Download 225exam1solutions

Math 225 – Financial Mathematics
September 29, 2004
name______________________________
Exam 1 – Due by 11:10am Friday, 10/1/2004
You have two hours to take this exam. You may take it when and where you like.
You may use a calculator but not a computer, references, or consultation (except with me,
[email protected] or cell 610-220-4382). You may use this paper, your own
paper, or both. It isn’t necessary to turn in scratch paper.
Return this exam in class, to my office, or to the mathematics office by 11:10 am Friday.
This exam has 7 numbered problems on 4 numbered pages.
Enjoy!
It is never necessary to reduce your answer to a number. An explicit formula (that is, all
numbers with no variable or function names) is always sufficient, if it’s correct and can
actually be evaluated. (Of course, you can usually be more confident of your answer if
you can see that it is a sensible number.)
1. (15 pts.) Let A and B be independent events with
P ( A ) = 0.10,
P ( B ) = 0.60.
a. What is P ( AB ) ?
0.06
b. What is P ( ABC) ?
0.04
c. What is the probability that neither A nor B occurs ?
0.36
1
2. (25 pts.) Let X be a random variable with the following distribution function:
a
2
3
4
5
P(X=a)
0.10
0.20
0.20
0.50
a. What is P ( 3 ≤ X ≤ 5 ) ?
0.20 + 0.20 + 0.50 = 0.90
b. What is E ( X ) ?
2(0.10) + 3(0.20) + 4(0.20) + 5(0.50) = 4.1
c. What is E ( X2 ) ?
22(0.10) + 32(0.20) + 42(0.20) + 52(0.50) = 17.9
d. What is the variance of X ?
Var(X) = E(X2) – E(X)2 = 17.9 – 4.12 = 1.09
e. What is the standard deviation of X ?
X =
Var(X )  1.044
2
3. (20 pts.) A certain contract gives you the right to receive $1000 per year for 5 years,
starting one year from now. (That means that the final payment, the fifth payment, is also
for $1000 and occurs exactly five years from now.)
For this problem, assume that anyone can borrow or lend any amount at an interest rate of
4% (continuously compounded).
a. What is the present value of the final payment ?
$1000 e-(0.04)5  $818.73
b. Write an explicit formula for the present value of the entire contract.
[ $1000 e-0.04 + $1000 e-0.08 + $1000 e-0.12 + $1000 e-0.16 + $1000 e-0.20 ]
c. Is the present value of the contract…
(a) less than $4800 ?
(b) between $4800 and $5200, inclusive ? or,
(c) more than $5200 ?
4. (10 pts.) If the price of a share of stock changes from $40 yesterday to $50 today, what
is the (logarithmically-defined) daily return for today ?
(That is, what is L(t)? Assume that there are no dividends, splits, mergers, or other such
changes.)
Typo on my part. The return is R(t).
L(t) = ln(50) = 3.912…
Logarithmically-defined daily return
= R(t) = ln(50)-ln(40) = ln(1.25) = 22.31 %
Or: A(t) is clearly 25%, so R(t) = ln(1+A(t)) = 22.31 %.
3
5. (10 pts.) A continuously compounded interest rate of 4% ( or 0.04 ) produces the same
result as an annually compounded rate (“effective rate”) of R. What is R ?
(If your answer is a number, it should be within 0.0001 of the exact value.)
R = e0.04 – 1 = 4.08 %
6. (10 pts.) A certain bond guarantees a payment of $100 exactly 2 years from now (and no
other payments). According to the newspaper, the bond can be bought or sold on Wall
Street for $ 96.00. What is the yield on the bond ?
r satisfies 96.00  100 e r 2 , so
r = -(1/2)ln(0.96) = 2.0411 %.
7. (10 pts.) Suppose that the forward interest rate curve, r(t), is given by the function
0.06 if t  4.0 years,
r (t )  
0.04 if t  4.0 years.
What is the present value (at time 0) of a guaranteed payment of $1 at time 6 years?
That is, what is P ( 6 ) ?

P (6)  exp  
6
s 0
r (s )ds

 exp   (0.06)  4  (0.04)  2    e 0.32  0.726
(end of exam)
4