Brosius Flashcards
Estimate y (losses at future eval) using the Link Ratio method (CL)
L(x) = cx
c is the selected link ratio (LDF)
Estimate y using the Budgeted Loss method
L(x) = k
k is a constant (plan)
Estimate y using the Least Squares Method
L(x) = a + bx
a = ybar - bxbar
b = (xybar - xbarybar) / (x^2bar - xbar^2)
When a = 0, L(x) = bx (link ratio method)
When b = 0, L(x) = a (budgeted loss method)
When b = 1, L(x) = a + x (BF metod)
Briefly explain how the Least Squares method is more flexible than the BF method
LS allows b to vary according the the data (b is not constrained to 1 as in BF)
LS allows for negative development
Name 2 situations which result in problems for estimating parameters for the Least Squares method
- Significant changes to the nature of loss experience in book of business
- Normal sampling error will lead to variance in a and b estimates
What are the problems if a or b are negative? What are the possible corrections?
If a is negative: estimate of developed losses (y) will be negative for small values of x
Substitue for link ratio method instead
If b is negative, estimate of y decreases as x increases
Substitute for the budgeted loss method instead
What adjustments should be made to data prior to using LS development?
- If incurred losses: correct data for inflation to put losses on a constant-dollar basis
- If there is significant growth in the book: divide losses by an exposure basis to correct the distortion
Describe the Simple Model and the best estimate of ultimate claims under that model.
Simple Model:
Y is either 0 or 1 with equal probability.
If there is a claim (Y=1), there is a 50% chance it’s reported by year YE(x).
yhat = 1/3 + 2x/3
Only the LS method is compatible
Describe the Poisson-Binomial Model and the best estimate of ultimate claims under that model.
Poisson-Binomial model:
Y is Poisson withe mean mu
Any given claim has probability d of being reported by YE
yhat = x + mu(1-d)
This is the same form as both LS and BF (b = 1)
Describe the Negative Binomial - Binomial Model and the best estimate of ultimate claims under that model.
NB - Binomial Model:
Y is NB with parameters (r,p)
Any given claim has probability f of being reported by YE
yhat = (1-d)(1-p)(x+r) / (1 - (1-d)(1-p))
Only the LS method is compatible
State 3 advantages of the best linear approximation (Bayesian Credibility)
- Simpler to compute
- Easier to understand and explain
- Less dependent upon the underlying distribution
Calculate the bet linear approximation to Q using the Development Formula 1
L(x) = (x-E(X))*Cov(X,Y)/V(X) + E(Y)
With empirical data:
L(x) = (x-Xbar)(XYbar - XbarYbar)/(X^2bar - Xbar^2) + Bar
If Cov(X,Y) smaller than V(X), large reported amount leads a decrease in reserve (budgeted loss method)
If Cov(X,Y) = V(X), large reported amount should not affect the reserve (BF method)
If Cov(X,Y) greater than V(X), large reported amount leads an increase in reserve (link ratio method)
When is the LS method appropriate to use? When is it inappropriate?
Appropriate if we have a series of data where we can assume stable distributions for Y and X.
We assume fluctuations are driven by random chance.
Inappropriate if YtoY changes are due to systematic changes in the book of business (e.g. mix shift).
Other methods such Berquist-Sherman may be better.
Define EVPV
Expected Value of the Process Variance
E(V(X given Y))
Variability resulting from the loss reporting process
Define VHM
Variance of the Hypothetical Mean
V(E(X given Y))
Variability resulting from the loss occurrence process
Calculate the best linear approximation to Q using Development Formula 2
L(x) = Zx/d + (1-Z)E(Y)
d = E(X/Y)
Z = VHM / (VHM+EVPV)
VHM = (E(X/Y)*sd(Y))^2
EVPV = sd^2(X/Y) * (sd^2(Y) + E(Y)^2)
If EVPV = 0 we given full weight to the link ratio estimate (fixed reporting case)
If VHM = 0 we give full weight to the budgeted loss estimate (fixed prior case)
Z = bd = b/c leads Development Formula 1
Define the caseload effect
When the caseload is low, a claim is more likely to be reported in a timely fashion than when caseloads are high.
Expected development ratio: E(X given y)/y decreases as y increases.
Calculate the best linear approximation to Q using Development Formula 3 (allowing for the caseload effect)
L(x) = Z(x-x0)/d + (1-Z)E(Y)
E(X given Y=y) = dy + x0
Z = VHM / VHM+EVPV
Development ratio: d + x0/y which decreases as y increases
When x0 = 0 we obtain Development Formula 2
Based on Hugh White’s Question, if reported losses are higher than expected, what are the 3 different responses to estimated loss reserves and the reasoning behind each response?
- Reduce the reserve by a corresponding amount
You believe loss reporting is accelerating, this is the Budgeted Loss method (fixed prior case) - Leave the reserve at the same % of expected losses
You believe there was a random fluctuation (e.g. large loss), this is the BF method - Increase the reserve proportionally to the increase in actual reported loss
You are not very confident in E(Y), this is the link ratio method
4 conclusions on the Least Squares Method
- It is easy to implement and uses accessible data
- It works well for developing losses for small states or lines that are subject to serious fluctuations
- It can lead you astray if corrections are not made to account for significant exposure changes or other shifts in loss history
- It is subject to sampling error since parameters are estimated from observed data
