Here is an e-mail from a reader:

I’m curious about the intercept of the PB-ROE line. In your examples (http://alephblog.com/2012/02/25/thinking-about-the-insurance-industry/), only life had a negative intercept; the others were all positive. Here’s the implication:

 PB = a + b ROE (where a and b are the intercept and slope).

If I divide both sides by ROE I get:

PE = a / ROE + b

Taking the differential:

dPE = – (a / ROE^2) dROE

So if the intercept is positive, an increase in ROE results in a lower PE and vice versa.

So here are two questions:

1) Why would the relationship be negative? Is it because the higher ROE is achieved via leverage, is therefore riskier, and requires a lower PE? 

2) Why is the intercept negative for life insurers but positive for the others?

First, you have to understand PB-ROE.  The idea is that there is a limiting factor to earnings with financial companies.  The earnings of financial companies is limited by its book capital.  I think this is correct to a first approximation.  But different financial companies experience different financial results; they have different ROEs.  How sustainable are those different ROEs?  ROEs tend to revert to mean; competition drives that.  How fast ROEs revert to mean derives from the length of the businsess written.  Long tail exposures found in life companies mean that a higher ROE usually gives more kick to the P/B multiple.

As an aside, with industrials and utilities =, I often think that sales are the limiting factor, and so my equation becomes:

P/S = a + b * E /S + e  (E/S = profit margin)

Now to your math.  You have the first equation wrong, it should be:

PB = a + b * ROE + e , where e is a normally distributed error term, so if you did the division by ROE, it would be:

PE = a / ROE + b + e / ROE, which means my error term could no longer be normally distributed.  So, you can’t divide through by ROE.  Not legitimate.

Let’s try a different approach.  What if we modeled P/E as a function of B/E?

First, to me that doesn’t make sense, because the idea of capital as a limiting resource goes out the window.

Second, if you did that the a and b would be different, because regression minimizes the squared differences of the dependent variable (actual versus expected).

So, with respect to what I said above, I would not do the math your way. Dividing and differentiating by ROE neglects the meaning of the original equation.  All models are just that, models.  But we can’t go neglecting what they internally assume, and expect to get good results.

So, I can answer your second question, but not your first question.  When we estimate PB-ROE, often the equations with the highest slopes have the lowest intercepts.  What that describes are situations where the ROEs, if they are high, are expected to remain high, and thus produce much higher P/Bs.  Such would be true with long duration coverages like in the life industry.

The reason the life industry is different is that the companies with high ROEs are expected to maintain those high ROEs for a longer period of time, because coverages are long, and pricing adjusts slowly.  With other insurance coverages, pricing adjusts annually or nearly so.

For a short-tail P&C company with an ROE of zero, I would expect a P/B multiple that is positive, because pricing adjustments and mean-reversion are coming soon.  For a life company with a low ROE, the adjustment will happen slowly, or it may never happen.  Perverse dynamics kick in when a company with long-tail coverages finds itself earning very little to nothing.  There is the tendency to mis-estimate reserves, “because we can’t be making so little.”  The length of accruals allows a greater degree of subjectivity to be injected into the estimates.  Short accruals get validated every year.  Long accruals don’t get that validation, at least not in a way that a public investor can see.

If you were an actuary inside the long-tailed life insurer, you would get some data telling you that your assumptions were optimistic, right, or pessimistic.  But it takes a while to figure out whether the last few years are a deviation or a trend.  Good actuaries dig in, and look at the causes for claims, trying to see if the reasons for policyholders making claims matches up with the original estimate of what the subject population would be likely to die (or have disability or LTC claims) from.  Too many abnormal claims may imply that the business has been underwritten wrong, and needs to be adjusted.

That analysis takes some doing, because long-tail life coverages are low-frequency and high-severity.  That’s why the qualitative data may help, by giving clues long ahead of the flood of claims that you did not expect.

To summarize: Life is different because the coverages are long duration in nature, and ROEs don’t change so rapidly.