risk2020exercices

Risk and Decision
Centrale Marseille (3A MMEFi) – Aix-Marseille Universit´e (M2 AMSE & IMSA)
Dominique Henriet
A. Exercise 1
A financial institution is considered to be subject to a potential net loss ex. It is assumed that exfollows
a D-parameter exponential law: Pr [ex≤x] = F(x) = 1 −exp −x
D. One has dF (x) = 1 Dexp −x Ddx.
1. Recall the value of E(ex). What is the quantile function of ex?
2. Calculate and graphically represent the Lorenz function of . Remember that ln (t)is the derivative of
tln (t)−t.
The regulator wants to impose a capital reserve such that the probability of bankruptcy is less than
5/100.
3. Calculate the minimum reserve that gives a probability of bankruptcy of less than 5%. .
The financial institution decides to set up two subsidiaries, one of which is assigned to the cases of large
losses and the other to other cases. The losses are allocated as follows:
ex1=ex1[Dln(25),+∞[(ex) et ex2=ex1[0,D ln(25)[ (ex)
4. Graphically (and schematically) represent the probability distributions of ex1and ex2. (Dirac masses
are represented by a thick vertical arrows).
5. Deduct the loss CDF of each of the two subsidiaries.
6. Calculate the regulatory reserves of each of the two subsidiaries, comment.
7. Calculate the Expected Shortfall of the non-spin-off company at the 0.95 level: the average loss in the
5% worst cases.
It is now assumed that the regulator imposes a reserve equal to the Expected Shortfall.
8. Is there still any advantage in creating subsidiaries?
B. Exercise 2
Consider a newspaper vendor selling papers on the corner. Each morning, the vendor must decide how
many papers to buy at the wholesale price w. The vendor then sells the papers during the day at a retail
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price phigher than the wholesale price. At the end of the day, any unused papers can no longer be sold
and are sold back to the wholesaler at a price sless than the wholesale price. This exercise is aimed to
answer the question : how many papers should the vendor order?
We note ythe number of papers ordered and exthe number of papers demanded. exis random with a
CDF F. The number of papers sold is then min(ex, y). All along the exercise we consider “continuous”
numbers for the number of papers...
1. If ypapers are ordered, what is the (random) number of unsold papers at the end of the day?
2. Write the random income ewof the newboy in the form ew=amin(ex, y)−by
Let ez≡min(ex, y) . Let Gthe CDF of ez, and Hthe CDF of ew.
3. Show ∀α, G−1(α) = min(F−1(α), y). Give the expression of H−1.
We first assume that the newsboy maximizes the expected value of income
4Recall the expression of expected value using the quantile function
5. What is the optimal number y∗(function of p−wand p−s)
We now consider that the newsboy is risk-averse so that he uses the“tail conditional expectation” criterion
:1
βRβ
0H−1(α)dα
6. Compute the conditional expectation (consider the 2 diffrent cases F(y)≤βand F(y)> β)
7. What is the optimal number of papers ordered.
8. Does a more risk averse newsboy buy more or less papers?
C. Exercise 3 (predictive markets)
One consider the following “asset” : One unit pays 1 euro if D. Trump is reelected on 3rd November and
0 if not.
Individuals, i= 1...n, use expected utility criterion with CARA functions : w→ − exp −ρi, w.
If ibuys xiunits at price phe will get : ew=e δi−pxi, where e δiis the “subjective” Bernoulli variable
: with probability πi,e δiis equal to 1 and with probability (1 −πi), e δiis equal to 0. πiis a subjective
probability
1. Give the expression of the supply/demand function of i.
The total initial number of assets is obviously zero
2. What is the equilibrium price? Comment.
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