Probability tools, idiosyncratic and systematic factors. The purpose of this toy question is to use some probability tools, investigate the effect of a systematic factor, and to compute expected values, variances and probabilities, including some quantities conditional on the value of a systematic factor. Consider a single roll of a pair of 6-sided dice and a flip of two fair coins. Let Xi be the outcome for die i, i = 1, 2, and let Ci be the number of heads obtained by filling coin i, i = 1, 2. Define the following S1 = X1 + 4C1, S2 = X2 + 4C2 and T = S1 + S2 = X1 + X2 + 4C1 + 4C2; and S1 = X1 + 4C1, S2 = X2 + 4C1 and T = S1 + S2 = X1 + X2 + 8C1. Thus, S1 and S2 are independent random variables (since no coin in common between S1 and S2). Thus, there is no systematic factor influencing the values of S1 and S2 and T is determined only by idiosyncratic effects. S1 and S2 are not independent random variables as they both depend on C1, the systematic factor in this example. Hence T is the sum of two dependent random variables. Note that T can be written as a linear combination of 3 independent random variables (see above). In this example, consider high values of T and T as bad outcomes the higher the value the worse the outcome. In an analogy to finance, think of T or T as loss random variables, which means as they get larger, the loss gets larger and hence the outcome worse.

Probability tools, idiosyncratic and systematic factors, Risk Measures. This question is a continuation of Question 1 on Assignment 1. (a) Quantile of a distribution. In financial risk management, it is common to fix a probability threshold and then use it to get an estimate of the size of the loss that is exceeded with (at most) that probability - effectively estimating a quantile of a loss distribution. In this problem, this amounts to finding the smallest values of a and a~ that satisfy P(T>a)0.03andP(T~>a~)0.03. i. Compute a and a~. ii. Interpreting T and T~ as losses, which is riskier as determined by a and a~ ? Hint: Consider values of a and a~ starting with 20,19,18,17, and compute the required probabilities. (b) Average value above a threshold. In financial risk management, it is common to fix a value for a loss (a threshold) and then compute expected loss, given the loss exceeds the threshold value. i. Compute E[TT16] ii. Compute E[T~T~16] iii. Interpreting T and T~ as losses, which is riskier as determined by average above a threshold? Hint: to compute the conditional expectation, use the conditional distribution of T given T16. In Assignment 1, you computed P(T16). To get the conditional probability function use the definition of conditional probability and the fact that {T=t,T16}= {T=t} for t=16,17,18,19,20.Similar logic applies for T~. Probability tools, idiosyncratic and systematic factors, Risk Measures. This question is a continuation of Question 1 on Assignment 1. (a) Quantile of a distribution. In financial risk management, it is common to fix a probability threshold and then use it to get an estimate of the size of the loss that is exceeded with (at most) that probability - effectively estimating a quantile of a loss distribution. In this problem, this amounts to finding the smallest values of a and a~ that satisfy P(T>a)0.03andP(T~>a~)0.03. i. Compute a and a~. ii. Interpreting T and T~ as losses, which is riskier as determined by a and a~ ? Hint: Consider values of a and a~ starting with 20,19,18,17, and compute the required probabilities. (b) Average value above a threshold. In financial risk management, it is common to fix a value for a loss (a threshold) and then compute expected loss, given the loss exceeds the threshold value. i. Compute E[TT16] ii. Compute E[T~T~16] iii. Interpreting T and T~ as losses, which is riskier as determined by average above a threshold? Hint: to compute the conditional expectation, use the conditional distribution of T given T16. In Assignment 1, you computed P(T16). To get the conditional probability function use the definition of conditional probability and the fact that {T=t,T16}= {T=t} for t=16,17,18,19,20.Similar logic applies for T~