MERRILL FINCH INC. RISK AND RETURN a. (1)Why is T-bill’s return independent of the state of the economy? Do T-bill’s promise a completely risk-free return? Explain (2)Why are High Tech’s returns expected to move with the economy, whereas, Collections’ are expected to move counter to the economy? 1. The 5. 5% T-bill return does not depend on the state of the economy because the Treasury must redeem the bills at par regardless of the state of the economy; therefore, T-bills are risk-free in the default risk sense because the 5.
% return will be realized in all possible economic states. Consequently, this return is composed of the real risk-free rate, (i. e. 3%, plus an inflation premium, say 2. 5%). As the economy is full of uncertainty about inflation, it is unlikely that the realized real rate of return would equal the expected 3%.
For example, if inflation averaged 3. 5% over the year, then the realized real return would only be 5. 5% – 3. 5% = 2%, not the expected 3%. To simplify matters, in terms of purchasing power, T-bills are not riskless.Investors are fully aware of the changes within a portfolio of T-bills, and as rates declined, the nominal income will fall; and T-bills are exposed to reinvestment rate risk. In summary, it is concluded that there are no truly risk-free securities within the United States. 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.
2. High Tech’s returns move with, hence are positively correlated with, the economy, because the firm’s sales, and hence profits, will generally experience the same type of difficulties as the economy.If the economy is booming, so will High Tech. On the other hand, Collections is considered by many investors to be a hedge against bad times and high inflation, so if the stock market crashes, investors in this stock should do relatively well. Stocks such as Collections are thus negatively correlated with the economy. b. Calculate the expected rate of return on each alternative and dill in the blanks on the row for in the previous table. The expected rate of return is expressed as follows: .
Here Pi is the probability of occurrence of the state, ri is the estimated rate of return for that state, and N is the number of states.Here is the calculation for High Tech: High Tech= 0. 1(-27. 0%) + 0. 2(-7. 0%) + 0.
4(15. 0%) + 0. 2(30. 0%) + 0. 1(45.
0%) = 12. 4%. We use the same formula to calculate r’s for the other alternatives: T-bills= 5. 5%.
Collections= 1. 0%. U. S.
Rubber= 9. 8%. M= 10. 5%. c. You should recognize that basing a decision solely on expected returns is appropriate only for risk-neutral individuals. Because your client, like most people, 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. 1) Calculate this value for each alternative and fill in the blank on the row for in the table. (2) What type of risk is measured by the standard deviation? (3) Draw a graph that shows roughly the shape of the probability distributions for High Tech, U. S. Rubber, and T-bills. 1.
The standard deviation is calculated as follows: = . High Tech= [(-27. 0 – 12.
4)2(0. 1) + (-7. 0 – 12. 4)2(0. 2) + (15. 0 – 12.
4)2(0. 4) + (30. 0 – 12. 4)2(0.
2) + (45. 0 – 12. 4)2(0. 1)] ? = = 20. 0%.
Here are the standard deviations for the other alternatives: T-bills= 0. 0%. Collections= 13.
2%. U. S. Rubber= 18. 8%.M= 15.
2%. 2. The standard deviation is a measure of a security’s stand-alone risk.
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. 3. The data provided the most risky investment is High Tech and the least risky are T-bills. d.
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 alternative being considered has widely differing expected returns.Calculate the missing CVs and fill in the blanks on the row for CV in the table. Does the CV produce the same risk rankings as the standard deviation? Explain 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. CV= /. CVT-bills= 0. 0%/5. 5% = 0.
0. CVHigh Tech= 20. 0%/12. 4% = 1.
6. CVCollections= 13. 2%/1. 0% = 13. 2. CVU.
S. Rubber= 18. 8%/9. 8% = 1. 9. CVM= 15.
2%/10. 5% = 1. 4. When we measure risk per unit of return, Collections, with its low expected return, becomes the most risky stock.The CV is a better measure of an asset’s stand-alone risk than because CV considers both the expected value and the dispersion of a distribution—a security with a low expected return and a low standard deviation could have a higher chance of a loss than one with a high but a high . e.
Suppose you created a two-stock portfolio by investing $50,000 in High Tech and $50,000 in Collections. (1) Calculate the expected return (rp), the standard deviation (p), and the coefficient of variation (CVp) for this portfolio and fill in the appropriate blanks in the table. 2) How does the riskiness of this two-stock portfolio compare with the riskiness of the individual stocks if they were held in isolation? 1. 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. Since we have half of our money in each stock, the portfolio’s return will be a weighted average in each type of economy. For a recession, we have: rp = 0. 5(-27%) + 0. 5(27%) = 0%.
We would do similar calculations for the other states of the economy, and get these results: State| | Portfolio| Recession| | 0. 0%|Below average| | 3. 0| Average| | 7. 5| Above average| | 9. 5| Boom| | 12. 0| Now we can multiply the probability times the outcome in each state to get the expected return on this two-stock portfolio, 6. 7%.
Alternatively, we could apply this formula, r = wi ri = 0. 5(12. 4%) + 0.
