Math 395 Risk Theory

Instructor – James G. Bridgeman

Maximum Entropy Paper (K. Conrad)

EXCEL Example for Convolution (see page 145) EXCEL Example for Panjer Recursion (try convolution on this one first!)

EXCEL Example for Distribution Fitting (see page 145)

Take-home quiz due back 12-2 Solution
Spreadsheet (the spreadsheet contains useful illustration
of the EXCEL functions OFFSET and SUMPRODUCT)

Example of Compound Geometric and Panjer Recursion For Ruin Probabilities

Notes for Last Class (the
snow day) You
are responsible for these notes on the final! You can bring a copy to the open-book final.

**Cumulative
Assignments** (final)

Read Sec. 8.5 for background only

Sec. 8.3-8.4 and exerc. 8.13-8.18

Sec. 8.1-8.2 and exerc. 8.1-8.3, 8.6-8.7, 8.9-8.12

Sec. 7.3 (for background only)

Sec. 7.1-7.2

A take-home quiz on Ch. 6 (posted above) is due back 12-2

Sec. 6.8-6.10 (read these 3 for background only)

Sec. 6.5-6.7 and exerc. 6.29-6.35, 6.37-6.58

A take-home quiz on Sec. 4.6 and 5.6 and related classroom notes will be given on 11-4, due back 11-14

Sec. 6.1-6.4 and exer. 6.1-6.27

Sec. 4.6.10-4.6.11, 5.6 and exer. 4.56-4.60, 4.62,5.24-5.27

Sec. 4.6.7-4.6.9 and Exer. 4.41-4.47, 4.50-4.55

Calculate the coefficients of skewness and kurtosis for the Poisson, Neg. Binomial, and Binomial distributions.

Sec. 4.6.6 and exer. 4.45-4.48

Turn in the take home exam at class on 10-12

Sec. 4.6.1-4.6.5 and exer. 4.40, 4.42-4.44, 4.49, and use Faa’s formula to calculate the first 4 raw and central moments of the Poisson, Neg. Binomial, and Binomial distributions

Validate (comparing formulas is good enough) that the moment shifted log-logistic is a transformed beta (or, when γ=1 a generalized Pareto)

Determine the coefficient of skewness in example 5.15 (surface interpretation!)

Write down the formula for the 3^{rd} moment
analogous to Theorem 5.14 (surface interpretation!)

Sec. 5.4-5.5 and exer. 5.14-5.23 (surface interpretation!)

Sec. 5.1 – 5.3 and exer. 5.1 to 5.13 (in ch. 5 try to think in terms of the surface interpretation!)

Sec. 4.5 and exer. 4.37-4.39

Exer. 4.33-4.36

Sec. 4.3-4.4 and exer. 4.13-4.32 (keep a bookmark in appendix A!)

Sec. 4.1-4.2 and exer. 4.1-4.5, 4.7-4.9, 4.11-4.12

Sec. 3.3 and exer. 3.20-3.24

Sec. 3.1-3.2 and Exer. 3.1-3.19X

**Project Topics**:
(pick any two to submit by end of semester … topics will be added as we
go)

#1 Critique the “proof” given in class that vanishing of (y^k)S(y) as y goes to infinity implies existence of the k-th moment (assume non-negative support and nice behavior for y near 0)

#2 Make three dimensional visual illustrations for the
surface interpretation, including 2^{nd} and 3^{rd} moments and
the relation of e(d), e^{2}(d), and e^{3}(d) to E[X], E[X^{2}],
E[X^{3}], E[X_{^}d], E[(X_{^}d)^{2}], and E[(X_{^}d)^{3}].

#3 See handout on L_{X}(u)=E[x_{^}u] with a
series of questions

#4 Prove Faa’s Formula

#5 Work out a description of a family of severity distributions analogous to the transformed beta family, but based upon the log-Laplace distribution rather than the log-logistic that stands at the base of the transformed beta family.

#6 Compare the distributions that appear in the transformed beta and transformed gamma families if you replace the moment shifting transformation

f_{X}(x)=x^{(α-1)}f_{Y}(x)/µ΄_{Y(α-1)}
with another transformation that shifts moments in a different way (and with
different strings of constants):

f_{X}(x)=(α-1)x^{(α-2)}S_{Y}(x)/
µ΄_{Y(α-1)}

How do the resulting distributions differ from the gamma, transformed gamma, generalized Pareto, and transformed beta that arose from the original moment shifting formula?

#7 Work out the definitions and properties of an inverse logistic and reciprocal inverse logistic family of distributions, analogous to the inverse Gaussian and reciprocal inverse Gaussian presented in class.

#8 Work out a family of distributions based on transformations of the lognormal, analogous to the transformed beta family that was based on the loglogistic.

#9 Prove (or, if a formal proof eludes you, just illustrate and discuss the fact) that the negative binomial is like a Poisson with contagion; i.e. the negative binomial with parameters (r,β) gives the number of events in unit time if the probability of one event in infinitesimal time dt, conditional on exactly m events having occurred from time 0 to time t, is equal to dt(rβ)((1+m/r)/(1+βt)). Try to make a similar interpretation of the binomial distribution.

#10 Prove the Panjer recursion formula for an (a,b,1) primary distribution using Faa’s formula.

#11 Come up with a spreadsheet (or other programming)
algorithm to generate sets {j_{ k}} with ∑_{ (k=1, ∞)}
j _{k} k = n, n=0,1, 2, … Is this efficient enough to warrant
replacing Panjer recursion with direct use of Faa’s formula to calculate
compound distribution probabilities for (a, b, 0) primary distributions? Note that this would give you a calculation
technique anytime the probability generating function of the primary
distribution is known, whether or not there is a recursive feature to it. Is this an improvement versus brute force
convolution?

#12 Show (using probability generating functions) that a mixed Poisson distribution with infinitely divisible mixing distribution is also a compound Poisson, and give two specific examples of the phenomenon.

#13 Develop an approximation formula (based on the surface
interpretation) for E[(S-(j+1)·h)_{+}^{k}] in terms of
E[(S-j·h)_{+}^{k}] for k=2 and 3 that would apply in general,
whether or not S has been discretized.
Write down the more simplified recursion formula this gives when S has
been discretized.

#14 Make up examples using both a compound Poisson and a
compound Negative Binomial process (and pick one or more severity distributions
to give yourself examples) and study how much, if any, difference there is in
your approximation for the upper limit exp(-R^{~}u) on the discretely
observed ruin probability ψ^{~}(u) if you use a third order
approximation for the adjustment coefficient R^{~} rather than the
second order approximation that is usually used. In other words, quantify the contribution of
skewness in the severity to an estimate for the upper limit of the ruin
probability.

#15 Develop the logic and formulas for a compound negative binomial process in the continuously observed collective risk ruin model. Point out the most significant departures from the conclusions of the corresponding compound Poisson process.