APS 502 Financial Engineering I Final Exam December 17,
2014
Rules: Closed book and closed notes except for both sides of a 3 by 5
inch note card. A non-programmable and non-…nancial calculator is permit-
ted. NEATNESS COUNTS! so be legible. SHOW ALL WORK. YOU MUST
TURN IN THIS EXAM (questions) along with your answer booklet. An IM-
PORTANT part of the exam is interpretation of the exam problems and so NO
questions will be taken that ask for clari…cation of the exam problems!
n+1
A potentially useful formula: 1 + b + b2 + + bn = 1 1b b .
Problem 1 (10 points, 5 points each)
(a) Determine the length of time necessary for $1,000 to grow to $1,500 if
the nominal continuously compounded interest rate is 6%.
(b) An individual who plans to retire in 20 years has decided to put an
amount A in the bank at the beginning of each of the next 240 months, after
which she will withdraw $1,000 at the start of each of the following 360 months.
Assuming a nominal yearly interest rate of 6% compounded monthly, how large
does A need to be?
Problem 2 (10 points, 5 points each)
A 3 year US treasury bond has a face value of $1000 and an annual coupon
rate of 8% (coupon payments are semi-annual). The one year spot rate is
s1 = 2% and f1;2 = 4%, and f2;3 = 5%:
(a) Compute the price of the bond.
(b) Even though we do not know what the exact interest rate will be in the
future explain why forward rates are enough to compute the price of a bond.
Illustrate on the cash‡ow obtained from the bond above at the end of the second
year.
Problem 3 (20 points)
The correlation between securities A and B is 0.1 with expected returns
and standard deviations for each security given by the table (note that =
AB = A B )
Security i i
A 10% 15%
B 18% 30%
(A1) (14 points) Using the Lagrangian method …nd the proportions xA of
A and xB of B that de…ne a portfolio of A and B having minimum standard
deviation (short selling is allowed).
(A2) (3 points) What is the value of the standard deviation of portfolio in
(A1)?
(A3) (3 points) What is the expected return of the portfolio in (A1)?
Problem 4 (10 points)
Suppose there are only 3 risky assets in the market. The covariance matrix
and return vector is given as follows:
1
2014
Rules: Closed book and closed notes except for both sides of a 3 by 5
inch note card. A non-programmable and non-…nancial calculator is permit-
ted. NEATNESS COUNTS! so be legible. SHOW ALL WORK. YOU MUST
TURN IN THIS EXAM (questions) along with your answer booklet. An IM-
PORTANT part of the exam is interpretation of the exam problems and so NO
questions will be taken that ask for clari…cation of the exam problems!
n+1
A potentially useful formula: 1 + b + b2 + + bn = 1 1b b .
Problem 1 (10 points, 5 points each)
(a) Determine the length of time necessary for $1,000 to grow to $1,500 if
the nominal continuously compounded interest rate is 6%.
(b) An individual who plans to retire in 20 years has decided to put an
amount A in the bank at the beginning of each of the next 240 months, after
which she will withdraw $1,000 at the start of each of the following 360 months.
Assuming a nominal yearly interest rate of 6% compounded monthly, how large
does A need to be?
Problem 2 (10 points, 5 points each)
A 3 year US treasury bond has a face value of $1000 and an annual coupon
rate of 8% (coupon payments are semi-annual). The one year spot rate is
s1 = 2% and f1;2 = 4%, and f2;3 = 5%:
(a) Compute the price of the bond.
(b) Even though we do not know what the exact interest rate will be in the
future explain why forward rates are enough to compute the price of a bond.
Illustrate on the cash‡ow obtained from the bond above at the end of the second
year.
Problem 3 (20 points)
The correlation between securities A and B is 0.1 with expected returns
and standard deviations for each security given by the table (note that =
AB = A B )
Security i i
A 10% 15%
B 18% 30%
(A1) (14 points) Using the Lagrangian method …nd the proportions xA of
A and xB of B that de…ne a portfolio of A and B having minimum standard
deviation (short selling is allowed).
(A2) (3 points) What is the value of the standard deviation of portfolio in
(A1)?
(A3) (3 points) What is the expected return of the portfolio in (A1)?
Problem 4 (10 points)
Suppose there are only 3 risky assets in the market. The covariance matrix
and return vector is given as follows:
1
