Portfolio Analysis and Theory

Portfolio Analysis

Definitions

• A portfolio is the collection of securities an investor holds

• A portfolio weight is the proportion of total wealth allocated to a given security

Risk and Return for a Portfolio

• The expected portfolio return is the weighted average of the expected returns on component securities

• The risk of a portfolio is measured by the standard deviation of return

Diversification

• Portfolio standard deviation depends upon the correlation of the returns of component stocks

• If stocks are not perfectly correlated, the portfolio standard deviation is less than the sum of the component standard deviations

• Therefore, diversification reduces risk

Systematic and Unsystematic Risk

• The risk of a portfolio can be divided into systematic and unsystematic risk

• Systematic risk can be explained by factors that are common to all securities

• Unsystematic risk can be explained by factors that are unique to a given security

• Systematic means part of the system

Diversification and Risk

• Unsystematic risk can be reduced through diversification

• Systematic risk cannot be reduced through diversification

• Systematic risk is measured by Beta

Beta and the Market Model

• Beta is the slope of the regression of the historical stock return on the return of a market index, like the S&P 500, containing a large number of stocks

• Unsystematic risk is the variation in the stock’s return that cannot be explained by the variation in the market index

Market Model (continued)

• The systematic risk of a portfolio is the weighted sum of the systematic risk of each component

• You cannot reduce systematic risk through diversification

• You can only obtain low systematic risk by choosing securities with low systematic risk for your portfolio

Systematic Risk

• For very large portfolios unsystematic risk can be almost eliminated

• In this situation the risk each security contributes to the portfolio is approximately equal to its systematic risk or Beta

• Therefore, the relevant risk for an individual security held within a well-diversified portfolio is its Beta

• Remember, that for a portfolio the relevant risk is the standard deviation

Example

• Suppose you put all your wealth in a General Motors stock. Then the relevant risk is the standard deviation because your portfolio consists of a single security

• On the other hand, suppose your holdings of General Motors is a small fraction of a large diversified portfolio. Then the relevant risk is the Beta

Portfolio Theory

Risk Aversion

• Risk averse investors require compensation, in the form of extra return, for assuming financial risk

• The risk-free asset has zero risk and is usually assumed to be the one-year U. S. treasury bill

• A risk averse investor will hold a risky portfolio only if its expected return is greater than the risk-free rate

Risk Aversion (continued)

• Investors are only compensated for bearing systematic risk

• Investors are not compensated for bearing unsystematic risk because it can be eliminated by diversification

Example• Suppose investors A and B choose portfolios by

throwing darts at the Wall Street Journal. Assume that the average return and average standard deviation of stock in the paper is 10% and 14%, respectively. If investor A throws one dart, then his expected return and risk will be 10% and 14%. If investor B throws ten darts, her expected return will still be 10% but her portfolio standard deviation will be (probably) less because of the effect of diversification.

• Should A be compensated for assuming more risk?

Risk-Return Relationship

• There should be a positive relationship between expected return and the Beta measure of systematic risk.

• There should be no relationship between the expected return and unsystematic risk.

• Formal economic model is the Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model Assumptions

• risk aversion• rational behavior• investors choose a portfolio on the basis of expected

returns and standard deviation• investors have same expectations about the future

expected returns, standard deviations and correlations among stocks

• existence of risk-free security• perfect markets: no taxes, transaction costs, or

restrictions on short sales

The security market line (SML)

• SML is the relationship between the expected return on an asset and its Beta measure of systematic risk

• Under the CAPM, the relationship is linear

Beta

• The Beta of the risk-free asset is zero and of course its return is the risk-free rate

• The Beta of the market portfolio is one

• Any stock or portfolio with a Beta = 1 has the same expected return as the market

• Any stock with a Beta = 0 returns the risk-free rate

Beta (continued)

• A stock with a Beta > 1 has more systematic risk than the market and has an expected return that is greater than the market

• A stock with a Beta < 1 has less systematic risk than the market and an expected return less than the market.

Beta (continued)

• A stock with high systematic risk (Beta > 1) will on average (not always) go up by a greater percentage than the market index when the market goes up.

• And of course will on average go down by greater percentage when the market goes down.

• This is what systematic risk means. • So if you thought the market were going down,

you would buy stocks with low Betas

The Price of Risk

• If systematic risk is priced, than high beta stocks should have an average return, over the ups and downs in the market, higher than the market index.

• The high Beta stock has greater systematic risk because when the market goes down the high Beta stock will go down even more.

• However, in theory, when you average over the ups and downs the risk averse investor will earn a higher average return

Required Return

• An investor demands a positive expected return for two reasons: time value of money and a risk aversion.

• The return to compensate for the time value of money is the risk-free rate.

• The extra return to compensate a risk averse for bearing risk is called the risk premium.

• The required return equals the risk free rate plus the risk premium.

Security Market Line

• The market risk premium (the premium on the market index) by definition equals the expected market return minus the risk-free rate

• Under the CAPM, the risk premium on any security equals the Beta multiplied by market risk premium

Security Market Line (continued)

• The security market line relates the required return on a security to its Beta systematic risk. The required return equals the risk-free rate plus the Beta times the market risk premium

The benefits of diversification and portfolio analysis are well

established. The conclusions drawn from portfolio theory are

tentative and debatable

Alternative Explanation • A relatively small number of stocks have high

performance

• Most of the return on a diverisified portfolio can be explained by a small number of high performing stocks in the portfolio

• If you fail to diversify, then you run the risk of not holding any high performing stocks

• Your portfolio will have below average return

Alternative Explanation

• A relatively small number of stocks will perform badly

• Some will go under

• If you fail to diversify, then you might end up with a large fraction of wealth in a failed stock

• Your return distribution will have fat tails