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Exam Workshop: Estimating Parameters and Premiums From a Table

Learn to turn balanced risk-level data into structural parameter estimates, credibility weights, and premiums, while checking when estimates require a practical constraint.

Intermediate 11 min read

What You'll Learn

  • Calculate the pooled within-risk estimate of EPV from raw observations.
  • Estimate VHM by correcting the sample variance of risk means for within-risk variation.
  • Calculate K, credibility weights, and premiums for individual risks.
  • Apply a nonnegative VHM constraint and interpret its effect on credibility.
  • Check structural estimates and premiums for common denominator and weighting errors.

From a Table of Observations to Credibility Premiums

A credibility problem may provide observations by risk and period, rather than giving the structural parameters directly. The calculation then has several linked stages: summarize each risk’s experience, estimate the within-risk and between-risk variation, and use those estimates to blend each risk’s observed mean with the portfolio mean. A mistake in an early denominator can affect every premium that follows.

This tutorial treats a balanced portfolio: each of \(N\) risks has the same number \(n\) of observations, and all observations represent the same exposure unit. The target is the expected outcome for a future exposure unit of a risk. Unequal exposure amounts require exposure-weighted methods and are not covered here.

Definition: In the Bühlmann model, let \(\Theta_i\) represent the unobserved risk characteristic of risk \(i\), and let \(X_{ij}\) be its observation in period \(j\). Conditional on \(\Theta_i\), observations for a risk have mean \(m(\Theta_i)=E[X_{ij}\mid\Theta_i]\) and variance \(s^2(\Theta_i)=\operatorname{Var}(X_{ij}\mid\Theta_i)\). The EPV is \(v=E[s^2(\Theta)]\), the expected process variance within a risk. The VHM is \(a=\operatorname{Var}(m(\Theta))\), the variance of hypothetical means across risks.

For a balanced table, define the risk mean \(\bar X_i\) and the portfolio mean \(\bar X\). Under the usual model assumptions—risks are drawn from the same population, observations are conditionally independent with a common conditional mean and variance within each risk, and the relevant moments are finite—the standard moment estimates are:

$$ \bar X_i=\frac{1}{n}\sum_{j=1}^{n}X_{ij}, \qquad \bar X=\frac{1}{N}\sum_{i=1}^{N}\bar X_i, $$
$$ \hat v= \frac{\displaystyle\sum_{i=1}^{N}\sum_{j=1}^{n}(X_{ij}-\bar X_i)^2} {N(n-1)}, \qquad \hat a= \frac{\displaystyle\sum_{i=1}^{N}(\bar X_i-\bar X)^2} {N-1} -\frac{\hat v}{n}. $$

The first estimate pools each risk’s deviations from its own mean. The second starts with the sample variance of the risk means and subtracts the part of that variation attributable to observing only \(n\) periods per risk. The fitted ratio and the credibility premium for risk \(i\) are:

$$ \hat K=\frac{\hat v}{\hat a}, \qquad \hat Z=\frac{n}{n+\hat K}, \qquad \hat P_i=\hat Z\bar X_i+(1-\hat Z)\bar X. $$

These are plug-in estimates: the structural parameters are estimated from the table and then used in the Bühlmann credibility formula. If a procedure constrains \(\hat a\) to be nonnegative and the unconstrained estimate is negative, it replaces it with zero. In that case, the usual convention is \(\hat K=\infty\), \(\hat Z=0\), and the premium is the portfolio mean. A negative unconstrained estimate is not evidence of negative true VHM; it signals that this sample does not support positive estimated between-risk variation under this procedure.

Why the VHM Estimate Has a Correction

The risk means do not equal the risks’ underlying hypothetical means exactly. Each is an average of \(n\) observations, so it contains both true variation between risks and sampling variation within a risk. The correction in \(\hat a\) removes the latter.

Justification: Write \(\bar X_i=m(\Theta_i)+\bar\varepsilon_i\), where \(E[\bar\varepsilon_i\mid\Theta_i]=0\) and \(\operatorname{Var}(\bar\varepsilon_i\mid\Theta_i)=s^2(\Theta_i)/n\). Then \(\operatorname{Var}(\bar X_i)=a+v/n\): the covariance between \(m(\Theta_i)\) and \(\bar\varepsilon_i\) is zero by the conditional mean-zero property, while \(E[\operatorname{Var}(\bar X_i\mid\Theta_i)]=v/n\). Thus the sample variance of the risk means estimates \(a+v/n\); subtracting \(\hat v/n\) estimates the VHM.

