H
AAS
S
CHOOL
OF
B
USINESS
U
NIVERSITY
OF
C
ALIFORNIA
AT
B
ERKELEY
UGBA 103
Avinash Verma
U
NCERTAINTY
AND
R
ISK
IN
F
INANCE
This handout is intended to be a
brief
preview of the statistics that we shall be using
in the course. Throughout this note,
key words
and concepts are identified by
bold
font.
Our goal is twofold:
F
IRST
, we would like to
measure
or
quantify
the
risk
associated with future
values of financial variables, such as cash flows, prices, and returns.
S
ECOND
, we want to be able to
factor
that risk into the rate of discount. In
other words, having measured the risk, we’d like to
price
it. That is to say,
we are interested in working out the appropriate
reward
per unit of
measured risk that investors expect to receive.
This note deals only with the first goal. Capital Asset Pricing Model [CAPM], to be
discussed in class later, deals with the second goal.
1.
There is
uncertainty
about the future value of most financial
variables
, such as prices, cash flows, returns, dividends, exchange
and interest rates, etc. In Finance, we deal with a situation of
uncertainty by turning it into a situation of
risk.
This is
accomplished
by imposing
a
probability distribution
on the future
values that financial variables can assume. When we do that, we
treat these financial variables as
random variables
.
2.
A random variable is a variable that takes on different values based
on the outcome of a
random
or a
chance
event. These different
values are called
realizations
of the random variable. For example,
if we bet $1 on the event that toss of an unbiased coin will result in
“heads up,” then the amount we win at the end of a
single
toss,
denoted
~
X
, is a random variable that can assume two possible
values. The first possible realization
of
~
X
, denoted
X
1
, occurs if
we win, and is equal to $1. The other possible realization
of
~
X
,
denoted
X
2
, equals
1
$

, and occurs if we lose. Randomness of a
variable is often denoted by putting a
tilde
(~) on top of the symbol
for the variable. Of course, once the outcome is known, there is no
further uncertainty. Therefore, the realizations are not random,
which is the reason why there is no tilde above the symbols used to
denote them.
3.
Imposing a probability distribution involves two tasks:
Making a
mutually exclusive
and
collectively exhaustive
list of
events that can occur in future. The fact that events in the list are
mutually exclusive means that no two events can occur at the
same time. And if taken together or collectively, the events in the
list exhaust all future possibilities, then, clearly, at least one event
from the list must occur.
Uncertainty and Risk in Finance
1