5(1. 0%) = 6. 7%, which finds r as the weighted average of the expected returns of the individual securities in the portfolio. It is tempting to find the standard deviation of the portfolio as the weighted average of the standard deviations of the individual securities, as follows: p wi(i) + wj(j) = 0. 5(20%) + 0. 5(13. 2%) = 16. %.
However, this is not correct—it is necessary to use a different formula, the one for that we used earlier, applied to the two-stock portfolio’s returns. The portfolio’s depends jointly on each security’s and the correlation between the securities’ returns. The best way to approach the problem is to estimate the portfolio’s risk and return in each state of the economy, and then to estimate p with the formula. Given the distribution of returns for the portfolio, we can calculate the portfolios and CV as shown below: p = [(0. 0 – 6. 7)2(0.
1) + (3. 0 – 6. 7)2(0.
2) + (7. 5 – 6. 7)2(0.
4) + (9. – 6. 7)2(0. 2) + (12. 0 – 6. 7)2(0. 1)]? = 3. 4%.
CVp = 3. 4%/6. 7% = 0. 51. 2. Using either or CV as our stand-alone risk measure, the stand-alone risk of the portfolio is significantly less than the stand-alone risk of the individual stocks.
This is because the two stocks are negatively correlated—when High Tech is doing poorly, Collections 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. e. , in a 1-stock portfolio. f.
Suppose an investor starts with a portfolio consisting of one randomly selected stock.What would happen: (1) To the riskiness and to the expected return of the portfolio as more randomly selected stocks were added to the portfolio? (2) What is the implication for investors? Draw a graph of the two portfolios to illustrate your answer. 1.
The standard deviation gets smaller as more stocks are combined in the portfolio, while rp remains constant. Thus, by adding stocks to your portfolio, which initially started as a 1-stock portfolio, risk has been reduced. In the real world, stocks are positively correlated with one another—if the economy does well, so do stocks in general, and vice versa.Correlation coefficients between stocks generally range in the vicinity of +0.
35. A single stock selected at random would on average have a standard deviation of about 35%. As additional stocks are added to the portfolio, the portfolio’s standard deviation decreases because the added stocks are not perfectly positively correlated. However, as more and more stocks are added, each new stock has less of a risk-reducing impact, and eventually adding additional stocks has virtually no effect on the portfolio’s risk as measured by .In fact, stabilizes at about 20% when 40 or more randomly selected stocks are added.
Thus, by combining stocks into well-diversified portfolios, investors can eliminate almost one-half the riskiness of holding individual stocks. The implication is clear: Investors should hold well-diversified portfolios of stocks rather than individual stocks. By doing so, they can eliminate about half of the riskiness inherent in individual stocks. 2. g.
Should the effects of a portfolio impact the way investors think about the riskiness of individual stocks?If you decided to hold a 1-stock portfolio (and consequently were exposed to more risk than diversified investors were), could you expect to be compensated for all your risk; that is, could you earn a risk premium on the part of your risk that you could have eliminated by diversifying? 1. Portfolio diversification does affect investors’ views of risk. A stock’s stand-alone risk as measured by its or CV, may be important to an undiversified investor, but it is not relevant to a well-diversified investor.A rational, risk-averse investor is more interested in the impact that the stock has on the riskiness of his or her portfolio than on the stock’s stand-alone risk.
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. 2. If you hold a one-stock portfolio, you will be exposed to a high degree of risk, but you will not be compensated for it.If the return were high enough to compensate you for your high risk, it would be a bargain for more rational, diversified investors. They would start buying it, and these buy orders would drive the price up and the return down. Thus, you simply could not find stocks in the market with returns high enough to compensate you for the stock’s diversifiable risk. h. The expected rates of return and the beta coefficients of the alternatives supplied by Merrill Finch’s computer program are as follows: SecurityReturn ()Risk (Beta) High Tech12.
4%1. 32 Market10. 51. 00 U. S. Rubber 9. 80.
88 T-bills 5. 50. 00 Collections 1. 0(0.
87) 1) What is a beta coefficient, and how are betas used in risk analysis? (2) Do the expected returns appear to be related to each alternative’s market risk? (3) Is it possible to choose among the alternatives on the basis of the information developed thus far? Use the data given at the start of the program to construct a graph that shows how the T-bill’s, High Techs, and the market’s beta collections are calculated. Then discuss what betas measure and how they are used in risk analysis. 1. (Draw the framework of the graph, put up the data, then plot the points for the market (45 line) and connect them, and then get the slope as Y/X = 1. . ) State that an average stock, by definition, moves with the market.
Then do the same with High Tech and Tbills. Beta coefficients measure the relative volatility of a given stock vis-a-vis an average stock. The average stock’s beta is 1.
0. Most stocks have betas in the range of 0. 5 to 1. 5. Theoretically, betas can be negative, but in the real world, they are generally positive.
