OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO =default Bond B =default Bond A O O O O O O OOO OOO O O O O O OOO O O O O O O O O oo OOOO Nm DN 0 0 11 12. M2 16 17 18 19 20 21 22 23 24 25 26 27 28 31 22 11 35 2 2. Risk analysis. Download the dataset Bonds A and B from BU Brain. Two risky firms, A and B, finance their operations in part by continuously issuing short-term (30-day) zero-coupon commercial paper. The face value of each debt obligation is $1. The two time series represent 60 months (five years) of data for defaults on these debt obligations: a value of 1 indicates that a default occurred that month (the bondholder received nothing), and a value of 0 indicates that a default did not occur that month (the bondholder received the face value. You will use these data to estimate the probability that the bonds default this period. a. First, use the fundamental pricing equation to find the price of the bond issued by each firm (A and B), assuming that the representative investor is risk-neutral and has a monthly time discount preference of 8 = 0.95. Use this price to find the expected return on each bond, E[ra] and E[ro]. (Calculate the return with and without default, then take a probability-weighted average. The probability that each bond defaults (or doesn't default) can be estimated from the data.) Find the volatility of each bond's return, A and op. For simplicity, we can assume normally-distributed outcomes and use the variance formula, as in Homework 1. (2 points) b. You are the representative investor, it is the first of the month, and you have a portfolio consisting of one unit of each bond, both maturing at the end of the month. The value of the portfolio, P*, is equal to the sum of the bond prices, PA and Pg. Ignore the role of correlation between the defaults on the bonds (assume the defaults occur totally independently of one another) and find the expected return on the portfolio, E[rp*], and the volatility of the portfolio return, Op. To do this, you need to find four probabilities: (1) the probability that both bonds default; (2) the probability that only bond A defaults; (3) the probability that only bond B defaults; and (4) the probability that neither bond defaults. (4 points) c. Now, incorporate correlation into your forecast of the portfolio's risk. Calculate and report the correlation coefficient between the two bonds' defaults (use =CORREL). Find the same four probabilities as in part (b), but this time, start by sketching the 2 x 2 probability space, as in Bayes' Theorem. Then, find the expected return on the portfolio, E[rp"], and the volatility of the portfolio return, Op**. How do they compare to the expected return and volatility that you found in part (b)? Produce a bar plot in Excel showing the probabilities of these three outcomes from each of part (b) and part (c): (1) zero bonds default; (2) only one bond defaults (either A or B); (3) both bonds default. What do you take away from this? (4 points) d. On the first day of the month, firm B announces that is experiencing financial difficulties and will place a moratorium on its debt payments (i.e., it is certain to default on bond B at the end of the month). Based on this observation, what should be the new price of bond A? (4 points) OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO =default Bond B =default Bond A O O O O O O OOO OOO O O O O O OOO O O O O O O O O oo OOOO Nm DN 0 0 11 12. M2 16 17 18 19 20 21 22 23 24 25 26 27 28 31 22 11 35 2 2. Risk analysis. Download the dataset Bonds A and B from BU Brain. Two risky firms, A and B, finance their operations in part by continuously issuing short-term (30-day) zero-coupon commercial paper. The face value of each debt obligation is $1. The two time series represent 60 months (five years) of data for defaults on these debt obligations: a value of 1 indicates that a default occurred that month (the bondholder received nothing), and a value of 0 indicates that a default did not occur that month (the bondholder received the face value. You will use these data to estimate the probability that the bonds default this period. a. First, use the fundamental pricing equation to find the price of the bond issued by each firm (A and B), assuming that the representative investor is risk-neutral and has a monthly time discount preference of 8 = 0.95. Use this price to find the expected return on each bond, E[ra] and E[ro]. (Calculate the return with and without default, then take a probability-weighted average. The probability that each bond defaults (or doesn't default) can be estimated from the data.) Find the volatility of each bond's return, A and op. For simplicity, we can assume normally-distributed outcomes and use the variance formula, as in Homework 1. (2 points) b. You are the representative investor, it is the first of the month, and you have a portfolio consisting of one unit of each bond, both maturing at the end of the month. The value of the portfolio, P*, is equal to the sum of the bond prices, PA and Pg. Ignore the role of correlation between the defaults on the bonds (assume the defaults occur totally independently of one another) and find the expected return on the portfolio, E[rp*], and the volatility of the portfolio return, Op. To do this, you need to find four probabilities: (1) the probability that both bonds default; (2) the probability that only bond A defaults; (3) the probability that only bond B defaults; and (4) the probability that neither bond defaults. (4 points) c. Now, incorporate correlation into your forecast of the portfolio's risk. Calculate and report the correlation coefficient between the two bonds' defaults (use =CORREL). Find the same four probabilities as in part (b), but this time, start by sketching the 2 x 2 probability space, as in Bayes' Theorem. Then, find the expected return on the portfolio, E[rp"], and the volatility of the portfolio return, Op**. How do they compare to the expected return and volatility that you found in part (b)? Produce a bar plot in Excel showing the probabilities of these three outcomes from each of part (b) and part (c): (1) zero bonds default; (2) only one bond defaults (either A or B); (3) both bonds default. What do you take away from this? (4 points) d. On the first day of the month, firm B announces that is experiencing financial difficulties and will place a moratorium on its debt payments (i.e., it is certain to default on bond B at the end of the month). Based on this observation, what should be the new price of bond A? (4 points)
