Why Balanced-Portfolio Estimation Matters
In Bühlmann credibility, the credibility premium combines a risk’s observed experience with the portfolio mean. The weight assigned to the experience depends on two structural quantities: the expected process variance (EPV), measuring variation within risks, and the variance of hypothetical means (VHM), measuring variation between risks. In applications, these quantities are often unknown and must be estimated from repeated observations.
A balanced portfolio provides a direct way to separate the two sources of variation: every risk has the same number of observations, with equal exposure in each period. This tutorial develops the estimates for that setting, applies the finite-sample correction to VHM, and shows how to proceed from the estimates to risk-specific premiums. The methods are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.
The Structural Model and Estimators
Assume the risks are drawn from the same population and are independent of one another. Conditional on a risk’s risk parameter, its observations are independent across periods and have a constant conditional mean and variance. Define the hypothetical mean for a randomly selected risk as \(m(\Theta)=E[X\mid\Theta]\). The structural parameters are the population averages of the within-risk variance and the between-risk variance:
The balanced-data estimate of EPV pools the variation around each risk’s own mean. The uncorrected sample variance of the risk means is not, by itself, an estimate of VHM: each risk mean still contains sampling variation from its \(r\) observations. The VHM estimate must subtract that contribution.
Here \(s_{\bar X}^{\,2}\) is the sample variance of the \(n\) risk means. When the estimated parameters give a positive \(\widehat{\text{VHM}}\), the plug-in Bühlmann premium for risk \(i\) is
The credibility constant \(K\) is the ratio of within-risk to between-risk variation. A smaller \(K\) gives greater credibility to the risk’s own experience for a fixed number of periods. Because the table is balanced, total exposure for each risk is \(r\) units; the factor is therefore \(r/(r+K)\).
Why the VHM Correction Is Needed
The correction follows from decomposing a risk mean into its hypothetical mean and its sampling error. Let \(\mu_i=E[X_{ij}\mid\Theta_i]\). Conditional independence across periods gives \(\operatorname{Var}(\bar X_i\mid\Theta_i)=\operatorname{Var}(X_{ij}\mid\Theta_i)/r\). Applying the law of total variance,
The sample variance \(s_{\bar X}^{\,2}\) estimates \(\operatorname{Var}(\bar X_i)\), not just the variation of the hypothetical means. Its expected value therefore includes \(\text{EPV}/r\). Subtracting \(\widehat{\text{EPV}}/r\) removes that contribution and gives the usual unbiased structural estimate of VHM under the stated model.
For EPV, each risk’s observations are centered at its own sample mean. The expected sum of squared deviations for one risk is \((r-1)\) times its conditional variance. Averaging over risks gives \(n(r-1)\text{EPV}\); dividing the pooled sum by \(n(r-1)\) estimates the average within-risk variance.
Worked Example: Estimating Structural Parameters From a Balanced Table
A hypothetical portfolio records three years of equal-exposure losses per risk. The observations, in dollars, are shown below. Estimate EPV, VHM, \(K\), and the Bühlmann premium for Risk A.
| Risk | Year 1 | Year 2 | Year 3 | Risk mean |
|---|---|---|---|---|
| A | 8 | 10 | 12 | 10 |
| B | 10 | 12 | 14 | 12 |
| C | 12 | 14 | 16 | 14 |
| D | 14 | 16 | 18 | 16 |
Step 1: Estimate EPV. For each risk, the squared deviations from its mean are \(4,0,4\), so each within-risk sum of squares is \(8\). The pooled sum is \(4(8)=32\), and the denominator is \(n(r-1)=4(2)=8\). Therefore \(\widehat{\text{EPV}}=32/8=4\) dollars squared.
Step 2: Estimate VHM. The portfolio mean of the risk means is \(\bar X=(10+12+14+16)/4=13\). Their squared deviations sum to \(9+1+1+9=20\), so \(s_{\bar X}^{\,2}=20/(4-1)=20/3\). Apply the correction: \(\widehat{\text{VHM}}=20/3-4/3=16/3\).
Step 3: Calculate \(K\) and \(Z\). The estimated constant is \(\widehat K=4/(16/3)=3/4\). With \(r=3\), \(\widehat Z=3/(3+3/4)=3/(15/4)=4/5\).
Step 4: Calculate Risk A’s premium. Its experience mean is \(10\), and the portfolio mean is \(13\). Thus \(\widehat P_A=(4/5)(10)+(1/5)(13)=8+2.6=\$10.60\). The premium is between the risk mean and portfolio mean, as expected for a credibility-weighted average. Method and trap avoided: this is the pooled within-risk and bias-corrected between-risk method; using \(20/3\) as VHM without the correction would mistake sampling variation in the risk means for true between-risk variation.
From the Estimates to a Premium
The steps are best kept in order: calculate each risk mean, pool the within-risk sums of squares, calculate the sample variance of the risk means, correct that variance, and only then calculate \(K\), \(Z\), and the premium. The next example shows why the correction can be material even when the table is small.
Worked Example: A Small Portfolio With Different Within-Risk Variation
A hypothetical service contract portfolio has two years of equal exposure for each of three risks. Its annual claim costs are:
| Risk | Year 1 | Year 2 | Risk mean |
|---|---|---|---|
| A | 10 | 14 | 12 |
| B | 12 | 18 | 15 |
| C | 16 | 20 | 18 |
Step 1: Calculate EPV. The within-risk sums of squares are \( (10-12)^2+(14-12)^2=8\) for A, \( (12-15)^2+(18-15)^2=18\) for B, and \( (16-18)^2+(20-18)^2=8\) for C. The total is \(8+18+8=34\). Since \(n(r-1)=3(1)=3\), \(\widehat{\text{EPV}}=34/3\).
