# Answers to end of chapter questions and

Stand-alone risk is only a part of total risk and pertains to the risk an investor takes by holding only one asset.

Risk is the chance that some unfavorable event will occur. A probability distribution is a listing, chart or graph of all possible outcomes, such as expected rates of return, with a probability assigned to each outcome.

When in graph form, the tighter the probability distribution, the less uncertain the outcome. The coefficient of variation CV is equal to the standard deviation divided by the expected return; it is a standardized risk measure which allows comparisons between investments having different expected returns and standard deviations.

A risk averse investor dislikes risk and requires a higher rate of return as an inducement to buy riskier securities. A realized return answers to end of chapter questions and the actual return an investor receives on their investment. It can be quite different than their expected return. A risk premium is the difference between the rate of return on a risk-free asset and the expected return on Stock i which has higher risk. The market risk premium is the difference between the expected return on the market and the risk-free rate.

The expected return on a portfolio. The market portfolio is a portfolio consisting of all stocks. Correlation is the tendency of two variables to move together. A correlation coefficient of zero suggests that the two variables are not related to one another; that is, they are independent.

It is measured by the beta coefficient. This risk can be eliminated by proper diversification. The relevant risk of a stock is its contribution to the riskiness of a well-diversified portfolio. The security market line SML represents in a graphical form, the relationship between the risk of an asset as measured by its beta and the required rates of return for individual securities.

The slope of the SML reflects the degree of risk aversion in the economy. The greater the average investors aversion to risk, then the steeper the slope, the higher the risk premium for all stocks, and the higher the required return. The probability distribution for complete certainty is a vertical line. No, it is not riskless. If the actual inflation rate is greater than that expected, interest rates in general will rise to incorporate a larger inflation premium IP and the value of the portfolio would decline.

No, you would be subject to reinvestment rate risk. The product rM — rRF bj is the risk premium of the jth stock.

If bj is low say, 0. However, if bj is large say, 2. For example, assume that the risk-free rate is 6 percent, and the market risk premium is 5 percent. In practice, however, it may be impossible to find individual stocks that have a nonpositive beta.

In this case it would answers to end of chapter questions and be impossible to have a stock portfolio with a zero beta. Even if such a portfolio could be constructed, investors would probably be better off just purchasing Treasury bills, or other zero beta investments. The beta of the new portfolio is: A risk-averse investor would choose the portfolio over either Stock A or Stock B alone, since the portfolio offers the same expected return but with less risk. These can be calculated with a spreadsheet.

Stock X is undervalued, because its expected return exceeds its required rate of return. Because the funds are to be invested in a business at the end of one year, you have been instructed to plan for a one-year holding period.

Further, your boss has restricted you to the following investment alternatives, shown with their probabilities and associated outcomes. Disregard for now the items at the bottom of the data; you will fill in the blanks later. Market 2-stock economy prob. Bills Inds Men Foam portfolio portfolio Recession 0. For example, when the economy is below average, consumers purchase fewer mattresses than they would if the economy was stronger. However, if the economy is in a flat-out recession, a large number of consumers who were planning to purchase a more expensive inner spring mattress may purchase, instead, a cheaper foam mattress.

Alta Industries is an electronics firm; Repo Men collects past-due debts; and American Foam manufactures mattresses and other foam products.

Given the situation as described, answer the following questions. What are investment returns? Investment return measures the financial results of an investment. They may be expressed in either dollar terms or percentage terms.

Do t-bills promise a completely risk-free return? The 8 percent t-bill return does not depend on the state of the economy because the treasury must and will redeem the bills at par regardless of the state of the economy.

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The t-bills are risk-free in the default risk sense because the 8 percent return will be realized in all possible economic states. However, remember that this return is composed of the real risk-free rate, say 3 percent, plus an inflation premium, say 5 percent.

- State Portfolio Recession 3. We have a negative beta stock, hence a required return that is less than the risk-free rate.
- You should recognize that basing a decision solely on expected returns is only appropriate for risk-neutral individuals.
- For example, assume that the risk-free rate is 6 percent, and the market risk premium is 5 percent.
- When we measure risk per unit of return, Repo Men, with its low expected return, becomes the most risky stock. We would do similar calculations for the other states of the economy, and get these results.

