When we attempt to quantify risk, we begin with the
question: “What holding period returns (HPRs) are possible, and how likely are
they?" A good way to approach this question is to devise a list of possible
economic outcomes, or scenarios, and specify both the likelihood (probability)
of each scenario and the HPR the asset will realize in that scenario. Therefore,
this approach is called scenario analysis. The list of possible outcomes is
called the probability distribution of HPRs.
Probability is defined as “the chance that
a certain event will occur”. For example, when we toss a coin, there is a 50%
probability of getting a head and 50% probability of a getting a tail.
Probability Distribution is defined as a “list of all possible outcomes
of an event with associated probabilities”. Holding Period Return (HPR)
is defined as the total return that is gained by holding an asset for a certain
period of time.
The probability distribution lets us derive measurements
for both the reward and the risk of the investment. The reward from the
investment is its expected return, which you can think of as the
average HPR you would earn if you were to repeat an investment in the asset many
times. The expected return is also called the mean of the distribution of HPRs,
and often is referred to as the mean return.
To compute the expected return from the data provided, we
label scenarios by s and denote the HPR in each scenario as r(s),
with probability p(s). The expected return, denoted by E(r), is
then the weighted average of returns in all possible scenarios, with weights
equal to the probability of that scenario.
E(r)
=
є
p(s)r(s)
In the following example, we show how scenario analysis
can be used to derive the expected return of an investment.
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