APS 502 Financial Engineering Final Exam Fall 2017
Instructions: Closed book and closed notes except for both sides of a 3 by
5 inch notecard. Only a simple scienti…c non-…nancial calculator with no pro-
gramming capability is allowed. Please write neatly as this will aid in provid-
ing maximum partial credit. IMPORTANT: Interpretation of the exam
questions is part of the exam and so no questions will be taken that
ask for clari…cation of the question. You must turn in this ques-
tion sheet with your answer booklet or else your exam will NOT be
marked. SHOW ALL WORK.
Problem 1 (15 points, 5 points each)
Consider the following three zero coupon bonds
Bond Maturity (years) Price
1 1 97.56
2 2 95.18
3 3 92.86
Assume that the face value is $100 for all bonds. Suppose you have a liability
of 10 million dollars every year for the next three years (the liability is due at
the end of each year).
(a) What is the present value and duration of your liabilities?
Solution: Present value = $10 million (0:9756+0:9518+0:9286) = $28:56
million
(b) Suppose that you will use 20 million dollars to help o¤set the liabilities
and you will invest this amount now in some of the bonds above. In order to
protect against interest rate changes, which 2 bonds should you select to invest
in and what percent of your 20 million dollars should go into each of these
bonds?
Solution: The duration of the liability D = $10 million (0:9756 + 2
0:9518 + 3 0:9286)=28:56 = 1:98 years. Now PP = D y=(1 + y) so to
perfectly hedge interest rate risk we need P of liability = P of asset, then
(D of asset) (P of asset)=(D of liability) (P of liability) which gives
(D of asset) 20=1.98 28.56 so (D of asset)=2.83 years. To create the
bond portfolio to achieve such a duration solve
2x + 3(1 x) = 2:83 to get x = 0:17 which is the investment in 2 year
zero-coupon bonds and the 0.83 proportion in 3 year zero coupons.
(c) If the interest rates increase by 0.10%, how much will be the remaining
amount of liability?
Solution: If you invest in the bonds according to part (b) then you are
perfectly hedged and any reasonable change in interest rate will have no impact
on your liabiity.
Problem 2 (10 points, 5 points each)
You purchased a 3 year coupon bond one year ago. Its face value is $1000
and the coupon rate is 6% with annual coupon payments. At the time you
1
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, purchased the bond, its yield was 6.5%. Suppose that you sell the bond after
receiving the …rst coupon payment.
(a) What is the total rate of return from holding the bond for the year if the
yield remains at 6.5% when you sell it?
Solution: See the midterm solutions.
(b) What if the yield to maturity becomes 6% when you sell it?
Solution: See the midterm solutions.
Problem 3 (12 points)
Which of the following portfolios can’t be on the mean-variance e¢ cient
frontier? You must justify your answer mathematically and NOT through any
visual diagrams.
portfolio expected return standard deviation
Q 10% 15%
R 10.5% 16.5%
S 11.5% 18.5%
T 12.5% 20%
Solution: See the midterm solutions for sketch of solution.
Problem 4 (18 points, 6 points each)
Your current portfolio consists of three assets, the stock of International
Business Machines (IBM), the stock of General Motors (GM) and an investment
in the riskless asset. You know the following about the stocks ( ij denotes the
correlation between asset i and asset j and M denotes the market portfolio):
IBM;M = 0:60 GM;M = 0:80
2 2
IBM = 0:0900 GM = 0:0625
You also have the following data about the market portfolio M and the
riskless asset f :
M = 0:13 rf = 0:04
2
M = 0:04
Suppose that individuals can borrow and lend at risk-free rate rf and that
the Capital Asset Pricing Model (CAPM) describes expected returns on risky
assets. Your current portfolio has $200,000 invested in IBM, $200,000 invested
in GM, and $100,000 invested in the riskless asset.
(a) What are the expected returns on IBM stock and GM stock?
