Deriving D-Ratios from Severity Distributions

What the D-Ratio Measures

A primary-excess experience plan divides each ground-up loss at a specified split point. The primary part is capped at the split point, while any amount above it is excess. The D-ratio describes the expected share of ground-up losses assigned to the primary part under a specified severity distribution. It is a loss-weighted measure: it is not, in general, the proportion of claims below the split point.

Let $X$ be a nonnegative ground-up loss for one claim, and let $s>0$ be the primary-excess split point. Assume $0<E[X]<\infty$. The primary payment per claim is $\min(X,s)$. The excess payment per claim is $(X-s)_+=\max(X-s,0)$. These definitions assume that the split is applied to the loss amount on the same basis as the severity distribution, with no separate deductible or policy limit changing the amount being split.

Definition: The D-ratio at split point $s$ is the expected primary loss per claim divided by the expected ground-up loss per claim: $$ D(s)=\frac{E[\min(X,s)]}{E[X]}. $$ The complementary expected excess share is $1-D(s)=E[(X-s)_+]/E[X]$. The ratio is defined when $E[X]$ is positive and finite.

If the same claim frequency applies to both components, multiplying each per-claim expectation by that frequency gives expected aggregate primary and total losses; the frequency cancels in their ratio. Thus the severity distribution determines the D-ratio under these assumptions. There is no interest rate or mortality assumption in this calculation: losses are evaluated on a common dollar basis without discounting.

Deriving the D-Ratio from the Survival Function

Write $S_X(x)=P(X>x)$ for the survival function of the ground-up severity. For any nonnegative random variable with a finite mean, its expectation is the area under its survival function. The same idea applies to the severity capped at $s$, but only over the interval from zero to $s$.

E[\min(X,s)]=\int_0^s S_X(x)\,dx,\qquad E[X]=\int_0^\infty S_X(x)\,dx, \qquad D(s)=\frac{\int_0^s S_X(x)\,dx}{\int_0^\infty S_X(x)\,dx}.

To justify the first identity, for each fixed loss amount $X$, the quantity $\min(X,s)$ is the length of the interval from zero to $s$ over which the indicator $1_{\{X>x\}}$ equals one. In integral form,

\min(X,s)=\int_0^s 1_{\{X>x\}}\,dx.

Taking expectations and interchanging expectation and integration for this nonnegative integrand gives

E[\min(X,s)] =\int_0^s E[1_{\{X>x\}}]\,dx =\int_0^s P(X>x)\,dx =\int_0^s S_X(x)\,dx.

Applying the same argument over the whole nonnegative axis gives $E[X]=\int_0^\infty S_X(x)\,dx$, provided the mean is finite. Substitution into the definition yields the survival-function form of the D-ratio. This derivation does not require a density, so it also works for discrete severities.

Worked Calculations

Worked Example: Exponential Severity at a Given Split

A hypothetical property portfolio has ground-up claim severity $X$ that is exponentially distributed with mean $\theta=\$20{,}000$. The split point is $s=\$10{,}000$. Find expected primary loss per claim, expected excess loss per claim, and the D-ratio. No interest or mortality assumption applies; the severity amounts are undiscounted.

Method. For an exponential severity with mean $\theta$, the survival function is $S_X(x)=e^{-x/\theta}$, and $E[X]=\theta$. Therefore,

E[\min(X,s)] =\int_0^s e^{-x/\theta}\,dx =\theta(1-e^{-s/\theta}).

Substitute the parameters: $s/\theta=10{,}000/20{,}000=0.5$, and $e^{-0.5}=0.6065307$. Thus

E[\min(X,s)] =20{,}000(1-0.6065307) =20{,}000(0.3934693) =\$7{,}869.39.

The expected total loss per claim is $E[X]=\$20{,}000$. Since primary plus excess equals the ground-up loss claim by claim, expected excess is $\$20{,}000-\$7{,}869.39=\$12{,}130.61$. The ratio is

D(10{,}000)=\frac{7{,}869.39}{20{,}000}=0.3934695\approx 0.3935.

The expected components check: $\$7{,}869.39+\$12{,}130.61=\$20{,}000.00$. The common mistake is to report $1-e^{-0.5}=0.3934693$ as the primary dollar amount. That number is a dimensionless factor; multiplying by the mean severity gives the expected primary loss of $\$7{,}869.39$.

Worked Example: Discrete Severity with an Atom Above the Split

For a hypothetical equipment portfolio, severity $X$ has this distribution: $\$2{,}000$ with probability $0.50$, $\$8{,}000$ with probability $0.30$, and $\$20{,}000$ with probability $0.20$. The primary-excess split is $s=\$6{,}000$. Calculate the D-ratio and verify the expected excess. There is no discounting or mortality component.

Method. First calculate total expected severity directly from the probability mass function:

E[X] =0.50(2{,}000)+0.30(8{,}000)+0.20(20{,}000) =1{,}000+2{,}400+4{,}000 =\$7{,}400.

The capped primary amounts at the three possible severities are $\$2{,}000$, $\$6{,}000$, and $\$6{,}000$. Therefore,

E[\min(X,6{,}000)] =0.50(2{,}000)+0.30(6{,}000)+0.20(6{,}000) =1{,}000+1{,}800+1{,}200 =\$4{,}000.

The D-ratio is $4{,}000/7{,}400=0.5405405$, or $0.5405$ to four decimal places. For a check, the excess amounts are $\$0$, $\$2{,}000$, and $\$14{,}000$, so expected excess is

0.50(0)+0.30(2{,}000)+0.20(14{,}000) =0+600+2{,}800 =\$3{,}400.

