Answers _ Mathematics and Statistics for Financial Risk Management

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10/23/23, 1:31 PM Answers | Mathematics and Statistics for Financial Risk Management https://learning.oreilly.com/library/view/mathematics-and-statistics/9781118170625/OEBPS/9781118170625_epub_bm_06.htm#ans-exs-0009 1/40 Answers CHAPTER 1 1. a. y = 5 a. y = ln(1) – ln( e ) = 0 – 1 = –1 b. y = ln(10) + ln( e ) = ln(10) + 1 = 3.3026 2. Annual rate = 5.12%; semiannual rate = 5.05%; continuous rate = 4.99%. 3. 4. 5. 6. 7. 8. ln(ln(10)) = 0.8340.
10/23/23, 1:31 PM Answers | Mathematics and Statistics for Financial Risk Management https://learning.oreilly.com/library/view/mathematics-and-statistics/9781118170625/OEBPS/9781118170625_epub_bm_06.htm#ans-exs-0009 2/40 9. 10. 11. The bond will pay 10 coupons of $2, starting in a year's time. In addi- tion, the notional value of the bond will be returned with the final coupon payment in 10 years. The present value, V, is then: We start by evaluating the summation, using a discount factor of δ = 1/1.05 0.95: Inserting this result into the initial equation we obtain our final result: Note that the present value of the bond, $78.83, is less than the notional value of the bond, $100. This is what we would expect, given that there is no risk of default, and the coupon rate is less than the discount rate. CHAPTER 2
10/23/23, 1:31 PM Answers | Mathematics and Statistics for Financial Risk Management https://learning.oreilly.com/library/view/mathematics-and-statistics/9781118170625/OEBPS/9781118170625_epub_bm_06.htm#ans-exs-0009 3/40 1. Probability that both generate positive returns = 60% × 70% = 42%. Probability that both funds lose money = (1 – 60%) × (1 – 70%) = 40% × 30% = 12%. 2. 88%. The sum of all three events—upgrade, downgrade, and no change —must sum to one. There is no other possible outcome. 88% + 8% + 4% = 100%. 3. 50%. The outcomes are mutually exclusive; therefore, 20% + 30% = 50%. 4. 5. 6. 32.14%. By applying Bayes’ theorem, we can calculate the result: Even though the model is 90% accurate, 95% of the bonds don't default and it predicts that 10% of them will. Within the bond portfolio, the mod- el identifies 9.5% of the bonds as likely to default, even though they won’t. Of the 5% of bonds that actually default, the model correctly identifies 90%, or 4.5% of the portfolio. This 4.5% correctly identified is over- whelmed by the 9.5% incorrectly identified.
10/23/23, 1:31 PM Answers | Mathematics and Statistics for Financial Risk Management https://learning.oreilly.com/library/view/mathematics-and-statistics/9781118170625/OEBPS/9781118170625_epub_bm_06.htm#ans-exs-0009 4/40 7. Given the density function, we can find c by noting that the sum of probabilities must be equal to one: 8. First we check that this is a valid CDF, by calculating the value of the CDF for the minimum and maximum values of x : Next we calculate the PDF by taking the first derivative of the CDF: 9. We first calculate the CDF by integrating the PDF:
10/23/23, 1:31 PM Answers | Mathematics and Statistics for Financial Risk Management https://learning.oreilly.com/library/view/mathematics-and-statistics/9781118170625/OEBPS/9781118170625_epub_bm_06.htm#ans-exs-0009 5/40 We first try to find c using the fact that the CDF is zero at the minimum value of x , x = 0. As it turns out, any value of c will satisfy this constraint, and we cannot use this to determine c . If we use the fact that the CDF is 1 for the maximum value of x , x = e , we find that c = 1: The CDF can then be expressed simply as: 10. P (both bonds default) = 9%. P (one defaults) = 42%. P (neither defaults) = 49%. 11. We can start by summing across the first row to get W : In a similar fashion, we can find X by summing across the second row: To calculate Y , we can sum down the first column, using our previously calculated value for W : Using this result, we can sum across the third row to get Z :
10/23/23, 1:31 PM Answers | Mathematics and Statistics for Financial Risk Management https://learning.oreilly.com/library/view/mathematics-and-statistics/9781118170625/OEBPS/9781118170625_epub_bm_06.htm#ans-exs-0009 6/40 The completed probability matrix is: The last part of the question asks us to find the conditional probability, which we can express as: We can solve this by taking values from the completed probability matrix. The equity underperforms in 40% of scenarios. The equity underper- forms and the bonds are downgraded in 15% of scenarios. Dividing, we get our final answer, 37.5%. 12. The probability that a B-rated bond defaults over one year is 2%. This can be read directly from the last column of the second row of the ratings transition matrix. The probability of default over two years is 4.8%. During the first year, a B-rated bond can either be upgraded to an A rating, stay at B, be down- graded to C, or default. From the transition matrix, we know that the probability of these events is 10%, 80%, 8%, and 2%, respectively. If the bond is upgraded to A, then there is zero probability of default in the sec- ond year (the last column of the first row of the matrix is 0%). If it re-
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