Financial Econometrics – Slides-02Linear RegressionReview and Applications in Finance
Linear Regression Applications In Finance Review of Linear Regression model
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R. Ouysse Economics Financial Econometrics
Linear Regression Applications In Finance Review of Linear Regression model
Financial Econometrics
Linear Regression
Review and Applications in Finance
Economics1
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removed from this material.
R. Ouysse Economics Financial Econometrics
Linear Regression Applications In Finance Review of Linear Regression model
Linear Regression
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• A model where one variable Yt is linearly explained by a group of variables
(X1t, · · · , Xkt), t = 1, · · · , T
• Easy to Implement
• Versatile for financial data analysis
• Foundation for more advanced models
• General formulation
• Yt = �1 + �2Xt1 + �3Xt2 + · · ·+ �KXtK + µt, t = 1 · · · , T
• Yt: dependent variable
• Xt1, · · ·XtK : explanatory variables, regressors
• �1,�2, · · · ,�K : parameters (to be estimated)
• µt: error term
• T : number of observations
R. Ouysse Economics Financial Econometrics
Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model
Application 1: Capital Asset Pricing Model aka CAPM
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One of the most important problems of modern financial economics is the
quantification of the tradeo↵ between risk and expected return. Common
sense suggests risky investments (stock market) will generally yield higher returns
than investments free of risk!
• Markowitz (1995) casts the investor’s portfolio selection problem in terms
of expected return and variance of the return.
! Investors optimally hold a mean-variance e�cient portfolio: a portfolio
with the highest expected return for a given level of variance.
=) The E�cient Frontier & Capital Market Line
• Capital Asset Pricing Model is concerned with the pricing of assets in
equilibrium. In equilibrium, all assets must be held by someone.
�! How investors determine the expected returns—and thereby asset
prices—as a function of risk.
=) The Security Market Line
R. Ouysse Economics Financial Econometrics
Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model
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• Given that: some risk can be diversified, diversification is easy and
costless, and rational investors diversify
• There should be no premium associated with diversifiable risk.
• The question becomes: What is the equilibrium relation between
systematic risk and expected return in the capital markets?
• The CAPM is the best-known and most-widely used equilibrium model of
the risk/return (systematic risk/return) relation.
R. Ouysse Economics Financial Econometrics
Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model
CAPM Intuition
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What would be a ”fair” expected return on any stock?
• E(Rit) = Rft (risk free)+ Risk Premium
• Risk free assets earn the risk-free rate (think of this as a rental rate on
capital). The risk free compensate for time.
• If the asset is risky, we need to add a risk premium.
• The size of the risk premium depends on the amount of systematic risk for
the asset (stock, bond, or investment project) and the price per unit risk.
• Rit �Rft: Excess return
R. Ouysse Economics Financial Econometrics
Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model
CAPM Intuition Formalized
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E [Rit] = Rft +
Cov (Rit, Rmt)
V ar (Rmt)
[E [Rmt]�Rft]
E [Rit] = Rft + �i [E [Rmt]�Rft]
The expression above is referred to as the ”Security Market Line”.
• E [Rmt]�Rft Market Risk premiun (compensation for risk) or the price
per unit of risk
• �i: number of units of systematic risk
• �i > 1 (or < 1): the asset is more (less) risky than the market portfolio• �i < 0 : the asset is a hedge against the market portfolio• �i how sensitive the asset to market movementR. Ouysse Economics Financial EconometricsLinear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value modelCAPM FormalizedCopyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material.Three inputs are required:(i) An estimate of the risk free interest rate. The current yield on short termtreasury bills is one proxy. Practitioners tend to favor the current yield onlonger-term treasury bonds but this may be a fix for a problem we don’tfully understand.(ii) An estimate of the market risk premium, E [Rmt]�Rft. Expectations arenot observable. Generally use a historically estimated value.The market is defined as a portfolio of all wealth including real estate,human capital, etc. In practice, a broad based stock index, such as theS&P 500 or the portfolio of all NYSE stocks, is generally used.(iii) An estimate of beta.R. Ouysse Economics Financial EconometricsLinear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value modelCAPM: Econometric modelCopyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material.Let Xmt = Rmt �Rft and Xit = Rit �Rft and consider the econometricXit = ↵i + �iXmt + µit• The CAPM can be examined by testing H0 : ↵i = 0• If ↵i > 0, asset i beats the market by earning more than �iE [Xmt]
• This has been used to test the performance of mutual funds (application
in the Brooks textbook)
R. Ouysse Economics Financial Econometrics
Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model
CAPM: Application
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What detremines the expected return of an asset?
Example: Mobil (a US petroleum firm), 1978:01-1987:12 with T = 120.
Topic 2. Linear Regression & Applicationsin Finance
School’of’Economics,’UNSW’ Slides<02,’Financial’Econometrics’ 7’78 79 80 81 82 83 84 85 86 87MARKET RISKFREE MOBIL-.3 -.2 -.1 .0 .1 .2Scatter PlotDependent’Variable:’E_MOBIL ‘ ‘ Method:’Least’Squares ‘ ‘ Sample:’1978M01’1987M12 ‘ ‘ Included’observaOons:’120 ‘ ‘ Variable Coefficient Std.’Error t
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