Actuarial Assumptions

Study guide: Exam 7

Actuarial assumptions are the statements about the future, explicit or implicit, on which an actuary's method or model rests. In estimating unpaid claims they include the assumptions built into each reserving method (that past development patterns will repeat, that accident years are independent) and the parameters chosen to apply it (development factors, a tail factor, an expected loss ratio, a trend rate).

E\left[C_{i,k+1} \mid C_{i1}, \dots, C_{ik}\right] = C_{ik}\,f_k

• This is the first of the three assumptions Mack (1994) shows the chain ladder makes implicitly: expected cumulative loss C_{i,k+1} of accident year i is the current cumulative loss C_{ik} times a development factor f_k common to all years. The other two are independence of accident years and a variance proportional to C_{ik} (Mack Chain Ladder Model).
• What the standard requires. ASOP 43 §3.6.2 has the actuary consider the reasonableness of the assumptions underlying each method or model. Assumptions may be implicit or explicit, and may interpret past data or project future trends. They should have no known significant bias to underestimation or overestimation of the intended measure, and should not be internally inconsistent.
• Sensitivity. The actuary should consider the sensitivity of the estimate to reasonable alternative assumptions. When the effect would be material, the actuary should notify the principal and attempt to discuss it. Changes in assumptions with a material impact on an updated estimate are disclosed, with the reasons for them.
• Testing them. An implicit assumption can be tested against the data it was fitted to. Venter (1998) turns the chain ladder's assumptions into testable implications (Chain Ladder Assumptions): significant factors, linearity, stability, no correlation between development factors and no calendar-year effects. Residual diagnostics (residual plots, heteroscedasticity, normality) test the assumptions of a stochastic model such as the ODP bootstrap.
• A failed test points to the assumption to change, not only to a wider range: Venter's alternatives to the chain ladder (a constant added to the factor, Bornhuetter-Ferguson, Cape Cod) each replace the chain ladder's assumption about expected emergence with a different emergence pattern.

Sensitivity of Unpaid Claims to the Tail Factor Cumulative reported losses at the last age in the triangle total \$50{,}000K across accident years, and the rest of the unpaid claim estimate (case reserves and IBNR to that age) is \$18{,}500K. The actuary selects a tail factor of 1.03; a tail of 1.06 is also reasonable. How much does the choice move the unpaid claim estimate?

Answer

\begin{align*}
\text{Tail IBNR}_{1.03} &= 50{,}000 \times 0.03 \\
&= 1{,}500 \\
\text{Tail IBNR}_{1.06} &= 50{,}000 \times 0.06 \\
&= 3{,}000
\end{align*}

The unpaid claim estimate is \$20{,}000K with the selected tail and \$21{,}500K with the alternative, a difference of 7.5\%. If the actuary judges that material, ASOP 43 calls for notifying the principal and attempting to discuss the sensitivity, rather than only reporting the point estimate.

Is the Development Factor Significant? Following Venter, incremental reported loss from 12 to 24 months is regressed on cumulative loss at 12 months across accident years, with no constant. The estimated factor (the age-to-age factor less 1) is 0.35 with a standard deviation of 0.20. Does the factor pass Venter's significance test?

Answer Venter requires the absolute value of a factor to be at least twice its standard deviation:

\begin{align*}
2 \times 0.20 &= 0.40 \\
0.35 &< 0.40
\end{align*}

The factor is not significantly different from zero, so the chain ladder's assumption that the next period's development is proportional to the losses to date is not supported for this column. Venter treats this as a level of comfort rather than a strict statistical test, and the next step is to try the alternatives, such as a factor plus a constant or a Bornhuetter-Ferguson form, and compare their fit.