Balanced-Portfolio ANOVA for K

Why Use a Balanced-Portfolio ANOVA?

The Bühlmann parameter \(K\) compares process variation within a risk with variation in expected outcomes between risks. A balanced portfolio provides a direct way to estimate both components from repeated observations: the variation around each risk’s own average informs EPV, while the variation among risk averages informs VHM. The ANOVA calculation is useful because it keeps these two sources of variation distinct before forming their ratio.

Here, “balanced” means that the same number of observations is available for every risk. The observations could be annual claim costs for a hypothetical group of policyholders, provided the units have the same meaning and exposure across risks and periods. The estimates below are sample-based estimates of the Bühlmann components; the ratio of those estimates is a plug-in estimate of \(K\), not generally an exactly unbiased estimator of \(K\).

Definition: Let \(X_{ij}\) be the observation for risk \(i\) in period \(j\), with \(i=1,\ldots,r\) and \(j=1,\ldots,n\). There are \(r\) risks and \(n\) equally weighted observations per risk. Let \(\mu_i=E[X_{ij}\mid\text{risk }i]\). The expected process variance is \(\operatorname{EPV}=E[\operatorname{Var}(X_{ij}\mid\text{risk }i)]\), and the variance of hypothetical means is \(\operatorname{VHM}=\operatorname{Var}(\mu_i)\). Under the Bühlmann model, \(K=\operatorname{EPV}/\operatorname{VHM}\) when \(\operatorname{VHM}>0\).

Balanced ANOVA Quantities

Write \(\bar X_{i\cdot}\) for the average of the \(n\) observations for risk \(i\), and \(\bar X_{\cdot\cdot}\) for the average of all \(rn\) observations. The within-risk sum of squares measures fluctuation around each risk’s own average. The between-risk sum of squares measures fluctuation among the risk averages, scaled by the number of observations per risk.

\bar X_{i\cdot}=\frac{1}{n}\sum_{j=1}^{n}X_{ij}, \qquad \bar X_{\cdot\cdot}=\frac{1}{r}\sum_{i=1}^{r}\bar X_{i\cdot}

SS_W=\sum_{i=1}^{r}\sum_{j=1}^{n}(X_{ij}-\bar X_{i\cdot})^2, \qquad SS_B=n\sum_{i=1}^{r}(\bar X_{i\cdot}-\bar X_{\cdot\cdot})^2

The corresponding degrees of freedom are \(r(n-1)\) within risks and \(r-1\) between risks. Dividing each sum of squares by its degrees of freedom gives the within and between mean squares. Under the assumptions below, these mean squares estimate the components through their expected values.

MS_W=\frac{SS_W}{r(n-1)}, \qquad MS_B=\frac{SS_B}{r-1}

Result: For a balanced portfolio with \(r>1\) risks and \(n>1\) observations per risk, the ANOVA component estimates are \(\widehat{\operatorname{EPV}}=MS_W\) and \(\widehat{\operatorname{VHM}}=(MS_B-MS_W)/n\). When the estimated VHM is positive, the plug-in estimate is \(\hat K=MS_W/\widehat{\operatorname{VHM}}=nMS_W/(MS_B-MS_W)\).

Why the Mean-Square Formulas Work

The calculation relies on risks being independent draws from a common population of risks, and on observations being conditionally independent and identically distributed across periods for a given risk. Conditional means and variances may differ from one risk to another. Their population averages define EPV and VHM. No normality assumption is needed for the expected-mean-square identities, though normality is often used for classical ANOVA tests and confidence procedures.

For the within-risk sum of squares, subtracting a risk’s sample average removes one degree of freedom. Conditional on that risk, the expected sum of squared deviations from its sample average is \((n-1)\operatorname{Var}(X_{ij}\mid\text{risk }i)\). Averaging over the risk population and adding across \(r\) risks gives the first identity below.

For the between-risk mean square, each risk average has variance \(\operatorname{VHM}+\operatorname{EPV}/n\): the risk’s conditional mean varies across risks, and averaging \(n\) conditionally independent observations contributes conditional process variance divided by \(n\). The expected sample variance of the \(r\) risk averages is therefore that same quantity. Since \(SS_B\) is \(n\) times the sum of squared deviations of those averages, its mean square has the second expectation below.

E[MS_W]=\operatorname{EPV}, \qquad E[MS_B]=n\operatorname{VHM}+\operatorname{EPV}

Solving the second identity for VHM gives \((E[MS_B]-E[MS_W])/n\). Replacing the expected mean squares by their observed values yields the ANOVA estimates. This explains both why the within mean square estimates EPV and why the within component must be subtracted from the between mean square before dividing by \(n\).

