380 THE HANDBOOK OF PORTFOLIO MATHEMATICS
Let’s consider now what we are confronted with, mathematically, when
there are various outcomes involved, and those outcomes are a function of
a stake that is multiplicative across outcomes as the sequence of outcomes
is progressed through.
Consider again our two-to-one coin toss with f=.25:
+2,− 1 (Stream)
1 .5,. 75 (HPRs)There are four possible chronological permutations of these two scenarios,
as follows, and the terminal wealth relatives (TWRs) that result:
1. 5 × 1. 5 = 2. 25
1. 5 ×. 75 = 1. 125. 75 × 1. 5 = 1. 125
. 75 ×. 75 =. 5625
Note that the expansion of all possible scenarios into the future is like
that put forth in Chapter 6.
Now let’s assume we are going to consider that we are ruined if we
have only 60% (b=.6) of our initial stake. I will attempt to present this so
that you can recognize how intuitively obvious this is. Take your time here.
(Originally, I had considered this chapter as the entire text—there is a lot to
cover here. The concepts ofruinanddrawdownwill be covered in detail.)
Looking at the four outcomes, only one of them ever has your TWR dip
to or below the absorbing barrier of .6, that being the fourth sequence of
.75×.75. So we can state that in this instance, the risk of ruin of .6 equity
left at any time is^1
/
4 :
RR(.6)=^1
/
4 =.^25
Thus, there is a 25% chance of drawing down to 60% or less on our initial
equity in this simple case.
Any time the interim product<=RR(b), we consider that ruin has
occurred.
So in the above example:
RR(.8)=^2/
4 =50%
In other words, at anfvalue of .25 in our two-to-one coin-toss scenario
spectrum, half of the possible arrangements of HPRs leave you with 80% or
less on your initial stake (i.e., the last two sequences shown see 80% or less
at one point or another in the sequential run of scenario outcomes).