Paid Losses

Study guide: Exam 5

Paid Losses are the cumulative cash payments made on covered claims through a valuation date — the only loss figure that contains no estimate.

\text{Reported Losses} = \text{Paid Losses} + \text{Case Reserves}

\text{Ultimate}_{\text{paid}} = \text{Paid}_n \times \text{CDF}^{\text{paid}}_{n \to \text{ult}}

• Paid data is objective. It carries no adjuster judgment, so a paid triangle is immune to changes in case reserve adequacy — the single most common distortion in reported triangles.
• The price of that immunity is less information and more leverage. Paid losses lag reported losses, so paid development factors are larger at every maturity and a paid chain ladder at an early age multiplies a small, volatile number by a large factor.
• Paid triangles are not immune to everything: they are highly sensitive to changes in settlement rate. Speeding up claim closure inflates the paid diagonal and makes historical paid factors too high; the Berquist-Sherman paid adjustment exists to restate the triangle onto a common disposal-rate basis.
• Running paid and reported methods side by side is standard practice. Agreement is reassuring; divergence is a diagnostic. Paid ultimates above reported ultimates suggest case reserves are inadequate; the reverse suggests they are strengthening (or claims are closing slowly).
• Paid losses are also what the payment pattern is built from, which drives discounting, cash-flow testing and the value of float in the profit provision.

Paid and Reported Chain Ladder on the Same Year Accident year 2023 at 24 months: paid \$2{,}400{,}000, case reserves \$1{,}900{,}000. Selected cumulative factors to ultimate are 2.05 on paid and 1.28 on reported.

Compare the two estimates and interpret the difference.

Answer

\begin{align*}
\text{Reported} &= \$2{,}400{,}000 + \$1{,}900{,}000 \\
&= \$4{,}300{,}000 \\[6pt]
\text{Ultimate}_{\text{paid}} &= \$2{,}400{,}000 \times 2.05 \\
&= \$4{,}920{,}000 \\[6pt]
\text{Ultimate}_{\text{reported}} &= \$4{,}300{,}000 \times 1.28 \\
&= \$5{,}504{,}000
\end{align*}

The reported method is \$584{,}000 (12\%) higher. Two readings are consistent with that gap:

• Case reserves have been strengthened relative to history, so the reported diagonal is high and the historical reported factors over-develop it.
• Claims are closing more slowly, so the paid diagonal is low relative to history and the paid method under-develops.

The next step is not to average the two but to look at the diagnostics — average case outstanding and the ratio of closed to reported counts by maturity — and identify which story the data supports before selecting.

Why Paid Development Factors Are Larger A liability line reports the following cumulative percentages of ultimate at each maturity:

Maturity	Reported	Paid
12 mo	55\%	20\%
24 mo	78\%	45\%
36 mo	90\%	68\%
48 mo	96\%	85\%

Compute the CDFs to ultimate at 12 months and comment on the leverage.

Answer

\begin{align*}
\text{CDF}^{\text{rep}}_{12} &= \frac{1}{0.55} = 1.818 \\[4pt]
\text{CDF}^{\text{paid}}_{12} &= \frac{1}{0.20} = 5.000
\end{align*}

Now suppose the 12-month figure is misstated by \$100{,}000 — one large claim paid or reserved slightly early:

\begin{align*}
\text{Error}_{\text{reported}} &= \$100{,}000 \times 1.818 = \$181{,}800 \\
\text{Error}_{\text{paid}} &= \$100{,}000 \times 5.000 = \$500{,}000
\end{align*}

The paid method magnifies the same distortion nearly three times as much. This is the trade-off in a sentence: reported data is richer but contaminated by judgment; paid data is clean but thin. At early maturities both give way to the BF approach, which limits the leverage by applying development only to the unemerged portion.