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Unformatted text preview: Chapter 6: The Risk and Term Structure of
Interest Rates After this chapter you should be able to...
l.Understand the meaning of the risk structure of interest rates. 2.Be able to interpret through the supply and
demand model the risk premium for bonds of different default risk and the same maturity. 3.Understand the meaning of interest rate immen 4.Understand the meaning of the term structure
. HEW—mm“.
of interest rates. m 5.Understand and be able to interpret yield
curves. 6.Understand expectations theory, segmented
markets theory, liquidity premium theory, and
preferred habitat theory of the term structure of interest rates. 7. Be able to explain if the theories in (6) have
explained particular empirical facts. Risk Structure of Interest Rates Bonds with the same maturity have different
interest rates due to three considerations.
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1. Default risk: probability that the issuer of the
bond is unable or unwilling to make interest
payments or pay off the face value. 0 US. Treasury bonds are considered default
free (government can raise taxes). _.._.—"""_"  'sk premium: the spread between the
interest rates on bonds with default risk and the interest rates on (same maturity)
Treasury bonds To illustrate we will use the supply and demand
model for two bonds of the same maturity. One
bond is the defaultﬂee Treasury bond and the other a corporate bond. Both are zero coupons. IgbirJ‘iﬁ #8:; .3? {2" fqufgﬁgﬁ 33’} 5‘16: ?".‘9 Juana ’_ if. I. ,J
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1'3 0 351,: m (: IL,{?;N / 2 mike it, If»: {A (Tl Generally considered investment grade Short ieng _ {911$ ' 03m 0558 .= 38 ESE #23 Eu 3 fact so 3530 680 o> \(E, __w»_..‘m\\,.\.\xu. m§wﬁg Exam, M‘§& 3183999 map Alqluow ‘spuoq e1eJodJ03 — a $91139 AlqllJOLU ‘SbUOQ Mnseeu Ammew JéeAOL _ i—
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ﬁllIBnD o; 1q61H pue LungweJd xsm puog emJodJoo 2. Liquidity: the relative ease with which an asset can be converted into cash 0 Cost of selling a bond
0 Number of buyers/ sellers in a bond market 3. Tax considerations
‘_—____—"_'—'“——'—————_.4
0 One important example for the US. market is the interest payments on municipal bonds
are exempt fromws a w t I. 1': i «L ': 'Jr"3'[{';u."_r \ FIGURE 1: LongTerm Bond Yields, 1919—2008 Annual
Yield (%)
1 8 16 14 12 _
U.S. Government / LongTerm Bands I I State and Local Government
(Municipal) r 1920 1930 1940 1950 1 960 1970 1 980 1 990 2000 2010 Sources: Board of Governors of the Federal Reserve System, Banking and Monetary Statistics, 1941—1970; Federal Reserve:
wwwfederalreservegovfreleasesfhl Sfdatahim. Term Structure of Interest Rates Bonds with identical risk, liquidity, and tax
characteristics may have different interest rates because the time remaining to maturity is
different. W: a plot of the yield on bonds with
differing terms to maturity but the same risk,
liquidity and tax considerations. ° Upwﬁldﬁmﬂg_g_ long—term rates are above shortterm rates 0 Egg: short— and longterm rates are the same
0 Inverted: longterm rates are below short— term rates Facts Theory of the Term Structure of Interest
Rates Must Explain 1. Interest rates on bonds of different maturities move together over time / ,4. l... . . Interest Rate (%) 16 14
Threeto _ 12 FiveNear .
Averages 1 1o a 20Year Bond f4. 6 Averages "' ' 4 H I!
1. ,. ff?! .
