5.1 Definitions and Assumptions

Table 5.1: Tables of Definitions
Venter Notations Definitions Mack Notations (4.1)
\(w\) AYs \(i\)
\(d\) Dev period; \(d=0\) is age @ end of year 1 \(k\)
\(c(w,d)\) Cumulative losses for AY \(w\) age \(d\) \(c_{i,k}\)
\(c(w,\infty)\) Ultimate losses for AY \(w\) \(c_{i,I}\)
\(q(w, d+1)\) Incremental losses for AY \(w\) age \(d\) to \(d+1\)
\(f(d)\) Col parameters; (LDF - 1); applies to whole col \(f_k - 1\)
\(F(d)\) LDF to ultimate that applies to \(c(w,d)\)
\(h(w)\) Row parameters; applies to whole row

5.1.1 Mack’s Chainladder Assumptions

Same assumptions as the Mack 1994 but stated in Venter’s terminology

  • Venter refers these as assumptions needed for least-squares optimality to be achieved by the typical age-to-age factor method of loss development
Proposition 5.1 (Mack Assumption 1) \[\mathrm{E}[q(w,d+1) \mid \text{Data to } w+d] = f(d) \times c(w,d)\]

Remark.

  • Here \(f(d)\) is LDF - 1

  • Expected incremental development is \(\propto\) reported losses

  • See also (Prop 4.1) from Mack (1994)

Proposition 5.2 (Mack Assumption 2) \[c(w,d) \perp\!\!\!\!\perp c(v,g) \:\:\:\: \forall \: d,g,v,w \:\:\:\: : \:\:\:\: v \neq w\]

Remark.

  • Losses not in the same row are independent of each other

  • See also (Prop 4.2) from Mack (1994)

Proposition 5.3 (Mack Assumption 3) \[\mathrm{Var}[q(w,d) \mid \text{Data to } w+d] = a_{fun}\big(d,c(w,d)\big)\]

Remark.

  • Variance of incremental losses depends only on:

    1. Cumulative losses reported to date \(c(w,d)\)

    2. Age of the AY \(d\) (Does not vary by AY down the column)

  • Different \(a_{fun}(\cdots)\) leads to different \(\hat{f}(d)\) estimate

  • See also (Prop 4.3) from Mack (1994)

5.1.2 Variance Assumptions (for Chainladder)

Same as shown in Mack 1994 Table 4.1

Table 5.2: Relationships between weight, variance and residual (Venter)
Weight Description Variance \(a_{fun}\big(d,c(w,d)\big)\) LDF - 1 \(f(d)\)
1 Simple Average \(k(d)c(w,d)^2\) \(\dfrac{\sum_w 1 \frac{q(w,d+1)}{c(w,d)}}{\sum_w 1}\)
\(c(w,d)\) Weighted Average \(k(d)c(w,d)\) \(\dfrac{\sum_w c(w,d) \frac{q(w,d+1)}{c(w,d)}}{\sum_w c(w,d)}\)
\(c(w,d)^2\) Least Square \(k(d)\) \(\dfrac{\sum_w c(w,d)^2 \frac{q(w,d+1)}{c(w,d)}}{\sum_w c(w,d)^2}\)

  • \(c(w,\infty) = F(d)c(w,d)\)

    \(F(d) = \prod_{s \geq d} (1 + f(s))\)

  • Recall \(\dfrac{q(w,d+1)}{c(w,d)}\) are the empirical LDF - 1

Definition 5.1 (Chainladder Parameter definition) \[\mathrm{E}[q(w,d+1)] = f(d)c(w,d)\]

  • \(n = \sum \limits_{i=1}^{m-1} i = \dfrac{m(m-1)}{2}\) = predicted data point?

  • \(p=m-1\) since we don’t predict the first column

  • \(m =\) dimension