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This risk is often measured against some future liability. A pension fund, for example, might need to plan for an increase in pension liabilities based on actuarial projections. An individual investor may plan a series of investments to represent the future funding of his or her children s college tuition. The risk we are talking about here is the risk of coming up short over a speci ed period of time. All of the performance-based risks just mentioned and described in detail throughout the chapter are backward looking. They are based on historical performance as opposed to future projections. As a result, it is important to understand that they are meant to convey probabilities of future behavior based on past behavior. And as we all know, probability does not equal certainty. Performance Disclaimer: Past performance may not be indicative of future results. Most investment rms include a passage similar to this quotation in their marketing materials. While the exact wording may change from rm to rm, the message is the same. When we make an investment in a particular fund or with a particular investment manager/team, we also make assumptions as to the return we might expect over the life of the investment. Risk is the uncertainty that goes along with our return expectations or assumptions. Different asset classes have different return and risk expectations. For example, it has been demonstrated that over the long term small-cap stocks exhibit greater risk (based on the volatility of returns) than do large-cap stocks. As a result, we would demand a greater return from our small-cap investments than our large-cap investments based on the higher level of risk assumed. As a general rule, the higher the expected risk, the higher the expected return should be. Given the choice between two investments

Then, almost surely, there is no point t I such that Y (t) = u This implies the following: Proposition 6.12. Let Z = {Z(t) : t W } be a random eld, W an open subset of Rd , with values in Rd . Let u Rd . We assume the following:

3.68 3.60 3.54 3.47 3.42

An enumerated datatype holds a set of predefined values. For example, assume an admission application that must store different test results. Possible test results might be pass, fail, and hold. If the application stores these as numeric values, the person reviewing the results might not know what each value signifies. If they are stored as PASS, FAIL, and HOLD, anybody who views the results will know exactly what each label 445

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It is known that most holograms are redundant, and only a small part of the hologram is suf cient to generate the desired information. If the hologram is to be viewed with the unaided eye, the resolution of image points is determined by the size of the eye lens aperture, which is of the order of several millimeters [Born and Wolf, 1969]. One desires to have as much information as possible in a hologram of this size. When the hologram is recorded with a certain kind of wave, such as acoustical waves or microwaves, and reconstruction is done with another kind of wave, such as

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