Math 420 – Spring 2016
The Term Structure of Interest Rates
Problem Set 7
“Term structure of interest rates” refers to the phenomenon in which rates of interest differ depending
on the term of otherwise identical financial instruments. The graph that displays this relationship
between rates on interest and the term of the investment is often called a yield curve.
•
Normal Yield Curve : rates increase with the length of the investment.
•
•
Occasionally, the yield curve is inverted.
When the yield rates are constant regardless of the term of an investment, we say the yield
curve is flat. It may occur in periods of stability in which investors do not expect dramatic
changes in the economy, investment rates, or future inflation rates.
How are yield curves determined? The most basic yield curve is determined by the yields on
zero-coupon bonds of varying terms backed by the US Treasury. We can also construct a yield curve
based on corporate bonds.
Spot Rates
The interest rates on the yield curve are often called spot rates.
Notation:
st = spot rate for a term of length t, expressed as an annual effective rate for any value of t.
Forward Rates : an expected spot rate which will come into play in the future (i.e. a rate of interest
that can be earned on an investment made at a future point in time).
A set of current spot rates will imply a set of forward rates.
Unless told otherwise, forward rates are quotes as annual effective rates.
Notation: Let ft be the 1-yr forward rate which applies from year t to t + 1.
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Math 420 – Spring 2016
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The Term Structure of Interest Rates
Problem Set 7
Math 420 – Spring 2016
The Term Structure of Interest Rates
Problem Set 7
1. An insurance company invests in two $100 par value bonds with 5% annual coupons and each
maturing at par. The one year spot rate is 2.00%. The price and term of each bond is as follows:
Bond I: Term 2 years, price = 103.87
Bond II: Term 3 years, price = 104.32
Let s2 be the 2 year spot rate and s3 be the 3 year spot rate implied by those bond prices.
Calculate s2 + s3.
2.One-year implied forward rates on zero-coupon bonds are as follows:
•3.6% (year 0 to year 1)
•4.8% (year 1 to year 2)
•5.8% (year 2 to year 3)
Compute the effective annual interest rate for a 2-year zero-coupon bond.
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Math 420 – Spring 2016
The Term Structure of Interest Rates
Problem Set 7
3. Zero-coupon bond prices per $1 of maturity payment are as follows:
•0.965251 (1-year)
•0.919245 (2-year)
•0.878817 (3-year)
Compute the implied one-year forward rate, starting in 2 years.
4. You are considering purchasing one of two assets.
•Asset 1 has payments of 5,000 at the end of year 1, 10,000 at the end of year 3, and 15,000 at
the end of year 5. The price for Asset 1 today is 26,000.
•Asset 2 has payments of 12,000 at the end of year 4 and 20,000 at the end of year 5. The price
of the asset 3 years from now is 29,500.
If the current spot curve is below, what is 1f3 (the one year rate three years forward)?
Term
1
2
3
4
5
Spot Rate
3%
3.4%
s3
s4
4.25%
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Math 420 – Spring 2016
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The Term Structure of Interest Rates
Problem Set 7