School

Harvard University **We aren't endorsed by this school

Course

FINANCE APS502

Subject

Accounting

Date

Nov 1, 2023

Type

Other

Pages

3

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APS 502
Financial Engineering I
March. 2, 2023
Instructions: Closed book and closed notes except for one side of a 3 by 5 inch
notecard. Only a simple scienti°c non-°nancial calculator with no programming
capability is allowed. Please write neatly as this will aid in providing maximum
partial credit.
Show All Work.
IMPORTANT: Interpretation of the exam questions is part of the
exam and so no questions will be taken DURING the exam 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.
Problem 1 (21 points, 7 points each)
Suppose that you lend a client $500,000 and the client will pay you back over
the next 30 years using equal monthly payments. Interest will be compounded
monthly with a nominal rate of 8.5%.
(a) What is the e/ective annual rate on this loan?
(b) What is the monthly payment?
(c) How much does the client owe you immediately after the 60th monthly
payment?
Problem 2 (24 points, 8 points each)
Suppose you have the following three bonds. Assume all coupon payments
are annual and that the face value for all bonds is $100.
Bond
Maturity (years)
Yield
Coupon rate
1
1
4%
4%
2
2
4.2%
5%
3
3
4.8%
5%
(a) Determine the price of each bond.
(b) Determine the 1 year, 2 year, and 3 year spot rates. (Hint: The price
of each bond that you computed in (a) is also equal to the present value of the
cash ±ows of the bond where the present value of each cash ±ow is discounted
using the appropriate spot rate.)
(c) Using the spot rates from (b) and assuming expectation dynamics illus-
trate the Invariance Theorem.
Problem 3 (24 points, 6 points each)
You will be paying $10,000 a year in tuition expenses at the end of the next
two years. Bonds currently yield 6%.
(a) What is the present value and duration of your tuition obligation?
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(b) Suppose that you will invest in a single zero coupon bond to o/set the
tuition obligation. What maturity and face value for a zero-coupon bond would
immunize your obligation?
(c) Suppose you buy the zero coupon bond in (b) to immunize your tuition
obligation. What happens to your net position (that is, to the di/erence between
the value of the zero coupon bond and that of your tuition obligation) if yields
immediately increases to 9%. (Answer this by computing the new net position.)
(d) Repeat (c) but instead yields decrease immediately to 7%.
Problem 4 (16 points)
Recall the °shing problem from lecture i.e. you will make a decision to °sh
or not °sh at the start of three consecutive periods. If you °sh then you will
get 50% of the °sh in the lake and assume that you will get the °sh at the
start of the period.
If you °sh then the population of the °sh goes back to
90% of what it was at the start of the next period. If you don²t °sh then the
population of the lake doubles at the start of the next period. Assume that the
per period interest rate is 15% and assume that there are 20 °sh in the lake at
the start of the °rst period. Solve for the optimal °shing strategy using dynamic
programming. Draw all relevant °gures and show your work.
Problem 5 (15 points)
Steven is a bond trader that is currently trying to maximize his pro°t in the
bond market. Four bonds are available for purchase and sale (see Table 1). The
bid and ask price of each bond is shown in Table 2. Steve can buy up to 1000
units of each bond at the ask price or sell up to 1000 units of each bond at the
bid price. During each of the next three years, the person who sells a bond will
pay the owner of the bond the cash payments in Table 1. For example, if you
sell one unit of Bond 1 then you must pay the owner 100, 110, and 1100 at the
end of the °rst, second, and third years, respectively. (Of course, if you buy one
unit of Bond 1 then you will receive from the seller 100, 110, and 1100 at the
end of the °rst, second, and third years, respectively.)
Steve²s goal is to maximize his revenue from selling bonds less his payment
for buying bonds subject to the constraint that after each year²s payments are
received, his current cash position which is amount that is the di/erence between
cash (only coupons and face values) received from owning bonds and cash (only
coupons and face valiues) paid out due to selling bonds is non-negative. Assume
that cash payments are discounted, with a payment of $1 one year from now
being equivalent to a payment of 90 cents or 0.90 dollars now.
The selling
and purchasing of bonds occurs only at the start of the °rst year and there
is no selling or purchasing of bonds in later periods.
Formulate an LP that
maximizes net pro°t from buying and selling bonds subject to the constraints
previously described. De°ne all variables in your LP model and explicitly use
actual numerical values for all parameters.
Why do you think we limit the
number of units of each bond that can be bought or sold?
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Year
Bond 1
Bond 2
Bond 3
Bond 4
1
100
80
70
60
2
110
90
80
50
3
1100
1120
1090
1110
Table 1
Bond
Bid Price
Ask Price
1
980
990
2
970
985
3
960
972
4
940
954
Table 2
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