Math 325
Ch 10 – The Term Structure of Interest Rates
“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.
1.Normal Yield Curve : rates increase with the length of the investment. Explained by:
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Expectations Theory : a higher percent of business and individuals expect interest rates to rise
in the future than the percent that expect rates to fall.
Liquidity Preference Theory : assumes individuals and firms prefer to invest for short periods
rather than long periods so that they will have early access to their funds, i.e. remain “liquid.” A
higher rate of interest with a longer term is an incentive to induce investors to commit their
funds for longer times.
Inflation Premium Theory: assumes investors demand higher rates of interest on longer
investments to protect against uncertainity of future rates of inflation.
2. Inverted Yield Curve : short-term rates exceed long-term rates.
Possible explanation: In the bond market, short-term interest rates are heavily influenced by the
policies of the Federal Reserve Board, which may be setting high short-term rates to fight current
inflation or to remove excess liquidity from the market.
Long-term rates are determined by demand and supply in the bond market and may lower due to
expectations of lower inflation rates in the future, or the possibility of economic recession or slow
down.
3. Flat Yield Curve : it is not necessarily perfectly 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
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Math 325
Ch 10 – The Term Structure of Interest Rates
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.
Example 1. You are given the following selected values from the yield curve:
Term
1-yr
Spot Rate 7.00%
2-yr
3-yr
4-yr
5-yr
8.00%
8.75%
9.25%
9.50%
a) Find the PV of payments of $1000 at the end of each year for 5 years using these spot rates.
b) What level yield rate would produce an equivalent value?
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Math 325
Ch 10 – The Term Structure of Interest Rates
Spot Rates and Bond Yields
Key concept: A coupon bond can be decomposed into a series of zero-coupon bonds, each of which can
be valued precisely using its associated spot rate.
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Math 325
Ch 10 – The Term Structure of Interest Rates
Example 2. Suppose that the current term structure has the following spot rates (annual effective)
Term
½ -year
1-yr
1.5-yr
2-yr
Spot Rate
8%
9%
10%
11%
I)Find the price of a 2-yr $100-par value bond with
a) no coupons
b) 5% semiannual coupons
c) 10% semiannual coupons
II) Find the at-par yield rate for the bond.
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Math 325
Ch 10 – The Term Structure of Interest Rates
Example 3. You are given the following prices of $1000 par value bonds with 10% annual coupons.
Term
Price
1-yr
1028.04
2-yr
1036.53
3-yr
1034.47
Find the spot rates for t = 1,2, 3 that are implied by these bond prices.
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Math 325
Ch 10 – The Term Structure of Interest Rates
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 325
Ch 10 – The Term Structure of Interest Rates
Example 4. Given the following spot rates, find all one-year forward rates that can be determined from
this information
Term
Spot Rate
1-yr
6%
2-yr
6.25%
3-yr
7%
4-yr
7.5%
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Math 325
Ch 10 – The Term Structure of Interest Rates
Example 5. Consider the forward rates defined by
fk = 0.09 + 0.002k – 0.002k2
Find the 4-year spot rate.
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for k = 0, 1, 2, 3, 4