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EC372 Bond and Derivatives Markets
Topic #1: Bond Markets & Fixed Interest Securities
R. E. Bailey
Department of Economics
University of Essex
Outline
Contents
1
Bonds and bond markets
1
2
Zero-coupon (ZC) bonds
3
2.1
5
3
4
5
Real ZC bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coupon-paying bonds
6
3.1
Macaulay Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2
Valuation of bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Term structure of interest rates
8
4.1
8
Implicit forward rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What determines the term structure?
10
5.1
Pure Expectations Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2
Liquidity Preference Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.3
Preferred Habitat Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.4
Applying the Theories in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Reading: Economics of Financial Markets, chapters 12, 13
1
Bonds and bond markets
What’s special about bonds?
• A bond’s indenture is a contract that requires the issuer to take specified, deliberate actions
– typically to make a definite sequence of payments until a specified terminal date
– unlike equity, for which future payoffs (dividends) are discretionary
• Bonds are low-risk – the payoffs are contractual
– but risk, especially of default, remains!
1
• Most bonds can be summarised by just two numbers:
1. Interest rate – the bond’s yield to maturity
2. Time to maturity – time to the date at which the obligation ceases
• Understanding bond markets (suggested approach)
– Central bank fixes the short-term interest rate (monetary policy)
– Bond yields adjust (via price changes) to balance supply and demand for all bonds.
Risks associated with bonds
It is common to classify the risks from holding bonds into two categories, interest rate risk and basis
risk:
• Interest rate risk: bond prices change when the level of interest rates changes (immunization
strategies attack this)
• Basis risk (everything other than interest rate risk):
1. Credit risk: risk of default
Credit rating agencies assign grades, e.g. ‘AAA’ for bonds with negligible risk, that are
indicators of the degree of credit risk.
N OTE: the grades are estimates, not guarantees
2. Reinvestment risk: future coupons invested at unknown interest rates
3. Timing risk: changes in cash flow of a bond during its lifetime
4. Exchange rate risk: unforeseen fluctuations in exchange rates
5. Purchasing power risk: changes in the value of money
Credit Rating Agencies
Credit rating agencies (the most well known being Standard & Poor’s, Moody’s and Fitch) express
their grades slightly differently, e.g. Standard & Poor’s ‘BBB’ corresponds to Moody’s ‘Baa2’.
Presumably the agencies obtain the grades they quote in a systematic manner, i.e. from models
of the bond markets. (The models are proprietary – they don’t reveal them.) Hence, the grades are
forecasts, inevitably subject to error: the grades are not guarantees of bonds’ performance.
Agencies make their living from fees charged to the issues of bonds for which they assign grades,
thus raising concerns about ‘conflicts of interest’: bond issuers have an incentive to gain as higher
rating as possible for the bonds they issue. Hence questions may be asked about the independence
of the rating agencies from the companies the bonds issued by which they rate, the implication being
that the grades are too generous.
The Unit Time Period
While rates of return are conventionally measured at an annual rate, other relevant time intervals
need not coincide with a calendar year.
Returns may be compounded more or less frequently than once per year, investors may have planning
horizons longer or shorter than a year, and they may take the opportunity to revise their decisions
many or few times each year.
Unless explicitly noted, it should be assumed that the unit time interval corresponds to ‘one year’.
The complications that occur when it is necessary to consider intervals of different length will be
addressed as they arise.
2
What defines a bond?
