Actuarial Notes

Ultimate Loss

Study guides: Exam 5, Exam GI-101, Exam GI-301

Ultimate Loss is the total cost a cohort of claims will have when every one of them is settled and closed — the quantity every reserving method exists to estimate.

U = \text{Paid} + \text{Case Reserves} + \text{IBNR}

U = \text{Reported} + \text{IBNR}

  • Ultimate is an estimate until the last claim closes, which in a long-tail line can be decades. Only at that point does it become a fact, and by then it is of historical interest only.
  • The methods differ in how they estimate the unknown part, not in what they estimate: chain ladder scales the diagonal, BF adds an a priori-based estimate of what has not emerged, Cape Cod derives that a priori from the data, frequency-severity multiplies projected counts by projected severity.
  • Selecting an ultimate is a separate step from computing the methods. Standard practice is to run several methods on both paid and reported data, lay the results side by side by accident year, and select — leaning on a priori methods at immature ages and development methods at mature ones, with reasons recorded.
  • The ultimate loss ratio (U / \text{EP}) is the number that links reserving to pricing: it is the retrospective test of whether the year was priced adequately, and it is the input to the a priori for future years (Rate Level Change).
  • Ultimate must be stated on a defined basis — gross or net, with or without ALAE, before or after salvage and subrogation — and the basis must be the same on both sides of every comparison.

Selecting an Ultimate from Several Methods AY 2023 at 12 months: paid \$300{,}000, case reserves \$200{,}000, reported \$500{,}000; earned premium \$1{,}200{,}000; a priori ELR 83.3\%; reported \text{CDF}_{12} = 2.400; paid \text{CDF}_{12} = 4.100.

Compute the candidate ultimates and select.

Answer

\begin{align*}
\text{Reported CL} &= \$500{,}000 \times 2.400 = \$1{,}200{,}000 \\[4pt]
\text{Paid CL} &= \$300{,}000 \times 4.100 = \$1{,}230{,}000 \\[4pt]
\text{Expected claims} &= 0.833 \times \$1{,}200{,}000 = \$1{,}000{,}000 \\[6pt]
\text{BF (reported)} &= \$500{,}000 + \left(1 - \tfrac{1}{2.400}\right)\$1{,}000{,}000 \\
&= \$500{,}000 + \$583{,}000 \\
&= \$1{,}083{,}000
\end{align*}

The four estimates span \$1{,}000{,}000 to \$1{,}230{,}000. At 12 months the development methods rest on a 2.4 (or 4.1) multiple of a thin diagonal, so they carry the least weight; the expected claims estimate rests entirely on an ELR that may be stale.

A selection around \$1{,}140{,}000 — between BF and the chain ladders, closer to BF — is defensible, implying

\text{IBNR} = \$1{,}140{,}000 - \$500{,}000 = \$640{,}000

What matters more than the exact pick is that the paid and reported chain ladders agree (\$1{,}200K vs \$1{,}230K). That agreement says the diagonal is not distorted by case adequacy or settlement speed, which is what makes leaning toward the development methods reasonable at all.

Ultimates Moving Between Valuations Selected ultimates for one accident year at successive year-ends:

Valuation Selected ultimate
12 mo \$9{,}200{,}000
24 mo \$9{,}400{,}000
36 mo \$10{,}100{,}000
48 mo \$11{,}300{,}000

What does this pattern indicate?

Answer The estimate has risen at every valuation, and the increases are accelerating (+2.2\%, +7.4\%, +11.9\%). A well-estimated ultimate should move randomly around its final value, not drift consistently in one direction.

Consistent upward drift means the estimation process is systematically low, and the usual causes are:

  • development factors selected too low, often by excluding "unusual" high factors each year;
  • a tail factor that is too small — which bites at exactly these later maturities;
  • an a priori ELR that was too optimistic, holding BF estimates down while the data pulled upward;
  • a genuine calendar-year deterioration (social inflation, a legal change) affecting all open years.

The distinction matters: the first three are process failures to be corrected in the method, the fourth is a real-world change to be quantified and reflected in both reserves and pricing. Tracking this pattern across accident years is what actual-versus-expected monitoring is for, and finding it four years late — as here — is exactly what that monitoring exists to prevent.