THE EXPECTED RETURN OF A PORTFOLIO Now that we understand the important role that portfolios play in reducing unsystematic risk, we must stay mindful of this fact. This means it is important for companies to understand how portfolios work and the implications for the return the investors expect on the stock.

Portfolio Weights A portfolio weight is the fraction of the total investment in a portfolio held in each individual investment in a portfolio. That is, the fraction of each investment is proportional to its value compared to the overall total value of the portfolio. Remember, these are dollar amounts not the number of shares held. Also, the portfolio weights of each investment add up to 100%, so they represent the way money is divided between the different individual investments in a portfolio. Wi = Value of investment t / Total value of the portfolio.

Portfolio Returns Since we now know how to determine the portfolio weight, the return on the entire portfolio can be determined. The return on a portfolio is the weighted average of the returns on the investments in a portfolio, where the weights correspond to the portfolio weights. Each investment in a portfolio must be accounted for. Because they all must be accounted for, the return on the portfolio (Rp): Rp = w1R1 + w2R2 +…+wnRn

Expected Portfolio Return To determine this, one must use the historical average return of a security as the expected return. The expected return of a portfolio is the weighted average of expected returns on the investments in a portfolio, where the weights correspond to the portfolio weights: E[Rp] = w1E[R1] +w2e[R2] + . . . + wnE[Rn] This makes the expected portfolio return the return expected to be earned on a portfolio, given the expected returns and the relative amount investment in each investment.

VOLATILITY OF A PORTFOLIO This is where a company must examine the risk investors may see in their investments. This makes in a requirement that companies understand how to calculate risk in a portfolio. After diversification, the amount of risk that remains depends upon the degree to which the stocks share common risk. The volatility of a portfoliois a measurement of the standard deviations of a portfolio which gives the total risk.

Diversifying Risks Investments can have the same volatility and average return, but it is important that the act differently to market stimuli so that the risk is diversified. Making sure a portfolio contains different investments in different industries is a good way to see the investments act differently. So the two important things here are: 1. Combining stocks into a portfolio reduces risk through diversification. 2. The amount of risk that is eliminated in a portfolio depends upon the degree to which the stocks face common risk and move together.

Measuring Stock’s Co movement: Correlation This is where the degree to which stocks’ returns move together is examined. Correlation is a measure of the degree to which returns share common risk. It is calculated as the covariance of the returns divided by the standard deviation of each return. This is a measurement of between -1 to +1. The closer to -1, the more the stocks move opposite each other and the closer to +1 the more the stocks move together. Independent risks are uncorrelated, meaning they have no tendency to move together or opposite – a correlation of 0. A correlation table can be created in excel which will give the correlations between any number of parameters.

Computing a Portfolio’s Variance and Standard Deviation An important note here is that the expected return of a portfolio is equal to the weighted average expected return of its stocks, but the volatility of a portfolio is less than the weighted average volatility. As a result of this, it becomes clear that some volatility can be eliminated by diversifying. The variance can be computed by: Var(Rp) = w21SD(R1)^2 + w22SD(R2)^2 + 2w1w2Corr(R1,R2)SD(R1)SD(R2) The first and second chunks of the equation accounts for the risk of stock one and two respectfully, while the last one account for the adjustment for how much the two move together. The portfolio will have the greatest variance if the stocks have a perfect positive correlation of +1.

Volatility of a Large Portfolio More diversification can be gained by holding more than just two stocks. An equally weighted portfolio is one in which the same dollar amount is invested in each stock. Benefits of diversification is felt more when the first few stocks are addend. In fact, the greatest amount of risk is eliminated when a portfolio goes from one to two stocks.

MEASURING SYSTEMATIC RISK The goal in this chapter is to understand the impact of risk on a firm’s investors. By understanding how they view risk, a firm can quantify the relation between risk and required return and this produces a discount rate. However, because total risk includes unsystematic risk, a new approach must be used.

Role of the Market Portfolio An experience investor will diversify until all, or almost all, unsystematic risk has been eliminated. A fully-diversified portfolio is called an aggregate portfolio. This aggregate portfolio holds all shares outstanding of every risky security: a market portfolio. Market capitalization is used to determine the market portfolio. Market capitalization is: Market capitalization = Number of shares outstanding * Price per share. More generally, a market portfolio will consist of all risky securities in the market. Because stocks are held in proportion to their market capitalization, it is said that the market portfolio is value-weighted. Because this portfolio contains only systematic risk, it can be used to measure the amount of systematic risk of other securities in the market.