Calculation Process

Check balance and identify the unit.
Confirm there are \(r\) risks, \(n\) comparable observations per risk, and a consistent observation unit for the resulting \(K\).

Calculate risk averages and the grand average.
Find each \(\bar X_{i\cdot}\), then average those risk averages. Because the table is balanced, this is also the average of all \(rn\) observations.

Calculate both sums of squares and mean squares.
Use deviations from risk averages for \(SS_W\), and deviations of risk averages from the grand average for \(SS_B\). Apply the correct degrees of freedom.

Estimate EPV, VHM, and K.
Set EPV to \(MS_W\), subtract \(MS_W\) from \(MS_B\) and divide by \(n\) for VHM, then form the ratio if the estimated VHM is positive.

Worked Calculations

Worked Example: Four Risks Observed for Three Periods

A hypothetical insurer studies equal annual claim-cost observations for four risks over three periods. The observations, in dollars, are shown below. Estimate EPV, VHM, and \(K\). Do not round intermediate calculations.

Risk	Period 1	Period 2	Period 3	Risk average
A	8	10	12	10
B	12	14	16	14
C	16	18	20	18
D	20	22	24	22

Method. Apply balanced one-way random-effects ANOVA. The risk averages are \(10,14,18,22\), so the grand average is \((10+14+18+22)/4=16\). For each risk, the within-risk deviations are \(-2,0,2\), whose squared deviations sum to \(4+0+4=8\). Thus \(SS_W=4(8)=32\), and its degrees of freedom are \(4(3-1)=8\). Therefore \(MS_W=32/8=4\), giving \(\widehat{\operatorname{EPV}}=4\).

For the between-risk calculation, the risk-average deviations from 16 are \(-6,-2,2,6\), with squared deviations \(36,4,4,36\), totaling 80. Thus \(SS_B=3(80)=240\). Its degrees of freedom are \(4-1=3\), so \(MS_B=240/3=80\). The VHM estimate is \((80-4)/3=76/3=25.3333\). The plug-in \(K\) estimate is \(4/(76/3)=12/76=3/19=0.1579\), rounded to four decimal places.

Check. \(3(25.3333)+4\) is approximately 80, matching \(MS_B\) apart from displayed rounding. The calculation uses annual observations, so this \(K\) is expressed in annual-observation units. The common mistake to avoid is dividing \(MS_B-MS_W\) by the number of risks; the component formula divides by the number of periods, \(n=3\).

Worked Example: A Two-Period Portfolio

For a hypothetical portfolio, three risks have two equal-exposure observations each: Risk A has 8 and 12, Risk B has 13 and 15, and Risk C has 18 and 22. Estimate the variance components and \(K\).

Method. There are \(r=3\) risks and \(n=2\) periods. The risk averages are \(10,14,20\), and the grand average is \((10+14+20)/3=44/3\). Within each risk the two observations differ from their risk average by \(-2,2\) for A, \(-1,1\) for B, and \(-2,2\) for C. The within sums are \(8,2,8\), so \(SS_W=18\). The within degrees of freedom are \(3(2-1)=3\), giving \(MS_W=18/3=6\). Hence \(\widehat{\operatorname{EPV}}=6\).

The deviations of the risk averages from \(44/3\) are \(-14/3,-2/3,16/3\). Their squared deviations sum to \((196+4+256)/9=456/9=152/3\). Therefore \(SS_B=2(152/3)=304/3\), and \(MS_B=(304/3)/(3-1)=152/3\). The VHM estimate is \((152/3-6)/2=(134/3)/2=67/3=22.3333\).

Finally, \(\hat K=6/(67/3)=18/67=0.2687\), rounded to four decimal places. As a check, \(2(67/3)+6=152/3\), the calculated between mean square. This example also shows why the grand average need not be an integer: using rounded risk averages would distort the sums of squares.

Worked Example: Distinguishing the Between Mean Square from VHM

A second hypothetical table has four risks with two observations each: A has 6 and 10, B has 10 and 14, C has 14 and 18, and D has 18 and 22. Estimate EPV, VHM, and \(K\), and identify the tempting incorrect calculation.

Method. The risk averages are \(8,12,16,20\), and the grand average is \(14\). Each risk has within deviations \(-2,2\), contributing \(4+4=8\), so \(SS_W=4(8)=32\). The within degrees of freedom are \(4(2-1)=4\), giving \(MS_W=32/4=8\) and \(\widehat{\operatorname{EPV}}=8\).

