(Wiley Finance) David M

8 MODERN ASSET ALLOCATION FOR WEALTH MANAGEMENT
Higher Moment Motivation
It was just shown that MPT is a second order approximation to the full problem
we ideally want to solve. A natural question now is whether the chopped
off third and higher terms that MPT is missing is a critical deficiency. If it is,
then the maximization of the full utility function becomes a necessity. But
it is quite hard to generalize the answer to the question, since it is highly
dependent on the utility function of the client, and the expected moments
of the assets being deployed; and we will not have those answers for a few
more chapters still. Even then the answers will vary for every single client
and portfolio, and will even change over time as capital markets evolve.
At this stage, though, we are focused on motivating a compelling asset
allocation framework; hence the key question we must address is whether
there is a chance for higher order terms to come into play. To this end, it
should be noted that higher order terms do not exist in Eq. (1.4) unless
two criteria are met: (1) portfolio return distributions have third or higher
moments; and (2) our utility function has a preference regarding those higher
moments (i.e. it has non-zero third order derivatives or higher). By studying
higher moment properties of some typical investable assets and higher order
preferences embedded in typical utility functions, you will see that both conditions
are generally met, and higher order terms should be accounted for in
the process.
Before we tackle the first condition by investigating whether typical
assets have higher moment characteristics, let’s first build up our general
understanding of the third and fourth moments—skew and kurtosis, respectively.
The easiest way to think about higher moments is to start from the
most common distribution in mother nature, the normal distribution. 6 The
important thing to know about the normal distribution is that it is symmetric
about the mean (skew = 0), and its tails are not too skinny and are
not too fat (kurtosis = 3). Figure 1.2 shows a normal distribution of monthly
returns, with mean of 1% and volatility of 2%; this will serve as our baseline
distribution, to which we will now add higher moments.
The easiest way to intuit the effect of negative skew is to imagine a tall
tree, firmly rooted in the ground and standing up perfectly straight. If one
were to try to pull the tree out of the ground by tying a rope to the top of
the tree and pulling to the right, the top of the tree would begin to move
right and the roots on the left side of the tree would start to come out of the
ground while the roots on the right side would get compressed deeper into
6 Fun fact: the normal distribution is so common in our universe due to the Central
Limit Theorem, a mathematical proof that shows that a large sum of random variables,
no matter how they are individually distributed, will be normally distributed.