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Chapter 5 Understanding Risk Conceptual and Analytical Problems Consider a game in which a coin will be flipped three times. For each heads you will be paid $100. Assume that the coin comes up heads with probability ⅔. Construct a table of the possibilities and probabilities in this game. Compute the expected value of the game. How much would you be willing to pay to play this game? Consider the effect of a change in the game so that if tails comes up two times in a row, you get nothing. How would your answers to parts a-c change? Answer: a. The first step is to list all possible outcomes for the three coin tosses (heads (H), tails (T)).
Toss 1 Toss 2 Toss 3
H H H
H H T
H T H
H T T
T T H
T T T
T H H
T H T
We can see that there are four distinct options—3H and 0T, 2H and 1T, 1H and 2T, and 3T and 0H. Next, we will need to assign a probability to each of these four options, taking into account that the probability of the coin coming up heads is 2/3 (and so the probability of it coming up tails is 1/3). We do this by multiplying the probabilities of a certain outcome for each coin toss. For example, the probability of getting 3H outcomes is 2/3 × 2/3 × 2/3 = 8/27.
Possibilities Probability Outcome
1 1/27 0 heads, 3 tails
2 2/9 1 head, 2 tails
3 4/9 2 heads, 1 tail
4 8/27 3 heads, 0 tails
b. Expected Value = 1/27($0) + 2/9($100) + 4/9($200) + 8/27($300) = $200 c. A person who is risk-averse will want to pay less than $200; a person who is risk-neutral will be willing to pay $200. d.
Possibilities Probability Outcome Payoff
1 1/27 3 tails $0
2 2/27 Tails, heads, tails $100
3 2/27 Tails, tails, heads $0
4 2/27 Heads, tails, tails $0
5 4/9 2 heads, 1 tails $200
6 8/27 3 heads, 0 tails $300
Expected Value = 1/27($0) + 2/27($100) + 2/27($0) + 2/27($0) + 4/9($200) + 8/27($300) = $185 A person who is risk-averse will want to pay less than $185; a person who is risk- neutral will be willing to pay $185. Why is it important to be able to quantify risk? Answer: Core Principle 2 tells us that risk requires compensation. In order to determine what that compensation should be – to put a price on risk – we must have some measure of it. This facilitates the transfer of risk to those who are willing to bear it. You are the founder of IGRO, an Internet firm that delivers groceries. Give an example of an idiosyncratic risk and a systematic risk your company faces. As founder of the company, you own a significant portion of the firm, and your personal wealth is highly concentrated in IGRO shares. What are the risks that you face, and how should you try to reduce them? Answer: a. An idiosyncratic risk is that someone could create another Internet firm to deliver groceries, which would reduce IGRO’s share of the market. A systematic risk could be that people lose trust in the security of online transactions, in which case all firms offering purchases via the Internet would suffer. An example of a more widespread systematic risk is that the entire economy does poorly; if people’s incomes fall, they tend to buy less of most things, including groceries from IGRO (people’s overall food consumption would not be greatly affected, but they may return to buying groceries from a supermarket instead of online to save on delivery costs). b. You could suffer large losses if IGRO does poorly; your stock holdings could greatly decrease in value, and you could lose your job. You should try to diversify and invest in assets whose returns are not correlated with returns on IGRO stock. Assume that the economy can experience high growth, normal growth, or recession. Under these conditions, you expect the following stock market returns for the coming year:
State of the Economy Probability Return
High Growth 0.2 +30%
Normal Growth 0.7 +12%
Recession 0.1 -15%
Compute the expected value of a $1,000 investment over the coming year. If you invest $1,000 today, how much money do you expect to have next year? What is the percentage expected rate of return? Compute the standard deviation of the percentage return over the coming year. If the risk-free return is 7 percent, what is the risk premium for a stock market investment? Answer: a. Expected Value = 0.2($1,000)(1 + 0.30) + 0.7($1,000)(1 + 0.12) + 0.1($1,000)(1 – 0.15) = $1,129 Expected Percentage Return = 0.2(0.30) + 0.7(0.12) + 0.1(-0.15) = 0.129, or 12.9% Alternatively, ((1,129 – 1,000)/1,000) × 100 = 0.129, or 12.9% b. Standard Deviation = c. Risk Premium = 12.9% - 7% = 5.9% Using the information from the table in Problem 4, in dollar terms what is the value at risk associated with the $1,000 investment? Answer: Since the worst outcome is a loss of 15 percent, the value at risk is $150 ( = $1,000 × 0.15). 