1. Introduction
1.1. The classical concept of credibility
The centuryold concept of credibility presumes combining real data R (also referred to in actuarial literature as “current observations”, “subject experience”, or “sample mean”) with a hypothetical source H (“external information”, “collective experience”, “relevant experience”, or “prior mean”) through a compromise estimator
C=ZR+(1−Z)H,
where Z is the credibility factor (LongleyCook 1962; Klugman 1992; Hickman and Heacox 1999; Bühlmann and Gisler 2005; Klugman, Panjer, and Willmot 2012; Herzog 2015). This approach is a standard method of valuation of insurance contracts in a typical situation in which the actual history of an insured is insufficient to assure any reasonable estimation accuracy. In order to estimate the pure premium, a company needs a reliable predictor of the total loss, the amount it will have to pay in order to cover claims due to the occurrence of insured events such as traffic accidents or hospitalizations.
The total loss depends on the frequency and severity of insured events that are estimated from the past experience of the insured. When this experience is too short to produce a sufficient number of claims, the company supplements it with a prior distribution that is often based on the collective history of a risk group and other plausible sources. Similar situations often arise in business and finance, when new contracts are valuated in the presence of insufficient data or in a rapidly changing environment.
Credibility theory determines the minimal conditions under which a prediction for a cohort is fully credible. Full credibility (i.e. a credibility factor Z=1) is assigned in the case of a compromise estimator, C, that is computed from the actual data only. When full credibility is not possible, the compromise estimator is based on both the actual data and the (hypothetical) prior distribution information, and partial credibility is applied, with a credibility factor Z<1. Credibility factor Z thus plays a vital role in credibility estimation. It determines the portions of the compromise estimator, C, attributed to the real and to the hypothetical data.
The limited fluctuation credibility approach is a standard way of determining allowed values of Z. It requires that in order to deserve a credibility factor Z, the real data must satisfy the following condition,
P{ZR−E(R)>cE(R)}≤α,
for a given probability α and desired relative precision c.
Under the classical frequencyseverity model, when the total loss R=X=∑Ni=1Yi consists of a Poisson(λ) number of individual losses Yi, condition (1.2) is satisfied by a sufficiently large frequency of insured events λ. It follows from rather simple probability arguments that the frequency λ satisfying the inequality
λ≥λF=(zα/2c)2(1+γ2)
suffices for the full credibility, where γ=σ/θ is a coefficient of variation of losses Yi, θ=E(Yi) is the mean loss, σ is the standard deviation of losses, and zα/2 is the upper α/2quantile of the standard normal distribution. It is assumed here that λF is sufficiently large to allow normal approximation of X (Herzog 2015, chapter 5; Klugman, Panjer, and Willmot 2012, chapter 17).
When λ<λF, the partial credibility condition (1.2) is satisfied with frequency
λ≥λP=Z2λF.
Either full or partial credibility can always be assigned under these conditions because the hypothetical estimate H, equal to the prior mean μ with probability 1, is assumed without any error or uncertainty. When the history of real data is short, (1.2) always holds with a zero or very small credibility factor. One can always compensate for the lack of information contained in real data by using “infinite information” contained in the prior distribution because it is assumed to be known completely, without any error or any uncertainty.
1.2. Two pointed limitations of the classical concept
Limitation 1: Assumption of errorfree prior. The classical limitedfluctuation credibility approach accounts for uncertainty in the real data only, assuming no uncertainty in the hypothetical prior data – that is, considering the prior data as fully credible. Tindall and Mast (2003) and Atkinson (2019) pointed out that this assumption is misleading and unrealistic in practical settings.
In actuarial practice, it is unrealistic to expect to know prior parameters exactly, without any error or uncertainty. More typically, they are estimated from past experience, such as the frequency and severity of claims in a given risk group. As Tindall and Mast (2003) noted, in practice, the amount of confidence in the prior mean (called the “current expectations” in the Tindall and Mast paper) “drives the extent to which the actuary relies on credibility theory” (2003, 47). If the prior mean was obtained as “purely a guess”, then the actuary will rely more heavily on the current real data (i.e., what the authors call the “emerging experience”). “Another potential flaw is that the Prior Rate may be less than fully credible”, according to Atkinson (2019, 9). “Since the Limited Fluctuation method assumes that the Prior Rate has full credibility, it may receive more weight than it deserves.”
In this paper, we introduce uncertainty in the prior distribution parameters and express it in terms of a nonzero variance of the corresponding hyperprior (i.e., the secondlevel prior), which is the prior distribution of the prior mean of the loss model. We then revisit the limitedfluctuation condition under this generalized model, deriving corrected conditions for full and partial credibility. Several approaches are proposed; each of them provides a compromise estimator with a credibility factor determined by the relative uncertainty of real and hypothetical data. The standard criteria (1.3) and (1.4) for full and partial credibility both appear as a special case of a fully credible prior, when the hyperprior variance is 0.
