APS 502 Financial Engineering I March. 7, 2019
Rongwei Man
Name:_________________________________
1003100388
Student #:______________________________
Instructions: Closed book and closed notes except for one side of a 3 by 5 inch
index card of notes. Only a simple scientific non-financial calculator with no
programming and no memory capability is allowed. Please write neatly as this
will aid in providing maximum partial credit. IMPORTANT: Interpretation
of the exam questions is part of the exam and so no questions will be
taken DURING the exam that ask for clarification of the questions.
You MUST turn in this question sheet with your answer booklet or
else your exam will NOT be marked. SHOW WORK!
Problem 1 (12 points, 4 points each)
You will be buying a 400 square feet condo in Toronto for $600,000. You
will make a $150,000 down payment and borrow $450,000 from the bank. The
amount borrowed will be paid back in equal monthly payments over 25 years
Finitelife
with a nominal interest rate of 8.5%.
(a) What is the e§ective annual rate? EffectiveRate
(b) What is the monthly payment?
(c) How much do you owe the bank immediately after the 47th payment?
streams
Problem 2 (9 points)
Suppose we have the spot rates si ans sj where i < j and the forward rate
fi,j . Suppose that (1 + si )i (1 + fi,j )j!i < (1 + sj )j . Then, specify a sequence of spotrate
trades that represents an arbitrage strategy. Forwardrate
Problem 3 (12 points, 4 points each)
For each assertion below state whether it is TRUE, FALSE, or "it depends".
NO explanation is required.
(a) The term structure of interest rates must be always upward sloping
F
because longer maturity bonds are riskier.
(b) Bonds with higher coupon rates have more interest rate risk.
(c) To reduce interest rate risk, a pension fund with more assets than liabil-
F
ities (i.e. present value of its assets is larger than present value of its liabilities)
F
should invest in assets with longer duration than its liabilities.
Problem 4 (20 points, 4 points each except (c) is 8 points))
Consider the following two bonds. Bond 1 is a one year bond with a face
value of $100 with coupon rate of 7% with semi-annual coupons.
Bond 2 is 5 year bond with face value of $100 with a coupon rate of 9% and Duration of
has annual coupon payments. Assume that the yield is 9% for both bonds.
(a) Find the price P1 of Bond 1 and the price P2 of Bond 2. bond Portfolio
(b) Find the MacCaulay duration D1 of Bond 1 and the MacCaulay duration
D2 of Bond 2 by using the DEFINITION of MacCaulay duration. Immunization
1
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, (c) Suppose that you have a liability of $400 at the end of year 2, a $500
liability at the end of year 3 and a liability of $1500 at the end of year 4. Find
a bond portfolio of Bond 1 and Bond 2 that will meet the liabilities but is
immunized against changes in interest rate.
(d) Recall that bonds with shorter duration are less sensitive to changes in
interest. For the following assertion determine whether it is true or false and
provide a brief explanation. "It is always better to buy as many bonds
as possible with shorter duration."
Problem 5 (12 points)
A small pension fund has the following liabilities (in millions of dollars)
Lpofbond
year 1 2 3
liability 24 26 28
For example, the pension fund has to payout 24 millions dollars at the end
of year 1, 26 millions dollars after year 2, etc...
portfolio
The pension fund will use the following bonds to construct a bond portfolio construction
that they will form today and hold until all bonds expire and whose cash flows
problem
will o§set the liabilities.
Bond 1 2 3 4 5
Price ($) 102.44 99.95 100.02 102.66 87.90
Coupon (%) 5.625 4.75 4.25 5.25 0.00
Maturity (year) 1 2 2 3 3
Rating A B A A B
All bonds have a face value of $100 and coupon payments are made annually.
The rating quality is a meaure of the likelihood that the issuer of the bond will
pay coupon payments and face value. An ’A’ rating is better than a ’B’ rating.
Formulate the best linear programming model that you can (assume any
carry over of cash does NOT earn interest) that will construct the minimum
cost bond portfolio that will meet the liabilities subject to that no more than
50% of the bond portfolio’s (dollar) value can be in bonds with a rating of B.
Problem 6 (20 points)
You are asked to price some options on ABC stock. ABC’s stock price can
go up by 15 percent every year, or down by 10 percent.The risk free rate is 5
percent and the current price of ABC stock is $100 per share. Consider a 3
period (i.e. three year) binomial tree to model the price of ABC stock for parts
(a) and (b) below.
(a) (8 points) Price a European put option on ABC stock with a maturity
of 3 years and a strike price of $95.