• Maturity date, after which the bond ceases to exist
Let T denote the maturity date. If t is ‘today’, n = T − t is the bond’s time to maturity (its
‘life’)
• Bonds are commonly redeemed at a specified face value, m
– Callable bonds: issuer can terminate before T
– Convertible bonds: holders can exchange bond for another asset
– Perpetuities: no specified maturity date
– Sinking fund: issuer redeems bonds over an extended period
• Coupons: a sequence payments from the issuer to the holder
– ct+1 , ct+2 , . . . , cT , typically constant, c, paid twice per year
– Coupon rate ≡ c/m (typically used to define the coupon)
• Other sorts of bonds include:
– Floating rate bonds: coupons linked to another interest rate
– Index linked bonds: payoffs linked to the price level
Timing of coupon payments. Although coupons are almost always expressed at annual rates, their
payment is commonly split into instalments, typically made at six-monthly intervals. For example,
a 5% coupon means that 5 units of account (dollars, euros or whatever) are promised each year on
a bond with face value of 100 units. With six-monthly instalments, 2 12 % of the face value is paid
twice per year. Between the dates at which coupons are paid, the price at which a bond is traded is
‘dirty’ in the sense that the price reflects an element of accrued interest. It is possible to estimate the
implied value of accrued interest and to subtract it from the dirty price to obtain a clean price. Unless
explicitly noted, all bond prices are assumed clean, i.e. a correction has been made to eliminate the
effect of interest accrued since the last coupon payment.
2
Zero-coupon (ZC) bonds
Zero-coupon – ‘pure discount’ – bonds
• Bonds for which c = 0.
• Why are they important?
– The theory of bond markets relies on ZC bonds.
– They do exist, sometimes created by stripping coupons.
• Nominal ZC bonds: face value, m, fixed in units of money
• pn : market price for a ZC bond with n years to maturity.
• Spot yield yn on a ZC bond with n years to maturity:
pn = m/(1 + yn )n so that yn = (m/pn )1/n − 1.
• yn is the rate of return on the bond if held to maturity
3
• But suppose the bond is sold before maturity?
– Then it’s risky – because future bond prices are uncertain
– Known as interest rate risk
• Holding period yield rn,t+1 between t and t + 1 equals
rn,t+1 = (pn−1,t+1 − pn,t )/pn,t
Example: a bond with face value, m = 100, four years to maturity, n = 4, and with price, p4 = 83,
100 (1/4)
has a spot yield of approximately 4.77% ≈
−1 per annum (the return being compounded
83
once per year). p
6
m
-
y
Figure 1: A zero coupon bond’s price p as a function of its yield, y.
A ZC bond’s price, p is related to its yield via p = m/(1 + y)n (omitting the n subscript
on price and yield, for simplicity). The curve has a negative slope because a higher value
of y implies that the future payoff on the bond, m, is discounted at a higher rate, thus
resulting in a lower value for its net present value (which, by definition, is equal to the
bond’s price). Note that different values for n and m shift the curve: there is a separate
relationship for each bond.
Stripped bonds
To create a stripped bond, a financial intermediary purchases a coupon-paying bond and ‘repackages’ it in the form of a sequence of zero-coupon bonds, one for each coupon (each coupon of the
underlying bond becomes the face value of one of the stripped bonds) and one for the face value paid
at maturity. The trading of stripped government bonds has become active, with official sanction and
support, in several markets since the 1980s.
A coupon-paying bond can be viewed as a portfolio of zero-coupon bonds. For example, a
3-year bond, with face value $1,000, promising to pay a coupon of $40 every six months, can be
treated as a portfolio of five zero-coupon bonds each with face value $40, each maturing separately
at six-month intervals, and one 3-year zero-coupon bond with face value $1,040.
4
2.1
Real ZC bonds
The material on real ZC bonds is optional for 2016/17 (other topics are more important).
Real ZC bonds
• ‘Real’ ≡ face value protected against changes in the price level
In practice real bonds are called ‘Index Linked’ because their payoffs are linked to an index
of the general level of prices, such as as the Consumer Price Index, CPI, or the Retail Price
Index, RPI.
• Let Zn = factor by which price level increases in next n years
– annual inflation rate: πn = (Zn )1/n − 1.
The ‘annual inflation rate’ here is the average rate over the next n years: it reveals nothing
about the inflation rate for particular years, between the present and n years from now.
• Let pen = market price, the e denotes ‘real’ bond
• Payoff at maturity: m × Zn
• Nominal spot yield yen such that pen = m × Zn /(1 + yen )n
• Real spot yield yen∗ such that pen = m/(1 + yen∗ )n
• Let yen∗ = real spot yield,
∗
denotes ‘real’ yield
• Real spot yield on a real bond = certain
• Nominal spot yield on a real bond = uncertain
– because Zn is unknown today.
