Math 395 Risk Theory

Instructor – James G. Bridgeman

final exam final exam solutions

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 4-18 Solution Spreadsheet (note: the spreadsheet contains useful
illustration of the EXCEL functions OFFSET and SUMPRODUCT)

Example of Compound Geometric and Panjer Recursion For Ruin Probabilities

**Cumulative
Assignments** (final):

Exerc. 8.20-8.25

Sec. 8.3-8.5 and exerc. 8.13-8.19

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

Sec. 7.1-7.2, 7.3 (for background only)

Exer. 6.49-6.58

Sec. 6.7, Sec. 6.8-6.10 (read these 3 for background only)

Sec. 6.5, 6.6 and exerc. 6.26-6.27, 6.29-6.35, 6.37-6.48

A take-home test on the material in sec. 4.6 and 5.6 will be given on Mar. 18, due back Mar. 25

Sec. 4.6.11, 5.6, 6.1-6.4 and exer. 5.24-5.27, 6.1-6.25

Sec. 4.6.7-4.6.10 and exer. 4.41, 4.45-4.47, 4.50-4.60, 4.62

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

A take-home test on the material thru Sec. 4.5.3 and 5.5 will be given on Feb. 18, due back Feb. 25

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 and, finally, pick one 5^{th} moment
to calculate using Faa’s formula

Sec. 4.5.3, 5.1-5.5 (in ch. 5 think in terms of surface interpretation!), and exer. 4.37-4.39, 5.1-5.23 (think surface interpretation!)

Sec. 4.4, 4.5.1-4.5.2 and exer. 4.18-4.36 (keep a bookmark in App. A !)

Sec. 4.3 and exer. 4.13-4.17

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. 3.2 and exer. 3.16-3.19

Sec. 3.1 and exer. 3.1-3.15

Ch. 2 and exer. 2.1-2.5

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

#1 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}].

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

#3 Prove Faa’s Formula

#4 Work out the formulas and properties for an alternative version of the exponential (gamma) and log-logistic (beta) families of distributions using the alternative moment-shifting concept that defines the moment-shifted density in terms of the original decumulative distribution function.

#5 Try to work out summation formulas (or something to
replace them) for the (original definition) moment-shifted S_{X}(x). Try to leave a CTM term in the resulting
formula.

#6 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”.

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

#8 Prove that the negative binomial is like a Poisson with contagion; i.e. prove that the negative binomial (r, β) gives the number of events in unit time if the number of events in infinitesimal time dt, conditional on exactly m events having occurred from time 0 to time t, is dt(rβ)((1+m/r)/(1+βt)). Try to make a similar interpretation of the binomial distribution.

#9 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.

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

#11 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.

#12 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.

#13 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.