Problems Bsd
1. Define the following terms:
A. Probability
B. Conditional probability
C. Event
D. Independent events
E. Variance
2. State three mutually exclusive and exhaustive events describing the reaction of a company's stock price to a corporate earnings announcement on the day of the announcement.
3. Label each of the following as an empirical, a priori, or subjective probability.
A. The probability that U.S. stock returns exceed long-term corporate bond returns over a 10-year period, based on Ibbotson Associates data.
B. An updated (posterior) probability of an event arrived at using Bayes' formula and the perceived prior probability of the event.
C. The probability of a particular outcome when exactly 12 equally likely possible outcomes exist.
D. A historical probability of default for double-B rated bonds, adjusted to reflect your perceptions of changes in the quality of double-B rated issuance.
4. You are comparing two companies, BestRest Corporation and Relaxin, Inc. The exports of both companies stand to benefit substantially from the removal of import restrictions on their products in a large export market. The price of BestRest shares reflects a probability of 0.90 that the restrictions will be removed within the year. The price of Relaxin stock, however, reflects a 0.50 probability that the restrictions will be removed within that time frame. By all other information related to valuation, the two stocks appear comparably valued. How would you characterize the implied probabilities reflected in share prices? Which stock is relatively overvalued compared to the other?
5. Suppose you have two limit orders outstanding on two different stocks. The probability that the first limit order executes before the close of trading is 0.45. The probability that the second limit order executes before the close of trading is 0.20. The probability that the two orders both execute before the close of trading is 0.10. What is the probability that at least one of the two limit orders executes before the close of trading?
6. Suppose that 5 percent of the stocks meeting your stock-selection criteria are in the telecommunications (telecom) industry. Also, dividend-paying telecom stocks are 1 percent of the total number of stocks meeting your selection criteria. What is the probability that a stock is dividend paying, given that it is a telecom stock that has met your stock selection criteria?
7. You are using the following three criteria to screen potential acquisition targets from a list of 500 companies:
|
Fraction of the 500 Companies | |
|
Criterion |
Meeting the Criterion |
|
Product lines compatible |
0.20 |
|
Company will increase combined sales growth rate |
0.45 |
|
Balance sheet impact manageable |
0.78 |
If the criteria are independent, how many companies will pass the screen?
8. You apply both valuation criteria and financial strength criteria in choosing stocks. The probability that a randomly selected stock (from your investment universe) meets your valuation criteria is 0.25. Given that a stock meets your valuation criteria, the probability that the stock meets your financial strength criteria is 0.40. What is the probability that a stock meets both your valuation and financial strength criteria?
9. A report from Fitch data service states the following two facts:1
• In 2002, the volume of defaulted U.S. high-yield debt was $109.8 billion. The average market size of the high-yield bond market during 2002 was $669.5 billion.
• The average recovery rate for defaulted U.S. high-yield bonds in 2002 (defined as average price one month after default) was $0.22 on the dollar.
Address the following three tasks:
A. On the basis of the first fact given above, calculate the default rate on U.S. high-yield debt in 2002. Interpret this default rate as a probability.
B. State the probability computed in Part A as an odds against default.
C. The quantity 1 minus the recovery rate given in the second fact above is the expected loss per $1 of principal value, given that default has occurred. Suppose you are told that an institution held a diversified high-yield bond portfolio in 2002. Using the information in both facts, what was the institution's expected loss in 2002, per $1 of principal value of the bond portfolio?
10. You are given the following probability distribution for the annual sales of ElStop Corporation:
Probability Distribution for ElStop Annual Sales
Sales
Probability (millions)
A. Calculate the expected value of ElStop's annual sales.
B. Calculate the variance of ElStop's annual sales.
C. Calculate the standard deviation of ElStop's annual sales.
11. Suppose the prospects for recovering principal for a defaulted bond issue depend on which of two economic scenarios prevails. Scenario 1 has probability 0.75 and will result in recovery of $0.90 per $1 principal value with probability 0.45, or in recovery of $0.80 per $1 principal value with probability 0.55. Scenario 2 has probability 0.25 and will
1"High Yield Defaults 2002: The Perfect Storm,'' 19 February, 2003.
result in recovery of $0.50 per $1 principal value with probability 0.85, or in recovery of $0.40 per $1 principal value with probability 0.15.