Nominal and Real Spot Yields
Spot yields
Nominal ZC bond
Real ZC bond
Nominal
Real
certain (yn )
uncertain (yn∗ )
uncertain (e
yn )
certain (e
yn∗ )
Comparing Nominal and Real ZC bonds
• Nominal spot yield on a nominal bond yn such that pn = m/(1 + yn )n
• Real spot yield on a nominal bond yn∗ such that pn = m/Zn /(1 + yn∗ )n
• Suppose that investors are risk neutral
• Then yn = yen (nominal returns are equal for both bonds)
Equivalently, yn∗ = yen∗ (real returns are equal for both bonds)
• Then pen /pn = Zn .
5
• Implication: bond prices reveal market expectations of inflation
1/n
– Implied expectation of average annual inflation rate: πn = Zn − 1.
• But this result depends on investors’ risk neutrality
– if investors are not risk neutral, it need not hold.
Example: Suppose that the price of a 10-year nominal ZC bond with face value $100 is currently
p10 = $52, while the price for a real ZC bond is pen = $78. The nominal spot yield on the nominal
∗ , is (approximately)
bond, y10 , is (approximately) 6.8%, while the real yield on the real bond, ye10
1
2.5%. Thus, the expected annual inflation rate over the ensuing 10 years might be measured by
∗ , i.e. 6.8 − 2.5 = 4.3% (average annual rate for each of the next 10 years). But this ignores
y10 − ye10
risk aversion on the part of investors who are uncertain about future inflation rates.
3
Coupon-paying bonds
Coupon-paying bonds
• Coupon-paying bond: promise to pay c per period until date T = t + n, when the bond is
redeemed with a payment of m (face value).
• A coupon-paying bond is a portfolio of ZC bonds.
• Given p (market price), c, m and n, the yield to maturity, y, satisfies: p =
··· +
c
c
+
+
(1 + y) (1 + y)2
c+m
.
(1 + y)n
• Is y the rate of return if the bond is held to maturity?
– No, because there is no guarantee that future coupons received could be invested at rate y.
• For a coupon-paying bond, y is a defective (but standard) index of its return.
Yield to maturity on a coupon paying bond: The yield to maturity can be understood as the internal
rate of return on the bond.
Notice that both p and y depend upon: (i) the time to maturity, n; (ii) the coupon, c; and, (iii) the
face-value of the bond, m.
The prices (and yields) of bonds with different n, c and m will almost surely differ among one
another. While this is not explicit in the notation, it should be clear from the context.
For coupon-paying bonds — unlike those with c = 0 — an explicit algebraic formula for y is
unavailable except for special cases (in particular, when n < 5). But, if all coupons are non-negative,
a unique solution exists and can be found by numerical methods.
The yield to maturity on a coupon-paying bond does not have the same interpretation as the
spot yield on a ZC bond. In particular, it need not be the case that the rate of return from holding a
coupon-paying bond from the present until maturity equals its yield to maturity. Why not? Because
the stream of coupons received between the date of purchase and maturity may, of necessity, be reinvested at rates different from y. Indeed, the coupons might not be re-invested at all. Only if every
1
Formally, y10 =
100
52
1/10
∗
− 1 ≈ 0.068, and ye10
=
6
100
78
1/10
− 1 ≈ 0.025.
coupon is re-invested (from the date of its receipt until maturity) at rate y will the rate of return from
holding the bond until it matures equal the yield to maturity, y.
For coupon-paying bonds, the value of y is, at best, an approximation to the rate of return
from holding the bond from the present until it matures. Unless forward contracts are available to
guarantee the rates at which future coupons can be re-invested, then the rate of return on a couponpaying bond is inherently uncertain. The risk associated with the rates at which coupons can be
re-invested is known as re-investment risk.
3.1
Macaulay Duration
Macaulay Duration
• How much does a bond’s value, p, respond to y?
• Macaulay duration: an index of this responsiveness:
1
1·c
2·c
n · (c + m)
D=
+
+ ··· +
p (1 + y) (1 + y)2
(1 + y)n
• Differentiation shows that: D = −
(1 + y) ∂p
·
p
∂y
• Why bother?
