Exam (elaborations)
CAS Exam 7 Answer-Integrated Exam 2024/2025
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- CAS
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- University Of Illinois - Urbana-Champaign
CAS Exam 7 Answer-Integrated Exam 2024/2025
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CAS Exam 7 Answer-Integrated Exam2024/2025
If Claim Counts are Poisson(m), and
d = probability of reporting in first period
What is the Bayes Reserve? Answer: R(x) = mu(1 - d)
List the 3 Special Cases for Brosius Least Squares Answer: y = a + bx | Least Squares | Restriction
y = bx | Chainladder | a=0
y = a + x | Bornhuetter-Ferguson | b=1
y = a | Budgeted Loss | b=0
Claim Counts are NegBin(r, p)
d is the probability of reporting in the first period
x is the actual reported in the first period
What is the Bayes Reserve? Answer: R(x) = [s /(1 - s)](x + r)
s = (1 - d)(1 - p)
Negative Binomial Distribution Answer: Y ~ NegativeBinomial(r, p)
E[Y] = r(1 - p)/p
For Ultimate Losses Y, and Reported Losses X What is the Best Linear estimate of Y given x? Answer:
L(x) = [x - EX][Cov(X,Y)/Var(X)] + EY
,What are the formulas for a and b in Brosius Least Squares? Answer: b = (E(xy) - E(x)E(y)) / (E(x^2) -
E(x)^2)
a = E(y) - bE(x)
Estimate Ultimate Losses L(x) using a credibility formula? Answer: L(x) = Zx/d + (1 - Z)EY
Z = VHM/(VHM + EVPV)
VHM = VarY[E(X | Y)]
EVPV = EY[Var(X | Y)]
When is Brosius Least Squares Appropriate? Inappropriate? Answer: - Appropriate: When the
Distribution is the same across multiple years
- Inappropriate: Year to Year changes are due to systemic shifts, eg. inflation, legal environment
Calculate the Statistics used in Brosius Answer: b = (E(xy) - E(x)E(y)) / (E(x^2) - E(x)^2)
a = E(y) - bE(x)
LDF: c = E(y)/E(x)
Credibility: Z = b / c
,%Reported: d = 1 / c
Estimate Ultimate Losses L(x) using a credibility formula when given mean and variance of Ultimate
losses and the development pattern Answer: L(x) = Zx/d + (1 - Z)EY
Z = VHM/(VHM + EVPV)
VHM = (d^2) (sigmay^2)
EVPV = (sigmad^2)[(sigmay^2) + (EY)^2]
d = expected % paid to date
Definition of an exposure and an exposure base Answer: An exposure is the basic unit of risk
underlying the insurance premium.
The definition of what an exposure represents varies by line of business and sometimes coverage and is
called an exposure base.
Short Formula for the Benktander Reserve Answer: RGB = (qk)x(UBF)
qk = % Unreported at Age k
UBF = Ultimate losses from Bornhuetter-Ferguson Method
Hurlimann - Calculate the Collective Loss Ratio Reserve Estimate Answer: RColl = q (V * ELR)
, V = Premium (or other exposure base, used to calculate ELR)
This is similar to the BF method
Hurlimann - Calculate the Individual Loss Ratio Reserve Estimate Answer: Rind = C/p - C
p = % Reported
C = Paid to Date
This is similar to the CL method
Hurlimann - Credibility Options for Individual and Collective Methods Answer: Zi | Method
1 | Individual (CL)
0 | Collective (BF)
pk | Benktander
pk * ELR | Neuhaus
pk/(pk + sqrt(pk)) | Optimal Credibility
R = Z * Rind + (1 - Z) * RColl
Calculate % Reported, according to Hurlimann Answer: mk = sum( Losses in Column k) /
sum(Corresponding Premium)
ELR = Sum(mk)
pk = Percent Paid to Date by age k
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