Additive and multiplicative risk premiums with multiple sources of risk

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Abstract
This paper considers the risk premium for one risk when other background risk is present. In a mean-variance setting, we examine the conditions under which the risk premium will be negative. These conditions consider the variance of the considered risk in relation to the covariance of the considered risk with the background risk. We consider the cases of both multiplicative and additive risks. A breakdown of the total risk premium into several components is established by removing each source of risk consecutively. The case of multiplicative risks is shown to be quite different from the additive-risk case in that the individual's total willingness-to-pay for the removal of all risk on a sequential basis does not necessarily equal the willingness-to-pay for removal of all risk simultaneously.