Stock Market Indexes as the Market Portfolio Because the market portfolio contains all risky securities it must include all stocks, bonds, real estate, commodities, etc. both in the U.S. and around the world. This would be virtually impossible to do. So in practice a market proxy is used. A market proxy is a portfolio whose return should track the underlying, unobservable market portfolio. The most common market proxies are market indexes. These market indexes report the value of a particular portfolio of securities.

Dow Jones industrial average One market proxy is the Dow Jones. This is the most familiar stock index in the U.S. It consists of a portfolio of 30 large, industrial stocks.

S&P 500 Another market proxy is the S&P 500. This is a better representation of the entire U.S. stock market. It is a value-weighted portfolio of 500 of the largest U.S. stocks. It was the first widely publicized value-weighted portfolio and is the standard benchmark for professional investors.

Market Risk and Beta This is where an actual measurement of systematic risk starts to form. The relation between an individual stock’s returns and the market portfolio’s returns can be used to measure the amount of systematic risk present in a particular stock. If a stock’s returns do not depend on the market’s returns, then it has little systematic risk - they are not strongly reflected in returns. The specific measure of stock’s systematic risk is measured by beta. Beta is the expected percent change in the excess return of a security for a 1% change in the excess return of the market portfolio. The beta of the overall market portfolio is 1, so this represents the average exposure to systematic risk.

Estimating Beta form Historical Returns Beta represents the amount by which risks that affect the overall market are amplified or dampened in a given stock or investment. Securities whose returns tend to move one for one with the market on average have a beta of one. Securities that tend to move more than the market have higher betas, while those that move less than the market have lower betas. Beta can be found by plotting an investments return and the S&P excess return. The line-of-best it in this graph would represent beta.

CAPITAL ASSET PRICING MODEL The goal of this chapter was to compute the cost of equity capital for an investment. This is the best available expected return offered in the market on an investment of comparable risk and term. To compute the cost of capital, we need to know the relation between risk and expected return.

CAPM Equation Relating Risk and Expected Return Remember: only common, systematic risk determines expected returns. Any investment should come from two components: 1. A baseline risk-free rate of return that would demand to compensate for inflations and the time value of money, even if there were no risk of losing money. 2. A risk premium that varies with the amount of systematic risk in the investment. Expected return = Risk-free rate + Risk premium for systematic risk Expected return for investment I = Risk-free rate + Bi * Risk premium per unit of systematic risk The market risk premium in that last part of the equation is the historical average excess returns on the market portfolio. With this last piece the CAPM can now be calculated. E[Ri] = rf + Bi(E[Rmkt] – rf) This equation is used to find the expected return of any investment. The CAPM simply says that the return expected on any investment is equal to the risk-free rate of return plus a risk premium proportional to the amount of systematic risk in the investment. Investors will not invest in a security unless they can expect at least the return given in the last equation – required return. The CAPM is the main method used by most major corporations to determine the equity cost of capital.

Security Market Line This is the line that is created when the expected return is graphed against beta. Formally, thesecurity market line is the pricing implication of the CAPM, it specifies a linear relation between the risk premium of a security and its beta with the market portfolio.

CAPM and Portfolios Because the security market line applies to all securities, we can apply it to portfolios as well. The expected return of a portfolio should correspond to the portfolio’s beta Bp = w1B1 +w2B2 + . . . + wnBn. This says that the beta of a portfolio is the weighted average beta of the securities in the portfolio

IMPORTANT THINGS TO REMEMBER There are four important issues to the CAPM to remember as we finish this chapter: 1. Investors require a risk premium proportional to the amount of systematic risk they are bearing. 2. We can measure the systematic risk of an investment by its B. 3. The most common way to estimate of stock’s beta is to regress its historical returns on the market’s historical returns. The stock’s beta is the slope of the line that best explains the relation between the market’s return and the stock’s return. 4. The CAPM says that we can compute the expected, or required, return for investment using: E[Ri] = rf + Bi(D[Rmkt] – rf)

## Systematic Risk and the Equity Risk Premium

THE EXPECTED RETURN OF A PORTFOLIO

Now that we understand the important role that portfolios play in reducing unsystematic risk, we must stay mindful of this fact. This means it is important for companies to understand how portfolios work and the implications for the return the investors expect on the stock.

