variance of the hypothetical means
/ abbreviated VHM /
Picture all the drivers in a portfolio lined up by their true, hidden accident rates. A safe one truly causes 0.1 accidents a year; a reckless one truly 0.8. If everyone were identical, those true rates would all be the same and there would be nothing to learn from an individual's record. The fact that they spread out — that risks genuinely differ — is exactly what makes experience rating worthwhile. The variance of the hypothetical means, VHM, measures how widely those true per-risk means are spread.
Each risk has a hypothetical mean: the expected claim it would produce if you knew its type theta, written E[claims | theta]. As theta varies across the population of risks, this hidden mean varies too, and VHM = Var( E[claims | theta] ) is the variance of that variation — the 'between-risk' or 'signal' piece. Example: if half the drivers truly average 0.2 claims and half truly average 0.6, the hypothetical means take values 0.2 and 0.6 with equal probability, so their mean is 0.4 and VHM = 0.5(0.2 - 0.4)^2 + 0.5(0.6 - 0.4)^2 = 0.04.
VHM is the second structural parameter of Buhlmann credibility, and it is the one that justifies trusting an individual at all. A large VHM means risks really are different, so an individual's own record carries genuine signal — this pulls the Buhlmann constant k = EPV/VHM down and the credibility Z up. If VHM were zero, all risks would be identical, k would be infinite, Z would be zero, and you would (correctly) use only the collective mean. Estimating VHM from data is delicate because it is found by subtracting the within-risk noise from the total observed spread, which can occasionally produce a negative number that must be floored at zero.
Half a book of drivers truly average 0.2 claims a year, half truly average 0.6. The hypothetical means are 0.2 and 0.6 (overall mean 0.4), so VHM = 0.5(0.2-0.4)^2 + 0.5(0.6-0.4)^2 = 0.04.
VHM captures the spread of the TRUE per-risk means — the signal that makes individual data worth trusting.
Empirical estimates of VHM can come out negative, since it is total spread minus within-risk noise; convention floors it at zero, which forces Z = 0 (use only the collective). A negative estimate is a signal the risks may not differ enough to credibly distinguish.