Actuarial Notes

Chain Ladder Assumptions

Study guide: Exam GI-301

Chain Ladder Assumptions are the conditions under which the chain ladder's age-to-age factors give least-squares optimal reserves, meaning the minimum variance unbiased linear estimate of future emergence. Venter (1998) restates Mack's three assumptions for incremental losses q(w,d) and cumulative losses c(w,d):

E[q(w,d+1) \mid \text{data to } w+d] = f(d)\,c(w,d)

\text{Var}[q(w,d+1) \mid \text{data to } w+d] = a[d,\,c(w,d)]

  • The three assumptions. The first says expected emergence is proportional to losses emerged to date: a line through the origin, with the same factor for every accident year. The second, independence, says c(w,d) and c(v,g) are independent unless v = w; a strong diagonal breaks it. The third, the variance function, picks the estimator. Mack's a = k(d)\,c(w,d) gives the volume-weighted factor \sum q/\sum c. a = k(d)\,c(w,d)^2 gives the simple average of the ratios. A constant a gives \sum c\,q/\sum c^2 (Mack Chain Ladder Model). Here f(d) is the incremental factor, one less than the usual link ratio.
  • Six testable implications. The assumptions can't be tested directly, but these consequences can:
    1. f(d) is significant.
    2. It beats the alternative emergence patterns: linear with constant, f(d)c(w,d) + g(d); factor times parameter, f(d)h(w); and with a calendar year term, f(d)h(w)g(w+d).
    3. The model is linear: no runs of same-signed residuals against c(w,d).
    4. The factor is stable: no such runs against time.
    5. There is no correlation among columns (Development Factor Correlation Test).
    6. There are no particularly high or low diagonals (Calendar Year Effect).
  • Significance. Regress q(w,d+1) on c(w,d) with a constant a and a factor b. A factor is usually wanted at least twice its standard deviation (about a 4.5% chance for a Normal when the true value is zero), or 1.65 times (about 10%). In a cumulative-to-cumulative regression, test the factor's difference from 1.
  • The alternatives. In the parameterized Bornhuetter-Ferguson model, h(w) is proportional to each year's ultimate, giving 2n - 2 parameters for n years. The Cape Cod uses a single h, so it has as many parameters as the chain ladder. The additive chain ladder, whose g(d) is the average loss emerged at age d, fits exactly as the Cape Cod does.
  • Comparing fits. Use the sum of squared errors adjusted for p parameters and n observations. Venter's statistic is the first; the second approximates the AIC and the third ranks models as the BIC does:

\text{Adjusted SSE} = \frac{\text{SSE}}{(n-p)^2}

\text{AIC-type} = \text{SSE}\,e^{2p/n}

\text{BIC-type} = \text{SSE}\,n^{p/n}

  • The rule of thumb. "If the chain ladder doesn't work, try Bornhuetter-Ferguson." Venter endorses it when "doesn't work" means fails the assumptions of least-squares optimality and "try" means test the underlying assumptions of. If emergence is stable, using only the last few diagonals only adds estimation error.

Is the 12–24 Factor Significant? Cumulative paid at 12 months (x) and incremental paid from 12 to 24 months (y), in $000:

AY x y
1 1{,}000 2{,}100
2 1{,}500 1{,}900
3 2{,}000 2{,}300
4 2{,}500 2{,}000
5 3{,}000 2{,}200

Fit y = a + bx by least squares. Test a and b against twice their standard deviations, and say what the result implies.

Answer Here \bar x = 2{,}000 and \bar y = 2{,}100.

\begin{align*}
S_{xx} &= 2{,}500{,}000 \\
S_{xy} &= 150{,}000 \\
b &= 150{,}000/2{,}500{,}000 \\
&= 0.060 \\
a &= 2{,}100 - 0.060(2{,}000) \\
&= 1{,}980
\end{align*}

The residuals are 60, -170, 200, -130, 40, so SSE = 91{,}000 and s^2 = 91{,}000/3 = 30{,}333.

\begin{align*}
\text{sd}(b) &= \sqrt{30{,}333/2{,}500{,}000} \\
&= 0.110 \\
\text{sd}(a) &= \sqrt{30{,}333\left(\tfrac15 + \tfrac{2{,}000^2}{2{,}500{,}000}\right)} \\
&= 233.7
\end{align*}

The factor is only 0.060/0.110 = 0.5 standard deviations from zero, so it is not significant. The constant is 1{,}980/233.7 = 8.5 standard deviations, so it clearly is.

