# <u>Actuarial justification</u>

If it has been judged that the yield data should be de-trended it is appropriate and consistent to use de-trended data for calculation of both probable yields and premium rates. If, for example, there is judged to have been a large positive trend in yields over the last ten years but this trend is not removed for the probable yield calculations, the resulting threshold yields will be unreasonably low. Farmers would be offered products with low commercial premium rates that offered very little protection. Calculating probable yields based on de-trended threshold yields would result in products that offered a more typical degree of protection for a commensurately higher commercial premium rate.

The removal of (up to two) official calamity years in calculating the probable yield acts to increase the threshold yield and associated commercial premium rate and government subsidy rate for those insurance units in which droughts have been officially reported. The effect is therefore to target subsidies to districts in which droughts have been reported in the last seven years.

Appropriate smoothing of probable yields within districts decreases the standard error of average yield estimates, increasing equity and decreasing the potential threat from adverse selection. It is understood that probable yields are to be based on the most recent seven year average of nondrought year yields. Using seven years of data is likely to increase the stability of probable yields, relative to using three or five years of data as was required under the NAIS. The probable yield for a given insurance unit could be calculated as a weighted average of historical yields for that insurance unit only, as done for NAIS. However, if this was the case, a large part of the resulting variation in threshold yields within a district could be caused by natural statistical variation, rather than a variation in underlying agronomic characteristics; seven data points is not enough to obtain an accurate estimate of the mean yield unless the risk faced is very low. If premium rates were then set at the crop/district level farmers would be able to adversely select against the insurance program by participating when coverage is high or not insuring when coverage is low.

It is suggested that probable yields within a Threshold Yield Collective are smoothed through the use of within- Threshold Yield Collective credibility weighting. The formula for the probable yield is then: where Credibility Factor is calculated objectively and can range from 0% (no credibility assigned to yield history for insurance unit) to 100% (full credibility assigned to yield history for insurance unit). should be 0% when the crop/insurance unit average yield history is statistically uninformative, compared with the history for the full district. Correspondingly, should be 100% when the crop/district level average yields are not statistically useful for estimating the average yield.

Of particular interest is the credibility factor approach derived by Bühlmann (1967), which offers the best linear approximation to unconstrained Bayesian estimates.<sup>5</sup> Denoting the historical loss from product in year as this _Empirical Bayes Credibility Factor_ is given by where and denote the number of years of data and the number of products in the Threshold Yield Collective respectively, and

> 5 Textbook treatments include Herzog (1999) and Bühlmann and Gisler (2005).

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