Portfolio Weights

A portfolio weight is the fraction of the total investment in a portfolio held in each individual investment in a portfolio. That is, the fraction of each investment is proportional to its value compared to the overall total value of the portfolio. Remember, these are dollar amounts not the number of shares held. Also, the portfolio weights of each investment add up to 100%, so they represent the way money is divided between the different individual investments in a portfolio.

Wi = Value of investment t / Total value of the portfolio.

Portfolio Returns

Since we now know how to determine the portfolio weight, the return on the entire portfolio can be determined. The return on a portfolio is the weighted average of the returns on the investments in a portfolio, where the weights correspond to the portfolio weights. Each investment in a portfolio must be accounted for. Because they all must be accounted for, the return on the portfolio (Rp):

Rp = w1R1 + w2R2 +…+wnRn

Expected Portfolio Return

To determine this, one must use the historical average return of a security as the expected return. The expected return of a portfolio is the weighted average of expected returns on the investments in a portfolio, where the weights correspond to the portfolio weights:

E[Rp] = w1E[R1] +w2e[R2] + . . . + wnE[Rn]

This makes the expected portfolio return the return expected to be earned on a portfolio, given the expected returns and the relative amount investment in each investment.

VOLATILITY OF A PORTFOLIO

This is where a company must examine the risk investors may see in their investments. This makes in a requirement that companies understand how to calculate risk in a portfolio. After diversification, the amount of risk that remains depends upon the degree to which the stocks share common risk. The volatility of a portfolio is a measurement of the standard deviations of a portfolio which gives the total risk.

Diversifying Risks

Investments can have the same volatility and average return, but it is important that the act differently to market stimuli so that the risk is diversified. Making sure a portfolio contains different investments in different industries is a good way to see the investments act differently. So the two important things here are:

1. Combining stocks into a portfolio reduces risk through diversification.

2. The amount of risk that is eliminated in a portfolio depends upon the degree to which the stocks face common risk and move together.

Measuring Stock’s Co movement: Correlation

This is where the degree to which stocks’ returns move together is examined. Correlation is a measure of the degree to which returns share common risk. It is calculated as the covariance of the returns divided by the standard deviation of each return. This is a measurement of between -1 to +1. The closer to -1, the more the stocks move opposite each other and the closer to +1 the more the stocks move together. Independent risks are uncorrelated, meaning they have no tendency to move together or opposite – a correlation of 0. A correlation table can be created in excel which will give the correlations between any number of parameters.

Computing a Portfolio’s Variance and Standard Deviation

An important note here is that the expected return of a portfolio is equal to the weighted average expected return of its stocks, but the volatility of a portfolio is less than the weighted average volatility. As a result of this, it becomes clear that some volatility can be eliminated by diversifying. The variance can be computed by:

Var(Rp) = w21SD(R1)^2 + w22SD(R2)^2 + 2w1w2Corr(R1,R2)SD(R1)SD(R2)

The first and second chunks of the equation accounts for the risk of stock one and two respectfully, while the last one account for the adjustment for how much the two move together. The portfolio will have the greatest variance if the stocks have a perfect positive correlation of +1.

Volatility of a Large Portfolio

More diversification can be gained by holding more than just two stocks. An equally weighted portfolio is one in which the same dollar amount is invested in each stock. Benefits of diversification is felt more when the first few stocks are addend. In fact, the greatest amount of risk is eliminated when a portfolio goes from one to two stocks.

MEASURING SYSTEMATIC RISK

The goal in this chapter is to understand the impact of risk on a firm’s investors. By understanding how they view risk, a firm can quantify the relation between risk and required return and this produces a discount rate. However, because total risk includes unsystematic risk, a new approach must be used.

Role of the Market Portfolio

An experience investor will diversify until all, or almost all, unsystematic risk has been eliminated. A fully-diversified portfolio is called an aggregate portfolio. This aggregate portfolio holds all shares outstanding of every risky security: a market portfolio. Market capitalization is used to determine the market portfolio. Market capitalization is:

Market capitalization = Number of shares outstanding * Price per share.