Since there is uncertainty about inflation, it is unlikely that the realized real rate of return would equal the expected 3 percent. Thus, in terms of purchasing power, t-bills are not riskless. Also, if you invested in a portfolio of T-bills, and rates then declined, your nominal income would fall; that is, t-bills are exposed to reinvestment rate risk.

So, we conclude that there are no truly risk-free securities in the United States.

- Market 2-stock economy prob.
- What are investment returns?
- We have a negative beta stock, hence a required return that is less than the risk-free rate. Thus, by combining stocks into well-diversified portfolios, investors can eliminate almost one-half the riskiness of holding individual stocks.
- Here are the standard deviations for the other alternatives.
- We do not yet have enough information to choose among the various alternatives.

If the treasury sold inflation-indexed, tax-exempt bonds, they would be truly riskless, but all actual securities are exposed to some type of risk. Why are Alta Ind. If the economy is booming, so will Alta. On the other hand, Repo Men is considered by many investors to be a hedge against both bad times and high inflation, so if the stock market crashes, investors in this stock should do relatively well.

Stocks such as Repo Men are thus negatively correlated with move counter to the economy. Even Repo Men shares have positive but low correlation with the market. The expected rate of return, ris expressed as follows: Here is the calculation for Alta Inds.: You should recognize that basing a decision solely on expected returns is only appropriate for risk-neutral individuals.

Since your client, like virtually everyone, is risk averse, the riskiness of each alternative is an important aspect of the decision. One possible measure of risk is the standard deviation of returns. The standard deviation is calculated as follows: Here are the standard deviations for the other alternatives: What type of risk is measured by the standard deviation?

The larger the standard deviation, the higher the probability that actual realized returns will fall far below the expected return, and that losses rather than profits will be incurred. Draw a graph which shows roughly the shape of the probability distributions for Alta Inds, Am Foam, and T-bills.

Suppose you suddenly remembered that the coefficient of variation CV is generally regarded as being a better measure of stand-alone risk than the standard deviation when the alternatives being considered have widely differing expected returns. Calculate the missing CVs, and fill in the blanks on the row for CV in the table above. Does the CV produce the same risk rankings as the standard deviation?

The coefficient of variation CV is a standardized measure of dispersion about the expected value; it shows the amount of risk per unit of return. When we measure risk per unit of return, Repo Men, with its low expected return, becomes the most risky stock. To find the expected rate of return on the two-stock portfolio, we first calculate the rate of return on the portfolio in each state of the economy.

For a recession, we have: We would do similar calculations for the other states of the economy, and get these results: State Portfolio Recession 3. It is tempting to find the answers to end of chapter questions and deviation of the portfolio as the weighted average of the standard deviations of the individual securities, as follows: How does the riskiness of this 2-stock portfolio compare with the answers to end of chapter questions and of the individual stocks if they were held in isolation?

This is because the two stocks are negatively correlated--when Alta Inds is doing poorly, Repo Men is doing well, and vice versa. Combining the two stocks diversifies away some of the risk inherent in each stock if it were held in isolation, i. Suppose an investor starts with a portfolio consisting of one randomly selected stock.

What would happen 1 to the riskiness and 2 to the expected return of the portfolio as more and more randomly selected stocks were added to the portfolio? What is the implication for investors? Draw a graph of the two portfolios to illustrate your answer. Thus, by adding stocks to your portfolio, which initially started as a 1-stock portfolio, risk has been reduced. The average correlation between stocks is about 0. A single stock selected at random would on average have a standard deviation of about 35 percent.

Thus, by combining stocks into well-diversified portfolios, investors can eliminate almost one-half the riskiness of holding individual stocks. The implication is clear: In fact, individuals can hold diversified portfolios through mutual fund investments. By doing so, they can eliminate about half of the riskiness inherent in individual stocks. Should portfolio effects impact the way investors think about the riskiness of individual stocks? Stand-alone risk is composed of diversifiable risk, which can be eliminated by holding the stock in a well-diversified portfolio, and the risk that remains is called market risk because it is present even when the entire market portfolio is held.

If you decided to hold a 1-stock portfolio, and consequently were exposed to more risk than diversified investors, could you expect to be compensated for all of your risk; that is, could you earn a risk premium on that part of your risk that you could have eliminated by diversifying? If the return were high enough to compensate you for your high risk, it would be a bargain for more rational, diversified investors.