Caseload Effect

Caseload effect is the dependence of reporting speed on the number of claims. A claim is more likely to be reported in a timely fashion when the caseload is low, so the expected proportion reported by a given age, $E(X \mid Y = y)/y$ , falls as the number of claims $y$ grows. Brosius (1993) models it by giving the expected reported amount an intercept.

$E(X \mid Y = y) = d\,y + x_0$

$\frac{E(X \mid Y = y)}{y} = d + \frac{x_0}{y}$

$L(x) = Z\,\frac{x - x_0}{d} + (1 - Z)\,E(Y)$

$Y$ is the ultimate (claims incurred) and $X$ the amount reported at the evaluation age, with $d \neq 0$ and, presumably, $d$ and $x_0$ both positive. Setting $x_0 = 0$ gives proportional reporting, Brosius's Development Formula 2. The last block is his Development Formula 3, with the same credibility $Z = VHM/(VHM + EVPV)$ , where the VHM and EVPV are those of $X$ given $Y$ .
What it does to development factors. The expected factor to ultimate for a year with $y$ claims is $1/(d + x_0/y)$ . That is smaller for a light year than for a heavy one, so a single link ratio, which assumes the ratio is a constant $d$ , cannot fit both. Even with perfectly predictable reporting ( $EVPV = 0$ , $Z = 1$ ), the best estimate $(x - x_0)/d$ is a line with a negative intercept rather than a multiple of $x$ .
Why least squares still works. $L(x)$ is still linear in $x$ . Least squares development fits the line directly from past years, so it makes sense even when the development ratio varies with the caseload. $x_0$ and $d$ , which may be impossible to determine in practice, are never needed. The proof shifts $X$ by $x_0$ : $W = X - x_0$ has the same EVPV and VHM, so Formula 2 applies to $W$ .
The model implies $E(X \mid Y = 0) = x_0 > 0$ , which Brosius concedes may be undesirable. Adjuster caseloads are also among the claims processing conditions that change the timing of development.

Development Ratios and the Credibility Estimate Under a Caseload Effect For a small line, claims reported by year-end satisfy $E(X \mid Y = y) = 0.5\,y + 3$ . The number of claims incurred $Y$ has mean $20$ and variance $40$ , and $\mathrm{Var}(X \mid Y = y) = 0.25\,y$ .

(a) Find the expected development ratio for years with $10$ , $20$ and $40$ claims. (b) This year $15$ claims have been reported. Estimate the ultimate count with Development Formula 3, and compare the link ratio set at an average year.
Answer (a) The ratio is $d + x_0/y = 0.5 + 3/y$ :

Claims $y$ Reported ratio Factor to ultimate

$10$ $0.800$ $1.250$

$20$ $0.650$ $1.538$

$40$ $0.575$ $1.739$

Light years report faster, so one development factor cannot suit all three.

(b) The credibility weight:
 $\begin{align*} VHM &= \mathrm{Var}(0.5\,Y + 3) \\ &= 0.25 \times 40 \\ &= 10 \\ EVPV &= E(0.25\,Y) \\ &= 5 \\ Z &= \frac{10}{10 + 5} \\ &= 2/3 \end{align*}$ 
Then Formula 3 gives:
 $\begin{align*} L(15) &= \tfrac{2}{3} \cdot \frac{15 - 3}{0.5} + \tfrac{1}{3}(20) \\ &= 16 + 6.67 \\ &= 22.67 \end{align*}$ 
Check with Formula 1. Here $E(X) = 0.5(20) + 3 = 13$ , $\mathrm{Cov}(X, Y) = 0.5 \times 40 = 20$ and $\mathrm{Var}(X) = VHM + EVPV = 15$ :
 $\begin{align*} L(15) &= (15 - 13)\,\tfrac{20}{15} + 20 \\ &= 22.67 \end{align*}$ 
The link ratio of an average year is $20/13 = 1.538$ , which gives $15 \times 1.538 = 23.08$ . Two effects separate the estimates:

Inverting the reporting line gives $(15 - 3)/0.5 = 24$ . That is above $23.08$ because a heavier-than-average year reports a smaller share, so it needs a larger factor than the average year's.

Credibility then pulls the $24$ one-third of the way back to $E(Y) = 20$ , giving $22.67$ .

A least squares line through past years would approach $L(x) = \tfrac{4}{3}x + 2.67$ , without anyone knowing $d$ or $x_0$ .