Step 2: Calculate VHM. The grand mean is \((12+15+18)/3=15\). The sample variance of the risk means is \([ (12-15)^2+(15-15)^2+(18-15)^2]/2=18/2=9\). The correction is \(\widehat{\text{EPV}}/r=(34/3)/2=17/3\), so \(\widehat{\text{VHM}}=9-17/3=10/3\).
Step 3: Calculate the premium for Risk A. The constant is \(\widehat K=(34/3)/(10/3)=17/5\). Hence \(\widehat Z=2/(2+17/5)=10/27\). Risk A’s premium is \(\widehat P_A=(10/27)(12)+(17/27)(15)=(120+255)/27=125/9=\$13.89\), rounded to the nearest cent.
This result gives more weight to the portfolio mean than to Risk A’s own mean because the estimated within-risk variation is relatively large. Method and trap avoided: the VHM correction uses \(\widehat{\text{EPV}}/r=(34/3)/2\), not \(\widehat{\text{EPV}}\) itself.
Worked Example: Checking the Correction Before Credibility
A hypothetical balanced portfolio has four risks and three years per risk. Their means are \(30,32,34,36\), and the pooled within-risk sum of squares is \(20\). Find the parameter estimates and the premium for the risk with mean \(32\).
Step 1: EPV. The denominator is \(4(3-1)=8\), so \(\widehat{\text{EPV}}=20/8=5/2\).
Step 2: Sample variance of means and VHM. The grand mean is \((30+32+34+36)/4=33\). The squared deviations total \(9+1+1+9=20\), so \(s_{\bar X}^{\,2}=20/3\). The correction is \((5/2)/3=5/6\). Thus \(\widehat{\text{VHM}}=20/3-5/6=35/6\).
Step 3: Credibility and premium. \(\widehat K=(5/2)/(35/6)=3/7\), and \(\widehat Z=3/(3+3/7)=7/8\). For the risk with mean \(32\), the premium is \((7/8)(32)+(1/8)(33)=224/8+33/8=257/8=\$32.125\), or \(\$32.13\) to the nearest cent.
The portfolio mean is \(33\), so the premium lies between \(32\) and \(33\), as a weighted average should. Method and trap avoided: keeping the fractions through the calculation avoids rounding the estimated parameters before the final premium.
Common Errors and Interpretation
The calculations depend on equal exposure and a common number of observations per risk. They also presume enough data to calculate both within-risk and between-risk sample variation. If risks have unequal exposures, the balanced formulas above are not the appropriate estimation procedure.
- Using the wrong denominator for EPV. The pooled within-risk denominator is \(n(r-1)\), not \(nr-1\). Each risk mean is estimated separately, costing one degree of freedom per risk.
- Forgetting the VHM correction. The sample variance of risk means includes \(\text{EPV}/r\), so use \(s_{\bar X}^{\,2}-\widehat{\text{EPV}}/r\).
- Correcting by the wrong amount. Subtract \(\widehat{\text{EPV}}/r\), not the full EPV and not \(\widehat{\text{EPV}}/(nr)\).
- Using the wrong credibility factor. With balanced exposure \(r\), use \(r/(r+\widehat K)\). Do not substitute the number of risks for the number of periods.
- Rounding too early. Retain fractions or unrounded estimates through the calculation and round the final premium only.
The unconstrained VHM estimate can be zero or negative in a small or weakly differentiated portfolio. In that case, a positive \(K\) and the usual plug-in credibility factor cannot be obtained directly from these estimates. Report the issue rather than silently changing the estimate; any practical adjustment requires a stated convention.
Check Your Understanding
For each problem, use the balanced-portfolio formulas and retain exact values until the final answer.
- A hypothetical portfolio has three risks observed for two equal-exposure years: A has values \(4,8\); B has \(7,9\); C has \(10,14\). Estimate EPV and VHM. (Answer: \(\widehat{\text{EPV}}=6\), \(\widehat{\text{VHM}}=19/3\).)
- Four risks each have two observations: \(9,11\); \(11,13\); \(13,15\); and \(15,17\). Find the estimated EPV and VHM. (Answer: \(\widehat{\text{EPV}}=2\), \(\widehat{\text{VHM}}=17/3\).)
- A balanced portfolio has \(r=4\), estimated EPV \(12\), estimated VHM \(8\), a risk mean of \(50\), and portfolio mean \(46\). Find \(K\), \(Z\), and that risk’s premium. (Answer: \(K=1.5\), \(Z=0.7273\), premium \(48.91\), rounded to the nearest cent.)
- For a balanced portfolio with \(r=3\), the estimated EPV is \(6\) and the sample variance of the risk means is \(10\). Find the corrected VHM, \(K\), and \(Z\). (Answer: \(\widehat{\text{VHM}}=8\), \(K=0.75\), \(Z=0.8000\).)
- A balanced portfolio has \(r=2\), estimated EPV \(9\), and estimated VHM \(6\). A risk’s mean is \(40\), while the portfolio mean is \(34\). Calculate the credibility premium. (Answer: \(K=1.5\), \(Z=0.5714\), premium \(37.43\), rounded to the nearest cent.)