2 V  ThreeMonth Bills
_ _ (ShortTerm)
o 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2. When shortterm interest rates are low, yield
curves are more likely to have an upward slope;
when shortterm rates are high, yield curves are
more likely to slope downward and be inverted. 3. Yield curves almost always slope upward Let’s look at some theories. 10 _ E¥B?9£a_t19n_s_The9_ry_ o The interest rate on a long—term bond will equal
an averagehof the short—term interest rates that
people eggpth tooccur over the life of the long
term bond 0 Buyers of bonds do not prefer bonds of one
maturity over another; they will not hold
any quantity of a bond if its expected return
is less than that of another bond with a different
maturity  Bond holders consider bonds with different
maturities tobe ' "  r 6'" For example, suppose you had the choice
between a one year bond paying 5% return and a
two year bond. If you thought the return on a one
year bond the following year would be 7% then
you would be indifferent between a two year
bond paying 6% annual rate and a one year bond
purchase in each year. 11 More generally we can write these results as the
following for an investment of $1. Let, it = today's interest rate on a oneperiod bond
aim = interest rate on a oneperiod bond expected for next period
igt = today's interest rate on the twoperiod bond Then the expected return over the two periods
from investing $1 in the twoperiod bond and
holding it for the two periods is (1+i2t)(1+12t)—1=1+212tMar—1
= 2 a + (a )2 Since (lgt )2 is very small we can approximate the
expected return forholding the twoperiod bond for two periods 2152 i215) l2 If two one—period bonds are bought with the $1
investment then the two year return would be (1 +100 + Sim) — 1 I it + eit+l +it(eit+l) Since it( 6it“) is extremely we can approximate
the expected return to be it + eit+1 Therefore if they are perfect substitutes both
bonds will be held only if the expected returns equal I ___ I e. o _ o e
2 12t “— 1t + lt+1 01' 12t —' (1t + lt+1)/2 We can extend this to n periods by the following
expected return on the 11 period bond to equal, int = (it + eit+1 + + eit+n1) / n 13 Expectation y does the following: o Explains why the term structure of interest rates
changes at different times « Explains why interest rates on bonds with
differentmaturities movewtogether over time (fact 1) o Explains why yield curves tend to slope up
when shortterm rates are low and slope down when short—term rates are high (fact 2) 0 Cannot explain why yield curves usually slope
upward (fact 3) '14 Segmented Markets Theory 0 Bonds of different maturities are m? o The interest rate for each bond with a different
maturity is determined by the demand for and
supply of that bond 0 Investors have preferences for bonds of one
maturity over another 0 If investors generally prefer bonds with shorter
maturities that have less interest—rate risk, then
this explains why yield curves usually slope
upward (fact 3) I v: A? _,,J _ ,l I _
. .rlzl— [lb/(JHF ‘,. ' y. ia LII/1,31” :‘ ? d. \1; I "''1f 15 Liquidity Premium Theory
\_________~\ The interest rate on a long—term bond will equal
an average of shortterm interest rates expected to occur over the life of the longterm bond plus a
liquidity premium that responds to supply and
demand conditions for that bond Bonds of different maturities are Maori Maia
(ﬁeif WT fife9f This changes our expected return on the 11 period
bond to the following, int = lnt+ (it + eit+1 + m+ eit+n1) / 11 Where 1m is always positive and increases with
term maturity. It is the liquidity premium term. For Preferred Habitat theory, 1m, would be
called a preference term. 16 Preferred Habitat Theory says2
 Investors have a preference for bonds of one maturity over another 0 They will be Willing to buy bonds of different
maturities only if they earn a somewhat higher
expected return 0 Investors are likely to prefer shortterm bonds
over longer—term bonds 1? FIGURE 5: The Relationship Between the
Liquidity Premium (Preferred Habitat) and
Expectations Theory interest Liquidity Premium (Preferred Habitat) Theory
' Yield Curve \ Liquidity
Premium, inf Expectations Theory
Yield Curve 0 5 10 15 20 25 30
Years to Maturity, n 18 ﬂuidity Premium and Preferred Habitat
Theories say the following, 0 Interest rates on different maturity bonds
move togethQI over time; explained by the
alvc/ W term in the equation 0 Yield curves tend to slope upward when
shortterm rates are low and to be inverted
when short—term rates are high; explained by
the liquidity premium term in the ﬁrst case
and by a low expected average in the second case 0 Yield curves typically slope upward;
explained
by a larger liquidity premium as the term to maturity lengthens *l'cJz.r.:~r . . .  a: ..
 :.' ' " V t' 19 FIGURE 6: Yield Curves and the Market’s
Expectations of Future ShortTerm Interest Rates According to the Liquidity Premium (Preferred Hﬂmﬁ‘l‘gurq :6 in Yield to Yield to
Maturity Maturity
Term to Maturity Term to Maturity
(a) Future shortterm interest rates {b} Future shortterm interest rates
expected to rise W
h_"——.____——I
Yield to Yield to
Maturity Maturity
Term to Maturity Term to Maturity
(0) Future shortterm interest rates (at) Future shortterm interest rates
expected to fat! moderater mex ected to fat! sharpty
6i 4" ( we;
2 9h
cm Mort’0 at
_ e
mi 1', a Mr" 20 FIGURE 7 Yield Curves for US. Government
Bonds Interest Rate (%) 16
14
12 March 28, 1985
‘ May 16. 1930
10 8
March 3r 1997
6
February 6. 2006
4 ..
January 15, 2009a
2 /
5 10 15 20 Terms to Maturity (Years) 21 ...
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This note was uploaded on 12/21/2011 for the course BUS 100 taught by Professor Intro during the Fall '11 term at UCSB.
 Fall '11
 Intro

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