6. Car insurance companies sell a large number of policies. Explain how this practice minimizes their risk. Answer: Individual automobile accidents are uncorrelated in the sense that one person having an accident doesn’t have any effect on whether someone else will have one. By selling lots of insurance policies, the company is reducing risk by spreading it. Put another way, if each individual has a 1% chance of having an automobile accident each year, then on average one out of each 100 policyholders will make a claim in a given year. If the company sells 1,000,000 policies, then it can be reasonably sure they will face 10,000 claims. 7. Mortgages increase the risk faced by homeowners. a. Explain how. b. What happens to the homeowner’s risk as the down payment on the house rises from 10 percent to 50 percent. Answer: a. The mortgage is leverage for the homeowner, and leverage increases risk. b. From the formula in the Tools of the Trade we know that with 10 percent down, the leverage factor is 10 [1/ (1 – 90/100)], and with 50 percent down, it is 2 (1/ 1 – 50/100). A down payment of 50 percent reduces risk by a factor of 5 relative to a down payment of 10 percent. 8. Banks pay substantial amounts to monitor the risks that they take. One of the primary concerns of a bank’s “risk managers” is to compute the value at risk. Why is value at risk so important for a bank (or any financial institution)? Answer: The first concern of a bank’s management is to stay open. This means making sure that the risk of bankruptcy remains very small. That means focusing on the worst case, which is what value at risk does. Explain how liquidity problems can be an important source of systemic risk in the financial system. Answer: Lack of liquidity can make it difficult or impossible for certain firms to meet their obligations to other firms in the system. For example, if one firm cannot convert some assets to cash due to market liquidity problems, or if it cannot borrow due to funding liquidity problems, it may not be able to deliver on an obligation to another firm. This, in turn, may compromise the second firm’s ability to meet its obligations and so on, leading to system-wide problems. Give an example of systematic risk for the U.S. economy and how you might reduce your exposure to such a risk. Answer: A recession is one kind of systematic risk facing the U.S. economy. You could diversify your investments internationally. You could hedge against a U.S.-specific risk by investing in a country whose fortunes move in the opposite direction to those of the United States. Alternatively, you could reduce your risk by spreading your portfolio across a broad range of countries whose fortunes are independent of each other. For each of the following events, explain whether it represents systematic risk or idiosyncratic risk and explain why. Your favorite restaurant is closed by the county health department. The government of Spain defaults on its bonds, causing the breakup of the euro- area. Freezing weather in Florida destroys the orange crop. Solar flares destroy earth-orbiting communications satellites, knocking out cellphone service worldwide. Answer: This is idiosyncratic risk since it is unique to this particular establishment. This is a systematic risk that affects entire economies within the euro-area and beyond. This is idiosyncratic risk as only one of several orange-growing areas in the country is affected. For example, orange groves in California are not damaged by the Florida freeze. This is systematic risk as communications around the globe are disrupted, perhaps until new satellites can be constructed and put into orbit. You are planning for retirement and must decide whether to purchase only your employer’s stock for your 401(k) or, instead, to buy a mutual fund that holds shares in the 500 largest companies in the world. From the perspective of both idiosyncratic and systematic risk, explain how you would make your decision. Answer: From both perspectives, you should purchase the more highly diversified portfolio containing the 500 large companies. If you own only your company’s stock, your retirement savings could be permanently lowered by an idiosyncratic risk to the firm. If you hold the diversified portfolio, even if your employer is one of the 500 companies in the mutual fund, only a small portion of your savings is affected. In the event of systematic risk, the percentage decline in the price of your employer’s stock may be similar to that of the broad portfolio since all companies face the same challenge. In neither