– Because D has a time dimension
– D ≤ n: D measures ‘how soon’ bond’s payoffs are received
– For ZC bonds D = n: payoff only at maturity
– D is handy when designing immunization strategies intended to purge bond portfolios of
interest rate risk
Macaulay Duration: an example
Suppose that a bond with face value m = $100 pays a coupon c = $20 for two years, n = 2. If
y = 10%, then the bond price is
p=
20
20 + 100
+
≈ 117.36
(1 + 0.10) (1 + 0.10)2
(1)
and the Macaulay duration equals
1
D=
117.36
1 × 20
2 × (20 + 100)
+
(1 + 0.10)
(1 + 0.10)2
≈ 1.85
(2)
It should be remembered that the value of D depends upon n, c and m.
The main purpose of constructing D is to obtain a single number to measure the responsiveness of p
to y, allowing for differences in c and n among bonds.
An important aspect of D is its time dimension. In the example above, the time to maturity is
n = 2 years, while D ≈ 1.85 < 2. For a coupon-paying bond, D < n, always. Intuitively this is
because a portion of the payoff on the bond — the coupon of $20 at date 1 — is received before the
bond matures at date 2. It is as if the bond will mature shortly before date 2: the Macaulay duration
captures the precise sense of this notion.
7
3.2
Valuation of bonds
Valuation of bonds
• What is the ‘value’ of a coupon-paying bond for which there is no reliable market price?
• Method: set the value equal to the bond price if it were traded and in the absence of arbitrage
opportunities.
• Example: an n-year bond with coupon c face value m.
• Treat the bond as if it is a portfolio of ZC bonds, the prices of which can be observed.
• Then the bond’s value equals the value of the portfolio of ZC bonds with the same the sequence
of payoffs:
Bond value = p1 c + p2 c + p3 c + · · · + pn (c + m)
where pj is the price of a j-year ZC bond with m = 1.
• Problem: market prices for all the ZC bonds may not be observed
Simple example: Consider a zero-coupon (ZC) bond that pays $100 one year from the present. Also,
suppose that the risk-free rate of interest is 25%. In a frictionless market, the absence of arbitrage
opportunities ensures that the bond will be traded for $80. Why? Because only at this price is the
rate of return on the bond, (100/80) − 1, equal to the interest rate, 25%. At any other price, arbitrage
profits can be made: if the bond’s price is lower than $80, funds would be borrowed to buy the bond;
if the bond’s price exceeds $80, it would be sold short, the proceeds being lent at interest.
4
Term structure of interest rates
Term structure of interest rates
• Objective: to explain bond yields as a function of time to maturity.
– Central bank fixes the short-term interest rate (monetary policy)
– Bond yields adjust (via price changes) to balance supply and demand for all bonds.
• Yield curve: bond yields as a function of time to maturity.
• Must control for other factors, in particular coupons (that differ across bonds)
• Hence plot spot yields (on ZC bonds) as a function of time to maturity
4.1
Implicit forward rates
Implicit forward rates (ifr)
• ifr: future interest rates implied by today’s ZC bond prices.
• Purpose: another way of expressing the term structure
8
y 6
y 6
(a)
(b)
-
-
n
n
Figure 2: Yield curves.
Each point on the yield curve plots the spot yield on a zero-coupon bond as a function
of its time to maturity. A typical, positively sloped yield curve is shown in panel (a).
Alternatively, yield curves may be negatively sloped or constant across maturities (flat
yield curve). Also, it is possible for yield curves to be positively sloped over some
maturities and negative sloped over others, as the illustration in panel (b) shows.
• Definition: n−1 fn implicit rate between n − 1 and n years from today: (1 + yn−1 )n−1 (1 +
n
n−1 fn ) = (1 + yn ) where yn and yn−1 are spot yields on ZC bonds
• Hence:
n−1 fn
= (pn−1 /pn ) − 1
Relationship between spot yields and implicit forward rates:
There is an exact relationship between a sequence of spot yields, y1 , y2 , y3 , . . . , yn , . . . , and
implicit forward rates, 1 f2 , 2 f3 , 3 f4 , . . . , n−1 fn , . . .
