LOS I requires us to:
explain the use of conditional expectation in investment applications
a. Consider the following expectation’s tree:
In the above tree, there are two events S (having two expectations, i.e. S and S not happening, denoted by SC), and X (having possible values of X1, X2, ….Xn).
Now, if we have to calculate the conditional probability of X, given that S has already happened; we can calculate the same using the following equation:
This can be simplified to the following equation:
b. Similarly, we can also calculate the total expected value of X, we can do so by summing up the possibilities of X in both cases when S has happened and S has not happened. That is
Thus, in the following figure:
the total probability or expected value of S would be:
c. For example, if we extend the previous example to say that event X is succeeded by another event S having a 60% probability of S happening, and a 40% probability of S not happening. If S happens there is a 25% chance of event X1 and a 75% chance of X2. If S does not happens, there is a 60% chance of event X3 and a 40% chance of event X4. Consider the following table:
S |
P(S) |
X |
P(X) |
Xi |
S |
0.60 |
X1 |
0.25 |
2.60 |
X2 |
0.75 |
2.45 |
||
SC |
0.40 |
X3 |
0.60 |
2.20 |
X4 |
0.40 |
2.00 |
We can also draw the same on an event tree, as follows:
Now, the probability of individual values of X would be calculated as follows:
Thus the individual probabilities of values of X would be:
S |
X |
P(X) |
|
P(Xi) |
|
S1 |
0.6 |
X1 |
0.25 |
=0.6*0.25 |
0.15 |
X2 |
0.75 |
=0.6*0.75 |
0.45 |
||
S2 |
0.4 |
X3 |
0.6 |
=0.4*0.6 |
0.24 |
X4 |
0.4 |
=0.4*0.4 |
0.16 |
Thus the probability of X, given that S has already happened would be:
And the probability of X, given that S has not happened would be:
Thus, the total probability is:
d. Consider another example:
Suppose the probability of a bond default is 0.06. And if the bond makes a default its price at the end of the term would be 0. The risk-free rate on treasury bills is 5.8%. How much default risk premium should a bond earn?
Solution:
To calculate the same, we are given the following:
P(X) = 0.06, Rf = 0.058, & E(PB|X) = 0. We have to find the value of R-Rf.
Let the price of the bond, if it does not default be $ (1+R) (where R-Rf is the risk premium).
We can draw the expectation tree for the same as follows:
The value of the bond, after making the necessary provisions for the risk, should offer a return equal to the risk-free rate. Hence its price should be the face value plus the return at a risk-free rate. Thus, considering $ 1 face value:
Thus the risk premium is 6.75% or 0.675 basis points (i.e. 12.55%-5.8%).