The risk-average deviations are \(-6,-2,2,6\), whose squares total \(36+4+4+36=80\). Thus \(SS_B=2(80)=160\), with \(4-1=3\) degrees of freedom. This gives \(MS_B=160/3=53.3333\). The VHM estimate is \((160/3-8)/2=(136/3)/2=68/3=22.6667\). Therefore \(\hat K=8/(68/3)=24/68=6/17=0.3529\), rounded to four decimal places.

Common mistake and correction. A tempting but wrong calculation is to set \(\widehat{\operatorname{VHM}}=MS_B=53.3333\) and then report \(K=8/53.3333=0.1500\). The between mean square includes both process variation and between-risk variation. Correctly subtracting \(MS_W=8\) and dividing by \(n=2\) gives VHM \(22.6667\), not \(53.3333\). Check: \(2(22.6667)+8\) is approximately \(53.3333\).

Assumptions, Interpretation, and Exam Traps

The formulas require more than a rectangular-looking table. Each risk must have the same number of observations, and those observations must represent comparable units. The standard random-effects setup treats risks as independent draws from a common risk population. Within a risk, observations are conditionally independent with a common conditional mean and variance across periods. If exposure differs materially, or periods have different trend or conditions that have not been adjusted, raw observations may not be comparable; this balanced formula should not be applied without resolving that mismatch.

The estimates are in squared units of the observed quantity. For example, if observations are annual dollars, both estimated components are in dollars squared, while \(K\) is dimensionless as a ratio. Its practical interpretation is tied to the observation unit: changing the period or exposure definition changes the amount of process information represented by one observation and can change the corresponding \(K\).

A frequent exam error is confusing the degrees of freedom. The within denominator is \(r(n-1)\), not \(rn-1\); the between denominator is \(r-1\), not \(n-1\). Another error is omitting the factor \(n\) in \(SS_B\), or subtracting \(MS_W\) from \(MS_B\) but forgetting to divide by \(n\) when estimating VHM. Keeping the expected-mean-square identity visible is a useful safeguard:

MS_B\text{ estimates }\operatorname{EPV}+n\operatorname{VHM}, \quad\text{not VHM alone.}

Sampling variation can produce \(MS_B\leq MS_W\), which makes the raw VHM estimate zero or negative. A negative estimate is not a meaningful population variance; it signals that the observed between-risk mean square did not exceed the within-risk mean square. Report the raw calculation accurately and distinguish it from any separate boundary or adjustment convention. If the estimated VHM is zero, the ratio defining the plug-in \(K\) has a zero denominator and is not a finite estimate.

Key takeaway: In a balanced portfolio, \(MS_W\) estimates EPV, while \(MS_B\) estimates EPV plus \(n\) times VHM. Estimate VHM by \((MS_B-MS_W)/n\), then form \(K\) as the EPV-to-VHM ratio when the estimated VHM is positive.

Check Your Understanding

For each problem, use balanced-portfolio ANOVA and retain unrounded values until the final answer.

A hypothetical portfolio has three risks observed twice: \((2,6)\), \((6,10)\), and \((10,14)\). Calculate EPV, VHM, and \(K\). (Answer: EPV \(=8\), VHM \(=12\), \(K=0.6667\).)
Four risks each have three observations: \((4,5,6)\), \((6,7,8)\), \((8,9,10)\), and \((10,11,12)\). Calculate both mean squares and the component estimates. (Answer: \(MS_W=1\), \(MS_B=20\), EPV \(=1\), VHM \(=19/3=6.3333\), \(K=3/19=0.1579\).)
Three risks each have four observations: \((8,10,10,12)\), \((12,14,14,16)\), and \((16,18,18,20)\). Calculate EPV, VHM, and \(K\). (Answer: EPV \(=8/3\), VHM \(=46/3=15.3333\), \(K=4/23=0.1739\).)
A balanced portfolio has five risks, two observations per risk, \(MS_W=9\), and \(MS_B=39\). Find the estimated VHM and \(K\). (Answer: VHM \(=15\), \(K=0.6000\).)
A hypothetical annual portfolio has four risks and three annual observations per risk. Its calculated mean squares are \(MS_W=12\) and \(MS_B=48\). Find EPV, VHM, and \(K\), and state the unit associated with one observation. (Answer: EPV \(=12\), VHM \(=12\), \(K=1.000\), with one observation representing one year.)

What You'll Learn