(b) (8 points) Price a European call option on ABC stock with a maturity
of three years with a strike price of $100 but with a knock-out value of $125
2
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Rongwei Man
Name:_________________________________
1003100388
Student #:______________________________
Instructions: Closed book and closed notes except for one side of a 3 by 5 inch
index card of notes. Only a simple scientific non-financial calculator with no
programming and no memory capability is allowed. Please write neatly as this
will aid in providing maximum partial credit. IMPORTANT: Interpretation
of the exam questions is part of the exam and so no questions will be
taken DURING the exam that ask for clarification of the questions.
You MUST turn in this question sheet with your answer booklet or
else your exam will NOT be marked. SHOW WORK!
Problem 1 (12 points, 4 points each)
You will be buying a 400 square feet condo in Toronto for $600,000. You
will make a $150,000 down payment and borrow $450,000 from the bank. The
amount borrowed will be paid back in equal monthly payments over 25 years
Finitelife
with a nominal interest rate of 8.5%.
(a) What is the e§ective annual rate? EffectiveRate
(b) What is the monthly payment?
(c) How much do you owe the bank immediately after the 47th payment?
streams
Problem 2 (9 points)
Suppose we have the spot rates si ans sj where i < j and the forward rate
fi,j . Suppose that (1 + si )i (1 + fi,j )j!i < (1 + sj )j . Then, specify a sequence of spotrate
trades that represents an arbitrage strategy. Forwardrate
Problem 3 (12 points, 4 points each)
For each assertion below state whether it is TRUE, FALSE, or "it depends".
NO explanation is required.
(a) The term structure of interest rates must be always upward sloping
F
because longer maturity bonds are riskier.
(b) Bonds with higher coupon rates have more interest rate risk.
(c) To reduce interest rate risk, a pension fund with more assets than liabil-
F
ities (i.e. present value of its assets is larger than present value of its liabilities)
F
should invest in assets with longer duration than its liabilities.
Problem 4 (20 points, 4 points each except (c) is 8 points))
Consider the following two bonds. Bond 1 is a one year bond with a face
value of $100 with coupon rate of 7% with semi-annual coupons.
Bond 2 is 5 year bond with face value of $100 with a coupon rate of 9% and Duration of
has annual coupon payments. Assume that the yield is 9% for both bonds.
(a) Find the price P1 of Bond 1 and the price P2 of Bond 2. bond Portfolio
(b) Find the MacCaulay duration D1 of Bond 1 and the MacCaulay duration
D2 of Bond 2 by using the DEFINITION of MacCaulay duration. Immunization
1
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, (c) Suppose that you have a liability of $400 at the end of year 2, a $500
liability at the end of year 3 and a liability of $1500 at the end of year 4. Find
a bond portfolio of Bond 1 and Bond 2 that will meet the liabilities but is
immunized against changes in interest rate.
(d) Recall that bonds with shorter duration are less sensitive to changes in
interest. For the following assertion determine whether it is true or false and
provide a brief explanation. "It is always better to buy as many bonds
as possible with shorter duration."
Problem 5 (12 points)
A small pension fund has the following liabilities (in millions of dollars)
Lpofbond
year 1 2 3
liability 24 26 28
For example, the pension fund has to payout 24 millions dollars at the end
of year 1, 26 millions dollars after year 2, etc...
portfolio
The pension fund will use the following bonds to construct a bond portfolio construction
that they will form today and hold until all bonds expire and whose cash flows
problem
will o§set the liabilities.
Bond 1 2 3 4 5
Price ($) 102.44 99.95 100.02 102.66 87.90
Coupon (%) 5.625 4.75 4.25 5.25 0.00
Maturity (year) 1 2 2 3 3
Rating A B A A B
All bonds have a face value of $100 and coupon payments are made annually.
The rating quality is a meaure of the likelihood that the issuer of the bond will
pay coupon payments and face value. An ’A’ rating is better than a ’B’ rating.
Formulate the best linear programming model that you can (assume any
carry over of cash does NOT earn interest) that will construct the minimum
cost bond portfolio that will meet the liabilities subject to that no more than
50% of the bond portfolio’s (dollar) value can be in bonds with a rating of B.
Problem 6 (20 points)
You are asked to price some options on ABC stock. ABC’s stock price can
go up by 15 percent every year, or down by 10 percent.The risk free rate is 5
percent and the current price of ABC stock is $100 per share. Consider a 3
period (i.e. three year) binomial tree to model the price of ABC stock for parts
(a) and (b) below.
(a) (8 points) Price a European put option on ABC stock with a maturity
of 3 years and a strike price of $95.
(b) (8 points) Price a European call option on ABC stock with a maturity
of three years with a strike price of $100 but with a knock-out value of $125
2
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