The components reconcile: $\$4{,}000+\$3{,}400=\$7{,}400$. The trap is to use the probability of a claim at or below the split, $0.50$, as the D-ratio. That is a claim-count probability, not the proportion of expected dollars in the primary layer.

Worked Example: Heavy-Tailed Pareto-Type Severity

Consider a hypothetical liability portfolio with a Lomax (Pareto Type II) severity distribution. Its survival function is $S_X(x)=(1+x/\theta)^{-\alpha}$ for $x\geq 0$, with scale $\theta=\$10{,}000$ and shape $\alpha=2$. The split point is $\$10{,}000$. Find expected primary loss and the D-ratio. The finite-mean condition is $\alpha>1$, which holds; no interest or mortality assumption applies.

The expected severity is $\theta/(\alpha-1)=\$10{,}000$. Integrating the survival function up to the split gives

E[\min(X,10{,}000)] =\int_0^{10{,}000}\left(1+\frac{x}{10{,}000}\right)^{-2}dx.

Set $u=1+x/10{,}000$, so $dx=10{,}000\,du$. The integration limits become $1$ and $2$. Hence

E[\min(X,10{,}000)] =10{,}000\int_1^2 u^{-2}du =10{,}000[-u^{-1}]_1^2 =10{,}000\left(1-\frac12\right) =\$5{,}000.

Thus $D(10{,}000)=5{,}000/10{,}000=0.5000$. Expected excess is also $\$5{,}000$, so the components sum to the $\$10{,}000$ expected total. The calculation uses the survival function of the stated severity model; substituting a light-tailed model merely because it has the same mean would generally produce a different D-ratio.

How the Split Point Changes the Ratio

For a fixed severity distribution, increasing the split point cannot reduce the primary amount for any claim: if $s_2>s_1$, then $\min(X,s_2)\geq\min(X,s_1)$. Taking expectations and dividing by the same positive $E[X]$ shows $D(s_2)\geq D(s_1)$. The D-ratio is therefore nondecreasing in the split point.

When the severity has a continuous distribution, differentiating the survival-integral formula gives a more specific interpretation:

D'(s)=\frac{S_X(s)}{E[X]}.

A small increase in the split point adds primary expected loss at a rate equal to the probability that a claim exceeds the current split, divided by mean severity. As the split rises, fewer claims extend above it, so the marginal increase is generally smaller. For distributions with atoms, the ratio remains nondecreasing; the derivative statement applies where the integral is differentiable, rather than requiring a density-based formula at every point.

At $s=0$, the expected primary amount is zero and $D(0)=0$. As $s$ grows without bound, $\min(X,s)$ approaches $X$, and, for finite mean, the D-ratio approaches one. If all possible losses are at or below a finite split, the ratio is already one at that split.

The ratio also depends on the shape of severity, not just its mean. A distribution with more expected loss in its tail will generally have a smaller primary share at a given split than a less tail-heavy distribution with a comparable mean. This is why the loss basis, inflation level, and severity assumptions must be consistent with the split point. A split expressed in one dollar basis should not be paired unadjusted with severities measured on another.

Common Exam Traps

Confusing dollars with a factor. In the exponential example, $0.3934693$ is the factor $1-e^{-0.5}$; the expected primary dollars are $\$20{,}000$ times that factor, or $\$7{,}869.39$.
Using a claim-count proportion. The probability $P(X\leq s)$ counts claims. The D-ratio weights outcomes by their dollar amounts through $E[\min(X,s)]/E[X]$.
Applying the split incorrectly. The primary amount is $\min(X,s)$, not $X$ for claims below the split and zero otherwise. A claim above the split contributes the full split amount to primary.
Ignoring the mean condition. The ordinary D-ratio requires finite, positive $E[X]$. A severity model with an infinite mean does not yield a finite denominator for this definition.
Mixing bases. If expected severities are trended or otherwise adjusted, the split must be on the corresponding basis before calculating the ratio.

Key takeaway: Calculate the D-ratio as expected limited severity divided by expected ground-up severity. The survival-function form makes the role of the severity tail explicit, while the split-point result confirms that raising the primary cap increases, or leaves unchanged, the expected primary share.

Check Your Understanding

For each problem, use the stated severity basis and split point. Amounts are undiscounted; interest and mortality assumptions do not apply.

A hypothetical severity is exponential with mean $\$12{,}000$, and the split point is $\$6{,}000$. Find expected primary loss per claim and the D-ratio. (Answer: expected primary loss $\$4{,}721.63$; D-ratio $0.3935$)
Severity is $\$1{,}000$, $\$5{,}000$, or $\$15{,}000$, with probabilities $0.40$, $0.40$, and $0.20$, respectively. At a $\$4{,}000$ split, calculate expected total severity, expected primary severity, and the D-ratio. (Answer: $\$5{,}400$; $\$2{,}800$; D-ratio $0.5185$)
A Lomax severity has survival function $S_X(x)=(1+x/\theta)^{-\alpha}$, with $\theta=\$8{,}000$, $\alpha=3$, and split $s=\$8{,}000$. Find expected total severity and the D-ratio. (Answer: expected total severity $\$4{,}000$; D-ratio $0.7500$)
For exponential severity with mean $\$15{,}000$, compare the D-ratios at split points $\$5{,}000$ and $\$20{,}000$. (Answer: $0.2835$ and $0.7364$, respectively; the higher split has the higher D-ratio.)
A severity takes values $\$0$, $\$10{,}000$, and $\$30{,}000$ with probabilities $0.20$, $0.50$, and $0.30$. At a $\$20{,}000$ split, find the D-ratio and distinguish it from the probability of a claim not exceeding the split. (Answer: D-ratio $0.7857$; $P(X\leq20{,}000)=0.7000$)

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