Similarly, within each risk the deviations \(X_{ij}-\bar X_i\) measure conditional process variation. There are \(n-1\) residual degrees of freedom for each risk because that risk’s mean is estimated from its \(n\) observations. Across \(N\) risks, the pooled denominator is therefore \(N(n-1)\), not \(Nn\), \(N-1\), or the number of observations in the entire table.

Worked Calculations

Worked Example: Estimate the Parameters and Price One Risk

A hypothetical balanced portfolio has three risks, each observed for two periods. The observations are A: 6 and 8; B: 10 and 10; C: 12 and 14. Estimate EPV and VHM, calculate \(K\) and the credibility weight, and find risk C’s premium.

Method. Calculate the within-risk sum of squares and the sample variance of the three risk means, then apply the balanced Bühlmann formulas.

Risk means and portfolio mean. The means are \(\bar X_A=7\), \(\bar X_B=10\), and \(\bar X_C=13\). Therefore \(\bar X=(7+10+13)/3=10\).

EPV estimate. For A, the deviations from 7 are \(-1,1\), giving a squared-deviation sum of 2. For B, both deviations are 0, giving 0. For C, the deviations from 13 are \(-1,1\), giving 2. Thus the pooled within-risk sum is \(2+0+2=4\), and

$$ \hat v=\frac{4}{3(2-1)}=\frac{4}{3}=1.3333. $$

VHM estimate. The squared deviations of the risk means from 10 are \(9,0,9\). Their sample variance is \((9+0+9)/(3-1)=9\). The sampling-variation correction is \(\hat v/n=(4/3)/2=2/3\). Hence \(\hat a=9-2/3=25/3\).

Credibility and premium. The ratio is \(\hat K=(4/3)/(25/3)=4/25=0.16\). With \(n=2\), \(\hat Z=2/(2+0.16)=25/27\). Risk C’s premium is

$$ \hat P_C=\frac{25}{27}(13)+\frac{2}{27}(10) =\frac{345}{27}=\frac{115}{9} \approx 12.78. $$

The premium lies between risk C’s mean, 13, and the portfolio mean, 10, as a credibility premium should. The reported premium is rounded to two decimal places. The common mistake avoided here is dividing the within-risk sum of squares by the number of observations rather than by \(N(n-1)\).

Worked Example: A Negative Unconstrained VHM Estimate

A hypothetical portfolio has three risks, each observed twice: A: 4 and 10; B: 5 and 11; C: 6 and 12. Calculate the unconstrained parameter estimates and, using a nonnegative VHM constraint, determine the credibility weight and premiums.

Risk means. They are 7, 8, and 9, so the portfolio mean is \(\bar X=(7+8+9)/3=8\).

Unconstrained estimates. Each risk has deviations \(-3,3\) from its own mean, so its within-risk squared-deviation sum is \(9+9=18\). The pooled sum is \(54\), and

$$ \hat v=\frac{54}{3(2-1)}=18. $$

The sample variance of the risk means is \((1+0+1)/(3-1)=1\). The correction is \(\hat v/n=18/2=9\), giving \(\hat a=1-9=-8\).

Apply the stated constraint. Set the estimated VHM to zero. Under the nonnegative-constraint convention, \(\hat K=\infty\) and \(\hat Z=0\). Each premium is therefore \(\hat P_i=0\bar X_i+1\bar X=8\).

The result is consistent with the constrained method: with no positive estimated between-risk variation, the experience means receive no credibility weight. Do not take the absolute value of \(-8\), or continue with a negative \(K\); neither operation is the nonnegative VHM procedure.

Worked Example: Calculate Premiums for Every Risk

A hypothetical portfolio has four risks, each observed for three periods: A: 4, 5, 6; B: 6, 7, 8; C: 8, 9, 10; D: 10, 11, 12. Estimate the structural parameters and calculate the premiums for all four risks.

Risk means. The means are 5, 7, 9, and 11. The portfolio mean is \(\bar X=(5+7+9+11)/4=8\).