A. Compute the probability of each of the four possible recovery amounts: $0.90, $0.80, $0.50, and $0.40.
B. Compute the expected recovery, given the first scenario.
C. Compute the expected recovery, given the second scenario.
D. Compute the expected recovery.
E. Graph the information in a tree diagram.
12. Suppose we have the expected daily returns (in terms of U.S. dollars), standard deviations, and correlations shown in the table below.
|
U.S. Dollar Daily Returns in Percent | ||
|
Expected Return Standard Deviation |
U.S. Bonds German Bonds 0.029 0.021 0.409 0.606 |
Italian Bonds 0.073 0.635 |
|
Correlation Matrix | ||
|
U.S. Bonds German Bonds Italian Bonds |
U.S. Bonds German Bonds 1 0.09 1 |
Italian Bonds 0.10 0.70 1 |
Source: Kool (2000), Table 1 (excerpted and adapted).
Source: Kool (2000), Table 1 (excerpted and adapted).
A. Using the data given above, construct a covariance matrix for the daily returns on U.S., German, and Italian bonds.
B. State the expected return and variance of return on a portfolio 70 percent invested in U.S. bonds, 20 percent in German bonds, and 10 percent in Italian bonds.
C. Calculate the standard deviation of return for the portfolio in Part B.
13. The variance of a stock portfolio depends on the variances of each individual stock in the portfolio and also the covariances among the stocks in the portfolio. If you have five stocks, how many unique covariances (excluding variances) must you use in order to compute the variance of return on your portfolio? (Recall that the covariance of a stock with itself is the stock's variance.)
14. Calculate the covariance of the returns on Bedolf Corporation (Rb) with the returns on Zedock Corporation (Rz), using the following data.
|
RZ = 15% |
RZ = 10% |
II | ||
|
Rb |
= 30% |
0.25 |
0 |
0 |
|
RB |
= 15% |
0 |
0.50 |
0 |
|
Rb |
= 10% |
0 |
0 |
0.25 |
Note: Entries are joint probabilities.
Note: Entries are joint probabilities.
15. You have developed a set of criteria for evaluating distressed credits. Companies that do not receive a passing score are classed as likely to go bankrupt within 12 months. You gathered the following information when validating the criteria:
• Forty percent of the companies to which the test is administered will go bankrupt within 12 months: P(nonsurvivor) = 0.40.
• Fifty-five percent of the companies to which the test is administered pass it: P(pass test) = 0.55.
• The probability that a company will pass the test given that it will subsequently survive 12 months, is 0.85: P(pass test | survivor) = 0.85.
A. What is P(pass test | nonsurvivor)?
B. Using Bayes' formula, calculate the probability that a company is a survivor, given that it passes the test; that is, calculate P(survivor | pass test).
C. What is the probability that a company is a nonsurvivor, given that it fails the test?
D. Is the test effective?
16. On one day in March, 3,292 issues traded on the NYSE: 1,303 advanced, 1,764 declined, and 225 were unchanged. In how many ways could this set of outcomes have happened? (Set up the problem but do not solve it.)
17. Your firm intends to select 4 of 10 vice presidents for the investment committee. How many different groups of four are possible?
18. As in Example 4-11, you are reviewing the pricing of a speculative-grade, one-year-maturity, zero-coupon bond. Your goal is to estimate an appropriate default risk premium for this bond. The default risk premium is defined as the extra return above the risk-free return that will compensate investors for default risk. If R is the promised return (yield-to-maturity) on the debt instrument and Rp is the risk-free rate, the default risk premium is R — Rp. You assess that the probability that the bond defaults is 0.06: P(the bond defaults) = 0.06. One-year U.S. T-bills are offering a return of 5.8 percent, an estimate of Rp. In contrast to your approach in Example 4-11, you no longer make the simplifying assumption that bondholders will recover nothing in the event of a default. Rather, you now assume that recovery will be $0.35 on the dollar, given default.
A. Denote the fraction of principal recovered in default as G. Following the model of Example 4-11, develop a general expression for the promised return R on this bond.
B. Given your expression for R and the estimate of Rp, state the minimum default risk premium you should require for this instrument.
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