Chapter 6
The Risk and
Term Structure of
Interest Rates
© 2008 Pearson
6.1
Objective
We have calculated the Rate of Returns on
Bonds.
Now, use the Term Structure of YTM of
various-term Bonds for forecast of the
economy over the long-run.
2
The idea is that
• The long term bond’s YTM reflects the market
player’s expectations of many things, including
the macroeconomic changes.
• Thus YTMs of bonds with varying terms, such as
YTM of one year bond, YTM of 2 yr bond, and
YTM of 10 yr bond, have different time
perspectives.
• The differences between these YTMs reveal the
market player’s expectation of some changes in
macroeconomic variables.
3
1. Recall:
Returns on Bonds: YTM
1) Definition
• Effective Annual Rate of Returns
= Yield to Maturity (YTM)
= Nominal annual Interest Rate
+ Annual Capital Gains as percentage of
Average Price of Bonds
4
2) Approximation Formula for
Yield to Maturity
Coupon Payment  Annual Changes in Price
Average Price
where Average Price = (Purchase Price + 100)/2;
And Annual Changes in Prices = (Face Value – Purchase
Price) / Maturity Period
5
*Example
• suppose that newspaper on March 1, 2004
Issue
ABC Co.
Coupon Rate
10%
Maturity Date
Bid/Ask
1 March 08/09
92
Yield
?
Yield to Maturity = (10 + 8/4) / 96 x 100 = 11.46%
* ‘/09’ means that the bonds are extendable for a year.
6
2. Different Rates of Returns
on Different Bonds
YTM = Core/basis YTM
+ Various Risk Premiums
7
1) The Core Part of YTMs
Competition in the financial market leads
to the equalization of YTM of various
bonds and other financial assets as long as
have the same risk characteristics.
8
2) Different Risk Premiums
Differences in YTM for Different Bonds
= Compensations for differences in Risk
= Differences in Risk Premium
9
3) What risks?(recall)
a) Default/Credit Risk –The lower the Bond Rating, the higher the risk of default
= The higher the risk premium = The higher the YTM should be in order to
induce the people to hold the bonds of more risk
-> Bond Rating show the level of this risk
b) Liquidity Risk
c) Inflation/Macroeconomic Risk
d) (Financial) Market Risk
(1) Mean Variance Theorem – The higher the SD (=variance) of the price and
returns over time, the higher the rate of return should be ;
(2) Capital Asset Pricing Model – The higher the Correlation of the rate of
return of a bond with the overall Market Index, the higher the rate of return
should be.
e) Idiosyncratic Risk: specific to one particular bond; there is no risk premium
for this part.
10
3. Credit Risk and Bond Rating:
Bond Rating Services in the world
Moody’s
Standard & Poor’s
Aaa (“best quality”)
Aa (“high quality”)
A (“uppper-mediumgrade”)
Baa (“medium grade”)
_____________
Ba (“Speculative”)
B
Caa
Ca
C
AAA (“extremely strong”)
AA (“very strong”)
A (“strong”)
BBB (“adequate” or “fair”)
_________
BB(“uncertain” or
“speculative”)
B
CC (“extremely vulnerable”)
Canadian
Bond Rating
Services –
Combined
with S & P’s
11
**Junk Bonds?
• Bonds are generally classified into two groups "investment grade" bonds and "junk" bonds.
Investment grade bonds include either Standard &
Poor's (AAA, AA, A, BBB) or Moody's (Aaa, Aa,
A, Baa).
• The term "junk" is reserved for all bonds with
Standard & Poor's ratings below BBB and/or
Moody's ratings below Baa.
• Investment grade bonds are generally legal for
purchase by banks; junk bonds are not.