Thus, for example:
3 f4 =
(1 + y4 )4
− 1,
(1 + y3 )3
and generally:
n−1 fn =
(1 + yn )n
−1
(1 + yn−1 )n−1
Hence: The term structure of interest rates can be expressed either as a sequence of spot yields
or, equivalently, as a sequence of implicit forward rates.
Interpretation:
Spot yield yn is the average annual rate of return between t and t + n;
Implicit forward rate, n−1 fn is the implied rate of return between t + n − 1 and t + n.
N OTE: The words ‘implicit’ and ‘implied’ appear because the forward rates are not observed
directly from bond prices (as distinct from spot yields which are).
• Remarks:
1. Can be extended to cover time spans other than a year
2. Spot yields can be viewed as averages of implicit forward rates
3. Normally vary over time (should include t subscripts)
4. Obtained from ZC yields, not yields on coupon-paying bonds
• Implicit rates become explicit if forward markets in bonds exist
– a consequence of the arbitrage principle
9
Example: Suppose that p5 = 60 and p4 = 66 for two ZC bonds each with a face value of $100. The
implicit forward rate on a 1-year bond beginning four years from the present is 4 f5 = (66/60) − 1 =
10%. An investor who expects the 1-year interest rate four years from now to exceed 10% could
make a speculative profit by selling (i.e. effectively by issuing) 5-year bonds, and investing the funds
in 4-year bonds. After four years (when the 4-year bonds mature) the payoff would be invested at
the 1-year interest rate. If the investor’s expectation turns out to be correct, a profit would be made
after five years (beginning today) following redemption of the 5-year bonds (i.e. payment of the face
value of the bonds sold, or issued, today). This investment strategy is risky: there is no guarantee
that the investor’s expectation of the interest rate between years four and five will be realized.
The example shows why implicit forward rates may be important for an investor. They enable
inferences about interest rates applicable for loans over intervals in the future. This information
is relevant for determining which bonds should be sold or purchased, depending on the investor’s
expectations or future commitments.
5
5.1
What determines the term structure?
Pure Expectations Hypothesis
Pure Expectations Hypothesis (PEH)
• PEH: today’s bond yields depend on expected future yields
• Heroic assumption: future bond yields known with certainty
• Hence: rate of return on a 2-year bond equals rate on two 1-year bonds:
(1 + y2,t )2 = (1 + y1,t )(1 + y1,t+1 )
• Note: y1,t+1 is an expected rate, not directly observed
• To make PEH operational requires a theory of expectations, i.e., what determines y1,t+1 .
Informal argument for the PEH: Consider a world in which 1-year (‘short-term’) and 2-year
(‘long-term’) zero-coupon (ZC) bonds are traded. Suppose initially that the yields on both
bonds happen to be equal. If the yield on 1-year bonds is expected to rise in the future,
investors may prefer to hold 1-year bonds so that, when they mature, the proceeds can be
re-invested in 1-year bonds commencing next year, thus benefitting from the higher expected
yield in the future. This preference would be expressed by investors selling 2-year bonds and
buying 1-year bonds, hence leading to an equilibrium in which the price of 2-year bonds is
lower, and the price of 1-year bonds is higher, than otherwise. Given the inverse relationship
between yields and prices, the equilibrium yield on 2-year bonds becomes higher than the
yield on 1-year bonds. Hence, the theory predicts that the yield curve has a positive slope if
investors expect interest rates to rise.
Conversely, if the 1-year bond yield is expected to fall in the future, the expectations
hypothesis predicts that the yield curve will be negatively negative sloped. It is predicted to be
flat if the 1-year yield is expected to remain at its current level. More complicated shapes are
consistent with the theory for more complicated patterns of expected future bond prices.
I MPORTANT: y1,t+1 denotes the expected spot yield on a 1-year bond beginning at date t + 1,
where the expectation is formed at date t, while y1,t and y2,t are observed – they are the spot
yields at date t on 1-year and 2-year bonds, respectively.
