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Transcript
QMBU 301 – Exercise Problems Problem 1) As accounts manager in your company, you classify 75% of your customers as "good credit" and the rest as "risky credit" depending on their credit rating. Customers in the "risky" category allow their accounts to go overdue 50% of the time on average, whereas those in the "good" category allow their accounts to become overdue only 10% of the time. What percentage of overdue accounts are held by customers in the "risky credit" category? Problem 2) A rare genetic disease is discovered. Although only one in a million people carry it, you consider getting screened. You are told that the genetic test is extremely good; it is 100% sensitive (it is always correct if you have the disease) and 99.99% specific (it gives a false positive result only 0.01% of the time). Having recently learned Bayes' theorem, you decide not to take the test. Why? Bayes' Theorem states that for events X and Y: P(X|Y)=P(Y|X)*P(X)/P(Y). We want to know the probability of being healthy(X) given the positive test(PT) results(Y). According to the Bayes' Theorem, P(healthy|PT)=P(PT|healthy)*P(healthy)/P(PT). From the problem we know that P(healthy)=1-0.000001=0.999999 and getting a false positive P(PT|healthy)=0.0001. The only unknown in the formula above is the probability of having a positive test P(PT). It can be calculated using the definition of marginal probability P(Y)=P(Y|Z1)*P(Z1)+...+P(Y|Zn)*P(Zn), where Zi, i=1...n are all possible events. In our case there are only two possible events: "being healthy" and "being sick". Therefore P(PT)=P(PT|healthy)*P(healthy)+P(PT|sick)*P(sick). From the problem we know that P(PT|sick)=1.0 (test is always correct in presence of the disease) and P(sick)=0.000001. Substituting the numbers into the formula we get P(PT)=0.0001*0.999999+1.0*0.000001=0.000101. Finally, P(healthy|PT)=0.0001*0.999999/0.000101=0.990098, that is very close to 1. So, the probability of still being healthy given that the results of the test turned positive is above 99%. That is a good reason for not taking the test. Problem 3) Suppose you are thinking about investing your $100 in two stocks with $50 in each. Stock 1 is expected to bring in 10% return next year and has a standard deviation of 21%. On the other hand, stock 2 is expected to earn 30% but has a standard deviation of 34%. (a) Assuming that these two stocks are independent of each other, compute the probability that your investment shrinks your initial $100. (b) Assuming that these two stocks are negatively correlated, with a correlation of -0.837, compute the probability that your investment shrinks your initial $100. Hint: Use the empirical rule. Return from Investment = 50*(Stock 1 return) + 50*(Stock 2 return) E[Return from Investment] = 50*E[Stock 1 return)]+ 50*E[Stock 2 return] = 50*0.1 + 50*0.3 = $20 You expect a return of $20 from this investment. (a) If two stocks are independent Var(Return from Investment) = 502*Var(Stock 1 return) + 502*Var(Stock 2 return) = 2500*(0.21)2 + 2500*(0.34)2 = 399.25 Std dev(Return from Investment) = 399.25 $20 The probability of the initial investment shrinking means that the return from the investment is less than or equal to zero. Since the expected return is equal to $20, zero is one standard deviations away from $20. Hence, due to the empirical rule (stating that the interval around the mean with one standard deviation distance covers approximately 68% of the data), the probability of shrinking the investment is approximately 16%. (b) If two stocks are negatively correlated Var(Return from Investment) = 502*Var(Stock 1 return) + 502*Var(Stock 2 return) + 2*50*50*Cov(Stock 1, Stock 2) Cov(Stock 1, Stock 2) = Corr(Stock 1, Stock 2) * Stdev(Stock 1) * Stdev(Stock 2) = -0.837 * 0.21 * 0.34 = -0.0598 Var(Return from Investment) = 2500*(0.21)2 + 2500*(0.34)2 + + 502*Var(Stock 2 return) + 2*50*50*Cov(Stock 1, Stock 2) = 2500*(0.21)2 + 2500*(0.34)2 + 5000*(-0.0598) = 100.44 Std dev(Return from Investment) = 100.44 $10 The probability of the initial investment shrinking means that the return from the investment less than or equal to zero. Since the expected return is equal to $20, zero is two standard deviations away from $20. Hence, due to the empirical rule (stating that the interval around the mean with two standard deviations distance covers approximately 95% of the data), the probability of shrinking the investment is approximately only 2.5%. Therefore, we have reduced the risk considerably by including two negatively related stocks in our portfolio.