to achieve a certain expected change. The last measure of interest leads us to generalize the notion of the Sharpe ratio to the asset-liability framework. We define the risk-adjusted change in surplus (RACS) as RACS _Et[st+1-St(l+Rf)]_Et[st+1-St(l+Rf)] (io5) ot[st+1-St(l+Rf)] ot[st+1] where the second equality follows from the fact that St is known at time t. Here we assume the risk-free rate is constant through time, R=Rf. We claim that the RACS is the natural extension of the Sharpe ratio to an asset-liability framework. To see this, let RA t denote the return at time t on the asset portfolio and rewrite the last expression as Et\At(l+RAt+1)-Lt(l+RUt+1)-(At-Lt)(l+Rf)} RACSt = -L^------------- ^-1------ ^- ' / --------- ^-------- - (10-6> and note that in the absence of any liabilities (L( = 0), the RACS becomes £;[4K+1-,)]^4, ] ^ o([A(l+^+i)] 4R^+i] The last expression is the Sharpe ratio of the asset portfolio. Our new measure, the RACS, therefore has the nice property that it simplifies to the Sharpe ratio in the absence of liabilities. For this reason, it is a natural extension of the Sharpe ratio to the asset-liability framework. Whereas the Sharpe ratio evaluates investments relative to cash, the RACS evaluates them relative to liabilities. How does one interpret the RACS? The numerator measures the dollar return on the surplus that is expected in excess of the risk-free rate of return. The denominator measures the risk in the same quantity. Consider a fund with positive surplus and a perfectly known liability stream (i.e., no noise in the liabilities). One possible investment strategy for the fund is to purchase a portfolio of bonds to exactly match its future liabilities and to invest the remaining surplus into a risk-free asset. This strategy is completely risk-free and will produce a return of (1 + Rf) on the surplus with no volatility. If the fund undertakes any other investment strategy, the RACS measures how much the fund is being compensated for taking risk relative to the risk-free strategy. Next consider a fund with a deficit but whose liabilities are also known with certainty. If we assume that the fund can borrow at the risk-free rate, the fund can borrow the amount of its deficit at the rate R, and purchase a portfolio of bonds to exactly match its future liabilities. This strategy produces no volatility in the deficit and locks in a proportional increase of Rf in the deficit. For this fund, the RACS also measures how much it is being compensated for taking risk relative to the risk-free strategy of locking in an increase of Rf in the deficit.