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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.