 # Portfolio Theory & Capital Asset Pricing Model

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Portfolio Theory & Capital Asset Pricing Model
Chapter 4 Portfolio Theory & Capital Asset Pricing Model

How do we measure the risk/return tradeoff for an individual security?
We calculate the expected return on the security and the standard deviation/variance of this return How does the risk/return relationship change when we combine two different securities into a portfolio? This depends on the covariance between the returns on the two securities We can use this to calculate the feasible set and the efficient frontier of the risk/return outcomes

What happens when we combine many different securities into a portfolio?
The risk of the portfolio is more dependent on the covariance between the securities than on their variances The total risk of any individual security can be separated into its portfolio (systematic) risk and its idiosyncratic (diversifiable) risk

What happens when we are able to combine a portfolio of risky assets with risk-free bonds?
We find there is a single optimal portfolio of risky assets to hold; this is the same for all investors if they have homogeneous expectations Individual investors satisfy their risk preference by choosing how much to invest in this market portfolio relative to the riskless asset

How do we measure the risk of an individual security when investors hold a diversified portfolio of securities (the market portfolio)? Because investors are well-diversified, they do not care about the diversifiable risk of an individual security Instead, they care about what an individual security contributes to the risk of the market portfolio This contribution is measured by the security’s beta

How can we use all of this to determine the return investors require from a security given its risk (beta)? The expected return on the market as a whole can be separated into the risk-free rate and a risk premium The expected return on any individual security is the risk-free rate, plus the market risk premium times the security’s beta This relationship between risk and return is known as the capital asset pricing model (CAPM)

Expected Return, Variance, and Covariance
Consider the following two risky asset world. There is a 1/3 chance of each state of the economy and the only assets are a stock fund and a bond fund.

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance
The relationship between the returns on the two funds is measured by their covariance:

Expected Return, Variance, and Covariance
Like the variance of an individual security’s return, the covariance between the returns of two securities is difficult to interpret because it is in squared units. The correlation coefficient normalizes this measure by dividing the covariance by the standard deviations of the two securities: The correlation coefficient always lies between -1 and +1

The Return and Risk for Portfolios
Note that stocks have a higher expected return than bonds and higher risk. Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks.

The Return and Risk for Portfolios
The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:

The Return and Risk for Portfolios
The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:

The Return and Risk for Portfolios
The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:

The Return and Risk for Portfolios
The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio.

The Return and Risk for Portfolios
Notice, however, that the variance of the portfolio is not equal to the weighted average of the variances: This is because the two funds do not move in lock step with one another.

The Return and Risk for Portfolios
Observe the decrease in risk that diversification offers. An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than stocks or bonds held in isolation.

The Efficient Set for Two Assets
We can consider other portfolio weights besides 50% in stocks and 50% in bonds …

The Efficient Set for Two Assets
Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less. These comprise the efficient frontier.

Two-Security Portfolios with Various Correlations
return 100% stocks  = -1.0  = 1.0  = 0.2 100% bonds

Portfolio Risk/Return Two Securities: Correlation Effects
Relationship depends on correlation coefficient -1.0 < r < +1.0 The smaller the correlation, the greater the risk reduction potential If r = +1.0, no risk reduction is possible

Portfolio Risk as a Function of the Number of Stocks in the Portfolio
In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not. Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk Nondiversifiable risk; Systematic Risk; Market Risk n Thus diversification can eliminate some, but not all of the risk of individual securities.

The Efficient Set for Many Securities
return return Individual Assets P P Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios.

The Efficient Set for Many Securities
return minimum variance portfolio Individual Assets P Given the opportunity set we can identify the minimum variance portfolio

The Efficient Set for Many Securities
return efficient frontier minimum variance portfolio Individual Assets P The section of the opportunity set above the minimum variance portfolio is the efficient frontier.

Optimal Risky Portfolio with a Risk-Free Asset
efficient frontier return 100% stocks rf 100% bonds In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills

Market Equilibrium return CML M rf P
efficient frontier return CML 100% stocks M rf 100% bonds P With the capital allocation line identified, all investors choose a point along the line—some combination of the risk-free asset and the market portfolio M. In a world with homogeneous expectations, M is the same for all investors.

The Separation Property
return CML efficient frontier M rf P The Separation Property states that the market portfolio, M, is the same for all investors—they can separate their risk aversion from their choice of the market portfolio.

The Separation Property
return CML efficient frontier M rf P Investor risk aversion is revealed in their choice of where to stay along the capital allocation line—not in their choice of the line.

Market Equilibrium CML return rf 
100% stocks Balanced fund rf 100% bonds Just where the investor chooses along the capital market line depends on his risk tolerance. The big point is that all investors have the same CML.

Market Equilibrium CML return rf 
100% stocks Optimal Risky Porfolio rf 100% bonds All investors have the same CML because they all have the same optimal risky portfolio given the risk-free rate.

The Separation Property
CML return 100% stocks Optimal Risky Porfolio rf 100% bonds The separation property implies that portfolio choice can be separated into two tasks: (1) determine the optimal risky portfolio, and (2) selecting a point on the CML.

Optimal Risky Portfolio with a Risk-Free Asset
CML1 CML0 return 100% stocks Second Optimal Risky Portfolio First Optimal Risky Portfolio 100% bonds By the way, the optimal risky portfolio depends on the risk-free rate as well as the risky assets.

Definition of Risk When Investors Hold the Market Portfolio
Researchers have shown that the best measure of the risk of a security in a large portfolio is the beta (b) of the security. Beta measures the responsiveness of a security to movements in the market portfolio.

Slope = bi Ri = a i + biRm + ei Estimating b with regression
Characteristic Line Security Returns Slope = bi Return on market % Ri = a i + biRm + ei

Estimates of b for Selected Stocks
Beta Bank of America 1.55 Borland International 2.35 Travelers, Inc. 1.65 Du Pont 1.00 Kimberly-Clark Corp. 0.90 Microsoft 1.05 Green Mountain Power 0.55 Homestake Mining 0.20 Oracle, Inc. 0.49

The Formula for Beta Clearly, your estimate of beta will depend upon your choice of a proxy for the market portfolio.

Relationship between Risk and Expected Return (CAPM)
Expected Return on the Market: Expected return on an individual security: Market Risk Premium This applies to individual securities held within well-diversified portfolios.

Expected Return on an Individual Security
This formula is called the Capital Asset Pricing Model (CAPM) Expected return on a security = Risk-free rate + Beta of the security × Market risk premium If bi = 0, then the expected return is RF. If bi = 1, then

Relationship Between Risk & Expected Return
1.0

Relationship Between Risk & Expected Return
1.5

Summary and Conclusions
This chapter sets forth the principles of modern portfolio theory. The expected return and variance on a portfolio of two securities A and B are given by By varying wA, one can trace out the efficient set of portfolios. We graphed the efficient set for the two-asset case as a curve, pointing out that the degree of curvature reflects the diversification effect: the lower the correlation between the two securities, the greater the diversification. The same general shape holds in a world of many assets.

Summary and Conclusions
The efficient set of risky assets can be combined with riskless borrowing and lending. In this case, a rational investor will always choose to hold the portfolio of risky securities represented by the market portfolio. Then with borrowing or lending, the investor selects a point along the CML. return CML M M rf P

Summary and Conclusions
The contribution of a security to the risk of a well-diversified portfolio is proportional to the covariance of the security's return with the market’s return. This contribution is called the beta. The CAPM states that the expected return on a security is positively related to the security’s beta:

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