6.5 Cape Cod Method (Method 1)

Requires exposure base:

• e.g. on-level and trended EP, original loss projection, trended # of vehicles, claim counts, etc

• We want an exposure base that allow us to assume a constant ELR across AYs

Estimate \(\theta\), \(\omega\) and \(ELR\) with MLE

• Recall \(ELR\) estimate from prior section

\[ELR = \sum_{AY} \dfrac{\text{Losses Paid to Date}}{\underbrace{G(x)}_{\text{Expected portion paid}} \times Premium}\]

• Do not truncate when calculating \(ELR\)

• Calculate \(\mu\)

\[\mu = Premium_{AY} \times ELR \times [G(y) - G(x)]\]

• Calculate the log likelihood for the sum of the whole triangle

Calculate \(\sigma^2\) for residual or reserve process variance

\[\sigma^2= \dfrac{1}{n-p}\sum\limits_{i \in \Delta}^n\dfrac{(c_i - \mu_i)^2}{\mu_i}\]

• \(n =\) all data points

  (Not just the predicted like in Venter)

• \(p = 3\)

Check if the assumption of one expected LR is reasonable by looking for any upward or downward trends in the ultimate LR

• Since we’re assuming a single \(ELR\)

6.5.1 Reserve Estimate

Untruncated

• Reserve = On-level Premium \(\times \: ELR \: \times [1 - G(x)]\)

Truncated @ age \(x_t\)

• Reserve = On-level Premium \(\times \: ELR \: \times [(G(x_t) - G(x)]\)

  • Or, Reserve = On-level Premium \(\times \: ELR \: \times G(x_t) \: \times \:[(1 - G'(x)]\)

Similar story for the process variance and parameter variance as the LDF method

• The Covariance matrix \(\Sigma\) is smaller just \(3 \times 3\)

• Parameter variance is smaller than the LDF method since we have more information (exposure) in the Cape Cod method