Term Structure of Interest Rates
FNCE 4070
Financial Markets and Institutions
Same Maturity Different Yields
Default Risk
• The spread between the interest rates on
bonds with default risk and default-free
bonds, called the risk premium, indicates how
much additional interest people must earn in
order to be willing to hold that risky bond.
• A bond with default risk will always have a
positive risk premium, and an increase in its
default risk will raise the risk premium.
Credit Rating Agencies
• These companies rate securities according to
how likely they are viewed to default.
• A rating of Baa (BBB) or above is referred to as
Investment Grade
• A bond with a rating below this is known as a
junk bond or high yield bond.
Bond Ratings
Liquidity
• Another attribute of a bond that influences its
interest rate is its liquidity; a liquid asset is one
that can be quickly and cheaply converted into
cash if the need arises. The more liquid an
asset is, the more desirable it is (higher
demand), holding everything else constant.
Income Tax Considerations
• Interest payments on municipal bonds are
exempt from federal income taxes, a factor
that has the same effect on the demand
for municipal bonds as an increase in their
expected return.
• Treasury bonds are exempt from state and
local income taxes, while interest
payments from corporate bonds are fully
taxable.
Yield Curve and Term Structure
• Yield Curve – a collection of yields to maturity
on bonds with differing maturities but the
same risk, liquidity and tax considerations
• Term Structure – the relationship between the
different yields to maturity for a given yield
curve.
Criteria for Term Structure Theories
• The goal of a term structure theory is to explain
– Why interest rates on bonds of different maturities
move together
– When short-term interest rates are low, yield curves
are more likely to have an upwards slope; when short
term rates are high, yield curves are more likely to
have a downwards slope and be inverted.
– Yield curves almost always have an upwards slope.
Yield Curve Models
• Interest rates for different maturities
move together.
Term Structure Theories
• Expectations theory
– The interest rate on a long-term bond will equal an average of the
short-term interest rates that people expect over its life.
• Market Segmentation theory
– The markets for different-maturity bonds as completely separate and
segmented
– An extension of this theory is the Preferred Habitat theory which
proposes that investors and borrowers may stray away from their
preferred market segment if there are relatively better rates to
compensate them
• Liquidity Premium theory
– The interest rate on a long-term bond will equal the average of the
short term interest rates expected to occur over the life of the long
term bond plus a liquidity premium that responds to the supply-anddemand conditions for that bond.
Discount Factor
• A discount factor is the value today of 1 dollar
paid at a specific time in the future.
PV(cashflow) = cashflow ´ df
• In our standard interest rate formula we have
df (r, t) =
• Note
1
(1+ r )
t
df (t0, t2 ) = df (t0, t1 ) * df ( t1, t2 )
Treasury Note
• Maturity 2-10 years
• Auctioned monthly in the same way as Treasury
Bills
• Semi-annual coupons of ½ the interest rate
• These are auctioned using a semi-annual yield to
maturity
• The coupon rate on the bond is set to the largest
rate divisible by 0.125% such that the premium
on the bond is less than par. The rate is floored at
0.125%.
Expectations Theory
• 1 – Interest rates move together – as long term rates are an
average of short term rates movements in short term rates
will imply movements in long term rates
• 2 – Yield curves tend to have an upward slope when short
term rates are low and when short term rates are high the
yield curve is generally inverted. If one thinks that rates
revert to some sort of mean then if short term rates are
low we will see long term rates much higher and vice versa.
• 3 – The theory does not explain why yield curves are
generally upward sloping. The typical upward slope of yield
curve implies that short-term interest rates are generally
expected to increase.
Liquidity Premium theory
• This theory extends the Expectations theory to
account for the fact that yield curves are
generally upward sloping.
– If the expectations theory were true then short
term interest rates would generally rise.
• The Liquidity premium theory adds a positive
premium to compensate investors for the
additional risk of longer term bonds.
Liquidity Premium Theory
• Approximation:
int =
e
i et + it+1
+
+ it+e (n-1)
n
+ lnt
Where int is the interest rate on an n-period
bond starting at time t.
• More precisely:
(1+ int )
n
= (1+ ite ) (1+ i et+1 )
(
)
1+ it+e (n-1) + lnt
Important Point
• The liquidity premium in the two formulas on
the previous slide do not have the same
meaning.
• IF YOU USE THE FIRST FORMULA FOR THE
HOMEWORK OR MIDTERM YOU WILL
GENERALLY GET THE ANSWER WRONG!!!!
–
My daughter chose the formatting
Liquidity Premium Theory
• Explains the third fact that yield curves are
generally upward sloping
– Even if short term rates are expected to stay high
on average you expect longer term rates to be
higher
– A Yield curve will only become inverted if short
term rates are expected to decrease very rapidly.
Forecasting Interest Rates
• Expectations Theory
1+ i0,2 )
(
i1,2 =
-1
(1+ i0,1 )
n+1
1+ i0,n+1 )
(
in,n+1 =
-1
n
(1+ i0,n )
2
Forecasting Interest Rates
• Liquidity Premium Theory
1+ i0,2 ) - l0,1
(
i1,2 =
-1
(1+ i0,1 )
n+1
1+ i0,n+1 ) - l0,n+1
(
in,n+1 =
-1
n
(1+ i0,n ) - l0,n
2