Long Tail Lines

Study guide: Exam 5

Long Tail Lines are lines of insurance whose claims take many years to be reported, litigated and paid — so that at any valuation date most of the ultimate cost is still an estimate rather than a fact.

\text{Reported \%} = \frac{1}{\text{CDF}_{n \to \text{ult}}}

• Typical long-tail lines: medical malpractice, workers compensation (especially permanent total disability), general and products liability, excess casualty, environmental and asbestos exposures.
• Two lags compound. The reporting lag — latent occupational disease, defects discovered years later, injuries whose severity is not initially apparent — creates pure IBNR; the settlement lag from litigation creates IBNER on claims already known.
• Because the CDF at early maturities is large, estimates are highly leveraged: a small proportional change in a factor moves ultimate by a large dollar amount. A tail factor beyond the observed triangle is unavoidable and is often the single biggest judgment in the analysis.
• Long-tail reserves are also exposed to calendar-year forces that no accident-year pattern anticipates: social and economic inflation, legislative reform, judicial reinterpretation. These hit every open accident year at once.
• The long payout pattern means large held reserves, substantial investment income on them, and an underwriting profit provision that may legitimately be lower than a short-tail line's — a fully developed answer to "why does workers compensation target a higher combined ratio".
• No single method dominates. Practice is to run chain ladder, BF, expected claims and frequency-severity on both paid and reported data, and to weight toward the a priori methods at immature ages.

Leverage in the Tail Factor A medical malpractice accident year has \$3{,}000{,}000 reported at 36 months. Two actuaries select tail factors of 3.50 and 3.75 against the same age-to-age factors.

Quantify the disagreement.

Answer

\begin{align*}
\text{Ultimate}_A &= \$3{,}000{,}000 \times 3.50 = \$10{,}500{,}000 \\
\text{Ultimate}_B &= \$3{,}000{,}000 \times 3.75 = \$11{,}250{,}000
\end{align*}

\begin{align*}
\text{IBNR}_A &= \$7{,}500{,}000 \\
\text{IBNR}_B &= \$8{,}250{,}000
\end{align*}

A 7.1\% difference in the factor is a 10.0\% difference in IBNR — \$750{,}000 on a single accident year, from a judgment about development beyond any data either actuary has. Multiply across ten open accident years and the tail selection alone can move the balance sheet by more than the entire reserve of a short-tail line.

Why the Chain Ladder Fails Early on a Long-Tail Year A general liability accident year has these reported figures. Selected age-to-age factors are 12–24: 2.40, 24–36: 1.45, 36–ult: 1.60.

Maturity	Reported
12 months	\$800{,}000
12 months (one large claim reserved late)	\$1{,}100{,}000

Compare the chain ladder estimate under each figure, and comment.

Answer \text{CDF}_{12 \to \text{ult}} = 2.40 \times 1.45 \times 1.60 = 5.568

\begin{align*}
\text{Ultimate (as reported)} &= \$800{,}000 \times 5.568 \\
&= \$4{,}454{,}400 \\[4pt]
\text{Ultimate (with the claim)} &= \$1{,}100{,}000 \times 5.568 \\
&= \$6{,}124{,}800
\end{align*}

A single \$300{,}000 case reserve entered a few weeks earlier changes the ultimate estimate by \$1{,}670{,}000 — because the chain ladder multiplies whatever is on the diagonal by 5.568.

That sensitivity is the argument for the BF at immature maturities: BF applies the development factor only to the unreported portion and anchors the rest on an a priori expectation, so a one-off reserve entry moves the answer by its own amount rather than by five times its amount. See Large Loss for the parallel treatment in ratemaking.