12
***Some Canadian Examples:
Source:
http://www.standardandpoors.com/RatingsActions/RatingsLists/CanadianIssuers/index.ht
ml
•
•
•
•
•
•
•
•
•
•
•
•
•
Ontario Government AA
Quebec Government A
York Municipality AAA
Rogers Cable systems BB
Xerox Canada BBB
New Foundland A
Nova Scotia A
New Brunswick AA
Nortel Network A
Pacific Northern Gas BB
Air Canada BB
Alberta Government AAA or AA
Canadian Government AAA
13
5. Bonds issued
i)by the same borrower
ii) for a Variety of Terms to Maturity have
different YTM: Term Structure of Interest Rates
Thus, default risk is controlled (held constant), but the macroeconomic risk
varies from one year to another:
Eg) Same Government Bonds with Different Maturity Dates
Ontario Government Bonds which will mature and be repaid in six months;
Ontario Government Bonds which will mature and be repaid in one year;
Ontario Government Bonds which will mature and be repaid in two years;
14
4. Inflation Risk, and Term
Structure and Yield Curve
Go back to previous example.
1) Liquidity Risk Premium Theory
-> Simply, the longer the term, the higher the liquidity risk
and thus the higher the risk premium and the YTM
2) Inflation Expectation Theory
-> (Future) Monetary/ Economic Conditions change.
15
1) Liquidity Risk Premium Theory
Definition of liquidity: the speed at which you can
turn one asset into another asset, mainly bonds
into cash
The longer the term to maturity, the longer the
commitment and thus the higher the risk premium
should be.Otherwise, no one will hold the longterm bonds.
In general but not always, long-term bond’s YTM
should be always higher than Short-bond’s YTM 16
*Some argues against the
Liquidity Premium Theory
= Preferred Habitat Theory (Market Segmentation)
-> Risk Premium is not necessarily larger for long-term maturity
-> The risk premium depends on what you consider to be risky
For instance, the pension fund manager prefer the long-term investment to
short-term investment. Unless the short-term investment carries a
higher YTM, he would not invest on it.
But the PHT is not necessarily true
at the Aggregate Level, on Average
or In General.
17
2) Expectations Theory
The long-run interest rate is the average of the
current actual interest rate and the expected
future short-run interest rates, which
reflect the contemporary monetary
conditions, which are related to the most
important macroeconomic conditions,
National Income(Business Cycles) and
Price Level(inflation).
18
So, by comparing the YTMs of different
terms, some being short and others being
long, we can extract the market player’s
expectations about the monetary
conditions.
• We can compare one-year bond’s YTM and
10-yr-bond’s YTM
• Long-term bond’s YTM could be either
higher or lower than Short-term bond’s
YTM
19
Expectations Theory of
Term Structure
a) L.T. YMT reflects the present and future S.T. YTMs.
b) Formula
Rn: YTM of bond with n years to maturity date;
R1: current YTM of bond with 1 year to maturity date;
E n-1 1: expected YTM on a 1-year bond for one year starting n-1
years from today.
Rn 
R1  E
1
1
E
2
1
 ......  E
n 1
1
n
20
*Example I: Expectations Hypothesis
**Simplifying Assumptions: no credit risk difference; no liquidity risk
premium difference
• We assume the same risk for the short-term and the long-term
bonds- Ignore liquidity risk.
• Suppose that the current and the expected yields on a 1-year
short-term bonds are as follows.
R1
E11
E21
E31
E41
14%
13%
12.5%
12%
11.5%
•What would be the actual rate of returns on the long-term bonds?:
R2 = ?
R3 = ?
R4 = ?
R5 = ?
21
The Answer to Example I
• R1= 14%; R2 = 13.5%; R3 = 13.2%;
R4 = 12.9%; R5 = 12.6% “Term Structure”
• Note: As E goes down, R goes down too.
• Yield Curve is a graphic representation of the Term Structure
%
0 1 2 3
……………… yrs : Terms to Maturity
22
*Example II: Expectations
Hypothesis
• The current annualized short-term interest rate or
YTM is 10% for a one-year bond this year.
• The YTM for a two-year bond is 12%.
• What is the short-term annual interest rate which
the financial market expected for the next year?
23
Answer to Example II
Bonds
YTM
R2
12 %
Competition in the financial
market will ensure the equality
between Options I and II.