A new third scenario then emerges, in addition to full and partial credibility. In cases in which the degree of uncertainty is too high in both the real and the hypothetical data, even a partial credibility sometimes may not be possible. For example, this situation takes place when a totally new type of events is insured, with insufficient data as well as little or no past experience.
Limitation 2: Agreement and homogeneity. Tindall and Mast (2003) also mentioned another limitation of the standard credibility practice. They emphasize a typical heterogeneity within the insured groups. Credibility depends on the degree of agreement between the actual data and the prior distribution. “The closer the data lean toward homogeneity and comparability with current expectations, the higher the credibility is likely to be” (Tindall and Mast 2003, 47). At the same time, the authors noted that “experience, however, rarely matches contemporary expectations” (47).
Actuarial Standards of Practice No. 25 requires that “in carrying out credibility procedures, the actuary should consider the homogeneity of both the subject experience and the relevant experience. Within each set of experience, there may be segments that are not representative of the experience set as a whole” (Actuarial Standards Board 2011, 4). Indeed, as long as variability exists within a risk group, it is possible for a given insured to have distribution parameters different from those of the relevant risk group. For example, a driver with the same driving record, type of a vehicle, and family structure as the rest of the risk group may still be on the safer side or on the riskier side of the group. As a result, E(X) for the given insured is no longer equal to the hyperprior mean ν=E(μ). That is, the real data and the hypothetical data may have different means.
We address this issue in Section 2 by accounting for heterogeneity – different distributions of real data within each group. In Section 2, we assume “typical” or “random” members of risk groups, for whom the prior mean agrees with the actual expectation. An example of a perfect agreement is shown in Figure 1, lefthand side. Then, in Section 3, we consider insureds with some deviation from the prior mean, as depicted in Figure 1, righthand side. Since the prior mean is no longer assumed to be fully known with no error, such deviations occur inevitably. The impact of a disagreement between the real data and the prior mean is apparent. In the extreme case, when discordance between the current experience and expectations is too high, there may be no room even for a partial credibility. The last two sections of the paper contain an illustration of the proposed methods with specific scenarios, followed by a summary and conclusions.
2. Limited Fluctuation under Uncertainty in Hypothetical Data
Actuarial Standards of Practice No. 25 state, “The actuary should use an appropriate credibility procedure when determining if the subject experience has full credibility or when blending the subject experience with the relevant experience. The procedure selected or developed may be different for different practice areas and applications” (Actuarial Standards Board 2011, 3). We propose three limitedfluctuation methods that account for uncertainty in hypothetical data, and then we discuss the situations appropriate for each method.
To quantify this uncertainty, we assume that the prior mean, μ, of the total loss, X, is no longer deterministic. Rather, it has an approximately normal distribution, with mean ν and variance τ2. Sample mean and sample variance of losses within the risk group are often used to generate an estimate of μ, in which case τ2 is inversely proportional to the group size. To summarize, we have
Total loss:X=∑Ni=1Yi≈Normal,Frequency of claims:N∼Poisson(λ),Severity parameters:θ=E(Yi),σ2=Var(Yi),γ=σ/θ,Prior distribution:μ∼Normal(ν,τ2).
Under this general setting, the distribution of the total loss, X, has mean and variance expressed as:
E(X)=EE{∑N1YiN}=E(Nθ)=λθ,Var(X)=EVar{∑N1YiN}+VarE{∑N1YiN}=E(Nσ2)+Var(Nθ)=λσ2+λθ2.
Parameters λ, θ, and σ are generally unknown. Thus, the chosen prior distribution and the H component of the compromise estimator (1.1) may agree or disagree with the actual distribution of X.
By agreement, we understand that the chosen prior distribution of μ has a mean, ν, that actually equals the mean of X. Notice that this does not always have to be the case. On one hand, actuaries have the real data. On the other hand, they choose a prior distribution, or they obtain it from some data source such as based on the past experience of a relevant risk group.
In case of an agreement, E(X) =λθ =EE{Xμ} =E(μ) =ν. This can be understood as an unbiased choice of the prior distribution. For example, suppose the prior distribution is generated by the relevant risk group, and the given insured is a typical member of the associated risk group, in which case the expected total loss is E(X)=ν. This is justified when an insured is selected from a group at random. Then the expected total loss agrees with the expectation across the risk group, as on Figure 1, left.
Consider a general situation in which n years (or insured periods) of data are available, X1,…,Xn. The real component of the compromise estimator will then consist of the sample mean R=ˉX.
We now have two estimators of the expected total loss E(X) that complement each other  the sample mean ˉX and the prior mean μ. Each of them can be used to estimate E(X), but it is most efficient to combine them, in accordance with the credibility and Bayesian principles. Combining them results in the compromise estimator C.
In this respect, we introduce three limitedfluctuation credibility conditions that differ in their interpretation of precision while estimating the expected total loss. These conditions restrict deviations of estimates from the expected total loss in three different ways, forcing the sample mean and the prior mean to be close to E(X) separately, as in (2.3), or jointly, as in (2.5), or having the compromise estimator, C, close to E(X), as in (2.6).
