Download Review of Statistics in Finance

SEATTLE UNIVERSITY
Prof. Ken Shah
A BRIEF REVIEW OF STATISTICS IN FINANCE
In finance, we are typically concerned with what will happen in the future, not what has
happened in the past. Statistics aids us in forming expectations about the future. What has
happened in the past is only useful so far as it is likely to tell us something about the future.
Let x represent a random event in the future (e.g. future price or return on a share of stock).
Suppose x can take may take n possible values, say, x1, x2, ... , xn, each with a probability p1, p2,
... , pn. All the possible n values make up a population.
The expected value of x is defined as:
n
Expected Value = E(x) =
 p x
i
i
1
The expected value measures the most likely outcome and is one measure of central tendency.
However, it should be noted that expected value may or may not be an element in the population.
For example, say a share price one year from now can be either $75 or $125, each with .5 or 50%
probability. Although the expected value in this case is $100, the actual outcome (by definition)
will be either $75 or $125, not $100. In this case, one way to interpret expected value is that if i)
the above situation presented itself many, many times, ii) each time one bought 1 share of the
stock, and iii) one waited a year and took note of the actual value, added them all up and divided
by the number of shares bought, then iv) it is most likely that the 'average' price for the stock next
year is $100.
Where do the probabilities come from? Often, as in scenario analysis, the probabilities are
subjective. That is to say, an individual makes an informed guess about what they are. For
example, a financial analyst may simplify the problem at hand to considering an economy in
boom, normal growth, or recession. He/she may subjectively attach a probability to each of the
three outcomes as .25, .5, and .25.
Often, however, it is assumed that what has happened in the past is likely to be a good guide to
what will happen in the future. In this case, one takes a random sample of past realizations of the
random variable x, and estimates the expected value. For instance, the expected value can be
estimated by the sample average:
Estimate of E(x) = Mean or Average = x =
1 n
 xi
n 1
It is important to realize that when using estimates from past values (as is often done in finance),
we are implicitly assuming that what has happened in the past is likely to continue in the future.
Page 2
That is, there are no major shocks or 'unusual' events expected that would render the past
experience useless.
Variance measures the dispersion around an expected value. It can be thought of as an expected
value of squared surprises from the expected value. Surprises are defined as the (actual value –
expected value). Larger values of variance implies greater likelihood of an actual outcome to be
farther away from its expected value. Risk averse investors in general do not like surprises
unless they are properly compensated for it. In other words, investors would prefer less variance
to more, holding other things constant (ceteris paribus).
Variance = 2 = E xi  E( x ) =
2
n
 p  x
i
 E ( x)
2
i
1
An estimate of variance from a sample of past value is calculated as:
n
  xi  x 
n 1
Estimate of Variance =  2  1
2
1
A 'hat' ^ usually denotes that it is an estimate. For convenience, we often don't bother putting a ^,
but the it is clear from the context or the formula used. (In general, whenever one is multiplying
probabilities to get an expectation of something, then it is not an estimate, but when one sees 1/n,
1/n-1, 1/n-2, in a formula, it is an estimate from a random sample).
Standard deviation is an alternative measure of dispersion. It is simply the square root of the
variance and is denoted by . The advantage of standard deviation is that it is easier to interpret
since it has the same units as x. Sometimes in finance, however, variance is algebraically easier
to use when solving an equation. We will use both standard deviation and variance depending on
the algebraic convenience. They both capture the same underlying concept.
Standard Deviation =  =  2
Covariance measures the tendency of two random variables x and y to move together.
Mathematically, it is defined as the expected value of the product of deviations of x and y from
their expected values respectively:
Covariance between x and y = Cov(x,y) = E xi  E x yi  E y
=
1 N
 pi  xi  E xi yi  E  yi 
N 2 1
From n past values of x and y pairs, covariance is estimated as:
Estimate of covariance = Cov(x,y) =
1 n
  x i  x  yi  y 
n2 1
Page 3
Notice that a positive covariance means that, on average, when a particular value of x is above its
average  x  , that is to say the first [] in {} is positive, then y also tends to be above its average
 y  , i.e., the second [] in {}also tends to be positive. For negative covariance, when a particular
value of x is above its average  x  , that is to say the first [] in {} is positive, then y, on average,
tends to be below its average  y  , or vice versa.
Covariance, although it is sometimes easier to use algebraically, is difficult to interpret since its
units are dependent on the units of x and y. Thus, it can take any value, and it is sometime
difficult to judge whether a particular value of covariance is 'large' or 'small'. An alternative
standardized measure of the tendency of co-movement is the correlation coefficient. It is simply
the Cov(x,y) divided by the standard deviation of x and standard deviation of y and is denoted by
 X ,Y .
Correlation coefficient of x and y =  X ,Y =
Cov( x, y)
x  y
The correlation coefficient is easier to interpret because its value always lies between (and
including) ±1. It has no units. If  = 1 then we say there is perfect positive correlation between x
and y. In other words, x and y always move in the same direction. If  = -1, then we say there is
perfect negative correlation, i.e., x and y always move in the opposite direction. If  = 0, then
there is no association between x and y, and we can say that a movement of x in one direction
does not tell us anything about the direction of the movement in y.
Finally, beta is yet another measure of an association between two random variables. It measures
the sensitivity of a variable y to another variable x. Sometimes, we have reason to believe that
movements in x influences some or all of the movements in y, and in that case we measure the
extent of this influence by beta. For example, in finance, it is typical to assume that returns on a
particular stock is influenced to some degree by the overall 'market' returns.
Beta is estimated from past values of x and y by a Simple Linear Regression of y on x as in
y =  + x.
It is convenient to use a spreadsheet like Excel to estimate beta from a sample of past values of x
and y. When y (left-hand-side variable) is regressed on x (right-hand-side variable) as in the
above equation, mathematically, beta is defined as:
Beta =  =
Cov( x , y )
 2X
In theory, beta can be any number, including negative numbers. In practice, we rarely observe
stock beta greater than 3 or 4 in absolute value. Notice again that a negative beta implies that the
covariance (or the correlation coefficient) between x and y must be negative, since variance of x
cannot be negative.