Chapter 11

Beta and the Risk Premium example

Consider a portfolio made up of Asset A and a risk-free asset. We can calculate some different possible portfolio expected returns and betas by varying the percentages invested in these two assets. For example, if 25 percent of the portfolio is invested in Asset A, then the expected return is: E(RP)=.25 × E(RA) + (1 − .25) × Rf. =.25 × 20% + .75 × 8% =11.0% Similarly, the beta on the portfolio, βP , would be: βP=.25 × βA + (1 − .25) × 0 =.25 × 1.6 =.40 Notice that, because the weights have to add up to 1, the percentage invested in the risk-free asset is equal to 1 minus the percentage invested in Asset A.

Portfolio

Group of assets such as stocks and bonds held by an investor

Why is some risk diversifiable?

If we hold a large portfolio, on the other hand, some of the stocks in the portfolio will go up in value because of positive company-specific events and some will go down in value because of negative events. The net effect on the overall value of the portfolio will be relatively small, however, as these effects will tend to cancel each other out. Now we see why some of the variability associated with individual assets is eliminated by diversification. When we combine assets into portfolios, the unique, or unsystematic, events—both positive and negative—tend to "wash out" once we have more than just a few assets.

is possible for the percentage invested in Asset A to exceed 100 percent

The answer is yes. The way this can happen is for the investor to borrow at the risk-free rate. For example, suppose an investor has $100 and borrows an additional $50 at 8 percent, the risk-free rate. The total investment in Asset A would be $150, or 150 percent of the investor's wealth. The expected return in this case would be: E(RP)=1.50 × E(RA) + (1 − 1.50) × Rf =1.50 × 20% − .50 × 8% =26.0% The beta on the portfolio would be: βP=1.50 × βA + (1 − 1.50) × 0 =1.50 × 1.6 =2.4

Calculating the variance --> st. deviation

To calculate the variances of the returns on our two stocks, we first determine the squared deviations from the expected returns. We then multiply each possible squared deviation by its probability. We add these up, and the result is the variance. The standard deviation, as always, is the square root of the variance.

Unsystematic Risk

a risk that affects at most a small number of assets. Also, Because these risks are unique to individual companies or assets, they are sometimes called unique or asset-specific risk

systematic risk principle

the expected return on a risky asset depends only on that asset's systematic risk Because unsystematic risk can be eliminated at virtually no cost (by diversifying), there is no reward for bearing it. Put another way: The market does not reward risks that are borne unnecessarily.

Portfolio weights

the percentage of a portfolio's total value that is invested in a particular asset For example, if we have $50 in one asset and $150 in another, then our total portfolio is worth $200. The percentage of our portfolio in the first asset is $50/200 = .25. The percentage of our portfolio in the second asset is $150/200, or .75. Our portfolio weights are thus .25 and .75. Notice that the weights have to add up to 1.00 since all of our money is invested somewhere.

expected return on a portfolio E(rp)

the weighted average of the expected returns on the assets held in the portfolio E(RP)===.50 × E(RL) + .50 × E(RU) .50 × 25% + .50 × 20% = 22.5% Rl & Ru = different assets expected returns

Measuring Systematic Risk

- How do we measure systematic risk? — We use the beta coefficient - What does beta tell us? — A beta of 1 implies the asset has the same systematic risk as the overall market — A beta < 1 implies the asset has less systematic risk than the overall market — A beta > 1 implies the asset has more systematic risk than the overall market

Systematic Risk

A risk that influences a large number of assets. Also, market risk.

Under what conditions will an announcement have no effect on common stock prices?

If the announcement is not a surprise and more of a confirmation of what we already knew.

What happens to the standard deviation of return for a portfolio if we increase the number of securities in the portfolio?

It decreases.

How do you calculate a portfolio beta?

Multiply weight of return times beta, add all together.

What are the two basic parts of a return?

Normal or expected return from the stock and the uncertain or risky part.

expected portfolio return

Weighted average of the expected returns on the stocks in a portfolio E(RP)=.10 × E(RA) + .20 × E(RB) + .30 × E(RC) + .40 × E(RD) =.10 × 8% + .20 × 12% + .30 × 15% + .40 × 18% =14.9%

REWARD TO RISK EX.

What is the slope of the straight line in Figure 11.2A? As always, the slope of a straight line is equal to "the rise over the run." In this case, as we move out of the risk-free asset into Asset A, the beta increases from 0 to 1.6 (a "run" of 1.6). At the same time, the expected return goes from 8 percent to 20 percent, a "rise" of 12 percent. The slope of the line is thus 12%/1.6 = 7.50%. Notice that the slope of our line is just the risk premium on Asset A, E(RA) − Rf, divided by Asset A's beta, βA: Slope=E(RA) − RfβA =20% − 8%/1.6 =7.50% Slope What this tells us is that Asset A offers a reward-to-risk ratio of 7.50 percent.2 In other words, Asset A has a risk premium of 7.50 percent per "unit" of systematic risk.

Why can't systematic risk be diversified away?

because, by definition, a systematic risk affects almost all assets to some degree. As a result, no matter how many assets we put into a portfolio, the systematic risk doesn't go away. Thus, for obvious reasons, the terms systematic risk and nondiversifiable risk are used interchangeably.

The expected return on a risky asset depends on that asset's total risk. True or false

false. the expected run on assets depends only on that assets systematic risk

Portfolio Beta

if we had a large number of assets in a portfolio, we would multiply each asset's beta by its portfolio weight and then add the results up to get the portfolio's beta. βP=.10 × βA + .20 × βB + .30 × βC + .40 × βD =.10 × .80 + .20 × .95 + .30 × 1.10 + .40 × 1.40 =1.16

states of the economy

only two possible situations for expected return, (good outcome and bad outcome)

What is the principle of diversification?

spreading an investment across a number of assets will eliminate some, but not all, of the risk

principal of diversification

spreading an investment across a number of assets will eliminate some, but not all, of the risk

Beta

the amount of systematic risk present in a particular risky asset relative to that in an average risky asset

What does a beta coefficient measure?

the amount of systematic risk present in a particular risky asset relative to that in an average risky asset

risk premium

the difference between the return on a risky investment and that on a risk-free investment (Expected return − Risk-free rate)

reward to risk ratio

the slope of our line is just the risk premium on Asset A, E(RA) − Rf, divided by Asset A's beta, βA: E(Ra)-Rf/beta A = E(Rb)-Rf/beta B must be the same for all assets in the market

Expected return

the sum of the possible returns (boom/recession) multiplied by their probabilities