Question

• Assume that you recently graduated with a major in finance and just landed a job in the trust department of a large regional bank. Your first assignment is to invest $100,000 from an estate for which the bank is trustee. Because the estate is expected to be distributed to the heirs in approximately one year, you have been instructed to plan for a one-year holding period. Furthermore, your boss has restricted you to the following investment alternatives, shown with their probabilities and associated outcomes. The bank’s economic forecasting staff has developed probability estimates for the state of the economy, and the rate of return on each alternative is estimated under each state of the economy. High Tech, Inc., is an electronics firm; Collections, Inc., a firm collects past-due debts; and U.S. Rubber manufactures tires and various other rubber and plastic products.

• Given the situation as described, discuss the following questions:

  1. Calculate the expected rate of return on each alternative. Which alternative has the highest return and which one has the lowest return?
  2. Calculate the riskiness of each alternative using the standard deviation of returns. What type of risk does the standard deviation measure?
  3. Calculate the coefficient of variations (CVs) for the different securities and rank the alternatives based on the CVs. Does the CV measurement produce the same risk rankings as the standard deviation?
  4. Suppose you created a two-stock portfolio by investing $50,000 in High Tech and $50,000 in Collections. Calculate the expected return rP , the standard deviation ( δp), and the coefficient of variation (CVp) for this portfolio. How does the riskiness of this two-stock portfolio compare to the riskiness of the individual stocks if they were held in isolation?
  5. 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 randomly selected stocks are added to the portfolio? What is the implication for investors? Draw two graphs to illustrate your answer.

• Refer to the table below:

		Estimated Returns on Alternative Investments
State of Economy	Probability	T-Bills	High-Tech	Collections	US Rubber	Market Portfolio
Recession	0.1	8%	-22%	28%	10%	-13%
Below Average	0.2	8	-2	14.7	-10	1
Average	0.4	8	20	0	7	15
Above Average	0.2	8	35	-10	45	29
Boom	0.1	8	50	-20	30	43

Answer 1

Answer:

a.   (1)   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.

              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. Because there is uncertainty about inflation, it is unlikely that the realized real rate of return would equal the expected 3 percent. For example, if inflation averaged 6 percent over the year, then the realized real return would be only 8% - 6% = 2%, not 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. 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, and thus are positively correlated with, the economy, because the firm’s sales, and hence profits, generally will experience the same type of ups and downs 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 both 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 (move counter to) the economy.

b.   The expected rate of return,r̂, is expressed as follows:

Here Pr_i is the probability of occurrence of the it_h state, r_i is the estimated rate of return for that state, and n is the number of states. The calculation for High Tech is:

We can now add the 17.4% to the bottom of the table, and use the same formula to calculate r̂ for the other alternatives. Here they are:

c. (1)          The standard deviation is calculated as follows:

            Here are the standard deviations for the other alternatives:

             σ_T-Bills              =                   0.00%.

             σ_Collections          =                   13.36%.

             σ_U.S.Rubber         =                   18.82%.

             σ_M                   =                   15.34%.

(2)         The standard deviation is a measure of a security's (or a portfolio’s) total, or 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)         Probability distribution curves for High Tech, U.S. Rubber, and T-bills are shown here:

d.    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, Collections, with its low expected return, becomes the riskiest stock. The CV is a better measure of an asset’s total, or stand‑alone, risk than σ 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 σ but a high r̂.

e.     (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. Because 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: r_p = 0.5(-22%) + 0.5(28%) = 3%. We would do similar calculations for the other states of the economy, and get these results:

   State                                                         Portfolio

  Recession                                                   3.00%

  Below average                                           6.35

  Average                                                     10.00

  Above average                                           12.50

  Boom                                                          15.00

               Add these to the table to complete the last column.

Now multiply the probabilities times outcomes in each state to get the expected return on this two-stock portfolio—that is, 3.0%(0.1) + 6.35%(0.2) + 10.0%(0.4) + 12.5%(0.2) + 15.0%(0.1) = 9.57%.

Alternatively, we could apply this formula:

Which finds r̂_p as the weighted average of the expected returns of the individual securities in the portfolio.

The standard deviation of the portfolio is:

(2)     Using either σ or CV as our total risk measure, the total risk of the portfolio is significantly less than the total 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—that is, in a single-stock portfolio.

f.

This graph shows the probability distributions for a one-stock portfolio and a portfolio of many similar stocks. The graph shows that the standard deviation gets smaller as more stocks are combined in the portfolio, while r_p (the portfolio’s return) remains constant. Thus, by adding stocks to your portfolio, which initially started as a single-stock portfolio, risk has been reduced.

In the real world, stocks are positively correlated with one another—that is, the economy does well, so do stocks in general, and vice versa. Correlation coefficients between pairs of stocks generally range from +0.5 to +0.7. The graph below shows the relationship between portfolio size and risk.

A single stock selected at random would on average have a standard deviation of approximately 30 percent. 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 σ. In fact, σ stabilizes at about 15 percent 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. (Note: It is not completely costless to diversify, so even the largest institutional investors hold less than all stocks. Even index funds generally hold a smaller portfolio that is highly correlated with an index such as the S&P 500 rather than hold all the stocks in the index.)

The implication is clear: Investors should hold well-diversified portfolios of stocks rather than individual stocks. (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.