case would you expect to be worse off holding the mutual fund, but you would be better protected against idiosyncratic risk. For each of the following actions, identify whether the method of risk assessment motivating your action is due to the value at risk or the standard deviation of an underlying probability distribution. You buy life insurance. You hire an investment advisor who specializes in international diversification in stock portfolios. In your role as a central banker you provide emergency loans to illiquid intermediaries. You open a kiosk at the mall selling ice cream and hot chocolate. Answer: Life insurance only pays off when you die and your heirs are left without your regular paycheck, presumably the worst outcome. So this is an example of a value at risk assessment. You are taking steps to reduce idiosyncratic risk in a portfolio, attempting to minimize the variability of the portfolio’s value for a given expected return. It is an example of using the underlying probability distribution as the basis for the decision. Emergency loans to support the financial system are an attempt to avoid the worst outcome, and can be motivated by a value at risk assessment. You are trying to smooth out the earnings of the business across seasons. This is an example of using the seasonal probability distributions for both ice cream and hot chocolate sales to lower idiosyncratic risk. Which of the following investments in the following table would be most attractive to a risk-averse investor? How would your answer differ if the investor were described as risk-neutral?
Investment Expected Value Standard Deviation
A 75 10
B 100 10
C 100 20
Answer: A risk-averse investor requires a higher return for taking on more risk. This investor will also prefer an investment with a higher expected value given a certain level of risk. Therefore, a risk-averse investor will prefer Investment B, as it yields a higher expected value than Investment A and the same expected value as Investment C for a lower level of risk, as measured by the standard deviation. A risk-neutral investor is concerned only with the expected return of the investment and so would be indifferent between Investments B and C. Consider an investment that pays off $800 or $1,400 per $1,000 invested with equal probability. Suppose you have $1,000 but are willing to borrow to increase your expected return. What would happen to the expected value and standard deviation of the investment if you borrowed an additional $1,000 and invested a total of $2,000? What if you borrowed $2,000 to invest a total of $3,000? Answer: If you just invest your own $1,000, the expected value = 0.5($800) + 0.5($1,400) = $1,100 or 10% and the standard deviation = 300. If you borrow an additional $1,000, the expected value = 0.5($1,600-$1,000) + 0.5($2,800-$1,000) = $1,200 or 20%. You have doubled the expected return. The standard deviation = √(0.5(600-〖1,200)〗^2+0.5(1800-〖1,200)〗^2 ) = 600. The standard deviation has also doubled. If you borrowed $2,000 to invest a total of $3,000, the expected value = 0.5($2,400 - $2,000) + 0.5($4,200 - $2,000) = $1,300 or 30%. You have tripled the expected return versus the un-leveraged investment. The standard deviation = √(0.5(400-〖1,300)〗^2+0.5(2,200-〖1,300)〗^2 ) = 900. The standard deviation has tripled versus the un-leveraged investment. You can confirm your answer using the leverage ratio: In the first case, the owner’s contribution to the purchase is $1 for each $1 invested, so the leverage ratio is 1. In the second case, you contribute half of the cost of the total investment, so the leverage ratio is 1/0.5 = 2 – or $2,000/1,000 = 2. The expected value and standard deviation are increased by a factor of 2- or doubled. In the third case, you contribute one third of the cost of the total investment, so the leverage ratio is 1/0.3333 = 3.The expected value and standard deviation are tripled. Looking again at the investment described in Problem 15, what is the maximum leverage ratio you could have and still have enough to repay the loan in the event the bad outcome occurred? Answer: The bad outcome pays off $800 per $1,000 invested, so you lose $200 per $1,000 invested. Therefore, the maximum leverage ratio you could have is 5. Borrowing $4,000 would give a total investment of $5,000. In the event of the bad outcome, the payoff would be $800 ×5 - $4,000 – just enough to repay the loan. You would lose all of your own $1,000. Consider two possible investments whose payoffs are completely independent of one another. Both investments have the same expected value and standard deviation. If you have $1,000 to invest, could you