Cornell University
Fall 2009
Economics 3330: Problem Set 1 Solutions
1.
Derivatives
Solve for the first and second derivatives with respect to
x
for each of the following:
a. (1+x)
-t+2
:
-(t-2)(1+x)
-t+1
, (t-2)(t-1)(1+x)
-t
b. e
rx
: re
rx
, r
2
e
rx
2.
Expected Value and Variance
There are two securities: X and Y.
Security X takes the values -2, 4, and 10 in states L, M and H
with probabilities 0.25, 0.5, and 0.25, respectively.
Security Y takes the values -6, 4, and 14 with
the same probabilities
a.
What is the expected value of each security? E[X] = .25*(-2)+.5*4+.25*10=4. E[Y]=4.
b.
What is the variance of each security’s value? Var[X] = .25*(-2-4)
2
+ .5*(4-4)
2
+.25*(10-4)
2
=
18.
Var[Y] = 50.
c.
Which security would you expect to be more expensive?
Explain your answer.
Since the securities have the same return, but Y has a higher variance, one would usually
expect X to be more expensive.
d.
Now suppose that there is a trader who can purchase either one or the other of the securities.
The trader will receive 20% of the value of the security above zero as compensation; i.e., the
trader receives a portion of the gains but none of the losses.
What is the expected value
received by the trader for each security?
Explain why it is possible that the trader would be
willing to pay more for security Y than X.
The expected returns for a trader holding X and Y are 0.9 and 1.1, respectively.
The variance
of holding Y is higher is well.
The trader would be willing to pay more for Y than X if the
additional return outweighed the increased variance.
Now suppose that Security A takes the values -12, 3, and 4 in states L, M, and H with
probabilities 0.1, 0.6, and 0.3, respectively.
Security B takes the values 0, 1, and 5 with the same
probabilities.
e.