Betas are calculated as the slope of the “characteristic” line, which is the regression line showing the relationship between a given stock and the general stock market.As explained in Web Appendix 8A, we could estimate the slopes, and then use the slopes as the betas. In practice, 5 years of monthly data, with 60 observations, would generally be used, and a computer would be used to obtain a least squares regression line. 2. The expected returns are related to each alternative’s market risk—that is, the higher the alternatives rate of return the higher its beta. Also, note that T-bills have zero risk. 3. We do not yet have enough information to choose among the various alternatives.
We need to know the required rates of return on these alternatives and compare them with their expected returns. . The yield curve is currently flat; that is, long-term Treasury bonds also have a 5. 5% yield. Consequently, Merrill Finch assumes that the risk-free rate is 5. 5%. (1) Write out Security Market Line (SML) equation, use it to calculate the required rate of return on each alternative, and graph the relationship between the expected and required rates of return. (2) How do the expected rates of return compare with the required rates of return? (3) Does the fact that Collections has an expected return that is less than the T-bill rate make any sense?Explain (4) What would be the market and the required return of a 50-50 portfolio of High Tech and Collections? Of High Tech and U.
S. Rubber? 1. Here is the SML equation: ri = rRF + (rM – rRF)bi. Merrill Finch has estimated the risk-free rate to be rRF = 5. 5%. Further, our estimate of rM = M is 10.
5%. Thus, the required rates of return for the alternatives are as follows: High Tech:5. 5% + (10. 5% – 5.
5%)1. 32 = 12. 10%. Market:5. 5% + (10. 5% – 5. 5%)1. 00 = 10.
50%. U. S. Rubber:5. 5% + (10. 5% – 5.
5%)0. 88 = 9. 90%. T-bills:5. 5% + (10.
5% – 5. 5%)0 = 5. 50%. Collections:5. 5% + (10. 5% – 5.
%)-0. 87 = 1. 15%.
2. We have the following relationships: ExpectedRequired ReturnReturn Security ()(r)Condition High Tech12. 4%12. 1%Undervalued: > r Market10. 510. 5Fairly valued (market equilibrium) U. S. Rubber9.
89. 9Overvalued: r > T-bills5. 55. 5Fairly valued Collections1.
01. 2Overvalued: r > The T-bills and market portfolio plot on the SML, High Tech plots above it, and Collections and U. S.
Rubber plot below it. Thus, the T-bills and the market portfolio promise a fair return, High Tech is a good deal because its expected return is above its required return, and Collections and U.S. Rubber have expected returns below their required returns.
3. Collections are an interesting stock. Its negative beta indicates negative market risk—including it in a portfolio of “normal” stocks will lower the portfolio’s risk. Therefore, its required rate of return is below the risk-free rate. This means that Collections is a valuable security to rational, well-diversified investors. To see why, consider this question: Would any rational investor ever make an investment that has a negative expected return? The answer is “yes”—just thinks of the purchase of a life or fire insurance policy.
The fire insurance policy has a negative expected return because of commissions and insurance company profits, but businesses buy fire insurance because they pay off at a time when normal operations are in bad shape. Life insurance is similar—it has a high return when work income ceases. A negative-beta stock is conceptually similar to an insurance policy. 4. Note that the beta of a portfolio is simply the weighted average of the betas of the stocks in the portfolio. Thus, the beta of a portfolio with 50% High Tech and 50% Collections is: . bp= 0. 5(bHigh Tech) + 0.
(bCollections) = 0. 5(1. 32) + 0. 5(–0. 87) = 0. 225, rp = rRF + (rM – rRF)bp= 5.
5% + (10. 5% – 5. 5%)(0.
225) = 5. 5% + 5%(0. 225) = 6. 63% 6. 6%. For a portfolio consisting of 50% High Tech plus 50% U.
S. Rubber, the required return would be: bp = 0. 5(1. 32) + 0. 5(0. 88) = 1.
10. rp = 5. 5% + 5%(1. 10) = 11. 00%. j. Suppose investors raised their inflation expectations by 3 percentage points over current estimates as reflected in the 5. 5% risk-free rate.
What effect would higher inflation have on the SML and on the returns required on high-and low-risk securitiesSuppose instead that investors’ risk aversion increased enough to cause the market risk premium to increase by 3 percentage points. (Inflation remains constant. ) What effect would this have on the SML and on returns of high-and low-risk securities? Here we have plotted the SML for betas ranging from 0 to 2. 0. The base-case SML is based on rRF = 5. 5% and rM = 10. 5%. If inflation expectations increase by 3 percentage points, with no change in risk aversion, then the entire SML is shifted upward (parallel to the base case SML) by 3 percentage points.
Now, rRF = 8. 5%, rM = 13. 5%, and all securities’ required returns rise by 3 percentage points. Note that the market risk premium, rM – rRF, remains at 5 percentage points.
When investors’ risk aversion increases, the SML is rotated upward about the Y-intercept (rRF). rRF remains at 5. 5%, but now rM increases to 13. 5%, so the market risk premium increases to 8%. The required rate of return will rise sharply on high-risk (high-beta) stocks, but not much on low-beta securities.