Premium Earning Patterns for Multi-year Policies with Aggregate Deductibles Part 4

1.2 A First Example
Now we turn to the question of the indicated UEPR for multiyear policies.  For ease of exposition, let’s
first examine a simplified version of the problem.  We will assume that there is a maximum of one lossin each year, each loss is exactly $1000, and there is no investment income (i.e. all flows are
discounted at 0%).  We further assume that the probability that a loss occurs in any given year is 10%,
and  that different years are independent.  For this simplified setup we want to compute the pure
premium for the k
th
 loss during the next n years; we will denote this pure premium by PP(k,n).  It will
be convenient to have a name for a policy covering such a loss; for it we shall write Policy(k,n).
To illustrate:
PP(1,1) is the pure premium for a policy that pays $1000 if there is at least
2
 one loss during year 1, so
PP(1,1) = $1000 * 0.1 = $100.
PP(1,2) is the pure premium for a policy that pays $1000 if there is a loss during year 1 or year 2 (since
we discount flows at 0% it does not matter which).  The probability that there is no loss in two years is
0.9
2
 = 81%, so the probability of at least one loss is 19% and PP(1,2) = $190.
PP(2,2) is the pure premium for a policy that pays $1000 if there are at least two losses during years 1
and 2.  Since we are assuming at most one loss per year, this can only happen if there is exactly one
loss in each of years 1 and 2.  The probability of this is 0.10 * 0.10 = 1% and the pure premium is $10.
Suppose that you purchased both Policy(1,2) and Policy(2,2).  You would have full coverage for two
years.  In fact, your coverage would be identical to first purchasing Policy(1,1) and then one year later
purchasing a second Policy(1,1).  Your pure premium for the first set of policies would be $190 + $10
= $200, and for the second your pure premium would be $100 + $100 = $200 once more.  This is nocoincidence.  Identical coverages must have identical pure premiums.
In a frictionless world with no transaction costs, where risk carriers are willing to cede or assume risks
for their pure premiums, the following principle holds: If two sets of policies give identical coverage,
they must have the same premium charge.  (Below this simplified environment will be referred to as
“The Frictionless World”.)  If this were not so, a portfolio consisting of a long position (assumed risk)
and a short position (ceded risk) could be assembled which has positive net (pure) premium, but no net
risk.  This would violate the principle of no risk-free arbitrage.
3
 (Below this will be referred to as “The
No Arbitrage Principle”.)