In this section, we assume that a given insured is a typical member of the associated risk group, in which case the expected total loss is E(X)=ν. This assumption is justified, for example, when an insured is selected from a group at random. Then the expected total loss agrees with the expectation across the risk group.
2.1. Method 1: Fluctuations of real and hypothetical data
The standard form of the compromise estimator C=ZR+(1−Z)H contains two credibility factors. Factor Z shows the credibility of the real data, whereas (1−Z) corresponds to the credibility of the prior. Uncertainty in both R and H requires symmetric conditions on coefficients Z and (1−Z).
Thus, the classical condition (1.2) is now replaced by a combination of two conditions
{pR=P{ZˉX−E(X)>cE(X)}≤αRpH=P{(1−Z)μ−E(X)>kE(X)}≤αH
for chosen probabilities αR and αH and relative precision factors c and k. The first inequality in (2.3) is the classical limitedfluctuation condition on the permissible deviation of real data, X, from their expected value. The second inequality represents a similar condition on the hypothetical data.
Let us look at the probabilities in (2.3) that are depicted in Figure 2, with α=αR=αH=0.05, c=k=0.05, θ=200, σ=100, n=10, and different values of λ. Clearly, pR increases with Z, whereas pH decreases. Regions where both curves appear below the threshold αR=αH=0.05 contain the allowed credibility factors Z. For different frequency parameters λ, we notice three distinct cases.

Full credibility. When the expected frequency, λ, is sufficiently large, the solution set of (2.3) includes Z=1. The data are fully credible, and the pure premium can be estimated solely from the real experience (Figure 2, left).

Partial credibility. With a lower λ, solutions of system (2.3) form an interval, [Z1,Z2], with 0≤Z1≤Z2<1. Any credibility factor from this interval can be chosen. In practice, underwriters will often choose Z2. The interval does not contain Z=1, and therefore, the observed data are not fully credible (Figure 2, middle). Partial credibility applies in this case.

No credibility. Reducing λ even further, we enter the situation in which there are no solutions to (2.3) in 0≤Z≤1. This happens when the variability of both the real and the prior data is so high that no Zfactor can satisfy both limitedfluctuation conditions simultaneously.
In Figure 2 (right), the intersection point of the two curves appears above αR(=αH), and thus, conditions (2.3) are not satisfied together for any Z.
This case will never appear if the prior mean is (unrealistically) assumed to be exactly known. Such a 100% reliable estimate can always satisfy the limitedfluctuation condition. On the other hand, if both R and H are unreliable, then no compromise combination of them can “magically” become reliable!
2.1.1. Analytic solution
To solve inequalities (2.3) analytically, we use the mean and variance of losses derived in (2.2) based on the assumptions (2.1). Then, inequalities in (2.3) are equivalent to
cE(X)ZStd(ˉX)≥zαR/2 and kE(X)(1−Z)Std(μ)≥zαH/2.
Solving them for Z and using (2.2) and assumed agreement ν=E(X), we obtain a solution in the form of two inequalities,
{Z≤cE(X)zαR/2Std(ˉX)=cλθzαR/2√λ(θ2+σ2)/n=c√λnzαR/2√1+γ2Z≥1−kE(X)zαH/2Std(μ)=1−kνzαH/2τ
The interval of solutions
Z∈[Z1,Z2]={[1−kνzαH/2τ,c√λnzαR/2√1+γ2]if1−kνzαH/2τ≤c√λnzαR/2√1+γ2∅otherwise
represents all credibility factors that can be assigned.
Similarly to the classical credibility theory, a higher expected frequency λ of insured events makes the real data more credible by increasing the upper bound of (2.4). As seen in Figure 2, increasing λ moves an insured from the “no credibility” category into a “partial credibility” and further, into a “full credibility”. When the interval of (2.4) contains Z=1, full credibility can be assigned.
When the hypothetical mean μ is (unrealistically) considered fully credible, it corresponds to τ2=Var(μ)=0. In this case, the lower bound of (2.4) becomes −∞, converting (2.4) to the classical solutions for full and partial credibility (Klugman, Panjer, and Willmot 2012; Herzog 2015).
We can also see that the solution in (2.4) exists when the expected frequency of claims λ is high, reflecting informative real data, or when variance τ2 is low, reflecting informative hypothetical data. When both sources R and H lack information, the interval is empty, and there no solution.
2.2. Method 2: The jointly limited fluctuation
Here we require that the real and prior estimates attain the desired relative precision simultaneously with a high probability (1−α2). In other words,
p2=P{ZˉX−E(X)>cE(X)∪ (1−Z)μ−E(X)>kE(X)}≤α2.
Assuming independence of the real data and the prior parameters (such as independence of the given insured from all the other members of the risk group), condition (2.5) results in the inequality
p2=1−(1−pR)(1−pH)=1−{1−2Φ(−c√λnZ√1+γ2)}⋅{1−2Φ(−kν(1−Z)τ)}≤α2,
where Φ denotes the standard normal distribution function. Notice that probability p2 accounts for uncertainty in the present experience R as well as the hypothetical mean