benefit from dividing your funds between these investments? Explain your answer. Answer: Yes. Even though the investments have the same standard deviation, by spreading your $1,000 across both of them, you reduce your risk. Intuitively, you are adding combinations of possible payoffs that lie between the worst- and best-case scenarios and so the probability-weighted spread of the possible payoffs is smaller. Mathematically, the variance of the payoffs is halved. Suppose, as in Problem 17, that there were ten independent investments available rather than just two. Would it matter if you spread your $1,000 across these 10 investments rather than two? Answer: Yes. The gains from spreading would be larger if you spread the $1000 across ten investments. The risk, as measured by the variance of the payoffs, is inversely related to the number of independent investments. You are considering three investments, each with the same expected value and each with two possible payoffs. The investments are sold only in increments of $500. You have $1,000 to invest and so you have the option of either splitting your money equally between two of the investments or placing all $1,000 in one of the investments. If the payoffs from investment A are independent of the payoffs from investments B and C and the payoffs from B and C are perfectly negatively correlated with each other (meaning when B pays off, C doesn’t and vice versa), which investment strategy will minimize your risk? Answer: You should put $500 into each of B and C. Because one pays off when the other doesn’t, you eliminate your risk by hedging. Spreading your investment across A and either B or C would reduce your risk compared with investing all $1000 in one investment but would not eliminate it. In which of the following cases would you be more likely to decide whether to take on the risk involved by looking at a measure of the value at risk? You are unemployed and are considering investing your life savings of $10,000 to start up a new business. You have a full-time job paying $100,000 a year and are considering making a $1,000 investment in stock of a well-established, stable company. Explain your reasoning. Answer: You should be more concerned about the value at risk - a measure of the worse possible loss with a given probability - in case a. Experiencing that loss would likely have dire consequences. In the second case, even in the unlikely event that the investment lost all its value, the outcome would not be catastrophic. You have the option to invest in either country A or country B but not both. You carry out some research and conclude that the two countries are similar in every way except that the returns on assets of different classes tend to move together much more in country A– that is, they are more highly correlated in country A than in country B. Which country would choose to invest in and why? Answer: You should invest in country B as the benefits from diversification are greater than in country A. If everything else is equal, spreading your risk across different asset classes brings greater benefit when the correlation among the returns is lower. 22. In June 2016, the United Kingdom (U.K.) held a referendum on whether the country should remain a member of the European Union (EU). A decision to leave the EU was commonly referred to as Brexit and many expert commentators predicted that a vote for Brexit would diminish U.K. economic growth. Suppose you were a small business owner in the United Kingdom around this time. Ahead of the vote, as what kind of risk would you classify the possibility of a vote for Brexit? Why? Do you think a strategy to reduce this risk through hedging or spreading risk within the U.K. economy would be successful? Explain your answer. Answer: a. The prospect that the U.K. would vote to leave the European Union and suffer a loss of economic growth as a consequence is an example of a systematic risk; a risk that is economywide, rather than being specific to one firm or a small group of firms. The potential harm to economic growth would affect firms across the entire economy. b. Hedging and spreading risk are strategies that can be employed to manage idiosyncratic risks, but not systematic risk. Given the economy-wide nature of the Brexit risk, it is unlikely that you would find a way to hedge by assuming an opposing risk. For this reasons, the benefits of spreading risk within the U.K. economy would likely be limited 23. Consider again the U.K. referendum on membership of the European Union (EU) referred to in question 22, but this time from the perspective of an international investor. Assume that the downside