2
This study source was downloaded by 100000855420558 from CourseHero.com on 10-22-2022 21:43:45 GMT -05:00
https://www.coursehero.com/file/36501211/final-APS502-fall-2017-partial-solutionspdf/
Instructions: Closed book and closed notes except for both sides of a 3 by
5 inch notecard. Only a simple scienti…c non-…nancial calculator with no pro-
gramming capability is allowed. Please write neatly as this will aid in provid-
ing maximum partial credit. IMPORTANT: Interpretation of the exam
questions is part of the exam and so no questions will be taken that
ask for clari…cation of the question. You must turn in this ques-
tion sheet with your answer booklet or else your exam will NOT be
marked. SHOW ALL WORK.
Problem 1 (15 points, 5 points each)
Consider the following three zero coupon bonds
Bond Maturity (years) Price
1 1 97.56
2 2 95.18
3 3 92.86
Assume that the face value is $100 for all bonds. Suppose you have a liability
of 10 million dollars every year for the next three years (the liability is due at
the end of each year).
(a) What is the present value and duration of your liabilities?
Solution: Present value = $10 million (0:9756+0:9518+0:9286) = $28:56
million
(b) Suppose that you will use 20 million dollars to help o¤set the liabilities
and you will invest this amount now in some of the bonds above. In order to
protect against interest rate changes, which 2 bonds should you select to invest
in and what percent of your 20 million dollars should go into each of these
bonds?
Solution: The duration of the liability D = $10 million (0:9756 + 2
0:9518 + 3 0:9286)=28:56 = 1:98 years. Now PP = D y=(1 + y) so to
perfectly hedge interest rate risk we need P of liability = P of asset, then
(D of asset) (P of asset)=(D of liability) (P of liability) which gives
(D of asset) 20=1.98 28.56 so (D of asset)=2.83 years. To create the
bond portfolio to achieve such a duration solve
2x + 3(1 x) = 2:83 to get x = 0:17 which is the investment in 2 year
zero-coupon bonds and the 0.83 proportion in 3 year zero coupons.
(c) If the interest rates increase by 0.10%, how much will be the remaining
amount of liability?
Solution: If you invest in the bonds according to part (b) then you are
perfectly hedged and any reasonable change in interest rate will have no impact
on your liabiity.
Problem 2 (10 points, 5 points each)
You purchased a 3 year coupon bond one year ago. Its face value is $1000
and the coupon rate is 6% with annual coupon payments. At the time you
1
This study source was downloaded by 100000855420558 from CourseHero.com on 10-22-2022 21:43:45 GMT -05:00
https://www.coursehero.com/file/36501211/final-APS502-fall-2017-partial-solutionspdf/
, purchased the bond, its yield was 6.5%. Suppose that you sell the bond after
receiving the …rst coupon payment.
(a) What is the total rate of return from holding the bond for the year if the
yield remains at 6.5% when you sell it?
Solution: See the midterm solutions.
(b) What if the yield to maturity becomes 6% when you sell it?
Solution: See the midterm solutions.
Problem 3 (12 points)
Which of the following portfolios can’t be on the mean-variance e¢ cient
frontier? You must justify your answer mathematically and NOT through any
visual diagrams.
portfolio expected return standard deviation
Q 10% 15%
R 10.5% 16.5%
S 11.5% 18.5%
T 12.5% 20%
Solution: See the midterm solutions for sketch of solution.
Problem 4 (18 points, 6 points each)
Your current portfolio consists of three assets, the stock of International
Business Machines (IBM), the stock of General Motors (GM) and an investment
in the riskless asset. You know the following about the stocks ( ij denotes the
correlation between asset i and asset j and M denotes the market portfolio):
IBM;M = 0:60 GM;M = 0:80
2 2
IBM = 0:0900 GM = 0:0625
You also have the following data about the market portfolio M and the
riskless asset f :
M = 0:13 rf = 0:04
2
M = 0:04
Suppose that individuals can borrow and lend at risk-free rate rf and that
the Capital Asset Pricing Model (CAPM) describes expected returns on risky
assets. Your current portfolio has $200,000 invested in IBM, $200,000 invested
in GM, and $100,000 invested in the riskless asset.
(a) What are the expected returns on IBM stock and GM stock?
2
This study source was downloaded by 100000855420558 from CourseHero.com on 10-22-2022 21:43:45 GMT -05:00
https://www.coursehero.com/file/36501211/final-APS502-fall-2017-partial-solutionspdf/
