Sensitivity Of Experience Premiums To Credibility Parameters

Why Sensitivity Matters in Experience Rating

A credibility premium is a weighted average of a risk’s observed experience and a collective complement. The credibility weight depends on estimated variance components and, in an exposure-weighted model, on the amount of experience available. Consequently, changing a model assumption can change both the weight and the indicated premium. Sensitivity analysis makes that dependence explicit: it shows which assumptions matter, in which direction, and by how much.

The calculations here concern prospective expected loss costs. They hold the observed experience and collective complement fixed while changing a credibility parameter. If a model revision also changes the complement or the observed loss costs, those effects must be calculated separately. No interest or mortality assumptions are needed for these one-period expected loss-cost calculations.

Model and Notation

Definition: In the Bühlmann model, let $\Theta$ denote a risk’s latent risk parameter and let $X_1,\ldots,X_n$ be its observations in $n$ equal-exposure periods. Conditional on $\Theta$, the observations are independent and identically distributed, with conditional mean $\mu(\Theta)=E[X_j\mid\Theta]$ and conditional variance $\sigma^2(\Theta)=\operatorname{Var}(X_j\mid\Theta)$. Define the collective mean $m=E[\mu(\Theta)]$, the expected process variance $v=E[\sigma^2(\Theta)]$, and the variance of hypothetical means $a=\operatorname{Var}(\mu(\Theta))$. When $a>0$, the prior-strength parameter is $K=v/a$, and the credibility weight is $Z=n/(n+K)$. The premium per period is $P=(1-Z)m+Z\bar X$, where $\bar X$ is the risk’s arithmetic average experience.

The process variance $v$ measures variation around a risk’s own conditional mean. The between-risk variance $a$ measures how much conditional means differ across risks. A larger $v$, relative to $a$, makes a risk’s observed average less informative about its own mean; a larger $a$, relative to $v$, makes distinguishing risks more valuable.

For Bühlmann-Straub, observations can have unequal exposure. Let $w_j$ be the exposure in period $j$, and let $w=\sum_j w_j$. Under the usual exposure model, conditional expected loss is $w_j\mu(\Theta)$, conditional variance is $w_j\sigma^2(\Theta)$, and observations are conditionally independent. The exposure-weighted average is $\bar X_w=\sum_jX_j/w$. With $K=v/a$, the weight is $Z=w/(w+K)$, and the premium per unit of exposure is $P=(1-Z)m+Z\bar X_w$. Here $K$ is measured in the same exposure units as $w$.

These formulas require positive between-risk variance for a finite, positive $K$. If $a=0$, the model has no between-risk variation to support experience differentiation, so the usual finite-$K$ formula is not applicable; the collective mean is the appropriate premium under that model. If the observed experience equals the collective mean, changing $Z$ does not change the indicated premium.

How Variance Components Control Credibility

For fixed $n$, the Bühlmann weight can be written directly in terms of the variance components:

Z=\frac{n}{n+v/a}=\frac{na}{v+na}.

Taking partial derivatives while holding the other quantities fixed gives:

\frac{\partial Z}{\partial v}=-\frac{na}{(v+na)^2},\qquad \frac{\partial Z}{\partial a}=\frac{nv}{(v+na)^2},\qquad \frac{\partial Z}{\partial n}=\frac{av}{(v+na)^2}.

For positive $a$, $v$, and $n$, these signs show that increasing process variance reduces credibility, while increasing between-risk variance or the amount of equal-period experience increases it. The derivatives are local rates of change; they do not mean a finite change in a parameter produces a change of the same size in $Z$.

A useful scale-free version follows from $Z=n/(n+K)$. Its odds are $Z/(1-Z)=n/K=na/v$. Thus, holding the other quantities fixed, a proportional increase in $v$ reduces the credibility odds by the same proportion, while a proportional increase in $a$ or $n$ increases them by that proportion. In derivative form:

\frac{\partial Z}{\partial\log v}=-Z(1-Z),\qquad \frac{\partial Z}{\partial\log a}=Z(1-Z),\qquad \frac{\partial Z}{\partial\log n}=Z(1-Z).

Why the Bühlmann Weight Has This Form

A short variance argument explains the role of the two variance components. The average $\bar X$ has expected conditional variance $v/n$. Its conditional mean is $\mu(\Theta)$, whose variance across risks is $a$. By the law of total variance, the variance of $\bar X$ around the collective mean is therefore $a+v/n$. The best linear credibility estimate of $\mu(\Theta)$ based on $\bar X$ has coefficient equal to the covariance of $\mu(\Theta)$ and $\bar X$, divided by the variance of $\bar X$. Conditional expectation gives $\operatorname{Cov}(\mu(\Theta),\bar X)=a$, so:

Z=\frac{a}{a+v/n} =\frac{na}{na+v} =\frac{n}{n+v/a}.

This derivation uses the stated conditional independence and equal-exposure assumptions. It gives the Bühlmann linear credibility coefficient; it does not require the conditional distribution of each observation to be normal. The resulting credibility premium is a linear estimate, not a claim that every individual’s actual expected loss is known exactly.

Turning Weight Sensitivity into Premium Sensitivity

Write the premium as $P=m+Z(\bar X-m)$. If the experience and complement are held fixed, its response to any parameter $q$ that affects only $Z$ is:

\frac{\partial P}{\partial q} =(\bar X-m)\frac{\partial Z}{\partial q}.

The sign of the premium change depends on both the direction of the weight change and whether experience is above or below the complement. More credibility moves the premium toward the risk’s experience: upward when $\bar X>m$, downward when $\bar X<m$. This distinction is essential when interpreting a sensitivity result.

Worked Example: Separate Effects of Process and Between-Risk Variance

A hypothetical portfolio uses a Bühlmann model with $n=4$ equal-exposure years, $v=900$, and $a=300$. A risk’s average annual loss cost is $\bar X=\$1{,}200$, and the collective mean is $m=\$1{,}000$. Calculate the premium, then recalculate it if process variance alone rises by 10%, and if between-risk variance alone rises by 10%.

Method and baseline. The prior strength is $K=v/a=900/300=3$ years. Therefore $Z=4/(4+3)=4/7=0.5714$. The premium is:

P=1{,}000+\frac{4}{7}(1{,}200-1{,}000) =1{,}000+\frac{800}{7} =\$1{,}114.29.

Process variance rises by 10%. The revised value is $v=900(1.10)=990$, while $a=300$. Thus $K=990/300=3.3$ and $Z=4/(4+3.3)=4/7.3=40/73$. The premium is $1{,}000+(40/73)(200)=\$1{,}109.59$.

Between-risk variance rises by 10% instead. Return to the baseline and set $a=300(1.10)=330$, leaving $v=900$. Then $K=900/330=30/11$, and $Z=4/(4+30/11)=22/37$. The premium is $1{,}000+(22/37)(200)=\$1{,}118.92$.

Check and trap avoided. Higher process variance lowers the weight and moves this above-complement experience premium down toward \$1,000. Higher between-risk variance raises the weight and moves it up toward \$1,200. The common mistake is to reverse $v/a$: using $K=a/v=1/3$ at baseline would give $Z=12/13$ and a false premium of \$1,184.62. All premiums are rounded to the nearest cent.

Prior Strength and Exposure in Bühlmann-Straub

In Bühlmann-Straub, $K=v/a$ can be read as the amount of exposure whose credibility contribution is comparable to the model’s collective information. The weight $Z=w/(w+K)$ rises with exposure and falls with prior strength. Differentiating gives:

\frac{\partial Z}{\partial w}=\frac{K}{(w+K)^2}>0,\qquad \frac{\partial Z}{\partial K}=-\frac{w}{(w+K)^2}<0.

The interpretation of a changed $K$ needs care. If $K$ increases because estimated process variance increases while other quantities stay fixed, credibility falls. If $K$ increases because the estimated between-risk variance decreases, credibility also falls. A stated change in $K$ alone is enough to recalculate the weight, but identifying its cause explains the model assumption that changed.

Worked Example: Changing Exposure and Prior Strength

A hypothetical commercial account has total exposure $w=10$ units, prior strength $K=5$ exposure units, weighted average loss cost $\bar X_w=\$9{,}200$ per unit, and collective mean $m=\$8{,}000$ per unit. Find the baseline premium per unit and the separate effects of a 25% increase in $K$ and a 20% increase in exposure.

Baseline. The weight is $Z=10/(10+5)=2/3$. The premium is $8{,}000+(2/3)(9{,}200-8{,}000)=\$8{,}800.00$ per unit.

Increase prior strength by 25%. The new $K$ is $5(1.25)=6.25$. Holding $w=10$, $Z=10/(10+6.25)=8/13$. The premium is $8{,}000+(8/13)(1{,}200)=\$8{,}738.46$ per unit.

Increase exposure by 20% instead. Return to $K=5$ and set $w=10(1.20)=12$. Then $Z=12/(12+5)=12/17$, giving a premium of $8{,}000+(12/17)(1{,}200)=\$8{,}847.06$ per unit.

At the baseline, the local rates are $\partial Z/\partial w=5/15^2=1/45$ per exposure unit and $\partial Z/\partial K=-10/15^2=-2/45$ per exposure unit. Since $\bar X_w-m=\$1{,}200$, the corresponding local premium rates are $+\$26.67$ and $-\$53.33$ per unit for a one-unit increase in $w$ and $K$, respectively. The finite changes above are not obtained by multiplying these local rates by the full changes; the weight is nonlinear. The calculations are checked by confirming that the premiums lie between the complement and the observed average. Amounts are rounded to the nearest cent.

