Correlated risk refers to the simultaneous occurrence of many losses
from a single event. Natural disasters such as earthquakes, floods, and
hurricanes produce highly correlated losses: many homes in the affected
area are damaged and destroyed by a single event. An insurer will face
this problem if it has many eggs in one basket, such as providing
earthquake coverage mainly to homes in Los Angeles rather than
diversifying across the entire state of California.
To
illustrate the impact of correlated risks on the distribution of losses,
assume that there are two policies sold against a risk where p = 0.1, L
= $100, where p is the probability of a loss and L is the magnitude of
the loss. The actuarial loss for each policy is $10. If the losses are
perfectly correlated, then there will be either two losses with
probability of 0.1, or no losses with a probability of 0.9. On the other
hand, if the losses are independent of each other, then the chance of
two losses decreases to 0.01 (ie, 0.1 × 0.1), with the probability of no
losses being 0.81 (ie, 0.9 × 0.9). There is also a 0.18 chance that
there will be only one loss (ie, 0.9 × 0.1 + 0.1 × 0.9).
The
expected loss for both the correlated and uncorrelated risks is $20.
However, the variance is always higher for correlated than for
uncorrelated risks if each has the same expected loss. If a risk-averse
insurer faces a highly correlated loss from one event, it will want to
set a high enough premium to not only to cover its expected losses but
also to protect itself against the higher probability of experiencing
catastrophic losses due to the higher variance. Thus, risk-averse
insurers will always want to charge a higher premium for correlated
risks than uncorrelated risks.
This article is an edited version of
an entry in the “Encyclopedia of Quantitative Risk Analysis and
Assessment”, Copyright © 2008 John Wiley & Sons Ltd. Used by
permission.
