Tail Factor

Study guide: Exam 5

Tail Factor is the development factor applied beyond the last maturity observable in the triangle — the estimate of all emergence that will occur after the data runs out.

\text{CDF}_{\text{last age} \to \text{ult}} = f_{\text{last observed}} \times \text{Tail}

• The tail is the one factor with no data behind it, and in long-tail lines it is often the single largest assumption in the analysis. A triangle that runs ten years still misses decades of workers compensation medical payments.
• Methods for selecting one:
  • Industry benchmarks — published patterns (ISO, NCCI, Schedule P aggregates) for the same line, adjusted for differences in limits, mix and case reserving.
  • Curve fitting — fit an exponential or inverse-power curve to the observed age-to-age factors and extrapolate the decay. The inverse power curve gives a heavier tail than the exponential and is the more common choice in casualty.
  • Bondy method — set the tail equal to the last observed age-to-age factor (or a multiple of it), on the reasoning that development decays geometrically.
  • Paid-to-reported convergence — reported losses must eventually equal paid, so the ratio of the two at the oldest maturities implies how much paid development remains.
  • Judgment, documented, informed by the settlement characteristics of the line.
• The tail on a paid triangle is larger than on a reported one, since case reserves have already anticipated much of the reported development.
• A tail of exactly 1.000 is itself an assumption — that development is complete — and it is the wrong one for any casualty line. Understating the tail is a classic source of reserve deficiency, especially with latent exposures (asbestos, environmental, abuse claims).
• Because the tail multiplies the oldest accident years — the ones with the most dollars already reported — a small change in it moves a large reserve. Sensitivity testing the tail is not optional in a long-tail analysis.

Tail Selection Moves the Reserve The last observable age is 120 months, with \$2{,}000{,}000 reported. Two tail estimates are under consideration: 1.030 and 1.050.

Quantify the difference.

Answer

\begin{align*}
\text{Tail } 1.030: \; \text{Ultimate} &= \$2{,}060{,}000, \; \text{IBNR} = \$60{,}000 \\
\text{Tail } 1.050: \; \text{Ultimate} &= \$2{,}100{,}000, \; \text{IBNR} = \$100{,}000
\end{align*}

Two points of tail is \$40{,}000 on this year — a 67\% difference in its IBNR.

And the tail applies to every open accident year, not just the oldest. Across ten years of a mature book with \$20M reported beyond 120 months, the same two-point difference is \$400{,}000 of reserve, arrived at by judgment alone.

Fitting a Tail by Curve Extrapolation Observed age-to-age factors for a workers compensation reported triangle:

Interval	Factor
72–84	1.040
84–96	1.028
96–108	1.020
108–120	1.014

Estimate the tail beyond 120 months.

Answer Work with the excess over 1.0, which is what decays: 0.040, 0.028, 0.020, 0.014. Successive ratios are

\frac{0.028}{0.040} = 0.70 \qquad \frac{0.020}{0.028} = 0.71 \qquad \frac{0.014}{0.020} = 0.70

A clean geometric decay at r = 0.70. Extrapolating from the last excess of 0.014:

\begin{align*}
\text{Total remaining excess} &= 0.014 \times \frac{0.70}{1 - 0.70} \\
&= 0.014 \times 2.333 \\
&= 0.0327
\end{align*}

\text{Tail} \approx 1.033

The Bondy shortcut — take the tail as the last observed factor, 1.014 — gives a materially smaller answer, because it captures only one further period rather than the whole infinite remainder.

Two cautions. An exponential fit assumes the decay rate holds forever; an inverse power curve, which decays more slowly, would give a heavier tail and is the more conservative choice for a line with latent exposure. And the fit describes the observed pattern — if the last diagonals contain a case strengthening, the fitted decay inherits it.