Cumulative Development Factor

Study guide: Exam 5

Cumulative Development Factor (CDF, or age-to-ultimate factor) is the product of all selected age-to-age factors from a given maturity through to ultimate, including the tail.

$\text{CDF}_{n \to \text{ult}} = \left(\prod_{k \ge n} f_{k \to k+1}\right) \times \text{Tail}$

$\text{Ultimate} = C_n \times \text{CDF}_{n \to \text{ult}}$

The reciprocal is the percentage reported (or paid): $p_n = 1/\text{CDF}_n$ , and $1 - p_n$ is the percentage unreported. These are the quantities the BF, Cape Cod and Benktander methods actually use — the CDF and the emergence pattern are the same information in two forms.
CDFs are built backwards from the tail: start with the tail factor, multiply in each successive age-to-age factor moving toward younger ages. They decrease monotonically with maturity.
The CDF measures leverage. At a CDF of $4$ , a dollar of distortion on the diagonal becomes four dollars of ultimate — which is why the chain ladder is unreliable at immature ages in long-tail lines and why methods that develop only the unreported portion are preferred there.
A CDF is specific to the triangle it came from: paid CDFs differ from reported CDFs, count CDFs from dollar CDFs, and a CDF built on data capped at a retention differs from a gross one. Borrowing across bases is a common and consequential error.
Sensitivity to the tail rises sharply with the length of the tail: for a short-tail line the tail is a rounding difference, for a long-tail line it can be the largest single assumption in the analysis.

Building CDFs and Reading the Emergence Pattern Selected factors: $f_{12\text{–}24} = 1.500$ , $f_{24\text{–}36} = 1.200$ , $f_{36\text{–}48} = 1.050$ , tail $= 1.020$ .

Build the CDF table and the implied emergence pattern.
Answer Working backwards from the tail:
 $\begin{align*} \text{CDF}_{48} &= 1.020 \\ \text{CDF}_{36} &= 1.050 \times 1.020 = 1.071 \\ \text{CDF}_{24} &= 1.200 \times 1.071 = 1.285 \\ \text{CDF}_{12} &= 1.500 \times 1.285 = 1.928 \end{align*}$ 
Age CDF $\%$ reported $\%$ unreported

$12$ $1.928$ $51.9\%$ $48.1\%$

$24$ $1.285$ $77.8\%$ $22.2\%$

$36$ $1.071$ $93.4\%$ $6.6\%$

$48$ $1.020$ $98.0\%$ $2.0\%$

An accident year with $\$600{,}000$ reported at $12$ months has an ultimate of

$\$600{,}000 \times 1.928 = \$1{,}156{,}800$

of which $48.1\%$ — $\$556{,}800$ — is IBNR. Nearly half the estimate is an assumption at that maturity.

Age	CDF	$\%$ reported	$\%$ unreported
$12$	$1.928$	$51.9\%$	$48.1\%$
$24$	$1.285$	$77.8\%$	$22.2\%$
$36$	$1.071$	$93.4\%$	$6.6\%$
$48$	$1.020$	$98.0\%$	$2.0\%$

Paid and Reported CDFs Must Reconcile For the same accident year at $24$ months: paid $\$1{,}800{,}000$ with a paid CDF of $2.40$ ; reported $\$3{,}150{,}000$ with a reported CDF of $1.30$ .

Compare the estimates and diagnose.
Answer
 $\begin{align*} \text{Ultimate}_{\text{paid}} &= \$1{,}800{,}000 \times 2.40 = \$4{,}320{,}000 \\ \text{Ultimate}_{\text{reported}} &= \$3{,}150{,}000 \times 1.30 = \$4{,}095{,}000 \end{align*}$ 
A $5.5\%$ gap, with paid running higher. Both estimates rest on the same underlying claims, so the difference is information about the diagonal, not about the ultimate.

Implied case reserves are $\$3{,}150{,}000 - \$1{,}800{,}000 = \$1{,}350{,}000$ . The paid method says total future payments will be $\$2{,}520{,}000$ ; the reported method says $\$2{,}295{,}000$ . The reported method is therefore implicitly claiming that case reserves plus a modest IBNR will cover the run-off, while the paid method — which knows nothing about case reserves — projects more.

The reconciliation to perform: check whether case adequacy has weakened (which would make the reported figure too low and the paid method right), or whether settlement has accelerated (which would inflate the paid diagonal and make the paid method too high). The closure rate and average case outstanding decide between them; the gap itself does not.