Krepsism and Taoism

Prices are not probabilities

How Kreps’ model improves our understanding

Problems with Kreps’ assumptions

A model that eases these assumptions

Results compared:

• structure


• sample calculations

Is a Price a Probability?

Probability	Rates-on Line (without reinstatement)
0 < P < 1	0 < R < 1
Pr(not A) = 1 – Pr(A)	R(not A) = 1 – R(A)
P(A or B) ≤ Pr(A) + Pr(B)	R(A or B) ≤ R(A) + R(B)

Some actuaries like to treat rates-on-line as risk-adjusted probabilities

R(FL and CA) = R(FL) x R(CA)

or

R(FL or CA) = 1 – (1 – R(FL))(1 – R(CA))

assuming that Florida and California are independent events

A common calculation

R(2nd Event Cover) = R(1st Loss & 2nd Loss) = R(1st Loss) x R(2nd Loss) =
R(1 st Loss)^ 2

This is bad math and it is also wrong.

What is the Right Math?

Subsequent losses in a single policy term are not independent. A second loss is less likely than a first loss because part of the policy period is already expired. However, we can assume that the loss process is memory-less, that is, the relative frequency of a loss is proportional to the time remaining (A Poisson process).

Pr(1st Loss) = P

Pr(2nd Loss) = P – Pr(exactly 1 loss) = P – ln ((1/(1 – P))(1 – P)

If P = 10%, the conventional wisdom is

Pr(1st & 2nd Loss) = (10%)^2 = 1%

The right math is:

P – ln (1/(1 – P))(1 – P) = 10% – ln (10/9)(90%) = 0.52%

Unfortunately, the right math is still wrong! Prices are not probabilities.

Broker Estimates of Prices
Hypothetical Cedant, Real Brokers

Annual Layer and Retention ($ Millions)	Layer Penetration Recurrence Time (Years)	Estimated Price (ROL)	Pure Premium (ROL)	Standard Deviation (ROL)	Implied Loss Ratio
$10 xs $10	10	18%	10.0%	30.0%	55.5%
$10 xs $20	40	9.5%	2.5%	15.6%	26.3%
$30 xs $30	100	5%	1.0%	9.9%	20.0%
$40 xs $60	1,000	2.25%	0.1%	3.1%	4.4%

Krep’s Calculations:

p=µ +Rσ

R=czy/(1+y)

A sample calculation

y, target yield rate = 12%

c = correlation between contract and book = 80%

z = insurer’s target (and current) ratio of S.D. to surplus = 3.0

µ = contract’s expected losses = 10% of limit

Single limit, no reinstatement, no A.P.’s

Some simplifying assumptions

No taxes

No expenses

No '"banks”

No interest opportunities or costs

No market clout (insurer is a price taker)

Sample calculation, continued

Krep’s formula suggests a rate-on-line as follows:

p=µ +Rσ

R=czy/(1+y) = 80% x 3 x 12%/112% = 25.7%

If the layer is “skinny”, then σ = √µ(1-µ) = 30%

p=10% + (25.7%)(30%) = 17.7%

This is a close match to the 18% brokers' estimate

But:

Other things equal, an insurer prefers to reduce its probability of ruin.

If marginal results are very attractive, an insurer may choose to increase its probability of ruin.

The market capitalization rate applied to future profits depends on the kind and amount of business assumed.

Insurers do not directly examine the covariance between a proposed contract and the existing portfolios.

Insurers do not calculate unlimited means of the losses for their contracts.

Kreps’ process is circular.

Also:

I have never thought that allocating an insurer’s surplus means anything.

(I was surprised to find a very similar formulation without using this allocation.)

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