EE 518 : Midterm 1.30-3.20pm, Wednesday, October 14, 2020
No smart phones or computers are allowed.
Calculators are allowed but to be used for simple numerical and trigonometric calculations.
Results without clearly showing how they are obtained will be given zero credit.
Last Name (CAPITAL):
First Name (CAPITAL):
Student ID:
Check one: Remote[ ] Campus[ ]
Signature:
Problem Maximum Points Grade
1 20
2 20
3 30
4 30
Total 100
Page 1 of 5
EE 518 : Midterm 1.30-3.20pm, Wednesday, October 14, 2020
Problem 1
(a) (10 points) Let F (x) = ∫ x
0
sin(t)
t dt. Using Taylor’s series, show that
F (x) = x x
3
3 · 3! +
x5
5 · 5!
x7
7 · 7! + . . .
(b) (10 points) Use the Mean Value Theorem to show that
| sin(b) sin(a)| ≤ |b a|, for a, b ∈ R, a < b
Page 2 of 5
EE 518 : Midterm 1.30-3.20pm, Wednesday, October 14, 2020
Problem 2
1. (10 points) Let f, g : [a, b] → R be continuous and differentiable functions on [a, b] and let f(a) =
f(b) = 0. Prove that there is a point c ∈ (a, b) such that g′(c)f(c) + f ′(c) = 0.
Hint: Consider h(x) = f(x)eg(x)
2. (10 points) Find the forward rate r(0; t1, t2) given the zero rate curves r(0; t1) and r(0; t2), t2 > t1,
assuming discrete annual compounding, where t1 and t2 are measured in years.
Page 3 of 5
EE 518 : Midterm 1.30-3.20pm, Wednesday, October 14, 2020
Problem 3
(i) (20 points) Consider the stochastic differential equation
dXt = 1
t
X(t)dt+ αtdB(t), X(1) = 0
where α is a constant for t ≥ 1. Find an integral form of the solution X(t).
(ii) (10 points) Lest σXY be the covariance of two random variables X,Y . Prove that
σXY = E[XY ] E[X]E[Y ]
Page 4 of 5
EE 518 : Midterm 1.30-3.20pm, Wednesday, October 14, 2020
Problem 4
You plan to buy a particular car:
a) (5 points) You have the option to buy the car for $50K. You can put down $5K and finance the rest
for 5 years. At the end of the fifth year you want to sell the car. What is the monthly payment for the
car
b) (25 points) You also have the option to lease the car for 5 years by paying upfront $5K and then
making a monthly payment of $400 per month for 60 months. At the end of the 5 years you have the
option to return the car or buy it for the price of $32K. Analyze both options and compare them using
NPV to determine what the reselling price of the car in option (a) should be, for the option in (a) to
be better than the option in (b). Comment on your results.
In both options consider annual nominal rate of 4% compounded monthly.
Hint: ∑ni=1 xi = x 1 xn1 x , if x ∈ [0, 1)
Page 5 of 5
