UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Actuarial Science Program
DEPARTMENT OF MATHEMATICS
Math 210
Theory of Interest
Prof. Rick Gorvett
Fall, 2015
Homework Assignment # 7 (max. points = 10)
Due at the beginning of class on Thursday, November 12, 2015
You are encouraged to work on these problems in groups of no more than 3 or 4. However, each
student must hand in her/his own answer sheet. Please show your work – enough to show that
you understand how to do the problem – and circle your final answer. Full credit can only be
given if the answer and approach are appropriate. Please give answers to two decimal places –
e.g., xx.xx% and $xx,xxx.xx .
Note: Homework assignments are due at the beginning of the class. Assignments will not be
accepted after 9:45 am on the due date.
(1)
You purchase a 30-year bond for $1,150. The bond has annual coupon payments at a rate
of 7.5%, and has face / redemption / par values of $1,000. Your effective annual yield on
the bond is i %. Find i. (* See note below.)
(2)
A 20-year 10% semi-annual-coupon bond (meaning that a coupon of 10% / 2 = 5% is
paid every six months, because the “coupon rate” is generally denoted as an annual rate,
even if payments are twice per year) has a face (and redemption) value of $1,000. Find
the price of the bond, assuming a nominal annual interest rate of 12% convertible semiannually.
(3)
A 30-year 8% annual-coupon bond with a face value of $1,000 is purchased at a yield of
5.5%. Thus, the purchase price represents a premium above $1,000. Find the amount of
amortization of premium during the 12th year of the bond.
(4)
A 30-year 8% annual-coupon bond with a face value of $1,000 is purchased at a yield of
12%. Thus, the purchase price represents a discount from $1,000. Find the amount of
the accumulation of discount during the 18th year of the bond.
(5)
A 20-year 10% annual-coupon bond with a face value of $1,000 is purchased at a time
when the interest rate in the market is 7%. Immediately after the 8th coupon payment, the
interest rate in the market is 11%. The book value of the bond immediately after the 8th
coupon payment is BV, and the market value of the bond immediately after the 8th
coupon payment is MV. Find the difference, BV – MV.
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(6)
You purchase a 30-year 8% annual coupon bond with a face value of 1000, at a yield rate
of 9%. The bond is a callable corporate bond, with a call price of 1,050, and can be
called by the issuing corporation after five years. Immediately after the 9th coupon
payment, the issuing corporation redeems the bond. Determine the effective annual yield
you achieved on this 9-year investment. (* See note below.)
(7)
You are given the following term structure (yield curve) of interest rates:
Years
1
2
3
Annual Spot Rate
7.00%
6.30%
5.95%
Find the implied one-year forward rate two years from now (i.e., the interest rate
anticipated between time-points 2 and 3).
(8)
You are given the following term structure (yield curve) of interest rates:
Years
1
2
3
4
Annual Spot Rate
4.0%
4.8%
5.3%
5.6%
Find the implied 3-year forward rate one year from now (i.e., the expected 3-year spot
rate starting one year from now – express it as an annual rate).
(9)
You are given the following term structure (yield curve) of interest rates:
Years
1
2
3
4
Annual Spot Rate
5.0%
5.7%
6.1%
6.3%
You know that, two years from now (at time t = 2), you will want to purchase a two-year
10% annual coupon bond with a face value of $1,000. Find the expected price of that
bond at time t = 2. (Hint: Find the implied one-year forward rate two years from now,
and find the implied two-year forward rate two years from now.)
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(10)
A one-year zero-coupon bond has a price of 94.01. A two-year zero-coupon bond has a
price of X. A three-year zero-coupon bond has a price of 80.52. A three-year 8%
annual-coupon bond has a price of 101.47. All these bonds have face (and redemption)
values of 100. Find the price, X, of the two-year zero-coupon bond.
* (Note: you can determine the yield rate with a calculator, or by using the Solver tool in
Excel. Examples of how to perform this calculation on calculators which are permitted to
be used on the FM exam can be found as follows:
For the TI BA-35 calculator: on pages 14-15, “Finding Bond Yield on a Coupon Date,”
of the instruction manual: http://www.soa.org/files/pdf/FM-22-05.pdf
For the TI BA II Plus: on pages 20-21, “Finding the Bond Yield on a Coupon Date,” of
the instruction manual: http://www.soa.org/files/pdf/FM-23-05.pdf
Other calculators with similar capabilities are likely to use similar techniques.)
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