1.2 Best Estimate based on Bayes (with Theoretical Distribution)
Proposition 1.2 (Poisson - Binomial) Assume the loss reporting follows a theoretical distribution
\(Y =\) (Ultimate) # of claims incurred each year \(\sim Poi(\mu)\)
\(X =\) # of claims reported by year end \(\sim Bin(y,d)\)
- i.e. each claim have probability \(d\) of being reported in the first year
Ultimate claims = \(Q(x) = x + \mu(1-d)\)
Unreported claims = \(R(x) = \mu(1-d)\)
(Expected # of claims) \(\times\) (Expected % Unreported)
- Similar to the BF Method
Proof. \(Q(x)\) is the estimator of \(Y\). It’s the sum of all possible \(y\)’s \(\times\) probability of the result being \(y\) given \(x\)
\(\begin{align} Q(x) &= \sum \limits_{y = x}^{\infty} y \Pr(Y = y \mid X = x) \\ Q(x) &= \sum \limits_{y = x}^{\infty} y \dfrac{\Pr(Y = y)\Pr(X = x \mid Y = y)}{\sum_i \Pr(Y = i) \Pr(X = x \mid Y = i)} \\ \end{align}\)
And we get Ultimate claims = \(Q(x) = x + \mu(1-d)\)Proposition 1.3 (Negative Binomial - Binomial) Assume the loss reporting follows a theoretical distribution
\(Y =\) (Ultimate) # of claims incurred each year \(\sim NB(r, p)\)
\(Y =\) # of failures until \(r\) success with success probability = \(p\)
\(\mathrm{E[Y]} = \dfrac{r(1-p)}{p}\)
\(X =\) # of claims reported by year end \(\sim Bin(y,d)\)
Unreported claims = \(R(x) = \dfrac{s}{1-s}(x + r)\)
- \(s = (1-d)(1-p)\)
1.2.1 Comparing Loss Development Methods
Simulate loss based on one of the theoretical distribution
Apply the various loss development method and calculate their respective parameters (\(y = a + b x\))
Compare the estimated parameters with the true parameters based on the underlying theoretical distribution
Also compare the MSE from different methods