I. Is Beta of underwriting = 0?
Chapter 4 deals with the CAPM approach. Equation 4.7 develops the expected
underwriting margin in pricing as:
E(return on underwriting) = -k * (1-x) * risk-free return
+ beta of underwriting * [E(market
return) - risk-free return]
where x is the expense ratio, and k represents the funds generating
coefficient. The value k is described as Reserves / Premiums net of expenses,
i.e., R/[P * (1-x)], but it could also be described as the duration of the
In an article on the Part 10 syllabus, it states that one study indicated that
the beta of underwriting was a positive value, while another study indicated
it was negative. Without further research, let's suppose that beta is 0.
That means, of course, that underwriting profit margins are uncorrelated with
investment market returns.
In this case, the expected underwriting margin is:
E(r) = -k * (1-x) * risk-free return, or just
-duration * risk-free return.
It's interesting that this formula involves the risk-free return rather than
the embedded yield in the company's current assets or some form of risk
adjusted rate of return.
In another place on the syllabus, I recall that an article describes the loss
duration for the industry as about 2.3 years. If the risk-free rate is 5% for
a bond with duration of 2.3 years, wouldn't this formula be stating that the
expected underwriting margin for pricing should be -11.5% of premium? That
is, an acceptable combined ratio for the insurance industry would be 111.5%.
Of course, this formula ignores taxes, but D'arcy/Doherty's equation 4.11
could be used to develop the underwriting profit margin taking taxes into
consideration. At a glance, I'd guess that it reduces the target combined
ratio by 2% (to 109.5%).
Am I interpreting this article correctly? Does this result seem too high to
be an acceptable combined ratio for the industry?
II. Return on equity -- who gets the investment income?
Equation 4.2 (simplified) says that
E(roe) = constant1 * E(return on investments) - constant2 * E(return on
where E(return on underwriting) = -duration * risk-free rate
The expected return on investments is:
E(return on investments) = risk-free rate
+ Beta of investments * [E(market
return) - risk-free return]
Since the expected underwriting profit margin is a constant, then any increase
in investment income due to riskier investments (that is, a higher beta of
investments) accrues to the benefit of the owners rather than the
policyholders. In essence, the insurer "borrows" cash from the insured,
paying the insured a risk-free rate of return.
It would also seem that the insurer doesn't have complete freedom in investing
in risky securities because of the risk it has from its underwriting
operation. For that reason, the Beta on investments will be less than 1.0
(the market return). This is a second cost of borrowing cash from the
My question is whether this reasoning is valid.
III. What if Beta of underwriting isn't 0?
Suppose that the beta of underwriting isn't 0. The question is whether it is
positive or negative. It would make sense that the underwriting profit should
decrease if market yields or interest rates rise. This would be due to
insurers' desire to obtain cash to invest. That is, the market return and the
underwriting return should be negatively correlated. This suggests that the
beta of underwriting is likely to be negative, at least for coverages where
significant investable funds are generated.
If beta of underwriting is negative, then the expected underwriting margin
should be even lower than indicated above and the industry combined ratio
should be even higher than 109.5%.
IV. Is there a mistake in the formulas?
Equation 4.2 is essentially a disguised version of Ferrari's formula:
Total return on surplus = Investment yield * Assets
+ Underwriting profit margin *
Premium / Surplus
The important point is that the total return is on statutory surplus, not on
Several substitutions are then made in this formula using the CAPM model.
However, CAPM describes the return on a security's Market Value as a function
of the beta for that security. For that reason, do the formulas in this
chapter need to be adjusted for Market/Book ratios?
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