More generally, a market portfolio will consist of all risky securities in the market. Because stocks are held in proportion to their market capitalization, it is said that the market portfolio is value-weighted. Because this portfolio contains only systematic risk, it can be used to measure the amount of systematic risk of other securities in the market.

Stock Market Indexes as the Market Portfolio

Because the market portfolio contains all risky securities it must include all stocks, bonds, real estate, commodities, etc. both in the U.S. and around the world. This would be virtually impossible to do. So in practice a market proxy is used. A market proxy is a portfolio whose return should track the underlying, unobservable market portfolio. The most common market proxies are market indexes. These market indexes report the value of a particular portfolio of securities.

Dow Jones industrial average

One market proxy is the Dow Jones. This is the most familiar stock index in the U.S. It consists of a portfolio of 30 large, industrial stocks.

S&P 500

Another market proxy is the S&P 500. This is a better representation of the entire U.S. stock market. It is a value-weighted portfolio of 500 of the largest U.S. stocks. It was the first widely publicized value-weighted portfolio and is the standard benchmark for professional investors.

Market Risk and Beta

This is where an actual measurement of systematic risk starts to form. The relation between an individual stock’s returns and the market portfolio’s returns can be used to measure the amount of systematic risk present in a particular stock. If a stock’s returns do not depend on the market’s returns, then it has little systematic risk - they are not strongly reflected in returns. The specific measure of stock’s systematic risk is measured by beta. Beta is the expected percent change in the excess return of a security for a 1% change in the excess return of the market portfolio. The beta of the overall market portfolio is 1, so this represents the average exposure to systematic risk.

Estimating Beta form Historical Returns

Beta represents the amount by which risks that affect the overall market are amplified or dampened in a given stock or investment. Securities whose returns tend to move one for one with the market on average have a beta of one. Securities that tend to move more than the market have higher betas, while those that move less than the market have lower betas. Beta can be found by plotting an investments return and the S&P excess return. The line-of-best it in this graph would represent beta.

CAPITAL ASSET PRICING MODEL

The goal of this chapter was to compute the cost of equity capital for an investment. This is the best available expected return offered in the market on an investment of comparable risk and term. To compute the cost of capital, we need to know the relation between risk and expected return.

CAPM Equation Relating Risk and Expected Return

Remember: only common, systematic risk determines expected returns. Any investment should come from two components:

1. A baseline risk-free rate of return that would demand to compensate for inflations and the time value of money, even if there were no risk of losing money.

2. A risk premium that varies with the amount of systematic risk in the investment.

Expected return = Risk-free rate + Risk premium for systematic risk

Expected return for investment I = Risk-free rate + Bi * Risk premium per unit of systematic risk

The market risk premium in that last part of the equation is the historical average excess returns on the market portfolio. With this last piece the CAPM can now be calculated.

E[Ri] = rf + Bi(E[Rmkt] – rf)

This equation is used to find the expected return of any investment. The CAPM simply says that the return expected on any investment is equal to the risk-free rate of return plus a risk premium proportional to the amount of systematic risk in the investment. Investors will not invest in a security unless they can expect at least the return given in the last equation – required return. The CAPM is the main method used by most major corporations to determine the equity cost of capital.

Security Market Line

This is the line that is created when the expected return is graphed against beta. Formally, the security market line is the pricing implication of the CAPM, it specifies a linear relation between the risk premium of a security and its beta with the market portfolio.

CAPM and Portfolios

Because the security market line applies to all securities, we can apply it to portfolios as well. The expected return of a portfolio should correspond to the portfolio’s beta

Bp = w1B1 +w2B2 + . . . + wnBn.

This says that the beta of a portfolio is the weighted average beta of the securities in the portfolio

IMPORTANT THINGS TO REMEMBER

There are four important issues to the CAPM to remember as we finish this chapter:

1. Investors require a risk premium proportional to the amount of systematic risk they are bearing.

2. We can measure the systematic risk of an investment by its B.

3. The most common way to estimate of stock’s beta is to regress its historical returns on the market’s historical returns. The stock’s beta is the slope of the line that best explains the relation between the market’s return and the stock’s return.

4. The CAPM says that we can compute the expected, or required, return for investment using:

E[Ri] = rf + Bi(D[Rmkt] – rf)