7.44%-7.90% -12.29%0.49% 2.29%

-11.00% -10.88%8.64% 22.25%5.71% 7.87%

ReturnsMSFT

y = 0.9939x - 0.0035 R² = 0.31031

-30.00%

-20.00%

-10.00%

0.00%

10.00%

20.00%

30.00%

40.00%

-30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%

Sto

ck R

etur

ns

S&P 500 Returns

Security Characteristic Line

BETA ILLUSTRATIONSTwo examples: beta is similar for a conservative mutual fund and Internet video streaming service. Betas for different industries

HOW EQUILIBRIUM WORKS

• Since the optimal policy is to invest a fraction of wealth into the tangency portfolio, ultimately every investor will hold the tangency portfolio

• As a consequence, equilibrium will be achieved and the market value of each risky asset will become proportional to their weights in the tangency portfolio

• Another formulation: expected returns and variances will become equal to ones of the tangency portfolio

TANGENCY PORTFOLIO?? TOPIC REQUIRES SOME MPT KNOWLEDGE

It is a portfolio of risky assets (equity, commodity) allocated in such a way that it provides ‘the same’ return as a risk free asset (cash, bond) portfolio

Bonds are not strictly risk-free: they have interest rate and credit risks

41

Risk

Return

A

A

B

B

R

t

t

The tangency portfolio

Line parametrized by wt:

tt

ttttt

wRwRRww

1

Tangency portfolio

CAPM

CAPM IS A MODELLove based on outer beauty--of the elegant CAPM formulation,

of course--can be a good start but only love based on deep understanding and acceptance of weakness endures

WHAT CAPM PREDICTS

• CAPM is an equilibrium model and is based on expectations. That means that its use for prediction is theoretically justified

• It follows from the CAPM expectations formulation that the risk premium (excess return) of an individual asset in linear relationship with the risk premium of the market (times beta). The principle is scaleable to a portfolio of assets

• Hence a conclusion that, on average, the market will only compensate for taking on the systematic risk (adjusted by beta that reflects non-systematic risk but as a constant parameter)

PRACTICAL FORMULATION OF CAPM FOR MANAGEMENT

This formulation is used when estimating ‘cost of equity’ for the firm’s capital structure

Risk free rate can be found using the yield curve of US Treasuries (Bloomberg website)

Market rate of return is averaged return on a market index (e.g., Dow Jones or DAX)

Advice: use input from financial markets for corporate budgeting and investment planning but trading requires more sophisticated tools

PEDAGOGIC ISSUES

• An introduction to undergraduate students warrants step-by-step explanation and guidance through mathematics

• It is important make students accustomed to ‘the tools of the trade’ and generate linkages to the underlining knowledge while avoiding the fitting of CAPM to too many theories

• Important opportunity to do an application of the simple regression model and give understanding of ‘mechanics’

• Justification with equilibrium and MPT rather than utility

THRESHOLD CONCEPT ISSUES APPLIED TO CAPM PRESENTATION

• “The threshold concept approach is concerned with how students can be helped to acquire integrating ideas.” (Davies & Mangan 2007b)

• Flashing the CAPM formula on a slide and then doing some cost of capital examples would be the exactly wrong approach

• It is better of focus on intricacies of WACC and exercise practice closer to the time of examination, as a revision. However, past exam papers are to be released at the start

THRESHOLD CONCEPT ISSUES APPLIED TO CAPM PRESENTATION

• Demonstrate the workings of the model (statistics). Opaqueness will lead to rote learning of symbols’ definitions

• Show where the inputs can practically come from--the short-term part of US Treasury yield curve--stimulate informed guess and recognition based on prior knowledge

• Be honest about the model’s weaknesses and application of the outputs. Provide the idea of where the model was coming from and for what purposes it was derived and is used

To stimulate open mindset and memory retention: