Updating Credibility as Experience Accumulates

Introduction

A credibility estimate is not fixed once it has been calculated. As a risk accumulates exposure and new experience becomes available, its observed experience receives more weight relative to the collective mean. The central task is to update the quantities consistently: total exposure, the exposure-weighted experience mean, the credibility factor, and the resulting premium.

This tutorial considers updates under the Bühlmann-Straub model, with the collective parameters held fixed. It is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus. The focus is on the effect of additional exposure, not on re-estimating the collective parameters from a changing portfolio.

Definition: Let $X_j$ be the observed claim rate for a risk in period $j$, with exposure weight $w_j>0$. The total exposure after $n$ periods is $W=\sum_{j=1}^{n}w_j$, and the exposure-weighted experience mean is $M=\sum_{j=1}^{n}w_jX_j/W$. Let $m$ be the collective mean, $\text{EPV}$ the expected value of process variance per unit exposure, and $\text{VHM}$ the variance of hypothetical means. When these collective quantities are positive and fixed, the credibility constant is $K=\text{EPV}/\text{VHM}$, the credibility factor is $Z=W/(W+K)$, and the Bühlmann-Straub premium is $P=ZM+(1-Z)m$.

What Changes When Experience Is Added?

Suppose the risk has existing total exposure $W$ and experience mean $M$, then receives a new block of exposure $r>0$ with observed rate $x$. The old weighted experience total is $WM$, and the new block contributes $rx$. Therefore the updated exposure and experience mean are

W_{\text{new}}=W+r,\qquad M_{\text{new}}=\frac{WM+rx}{W+r}.

The updated credibility factor uses the new total exposure, while $K$, $m$, $\text{EPV}$, and $\text{VHM}$ remain fixed under the stated assumption:

Z_{\text{new}}=\frac{W+r}{W+r+K},\qquad P_{\text{new}}=Z_{\text{new}}M_{\text{new}}+(1-Z_{\text{new}})m.

For several added blocks $(r_k,x_k)$, add their exposure-weighted experience totals to the existing total. If $R=\sum_k r_k$, the updated mean is $(WM+\sum_k r_kx_k)/(W+R)$, and the updated credibility factor is $(W+R)/(W+R+K)$. The order in which the blocks are processed does not affect the final result.

Model assumption: The Bühlmann-Straub exposure interpretation assumes that, conditional on a risk’s hypothetical mean, the observed rates have that conditional mean and process variance that decreases in proportion to the inverse of exposure. The collective mean and structural variance components are treated as fixed during an update. If the underlying portfolio mix or collective parameters are re-estimated, this fixed-parameter update alone is not sufficient.

Why the Premium Has a Simple Updating Form

The premium can be written directly in terms of the weighted experience total and the credibility constant. Starting from $P=ZM+(1-Z)m$, substitute $Z=W/(W+K)$ and $M=(\sum_jw_jX_j)/W$:

P=\frac{W}{W+K}M+\frac{K}{W+K}m =\frac{WM+Km}{W+K}.

After adding exposure $r$ at rate $x$, the same expression gives

P_{\text{new}}=\frac{WM+rx+Km}{W+r+K}.

This also yields an informative update identity. The old premium is $P=(WM+Km)/(W+K)$, so $WM+Km=(W+K)P$. Thus

P_{\text{new}}=\frac{(W+K)P+rx}{W+K+r}.

The updated premium is therefore a weighted average of the old premium and the new observed rate, with weights $W+K$ and $r$, respectively. This identity explains why a new rate above the old premium raises the premium, while a new rate below it lowers the premium. The size of the change depends on the added exposure relative to $W+K$.

Record current experience.
Write down total exposure $W$ and experience mean $M$, or equivalently the weighted total $WM$.

Add the new weighted experience.
For each new block, add $r_k$ to exposure and $r_kx_k$ to the weighted total.

Recalculate the experience mean and credibility.
Use the new exposure total in both $M_{\text{new}}$ and $Z_{\text{new}}$; keep $K$ fixed.

Calculate and check the premium.
Use either $Z_{\text{new}}M_{\text{new}}+(1-Z_{\text{new}})m$ or $(W_{\text{new}}M_{\text{new}}+Km)/(W_{\text{new}}+K)$.

Worked Updates

Worked Example: Updating Through Two New Exposure Blocks

A risk currently has exposure $W=2$, experience mean $M=\$60$, collective mean $m=\$75$, and credibility constant $K=3$. It then receives exposure 2 with an observed rate of $\$90$, followed by exposure 1 with a rate of $\$30$. Find the experience mean, credibility factor, and premium after each addition and at the end.

Method. Maintain the exposure-weighted total, adding exposure times its observed rate for every block.

Initially, the weighted total is $WM=2(60)=\$120$. After the first addition, exposure is $2+2=4$, and the weighted total is $120+2(90)=\$300$. Therefore $M_1=300/4=\$75.00$, and $Z_1=4/(4+3)=4/7$. The premium is

P_1=\frac{4}{7}(75)+\frac{3}{7}(75)=\$75.00.

After the second addition, total exposure is $4+1=5$, and the weighted total is $300+1(30)=\$330$. Thus $M_2=330/5=\$66.00$, and $Z_2=5/(5+3)=5/8=0.625$. The final premium is

P_2=\frac{5}{8}(66)+\frac{3}{8}(75) =41.25+28.125 =\$69.375 \approx \$69.38.

Check. Using the direct formula, $P_2=(330+3(75))/(5+3)=555/8=\$69.375$, which rounds to $\$69.38$. The final experience mean is $\$66.00$, not $\$69.00$: the exposure-weighted total is $\$330$ over five exposure units. The common mistake this avoids is misadding the weighted experience or dividing by the wrong total exposure.

Worked Example: Tracking the Premium as the Risk’s Experience Develops

A hypothetical commercial equipment risk has exposure 4 at an observed rate of $\$100$. The collective mean is $m=\$80$, and $K=6$. New blocks arrive in sequence: exposure 3 at $\$92$, then exposure 5 at $\$74$. Calculate the credibility factor and premium initially, after the first block, and after both blocks.

Initially, $W=4$, $M=\$100$, and $Z=4/(4+6)=0.4$. Hence $P=0.4(100)+0.6(80)=\$88.00$.

After the first block, the weighted total is $4(100)+3(92)=400+276=676$, and total exposure is 7. Thus $M_1=676/7=\$96.5714$ to four decimal places, and $Z_1=7/(7+6)=7/13$. To avoid rounding the mean prematurely, calculate the premium from the weighted total:

P_1=\frac{676+6(80)}{7+6} =\frac{1156}{13} =\$88.9231\ \text{(to four decimals)}.

After the second block, the weighted total becomes $676+5(74)=676+370=1046$, and total exposure becomes $7+5=12$. Therefore $M_2=1046/12=\$87.1667$ to four decimal places, and $Z_2=12/(12+6)=2/3$. The final premium is

P_2=\frac{1046+6(80)}{12+6} =\frac{1526}{18} =\$84.7778\ \text{(to four decimals)}.

The first added block raises the premium from $\$88.00$ to $\$88.92$, while the next block lowers it to $\$84.78$. The final credibility factor is higher than the initial factor, but the premium falls because the added experience includes a sufficiently low rate. Increased credibility does not mean that the premium must increase.

Worked Example: Combining Unequal Period Exposures

A hypothetical small-business account has two prior periods: exposure 3 at a rate of $\$50$, and exposure 2 at a rate of $\$70$. Its collective mean is $m=\$64$, and its fixed credibility constant is $K=5$. It adds exposure 4 at $\$82$, followed by exposure 1 at $\$40$. Find the final experience mean, credibility factor, and premium.

The prior weighted total is $3(50)+2(70)=150+140=\$290$, with total exposure $3+2=5$. The new blocks add $4(82)=\$328$ and $1(40)=\$40$. The final weighted total is $290+328+40=\$658$, and the final exposure is $5+4+1=10$. Hence

M_{\text{new}}=\frac{658}{10}=\$65.80,\qquad Z_{\text{new}}=\frac{10}{10+5}=\frac{2}{3}.

The premium is

P_{\text{new}}=\frac{658+5(64)}{10+5} =\frac{658+320}{15} =\frac{978}{15} =\$65.20.

Check. The credibility-weighted calculation gives $(2/3)(65.80)+(1/3)(64)=43.8667+21.3333=\$65.20$. The two methods agree. An unweighted average of the four period rates would be inappropriate: the periods have different exposures, so their rates must contribute in proportion to exposure.

Common Mistakes and Interpretation

Taking a simple average of rates. When exposure differs, use $\sum_j w_jX_j/\sum_jw_j$. A rate observed over five exposure units contributes five times the weighted experience of a rate observed over one unit.
Updating the experience mean but not the credibility factor. Both depend on total exposure. Once exposure increases, calculate $Z_{\text{new}}=W_{\text{new}}/(W_{\text{new}}+K)$.
Changing $K$ during a fixed-parameter update. If the exercise says to hold collective parameters fixed, then $K=\text{EPV}/\text{VHM}$ does not change merely because this risk has new observations.
Applying exposure twice. If the supplied observation is already a rate per exposure unit, its contribution to the weighted total is $r x$, not $x$ alone and not $r^2x$.
Rounding intermediate values too early. Keep weighted totals and fractions exact where possible, and round the final monetary premium to the requested precision.

The credibility factor describes the weight assigned to the risk’s own experience, while the experience mean describes the level of that experience. As exposure grows with $K$ fixed, $Z$ increases. The premium’s direction, however, depends on how the added rate compares with the premium before the addition. The update identity makes this distinction explicit: the new rate pulls the old premium toward itself, with the added exposure determining the strength of the pull.

Key takeaway: For an update with fixed collective parameters, add exposure-weighted rates to the old weighted total, add exposures to the old total, and recalculate both the credibility factor and premium. The updated premium can be checked using $P_{\text{new}}=(W_{\text{new}}M_{\text{new}}+Km)/(W_{\text{new}}+K)$.

Check Your Understanding

For each problem, hold the stated collective parameters fixed and retain full precision until the final monetary rounding.

A risk has $W=6$, $M=\$45$, $m=\$50$, and $K=4$. It adds exposure 2 at a rate of $\$65$. Find the updated experience mean, credibility factor, and premium. (Answer: $M_{\text{new}}=\$50.00$, $Z_{\text{new}}=0.6667$, $P_{\text{new}}=\$50.00$.)
A risk has $W=3$, $M=\$120$, $m=\$100$, and $K=5$. It adds exposure 2 at $\$80$, then exposure 1 at $\$140$. Find the final experience mean and premium. (Answer: $M_{\text{new}}=\$110.00$, $P_{\text{new}}=\$105.45$.)
A risk has exposure 4 at an experience mean of $\$30$, with $m=\$36$ and $K=2$. It adds exposure 3 at $\$42$, then exposure 2 at $\$24$. Find the final total exposure, experience mean, credibility factor, and premium. (Answer: $W_{\text{new}}=9$, $M_{\text{new}}=\$32.6667$, $Z_{\text{new}}=0.8182$, $P_{\text{new}}=\$33.27$.)
A risk has exposure 5 and a weighted experience total of $\$450$. The collective mean is $\$75$, and $K=5$. It adds exposure 3 at a rate of $\$60$. Find the new experience mean and premium. (Answer: $M_{\text{new}}=\$78.75$, $P_{\text{new}}=\$77.31$.)
A risk currently has $W=4$, $M=\$90$, $m=\$80$, and $K=6$. What additional exposure is required for the credibility factor to equal $0.8$? If the added exposure has a rate of $\$70$, find the updated experience mean and premium. (Answer: additional exposure $=20$, $M_{\text{new}}=\$73.33$, $P_{\text{new}}=\$74.67$.)

What You'll Learn