EPV. Each risk has deviations \(-1,0,1\), so each within-risk sum of squares is \(1+0+1=2\). The total is 8, and

$$ \hat v=\frac{8}{4(3-1)}=1. $$

VHM. The squared deviations of the means from 8 are \(9,1,1,9\), summing to 20. Their sample variance is \(20/(4-1)=20/3\). The correction is \(\hat v/n=1/3\), so \(\hat a=20/3-1/3=19/3\).

Credibility factor. The ratio is \(\hat K=1/(19/3)=3/19\), and

$$ \hat Z=\frac{3}{3+3/19}=\frac{19}{20}=0.95. $$

Premiums. Each risk’s premium is \(0.95\) times its own mean plus \(0.05\) times 8. Thus A’s premium is \(0.95(5)+0.05(8)=5.15\); B’s is \(0.95(7)+0.05(8)=7.05\); C’s is \(0.95(9)+0.05(8)=8.95\); and D’s is \(0.95(11)+0.05(8)=10.85\). Each premium is between the corresponding risk mean and the portfolio mean. Values are shown to two decimal places.

Common Exam Traps and Checks

The calculation has several opportunities for a plausible-looking error. Keep the order of operations visible: first obtain the risk means, then pool within-risk variation, then correct the variance of risk means, and only then calculate \(K\), \(Z\), and premiums.

  • Using the wrong EPV denominator. The pooled within-risk degrees of freedom are \(N(n-1)\). For three risks observed twice, that is \(3(2-1)=3\), not 6.
  • Forgetting the VHM correction. The sample variance of risk means estimates \(a+v/n\), not \(a\) alone. Subtract \(\hat v/n\).
  • Using the wrong number of observations in the correction. In a balanced portfolio, use the common number \(n\) of observations per risk, not \(Nn\).
  • Reversing the credibility blend. The risk mean receives weight \(Z\); the portfolio mean receives weight \(1-Z\). The result must be between those two means when \(0\leq Z\leq1\).
  • Proceeding mechanically with a negative VHM. If the specified approach constrains VHM to be nonnegative, set it to zero and use the corresponding zero credibility weight; do not calculate a negative credibility factor.

A useful final check is to verify that the within-risk denominator matches the table and that premiums preserve the blend. When \(a\) is positive, \(K=v/a\) and \(Z=n/(n+K)\) should give a weight between zero and one. If the estimated VHM is large relative to EPV, the risk means receive more weight; if estimated within-risk variation dominates, they receive less.

Check Your Understanding

For each balanced portfolio, calculate the requested quantities before checking the final answer. Unless stated otherwise, use the unconstrained moment estimates when the estimated VHM is positive, and apply a zero VHM constraint if it is negative.

  1. A portfolio has three risks observed twice: A: 2 and 4; B: 4 and 6; C: 6 and 8. Find \(\hat v\), \(\hat a\), \(\hat K\), \(Z\), and risk C’s premium. (Answer: \(\hat v=2,\ \hat a=3,\ \hat K=2/3,\ Z=0.75\), premium \(6.50\).)
  2. Three risks are observed twice: A: 5 and 9; B: 6 and 10; C: 7 and 11. Find the unconstrained estimates, then apply the nonnegative VHM constraint and state each premium. (Answer: \(\hat v=8,\ \hat a=-3\) unconstrained; constrained \(\hat a=0,\ Z=0\), and each premium is \(8\).)
  3. Four risks are observed twice, with risk means 8, 10, 12, and 14. The within-risk pooled sum of squares is 18. Find the EPV estimate, the sample variance of risk means, and the unconstrained VHM estimate. (Answer: \(\hat v=18/[4(2-1)]=4.5\); sample variance of means \(=20/3\); \(\hat a=20/3-4.5/2=59/12\approx4.9167\).)
  4. A balanced portfolio has \(n=4\), \(\hat v=3\), and \(\hat a=6\). A risk’s observed mean is 109 and the portfolio mean is 100. Calculate \(K\), \(Z\), and its credibility premium. (Answer: \(K=0.5,\ Z=8/9\), premium \(=108\).)
  5. For a balanced portfolio with \(N=5\) risks and \(n=3\) observations per risk, the total within-risk squared-deviation sum is 40. The sample variance of risk means is 6. Find \(\hat v\) and \(\hat a\), and explain whether the nonnegative constraint applies. (Answer: \(\hat v=40/[5(3-1)]=4,\ \hat a=6-4/3=14/3\); the VHM estimate is positive, so the constraint does not apply.)

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