NOTES N
The Risk and Term Structure of Interest Rates
Historic Interest Rates
Why do bonds with same term to maturity have different rates of return?
=> Study the Risk Structure of Interest Rates
, 1-2-2002
Why do bonds with different term to maturity have different rates of return?
-> Study the Term Structure of Interest Rates
A: Risk Structure
Default Risk
= Issuer unable to pay either the interest or the face value
Investors get positive Risk Premium for holding risky assets
How To Gauge Default Risk: Credit Rating Companies (e.g. Moody, S&P)
Investment advisory firms that provide default risk information. That is:
how likely is the borrower to meet its financial obligations?
Three Theories
1. Expectations Theory explains 1 and 2, but not 3
2. Segmented Markets Theory explains 3, but not 1 and 2
3. Preferred Habitat Theory & Liquidity Premium
Combines features of both Expectations Theory and Segmented
Markets and explains all facts
B.1 Expectations Theory
Key Assumption:
Implication:
I n v e s tm e n t
G rade
Bonds of different maturities are perfect substitutes.
RETe on holding bonds of different maturities must be equal
in equilibrium
1: Example - Two investment strategies for one year horizon (invest $100)
1. Buy $100 of one-year bond and hold it until maturity (buy & hold strategy)
2. Buy
$100
of
6-month
bond.
When
it
matures
buy
another
six months bond. (roll-over strategy)
Junk B onds
Expected return from buy & hold strategy
Other factors related to the bond issuer:
Liquidity
= How quickly asset is converted to cash. Investors get
positive Liquidity Premium for holding less liquid assets
Information Costs = A fraction of return is lost due to costs of studying the
default risk of a bond
Income Taxes
= A fraction of return is lost due to taxes
No-Arbitrage Condition: iTaxFree = i (1 - tax rate)
B: Term Structure
Three Facts to be Explained:
1. Interest rates for different maturities move together
2. Yield curves tend to have steep slope when short rates are low and
downward slope when short rates are high
3. Yield curve is typically upward sloping
(it=today’s interest on the one-year bond):
(1 + it)x$100 – $100
RetB&H = —————————————————————— = it
$100
Expected return from roll-over strategy
(it1 = today’s interest on the 6-months bond,
iet2 = expected interest on the future 6-months bond)
(1 + it1/2)(1 + iet2/2)x$100 – $100
it1 + iet2 it1 x iet2
ReteB&H = —————————————————————————————————— = ———————— + ————————
$100
2
4
Since it1 x iet2 is very small, expected about return is about (it1 + iet2)/2
We have said that expected returns from both strategies must be equal, in
equilibrium (no-arbitrage is possible).
Therefore:
it
=
it1 + iet2
—————————
2
(1)
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More generally to find interest for an n-period bond (from a B&H strategy) as a
straight average of one-period bonds (from a roll-over strategy):
int
Interest rate
on long bond
=
it + iet+1 + iet+2 + ... + iet+n-1
———————————————————————————————
n
(2)
Average short rates expected to
occur over life of long bond
2: Numerical example on Expectations Theory
One-year interest rates over the next five years are 5%,6%,7%,8% and 9%, thus
Annual interest rate on two-year bond:
(5% +6%)/2 = 5.5%
Annual interest rate on three-year bond:
(5% +6% +7%)/3 = 6.0%
Annual interest rate on four-year bond:
(5% +6% +7% +8%)/4 = 6.5%
Annual interest rate for five-year bond: (5% +6% +7% +8% +9%)/5 = 7.0%
Annual interest rates for one to five year bonds: 5%, 5.5%, 6%, 6.5% and 7%.
Expectation Theory Explains why yield curve has different slopes:
1. When short rates expected to rise in future, the average of future short rates is above
today’s short rate: therefore yield curve is upward sloping
2. When short rates expected to stay same in future, average of future short rates are
same as today’s, and yield curve is flat
3. When short rates expected to fall will yield curve be downward sloping
Expectations Theory explains Fact 1 that short and long rates move together
1. Short rate rises are persistent
2. if it↑ today, it+1, it+2 etc.↑ tomorrow =>average of future short rates↑ => int↑
3. Therefore: if it↑ then int↑, i.e. short and long rates move together
Expectations Theory explains Fact 2 that yield curves tend to have steep slope
when short rates are low and downward slope when short rates are high
1. When short rates are low, they are expected to rise to normal level, and long rate =
average of future short rates will be well above today’s short rate: yield curve will
have steep upward slope
2. When short rates are high, they will be expected to fall in future, and long rate will
be below current short rate: yield curve will have downward slope
Doesn’t explain Fact 3 that yield curve usually has upward slope
Short rates as likely to fall in future as rise, so average of future short rates will not
usually be higher than current short rate: therefore, yield curve will not usually slope
upward
B.2 Segmented Markets Theory
Key Assumption: Bonds of different maturities are not substitutes at all.
Implication:
Markets are completely segmented: interest rate at each
maturity determined separately (not influenced by RETe on
assets with different maturity)
Explains Fact 3 that yield curve is usually upward sloping:
People typically prefer short holding periods, hence higher demand for short-term
bonds, which have higher price and lower interest rates than long bonds
Does not explain Fact 1 or Fact 2 because assumes long and short rates
determined independently (which is not the case)
B.3 Liquidity Premium or “Preferred Habitat” Theory
Key Assumption: Bonds of different maturities are substitutes, but are not
perfect substitutes (e.g. Coke & Pepsi)
Implication:
Modify Expectations Hypothesis with features of Segmented
Markets Theory. ☺
Investors prefer short rather than long bonds
=> must be paid positive term premium, knt , to hold long-term bonds.
Results in following modification of Expectations Theory:
int
=
it + iet+1 + iet+2 + ... + iet+n-1
——————————————————————————————— + knt
n
(3)
If knt >0 and is assumed to be increasing in n, then this is called the Liquidity
Premium Theory
3: Numerical Example of Liquidity Premium Theory
1. Exp. one-year interest rate over the next five years: 5%, 6%, 7%, 8% and 9%
2. Investors’ preferences for holding short-term bonds.
Term premiums for one to five-year bonds are: 0%, 0.25%, 0.5%, 0.75% and 1%
=> Interest rate on the two-year bond:
0.25% +(5%+6%)/2 = 5.75%
=> Interest rate on the five-year bond: 1.0%+(5%+6%+7%+8%+9%)/5=8%
Interest rates on one to five-year bonds:
5%, 5.75%, 6.5%, 7.25% and 8%.
Comparing with those for the expectations hypothesis, yield curves are more
steeply upward sloped here =>Liquidity Premium Theory Explains all 3 Facts:
Explains Fact 3 of usual upward sloped yield curve by preferred habitat for short bonds
(because of existence of positive term premium in case of long bonds).
Explains Fact 1 and Fact 2 using same explanations as expectations hypothesis because
it has average of future short rates as determinant of long rate
Use the Liquidity Premium Theory
Review
Risk Structure of interest rates
Term Structure of interest rates
Default Risk of a bond
Yield curve
Risk Premium of a bond
Expectations Theory of interest rates
Investment Grade vs. Junk
BondsSegmented Markets Theory of interest
rates
Information costs of a bond
Liquidity Premium Theory of interest rates
Liquidity Premium of a bond
Term Premium of a bond
No-Arbitrage Condition of tax free bonds
+ What facts the three theories above explain
Remember and know how to apply equations 2
and 3 (e.g. solve examples 2 and 3)
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