When we measure a value such as the amount of money
left in a portfolio after a fixed number of years, we have an idea
that with a certain probability the measured value will fall within a
certain range and that this probability depends on the range of
numbers we pick. How could we begin to understand what these
probabilities might look like? Suppose we have a portfolio with an
initial amount of investment money in it, and we are able to perform
the experiment of measuring its final value after some fixed number of
years over and over. We could then divide up the range between the
lowest final value and the highest final value into segments, and
count how many of the final values that were observed fell into each
"bin". If we then divide the number of observations in each bin by the
total number of observations, we get a picture of how likely it is
that our final portfolio value falls within a certain range.
For example in the picture above, we can observe that
the final portfolio value falls over 7.4 million dollars about 10% of
the time, since about 10% of the observations fall into this bin. And
what if we wanted to know about the probability that our measured
final value will be less than or equal to a certain key value that we
are concerned about? Again we can create a useful picture with our
experiment:
We can "accumulate" the bins in our first picture by
adding up all the bins to the left of each partitioning value, so that
when we look at our new plot, we can see that the portion of the bin
area to the left of each partitioning value (as compared with the
total bin area) corresponds to the likelihood that our final value
measurement comes out to be less than that partitioning value. For
instance, in our plot above we can see that our final portfolio value
is less than 1.23 million dollars about 50% of the time.
A value that can be measured and that can take on
different values when it is measured, such as portfolio value after a
given number of years, is commonly referred to as a random
variable. The first type of probability picture we created is called
the frequency function, because it shows us the frequency with which
our random variable falls in a certain range of numbers. The second
type of picture is called the cumulative distribution function because
it helps to picture the "accumulated" probability that our random
variable falls to the left of a certain value.
Monte Carlo simulation is the idea of using
statistical trials to get an approximate solution to a problem. There
is a random process (such as the generation of portfolio return) where
some parameters of the process are equal to the required quantities of
the problem. Since these parameters are not known exactly, many
observations are made so that the parameters of the process can be
determined approximately.
Experiments using Monte Carlo Simulation
Suppose we have a portfolio with a certain initial
value, and we wish to determine a "safe"aftertax spending level for the
portfolio to last some number of years with a certain percentage
confidence, let's say 85% confidence. That is, we want to determine
how much money we can spend yearly so that we are 85% sure that the
portfolio will not run out of money before the specified number of
years. For this simplified Applet we assume the portfolio is entirely in a taxed account. Suppose we are able to give these values for our portfolio:

Return: Average portfolio return; this is the gross return, before taxes and with no correction for inflation.

Volatility: The typical discrepancy between an annual return and the average annual return. For high yield portfolios this might be 1.5 times the return.

Inflation: Inflation Percentage

Tax: Tax Percentage

Confidence: Percentage confidence we want to have in our answer

# Years: The number N of years we want the portfolio to last

Initial Portfolio Value: starting amount in dollars
We can use the idea of Monte Carlo Simulation to model
the situation despite uncertainty. We will use a Java applet to help
us perform the experiment. There are certain quantities (random
variables) depending on the portfolio characteristics above which are
observed to closely follow a certain behavior in real life. Our applet
creates many observations of each random variable, computes the
frequency function in the way we discussed above, and returns a
picture of the simulated frequency function and cumulative
distribution function for the portfolio value after N years.
Note that the simulations performed by our applets in this primer use a
very basic model to get an idea of scenario outcomes.
The set of input parameters is extremely simplified when
compared with the more realistic portfolio and spending plan
characterizations used in Wagner's
RSP
products.
Try it! Start up the applet by
pushing the button below. (This applet requires your web
browser to support Java 1.1 : Netscape 4.06 or Explorer 4.0 and
above. Mac OS users must use Explorersorry. )
Enter the quantities for your portfolio, or press the
"Suggest Values" button to suggest some reasonably realistic
values. Then press the "Enter Values" button; a message will appear to
let you know if the portfolio characteristics you entered were
valid. You can clear your values and start over by pressing "Clear" if
desired. When your portfolio characteristics have been accepted, press
"Compute Safe Spending" and view the simulated results!
Now suppose we want to make a different sort of
estimate: we have a portfolio with a certain initial value, and we
wish to determine with 85% confidence how many years the portfolio
will last if we spend a certain percentage of the portfolio's initial
value each year. That is, for our given spending level we want to know
the number of years such that 85% of the time we expect that our
portfolio will not run out of money before that number of years has
passed. Let's say we are able to give these values for our portfolio:

Return: Average portfolio return; this is the gross return, before taxes
and with no correction for inflation.

Volatility: The typical discrepancy between an annual return and the
average annual return. For high yield portfolios this might be 1.5 times the
return.

Inflation: Inflation Percentage

Tax: Tax Percentage

Confidence: Percentage confidence we want to have in our answer

Spending Level: Amount spent yearly, as a percentage of the initial
portfolio value

Initial Portfolio Value: starting amount in dollars
We can use the same Monte Carlo Simulation idea to
model the situation despite uncertainty. We will use another Java
applet below to perform the second experiment. This time the applet
finds the number of years the portfolio is expected to last at the
given spending level. The number returned is an integer N; more
precisely, if the computation returns a value of N years, this
indicates that the portfolio would last between N and N+1 years. The
applet then returns a picture of the simulated frequency function and
cumulative distribution function for the portfolio value after N
years.
Try it! Start up the applet by
pushing the button below. (This applet requires your web
browser to support Java 1.1 : Netscape 4.06 or Explorer 4.0 and
above. Mac OS users must use Explorersorry. )
References
For a more detailed discussion of the mathematics behind the
computations performed in the applets, see the following article:
Bernard J. McCabe, Analytic Approximation for the Probability
That a Portfolio Survives Forever, Journal of Private Portfolio
Management, Vol.1, No. 4, Spring 1999.
Copyright Acknowledgement:
The applets contained in this page were developed using the charting
capabilities of KLGroup's JClass Chart product.
