Probability Concepts. Here are the sections that
we would go over. First, a brief introduction. Investment
decisions require an evaluation of risk, and just
to give you a simple example, when you purchase a stock,
obviously, you are concerned about what will happen
to the stock price when you purchase the stock, let's
say it is $50, you are concerned about the risk of the
stock price going down. Now, in order to try and understand
the risk of what you might think of is the probability
that the stock price will go down below $40,
the probability that the stock price would be between
40 and 50 and so on. So clearly then, probability
is important tool which can help us evaluate risk.
In this reading we will explore probability related
concepts, which are most relevant for investment
managers. Probability, Expected Value, and Variance.
We need to first understand a few basic definitions.
A random variable is an uncertain quantity or number.
So, if you buy a stock today for $50 at the end of the
month, the stock price will have a certain price,
we can think of that price as a random variable because
we don't know what that price will be. An Event is
a single outcome or a set of outcomes, so we can define
a event that the stock price is going to be less than
50 so this is a particular event. This event consists
of a set of outcomes. And what are those outcomes?
Those outcomes are stock price being equal to, say,
49.99, 49.98, and so on so all possible outcomes
such that the stock price is less than 50 represent
this event. We might have another event which says
that the stock price is going to be greater than or
equal to 50. So what we have done here is defined two
possible events and note that each event is essentially
a set of different outcomes. Mutually exclusive
events. What I have done here is defined mutually
exclusive events. What that means is that if event
1 occurs when an event 2 cannot occur, so clearly
we cannot have a situation where the stock prices,
both, less than 50 and greater than or equal to 50. So
these two events, event 1 and event 2, are mutually exclusive.
Exhaustive events means that between these two
events, all possibilities are covered and, notice
that the two events shown here are also exhaustive.
The probability of a given event has to be between
0 and 1. So if I say the stock price today is 50, then
the probability that the stock price will end up
being less than 50 at the end of the period has to be between
0 and 1 if I'm fairly sure this will happen. Then the
probability will be close to 1 if I am fairly sure
that this will not happen. Then the probability
will be close to 0, zero but it is not possible to have
a probability that is greater than 1 or a probability
that is less than zero. The sum of the probabilities
of mutually exclusive events must be equal to 1.
So if we know that these two events are mutually exclusive
and exhaustive, and we know that the probability
of the stock price being less than 50 is equal to 0.3,
then the probability of the stock price being
greater than 50 has to be 0.7, and note that the sum
of these two probabilities has to be equal to 1
because events E1 and E2 are mutually exclusive
and exhaustive. We now look at different ways of
estimating probability. One method is called, Empirical
Probability. This is where we calculate probability
using past data. Let us say that we are looking at
a particular stock market where there are 700 stocks
and off these 700 stocks, 350 pay dividends. Now, if the
question is the following, what is the probability
that the stock picked at random from this population
will be a dividend paying stock. And if we say that
the answer is 350 over 700, which is equal to 0.5, then
0.5 is a empirical probability because we have arrived
at this probability through analysis of the data.
A Priori Probability. This is calculated by using
formal reasoning and inspection. Just to give you
another simple example let us say, that the stock
price is currently $50 and we say that the probability
of the stock price going up is 50% and the stock price
going down or staying the same is also 50%, and we use
the reasoning that, given the current stock price,
there is an equal chance of the stock going up or down.
Now, this is based on reason. The reasoning might not
be correct, but at least there is a attempt at some
formal reasoning. Probability arrived at using
this method is called a, a priori probability. Subjective
Probability. This method is less formal and involves
personal judgment. Let us say that I have a high opinion
of the new CEO of company X, and I say that because I
believe the CEO is good, the probability that the
stock price will go up is 0.8. This probability is
based on my personal judgment about the quality
of the CEO and we would say that this is a subjective
probability. Here is a simple practice problem given
what you've just learned. The correct answer is
B. Probability Stated as Odds. Let's say that we
have an event and the probability of that event is
equal to one-third. Let's say the event is that the stock
price will go down, and we now the probability. How
can we express this probability as odds for E. What are
the odds that the stock price will go down? This is
the formula that will allow us to translate the probability
into odds. Probability of the event, one-third, divided
by 1 minus the probability of the event, so 1 - one-thirds
is two-thirds. one-thirds divided by two-thirds is simply equal to 1/2. So the odds are 1/2
and the way we write odds is by saying 1 to 2,
so the odds of the stock going down are 1 to 2, in other
words, a probability of one-thirds is exactly the same as
odds of 1 to 2. These are two different ways of saying the
exact same thing. If we can go from probability to
odds we should also be able to go from odds to probability.