The 12–24 emergence is not proportional to what has emerged by 12 months. It looks like a roughly constant amount, about $2.1M per accident year. That fails implication 1 and points to the additive or Cape Cod emergence pattern. The triangle should first be put on a common exposure and cost level so that a constant amount makes sense.

Chain Ladder Against the Additive Pattern Incremental incurred losses ($000):

AY Age 0 Age 1 Age 2 Age 3
0 1{,}000 2{,}000 1{,}200 400
1 1{,}600 2{,}100 1{,}100
2 700 1{,}900
3 1{,}300

Fit the chain ladder, with fitted increment f(d) \times the actual previous cumulative, and the additive chain ladder to the six increments at ages 1–3. Compare the adjusted SSE, and the reserves.

Answer Chain ladder. The incremental factors are f(0) = 6{,}000/3{,}300 = 1.8182, f(1) = 2{,}300/6{,}700 = 0.3433 and f(2) = 400/4{,}200 = 0.0952.

Cell Actual Fitted Residual
AY 0, age 1 2{,}000 1{,}818.2 181.8
AY 1, age 1 2{,}100 2{,}909.1 -809.1
AY 2, age 1 1{,}900 1{,}272.7 627.3
AY 0, age 2 1{,}200 1{,}029.9 170.1
AY 1, age 2 1{,}100 1{,}270.1 -170.1
AY 0, age 3 400 400.0 0

The SSE is 1{,}139{,}059.

Additive. g(1) = 2{,}000, g(2) = 1{,}150 and g(3) = 400 are the column averages. The residuals are 0, 100, -100, 50, -50, 0, so the SSE is 25{,}000.

Both models have p = 3 parameters for n = 6 observations, so both divide by (6 - 3)^2 = 9:

\begin{align*}
\text{Chain ladder} &= 1{,}139{,}059/9 \\
&= 126{,}562 \\
\text{Additive} &= 25{,}000/9 \\
&= 2{,}778
\end{align*}

The additive pattern fits far better. Age-1 emergence is about 2{,}000 whether a year started at 700 or 1{,}600, which the chain ladder's proportionality cannot reproduce.

Reserves. The chain ladder gives 457 + 1{,}225 + 4{,}090 = 5{,}772. The additive method gives 400 + 1{,}550 + 3{,}550 = 5{,}500. The two differ most for AY 3, whose above-average 1{,}300 the chain ladder multiplies through every future age.

Which Parameter Penalty? Venter's Reinsurance Triangle Venter fits three models to the n = 45 incremental losses of an excess casualty reinsurance triangle with ten accident years:

  • Bornhuetter-Ferguson: p = 18, SSE \approx 59.172M.
  • Cape Cod: p = 9, SSE \approx 97.730M.
  • Chain ladder: p = 9, SSE \approx 204.641M.

Rank them under each of the three adjustments.

Answer The multipliers are 1/(n-p)^2 with (45-18)^2 = 729 and (45-9)^2 = 1{,}296; e^{2p/n}, which is e^{0.8} = 2.2255 and e^{0.4} = 1.4918; and n^{p/n}, which is 45^{0.4} = 4.58443 and 45^{0.2} = 2.14113.

Model p SSE \text{SSE}/(n-p)^2 \text{SSE}\,e^{2p/n} \text{SSE}\,n^{p/n}
BF 18 59.172M 81{,}169 131.7M 271.3M
CC 9 97.730M 75{,}409 145.8M 209.3M
CL 9 204.641M 157{,}902 305.3M 438.2M

The chain ladder is last every time; its SSE is over twice the Cape Cod's with the same parameter count. The Cape Cod wins under (n-p)^2 and the BIC, and the Bornhuetter-Ferguson under the AIC, the most permissive of extra parameters. The data favour emergence as a percentage of ultimate over the chain ladder's proportionality, but which BF variant to use is a judgement the statistics inform without deciding.