risk for the U.K. economy from a Brexit vote would be much greater than for the rest of the global economy. Some countries could even benefit from a Brexit vote. a. Why might you demand a relatively higher return on U.K. investments ahead of the vote? b. What strategy might you employ to reduce the risk associated with your U.K. investments? Answer: a. Core principle 2 tells us that risk requires compensation. A higher return demanded on U.K. investments could reflect a higher risk premium required in the face of the additional perceived risk surrounding the vote. b. While the vote on Brexit represented a systematic risk from the perspective of the U.K. economy, from a global perspective it could be viewed as more of an idiosyncratic risk. Suppose, for example, that another country stood to gain from a Brexit vote as it became a relatively more attractive location for firms. An investor could then hedge U.K. investments that were likely to lose out with a Brexit vote by placing investments in the other country. Spreading risk across investments in different countries that were not highly correlated (even if they were not negatively correlated) also could help reduce risk. Data Exploration Plot the percentage change from a year ago of the Wilshire 5000 stock index at a monthly frequency (FRED code: WILL5000PR). Visually, has the risk of the Wilshire 5000 index changed over time? Answer: Visual examination of the plot below does not show any obvious change in the percentage volatility of stock prices. If this is approximately correct, then the variance or standard deviation of the index in percentage terms is a useful indicator of the risk. Wilshire Associates, Wilshire 5000 Price Index© [WILL5000PR], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/WILL5000PR. Another way to understand stock market risk is to examine how investors expect risk to evolve in the near future. The DJIA volatility index (FRED code: VXDCLS) is one such measure. Plot the level of this volatility index at a monthly frequency since October 1997 and as a second line, the percent change from a year ago of the Wilshire 5000 index (FRED code: WILL5000PR). Compare their patterns. Answer: The plot shows several spikes in the DJIA volatility index, usually in periods when the Wilshire 5000 index is falling. For example, the volatility index peaked at nearly 60% in the crisis of 2007-2009. In contrast, the volatility index is usually low when the stock market index is rising (see the period between 2003 and 2007). The causal relationship between expectations and market price swings can vary over time: falling stock prices encourage investors to revise upward their expectations of near-term market risks, but rising expectations of market risk also depress stock market prices. More information would be needed to distinguish these alternatives. Wilshire Associates, Wilshire 5000 Price Index© [WILL5000PR], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/WILL5000PR. Chicago Board Options Exchange, CBOE DJIA Volatility Index© [VXDCLS], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/VXDCLS. For the period since 2005, plot on one graph the 30-year conventional mortgage rate (FRED code: MORTG) and a measure of an adjustable mortgage rate (FRED code: MORTGAGE5US). Explain their systematic relationship using Core Principle 2. Answer: The plots of the 30-year and adjustable rates are: The adjustable rate is generally below the 30-year conventional rate since the borrower assumes the risk of rising interest rates. Since risk requires compensation, the borrower obtains a lower initial rate and assumes the risk of paying higher rates later on. Plot the difference since 1979 between the Moody’s Baa bond index (FRED code: BAA) and the U.S. Treasury 10-year bond yield (FRED code: GS10). Comment on the trend and variability of this “credit risk premium” (see Chapter 7) before and after the 2007-2009 financial crisis. Answer: The data plot is: Moody's Seasoned Baa Corporate Bond Yield©, copyright, 2016, Moody's Investor Services, retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/DBAA. Baa bonds are corporate issues with a higher probability of default than for government bonds, so the yield difference is an indicator of default risk. This “credit risk premium” rose sharply during the 2007-2009 financial crisis. It slid again after the crisis, but has remained somewhat elevated for several years compared to past periods of economic expansion indicates more difficult problems. Solution Manual for Money, Banking and Financial Markets Stephen G. Cecchetti, Kermit L. Schoenholtz 9781259746741, 9780078021749, 9780077473075