And this is how we do that. If the odds are expressed
as a to b, now the first item here is a, the second is b, and in our example,
a is 1 and b is 2, the implied probability of the event,
So P(E) is equal to a, which is 1 / (A + B), which is (1 + 2), in our
example. So 1 + 2 what we essentially have is one-thirds,
which is exactly what we started with. So it is a
fairly straightforward and mechanical exercise
to convert from probability to odds and from odds
back to probability. The odds against an event are
given by this expression. Notice, this is simply
the reciprocal of what we saw here and that makes sense
because you calculate odds against E instead of using
P(E), we use the equation 1 - P(E) which is essentially
the probability of not E and then in the denominator
instead of 1 - P(E), we do 1 - (1 - P(E)), and that is simply P(E).
All you need to do is remember that if the odds for an
event are given by this expression, the odds against
are given by the reciprocal and when you work backwards
the implied probability of not or the odds against
E are essentially b / (a + b). Let's see if you can do this
simple problem now. The correct answer is A. We will
now talk about Conditional vs. Unconditional probability.
Unconditional probability is the probability
of event regardless of other outcomes. So our event
here is A, this might be the probability that a stock
price goes up. So the event is stock price going up.
The probability of the stock price going up regardless
of any other events is called unconditional probability.
A conditional probability is written like this,
so we might say, 'what is the probability that the
stock price will go up assuming interest rates go
down?' So in this particular scenario, event A is that stock
price goes up and event B is that interest rates
come down. So we might say that the probability that
stock price goes up given that interest rates go
down is equal to 0.6. This is a conditional probability.
Because this probability is the probability of
something happening i.e., stock price going up, conditional
on another event, the condition in probability
terminology is expressed using this bar between
two events. Joint probability and the Multiplication
Rule. So let us first understand what we mean by joint
probability. The probability that both A and B will
occur is their joint probability. It is called
P(AB). So whenever you see AB without anything in
between that means we are talking about the joint
probability of two events A & B. Continuing with my
previous example, we might ask the question, what
is the probability that both stock price goes up
and interest rates come down. Remember event A was stock price
going up, event B was interest rates going down. So
the probability that both A and B happened is probability of
AB. That's how we express our joint probability.
We can compute the Joint probability using the Multiplication
Rule, which is the first formula that you are going
to learn. So the probability of both A and B happening
is equal to the probability of A given B. This is the
conditional probability of A. So the probability
of a conditional on B happening times the probability
of B. If you have this expression, then you can use
simple algebra to compute the probability of A given
B. So the conditional probability given a joint
probability and the probability of B. So if you get
a question on the exam where you are provided the
conditional probability and the probability of
B, you can then compute the Joint probability. On
the other hand, if you are given the joint probability
and the probability of B, then you can compute the
conditional probability from this expression.
I hope you can see that probability of AB can also
be written as probability of B given A times the probability
of A. Let us look at a simple example. So if the probability
that interest rates will decrease, and I'm just calling
that event D now, is equal to 40%, and the probability
that the stock price will increase is written as
P(S). What is the probability that the stock price
will increase and let's say that we are given that the
probability stock price will increase given that
interest rates decrease is 70%. This is a conditional
probability. The question is what is the joint probability?
The Joint probability that the stock price will
increase and interest rate will decrease. It is expressed like this, P(SD), and
we have learnt through the multiplication rule
that P(SD) is equal to P(S given D) times probability
of D, which is 0.7 from here times 0.4 from here
so our answer is 0.28 or 28%. Next we need to learn
the Addition Rule for Probabilities. The probability
that A or B occurs or both occur is given by this expression
so A or B means A happening, or B happening, or both
happening. This is equal to probability of A plus
probability of B minus the joint probability of. Again,
take a simple example, you have two stocks, A and B,
and you want to know, what is the probability of A going
up or B going up or both going up? If you are given that
the probability that A increases, this is the unconditional
probability of A going up is equal to 0.5. The unconditional
probability that stock B goes up is 0.7 and you are
also given the joint probability that both stocks
go up which is 0.3, then probability of A or B is computed
using the Addition Rule and we simply have
0.5 + 0.7 - 0.3 which is 0.9. See if you can do this question
now. The correct answer is A, you need to use the
Addition Rule and when you use the Addition rule
you should get 0.8.