Worked Example: Sensitivity When Experience Is Below the Complement

A hypothetical risk has $n=6$ equal-exposure periods, $v=1{,}600$, $a=400$, experience $\bar X=\$1{,}200$, and collective mean $m=\$1{,}500$. Calculate the baseline premium and the premiums when process variance alone doubles and when between-risk variance alone doubles. Also evaluate the local effect of proportional variance changes at baseline.

Baseline. $K=1{,}600/400=4$, so $Z=6/(6+4)=0.6$. Thus $P=1{,}500+0.6(1{,}200-1{,}500)=\$1{,}320.00$.

Double process variance. With $v=3{,}200$ and $a=400$, $K=8$, so $Z=6/14=3/7$. The premium is $1{,}500+(3/7)(-300)=\$1{,}371.43$.

Double between-risk variance instead. With $a=800$ and $v=1{,}600$, $K=2$, so $Z=6/8=0.75$. The premium is $1{,}500+0.75(-300)=\$1{,}275.00$.

At baseline, $Z(1-Z)=0.6(0.4)=0.24$. Therefore $\partial Z/\partial\log v=-0.24$ and $\partial Z/\partial\log a=0.24$. Multiplying by $\bar X-m=-\$300$, the local premium sensitivities are $+\$72$ per unit change in $\log v$ and $-\$72$ per unit change in $\log a$. For example, a small 1% increase corresponds approximately to a change of $0.01$ in the logarithm, or about $+\$0.72$ for process variance and $-\$0.72$ for between-risk variance. The exact doubling results are not these local approximations. The checks are consistent: less credibility pulls the below-complement premium up, and more credibility pulls it down. Amounts are rounded to the nearest cent.

Practical Interpretation and Common Traps

Sensitivity calculations support model review by separating assumptions from observed experience. A premium may be sensitive to estimated variance components even when the experience data themselves are unchanged. It is therefore useful to report both the revised weight and the resulting premium, rather than only saying that a parameter increases or decreases credibility.

Reversing the ratio: $K=v/a$, not $a/v$. Process variance belongs in the numerator because more within-risk noise requires more exposure to achieve a given weight.
Ignoring the direction of experience: an increase in credibility raises the premium only when experience exceeds the complement. Below the complement, it lowers the premium.
Treating a local derivative as a finite change: derivatives describe a local slope. For a substantial parameter change, recalculate $K$, $Z$, and $P$ using the revised values.
Mixing exposure units: $w$ and $K$ must use the same units. If $K$ is stated in policy-years, exposure must also be measured in policy-years.
Changing only the weight when the complement also changes: the derivative $(\bar X-m)\partial Z/\partial q$ holds $m$ and the experience fixed. Re-estimation that changes the collective mean requires recalculating the full premium.

Key takeaway: Credibility rises with exposure and between-risk variance, and falls with process variance and prior strength. The premium’s direction of movement depends on whether the risk’s experience is above or below the collective complement.

Check Your Understanding

For each problem, calculate the requested credibility weight or premium before checking the answer.

A Bühlmann model has $n=3$, $v=800$, and $a=200$. Calculate $K$ and $Z$. (Answer: $K=4$, $Z=3/7\approx0.4286$)
A risk has $n=8$, $v=1{,}200$, $a=300$, $\bar X=\$650$, and $m=\$500$. Calculate its Bühlmann premium. (Answer: $K=4$, $Z=2/3$, premium $=\$600.00$)
Under Bühlmann-Straub, $w=15$, $K=5$, $\bar X_w=\$200$, and $m=\$240$. Calculate the weight and premium per unit. (Answer: $Z=0.75$, premium $=\$210.00$)
A Bühlmann-Straub risk has $w=6$, $K=3$, $\bar X_w=\$1{,}300$, and $m=\$1{,}000$. If prior strength increases by 50% with the other inputs fixed, calculate both premiums and the premium change. (Answer: baseline $Z=2/3$, premium $=\$1{,}200.00$; revised $K=4.5$, $Z=4/7$, premium $=\$1{,}171.43$; change $=-\$28.57$)
A Bühlmann model has $n=5$, $v=1{,}000$, and $a=200$. Calculate the baseline weight and the weight if $a$ doubles while $v$ and $n$ remain fixed. Explain the direction. (Answer: baseline $K=5$, $Z=0.5$; revised $K=2.5$, $Z=2/3\approx0.6667$; credibility increases because between-risk variance is larger.)

What You'll Learn