By Stuart Jarvis, Frances Southall and Elliot Varnell

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We seek parameter values, α and β, that will solve this equation.   1 2    min  E  DK  α + ( P − β )    α      After some intermediate working we get the following expressions for optimum parameters α and β, which will minimize the expected frictional cost: β= α= E [ DP ] E [D] 1 2 E  DP 2  E [ D ] − E [ DP ] E [D] If the profit is purely deterministic we find that the β parameter reduces to the deterministic pure profit. The α parameter reduces to zero as shown in the expression below.

The debtholders will have first call on the company's assets if these assets are insufficient to cover both the debt and any shortfall within the scheme. This insolvency risk provides an incentive for the scheme's trustees to invest cautiously (in bonds). Finally, if the scheme is in sufficient surplus in one year's time, the scheme members' benefits may be increased. This could be modelled in a number of ways: we assume a simple formula, that liabilities will increase by x% of any surplus above y% (for example, 20% of any assets in excess of 150% of the liabilities) This benefit-leakage risk provides an incentive for the scheme's trustees to invest in risky assets (in the equity market, in this model).

R is a positive process, whose expected value under P is always one (since this is just the total probability, when measured under Q). Conversely, given any positive, unitexpectation martingale process, a new measure Q is obtained through the above construction (subject to technical constraints, see Duffie (1996) for details). 2 Change of numeraire A numeraire is just a unit of account. The numeraire normally used (implicitly) is cash in a particular currency. Prices could equally well be expressed relative to any 'currency', as long as it is always positive.

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