Morningstar Magazine - August/September 2018 - 48

Strategies

Filling in the Gaps
How smoothed discrete distributions
combine the advantages of discrete and
continuous models of returns.

QUANT U

Paul D. Kaplan

Quantitative investment models, such as optimization and simulation, embody specific
assumptions about the probability distribution
of returns. While the most common approach is to
assume that returns follow a normal or a lognormal distribution, that approach has important
limitations. Specifically, there is no flexibility
in the shape of the distribution. One way
of overcoming that limitation is to take a discrete
scenario-based approach. However, discrete
distributions have other limitations.
When Sam Savage and I developed the Markowitz
2.0 optimization framework, we introduced a
smoothed discrete distribution model that combines
the advantages of continuous distributions
with the advantages of scenario-based models.
(See the December/January 2015 issue of Quant
U.) In this issue of Quant U, I present the
smoothed discrete model in detail and illustrate
its advantages. But before I do, I present
a classification scheme for distributions to show
where the smoothed discrete model fits into
the grand scheme of things.
Types of Distributions
Discrete vs. Continuous
The main distinction between different distributions
is whether they are discrete or continuous. If a
random variable has a discrete distribution,
it can only take on specific values. For example, if
you toss a coin n times, the number of heads
follows a binomial distribution. The only values
that this random variable can take are 0, 1, 2,..., n.

Morningstar August/September 2018

In this case, the number of possible values is
finite. But there are discrete distributions where
the number of possible values is infinite,
but at discrete intervals. For example, a random
variable that follows a Poisson distribution
can take on any non-negative integer. (Poisson
distributions are used for modeling the number of
occurrences of something per unit of time,
length, volume, etc.)
Random variables that follow continuous
distributions can take on any value, possibly
limited to a range. For example, a random variable
that follows a normal distribution can take
any value, while a random variable following
a standard uniform distribution can take any value
between 0 and 1. The limit can be one-sided.
For example, the return on a long-only investment
can take on any value above negative 100%.
Parametric vs. Nonparametric
All of the distributions that I have discussed so
far are parametric: Each can be described
by a few parameters. For example, a binomial
distribution is described by two parameters: the
probability of getting heads on each coin toss
and the number of tosses. A Poisson distribution is
fully described by its mean. Both normal and
lognormal distributions can be described by their
means and standard deviations.
A limitation of normal and lognormal distributions
is that they do not model fat tails. In the
June/July 2014 issue of Quant U, I introduced a set
of parametric distributions, the Johnson family,
that generalizes both normal and lognormal
distributions by having skewness and kurtosis
as parameters.

While parametric distributions can be quite
useful in modeling investment returns and other
phenomena, they impose limitations on
distribution shape. Nonparametric distributions
do not have this limitation because they
can take on any shape. To form a nonparametric
distribution, the model builder creates a list
of scenarios and assigns a value and probability
to each. In investment analysis, the most
common way of creating a nonparametric
distribution is to take a historical return series,
treat each observation as a scenario,
and assign equal probabilities to them. (If the
application is forward-looking, the historical
returns should be adjusted to reflect forecasts.)
Four Types of Distributions
Because, on one hand, a distribution can be either
discrete or continuous and, on the other hand,
it can be either parametric or nonparametric,
there are four types of distributions. Here are the
distributions that I have discussed, plus the
smoothed discrete distribution that Savage and
I developed for Markowitz 2.0:
Classification of Probability Distributions
Parametric

Nonparametric

Discrete

Binomial,
Poisson

Scenario-Based

Continuous

Standard
Uniform,
Normal,
Lognormal,
Johnson

Smoothed
Discrete

I will now discuss the motivation for creating the
smoothed discrete distribution and the details of
how it works.
Smoothed Discrete Distributions
The Problem of Gaps
The limitation of the scenario approach is that
the distributions it creates have large gaps
between values, especially in the tails.
This is because the returns in the tails are rare.
Because one of the goals of Markowitz 2.0
is to effectively model return distributions
with fat tails, Savage and I needed a method for
filling in these gaps.

Table of Contents for the Digital Edition of Morningstar Magazine - August/September 2018