ABSTRACT
This paper contrasts high-risk, hedge fund trading,
with low-risk, mutual fund trading, in terms of their differing utility
functions. We envision hedge funds, led by informed traders who use information
to seek out investment opportunities, timing market conditions, with the
expectation that prices will move in their favor. Directional hedge funds act to
influence prices, while non-directional hedge funds do not act to influence
prices. We present utility functions based on steeply-sloping Laplace
distributions and hyperbolic cosine distributions, to describe the actions of
directional hedge fund traders. Less steeply-sloping lognormal distributions,
Coulomb wave functions, quadratic utility functions, and Bessel utility
functions are used to describe the investing style of non-directional hedge
fund traders. Flatter Legendre utility functions and inverse sine utility
functions describe the modest profit-making aspirations of mutual fund traders.
The paper’s chief contribution is to develop optimal prices quantitatively, by
intersecting utility functions with price distributions. Price distributions
for directional hedge fund returns are portrayed as sharp increases and
decreases, in the form of jumps, in a discrete arrival Poisson-distributed
process. Separate equations are developed for directional hedge fund strategies, including event-driven arbitrage, and
global macro strategies. Non-directional strategies include commodity
trading, risk-neutral arbitrage, and convertible arbitrage, with primarily
lognormal pricing distributions, and some Poisson jumps. Mutual funds are
perceived to be Markowitz portfolios, lying on the Capital Market Line, or the
International Capital Market Line, tangent to the Efficient Frontier of minimum
variance-maximum return portfolios.