1. Option I: Investing on a 2 yr bond
12% 
R1
10%
E11
x%
12% = 24 %
2. Option II: Rolling over investment on 1
yr bonds
10% +
X%
3. Returns on Option I = Returns Option II
Thus, X  14%
24
3) Monetarists’ view of Expectations
Theory of Term Structure
expectations of interest rate
=expectation of nominal interest rate (R = r
+ pe )
= expectations of inflation rate (= pe; because
r is constant)
= expectations of the rate of growth of money
(supply)
25
• Fisher Equation: R = r + pe
• Quantity Equation of Exchange:
- MV = P y
 D% M + D% V  D% P+ D% y , where D% P  p
• Combining the two: we get
RA = rA + pe A and R B = rB + pe B
RA - R B = (rA - rB ) + (pe A - pe B )
 A lower interest rate means a strict monetary policy
- Either a lower inflation
- Or a slower business
26
* Yield Curve: Graph of Term
Structure
Steep(er)
Typical(Normal)
• YTM(%)
Flat(tened)
Inverted
0 1
2
3
4
5
6
(Maturity Term:
Years)
27
**When we combine Expectations
Theory and Liquidity Premium Theory:
• Let’s suppose that the one-year interest rate over
the next five years is expected to be 5%, 6%, 7%,
8% and 9%. Investors’ preferences are such that
the liquidity premiums for one-year to five-year
bonds are 0%, 0.25%, 0.5%, 0.75%, and 1%
respectively. What is the actual interest rate on a
two-year bond and a five-year bond?
• Answer: 5.75% and 8%.
28
*Historical Examples of Yield Curves
in the U.S. economy
• Living Yield Curve
• http://fixedincome.fidelity.com/fi/FIHistoric
alYield
• http//www.smartmoney.com/onebond/index
.cfm?story=yieldcurve&nav=LeftNav
• http://www.smartmoney.com/Investing/Bon
ds/The-Living-Yield-Curve-7923/
29
Advanced Topics
30
1) Use of Term Structure of YTM for
Business Forecast
1) Term Structure tells strongly about the Future of Economy:
The difference between L.T. YTM(10 year bond) and S.T. YTM(one year
bond, or shorter) is is the most certain leading indicator of business
cycles.
- when L.T YTM minus S.T. YTM is negative (L.T. YTM < S.T. YTM), we
would soon see Recession:
“Inverted Yield Curve is observed just prior to Recession”
- When the difference between L.T. YTM and S.T. YTM grows (L.T. YTM >>
S.T. YTM), then the business gets better, and the growth rate of Y rises.
31
2) Empirical Evidence
• J. Haubrich and A. M. Dombrosky tested the predictive
power of the spread between the long-term and short-term
bond yields.
- Regression Model:
DY = a  b (R10-year – R3-month T-Bill)
- Data: 1961:1Q to 1995:IIIQ of U.S.
- Results:
D Y = 1.83  0.97 (R10-year – R3-month T-Bill)
b hat is 0.97 with its t-value=4.50, being very significant.
The spread has a substantial predictive power.
The yield curve emerges as the most accurate predictor of real
economic growth (better than more sophisticated leading
index of business cycles).
32
3)Two Explanations for Harvey
Campbell’s Inverted Yield Curve
1)
Economics Theory
Inflation Expectation Theory :
A lower long-term interest rate means that compared to
the current interest rate, either a lower rate of inflation, a
slower growth rate of money supply, or a slowing down of
the economy is expected for the future.
2) Finance Theory
Portfolio Substitution Theory:
•
When people expect a recession (not permanent), people
would like to shift their investment into a safe haven; from
short-term investment(bonds) to long-term investment (bonds)
•
Demand for, and Price of Long-term bonds go up
33
•
Yield of Long-term bonds falls – “Inverted Yield Curve”
4) New Empirical Evidence by Campbell
Harvey(1989):
- Use the term-structure interest rate spread (=long-term
interest rate – short-term interest rate) and other known
leading indicator for the regression forecasting the economic
growth
- Compared two predictors of the spreads and other Leading
indicators by their R2
- Two Regression Results
DY = a  b (R10-year – R3-month T-Bill)
R2 = 0.30 to 0.45
versus
DY = a  b (Stock Price Index as The Leading Indicator) R2 = -0.004 to
0.045
(Interpretation) The first is better than the second: bond market reveals more
information about future economic growth than stock market.
34