STOCHASTIC

as Carifio and Ziemba (1998), Carino et al. (1994, 1998) and Geyer et al.
(2005). This area also led to the modern notion of risk measures which were
developed using an Arrow-like axiomatic system in Artzner et al. (1999).
Subsequent work such as in Rockafellar and Ziemba (2000) and especially in
Follmer and Scheid (2002a, 2002b) rationalized the convex risk measures used
in the stochastic programming literature starting with Kusy and Ziemba (1986);
see also Acerbi (2004). These risk measures are theoretical improvements on
the value at risk (VaR). With VaR, one presupposes a cutoff loss level and a
confidence level such that one will not lose more than this amount with that
probability; see Duffie and Pan (1997) and Jorion (2000) for surveys. Hence it
is like a chance constraint and has the same drawbacks. For example, the
penalty is independent of the loss. Rockafellar and Uryasev (2000, 2002) show
that CVaR, which is linear in the penalty approximation of VaR, can be
computed endogenously and exogenously via a linear program. Our exercise
ME-26 shows this for the much simpler exogenous case, where the constraint
confidence level is specified in advance.
The Lintner and Vickson papers (written for this book) in Section 3 of
Part II discuss extensions of Tobin's 1958 separation theorem that evolves
when there is, as Tobin suggested, a risk free asset. So does Ziemba's stable
distribution paper in Part III that further extends the idea to fat tailed stable
distributions and Ziemba et al. (1974) which specifically shows how to compute
the two parts of the separation in Tobin's normal distribution world: the mutual
fund (i.e., market index) that is independent of the investor's concave utility
function and the optimal balance of cash and this mutual fund for any given
utility function. The first problem is deterministic in n variables and solved as a
linear complementary or quadratic programming problem. The second problem
is stochastic but has only one variable, the percent cash. Ross (1978) approaches
the analysis in a more general way than the normal, stable or other distributional
form and obtains theoretical conditions for separation but it is not clear on how
to find the separated portfolios.
Part III deals with static portfolio selection models and begins with papers
by Samuelson and Ohlson (written for this book) that show when meanvariance
analysis is optimal in cases other than normal distribution and quadratic
utility but with special distributions that converge properly. These results extend
to symmetric distributions. Pyle and Turnovsky's analysis of safety first
followed Roy's 1952 Econometrica paper that was a close alternative to
Markowitz's (1952) famous portfolio theory paper that ushered in modern
investment management and gave Markowitz the Nobel prize in economics in
XVI PREFACE AND BRIEF NOTES TO THE 2006 EDITION