Spatial Aggregation and Weather Risk Management

which pays $1 per ACDD. The tick is normalized to $1 for simplicity. For options, the
weight, or hedge ratio (contracts/acre), v, and strike price, K, are chosen by solving

∑( maxK- [ξ^de^t + Vk∏k ( Kk )],0})2
t

where π_k(K) is the profit of an ACDD call option with strike price K.

The hedging effectiveness of weather derivatives is evaluated by comparing
portfolios with and without derivatives and at different levels of aggregation using a
simple historical simulation. Hedging effectiveness is evaluated using hypothetical
ACDD derivatives written for the locations in Table 1.¹²

The criterion used to evaluate the change in risk exposure is the root mean square
loss (RMSL). RMSL is a simple function of SV
, .                                       1 Л \

(15)                         RMSL_k = J σ_ζj ² k -

к ×                                                  k τ к

where T=32 is the sample size, and σ²k is the SV from equations (13) and (14).

In addition to expected net losses, insurers may also be interested in the
magnitude of losses given an extreme event occurs. Thus, expected shortfall (ES) is also
reported (Dowd and Blake 2006).¹³ ES is the probability weighted average of the worst
α revenues. In the case of a discrete distribution, the ES is given by

1^α

(16)      ESα=—∑ (pth worst outcome) × (probability of pth worst outcome)

α p=0

and is reported for α= 6%, 9%. The ES measurements are calculated using a historical
simulation where each observation is assigned an equal probability of 1/T (T=32). Thus,
ES 6% equals the average of the two lowest valued observations, and ES 9% equals the
average of the three lowest observations. It can be interpreted as an expectation of yields

15

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