The PEH postulates that the relationship above holds between any pair of bonds. Thus:
(1 + y3,t )3 = (1 + y2,t )2 (1 + y1,t+2 ),
10
where y1,t+2 denotes the expected spot yield on a 1-year bond beginning at date t + 2, the
expectation being formed at date t, as before. Hence, the PEH implies that ‘long’ rates, such
as y3,t , can be understood as the average of expected spot rates on 1-year bonds. Substituting
above:
(1 + y3,t )3 = (1 + y2,t )2 (1 + y1,t+2 )
y3,t
= (1 + y1,t )(1 + y1,t+1 )(1 + y1,t+2 )
q
= 3 (1 + y1,t )(1 + y1,t+1 )(1 + y1,t+2 ) − 1
More generally:
yn,t =
q
n
(1 + y1,t )(1 + y1,t+1 )(1 + y1,t+2 ) · · · (1 + y1,t+n−1 ) − 1
The PEH in practice
In order to make the PEH testable – i.e., operational in practice – it is necessary to add
a theory of expectations (an explanation of what determines expectations). Without such a
theory the PEH is an identity (merely defines what the expectations would be to satisfy the
assumed relationships) – it is vacuous. This has the important implication that every test of
the PEH is a joint test of the PEH and the theory of expectations. It is not possible to test the
two separately.
Equivalent forms:
1. Local expectations hypothesis: expected return on every bond equals the short-term rate.
2. Return to maturity: for any horizon all bonds have the same expected payoff
3. Yields to maturity are equal for all bonds for any given horizon
4. Unbiased expectations: implicit forward rates equal expected yields
To show this, compare the statement of PEH with the definition of implicit forward rates:
PEH: (1 + y1,t )(1 + y1,t+1 ) = (1 + y2,t )2
Implicit forward rate: (1 + y1,t )(1 + 1 f2 ) = (1 + y2,t )2
(1 + y2,t )2 (1 + 2 f3 ) = (1 + y3,t )3
..
.
(1 + yn−1,t )n−1 (1 + n−1 fn ) = (1 + yn,t )n
Thus, y1,t+1 = 1 f2 , y1,t+2 = 2 f3 . . . or, more generally: y1,t+n−1 = n−1 fn .
N OTE : This is just a way of expressing the PEH. It does not in itself provide evidence in
favour or against the PEH.
• PEH prediction: all bond yields are equal over same holding period – but this does not predict
the yield curve
PEH with uncertainty
• Now abandon the heroic assumption: allow uncertainty
• Replace future bond yields with their expectations.
• But why? What about risk aversion? Whose expectations?
11
• When future bond yields are replaced with their expectations the various forms of the PEH are
no longer equivalent
• Allowing explicitly for uncertainty makes predictions of the PEH ambiguous
• Conclusion: in the presence of uncertainty the PEH is incomplete
– needs further assumptions.
5.2
Liquidity Preference Theory
Liquidity preference (risk premium) theory
• Extend the PEH to allow for risk aversion.
• Risk (or term) premium, `n , on n-year bond: `n = Et [rn,t+1 ] − r1,t+1
• Let Et [rj,t+1 ] = expected return on j-year bond from t to t + 1.
• Prediction: Et [rn,t+1 ] − `n = . . . = Et [r2,t+1 ] − `2 = r1,t+1
• Hicks’s theory: “a constitutional weakness at the long end of the market”
• Formally: `n > `n−1 > . . . > `2 > 0.
Hicks’s theory predicts that term premia have a positive relationship with time to maturity:
the expected holding-period yield on a long-term bond includes a higher premium than that on a
medium-term bond, which includes a higher premium than on a short-term bond.
In this theory there is an asymmetry between the preferences of lenders and borrowers: lenders
prefer to hold short-term bonds while borrowers prefer to issue long-term bonds. Here is how Hicks
expressed it:
[Bond markets] may be expected to have a constitutional weakness on one side, a weakness which offers an opportunity for speculation. If no extra return is offered for long
lending, most people (and institutions) would prefer to lend short . . . But this situation
would leave a large excess of demands to borrow long which would not be met. Borrowers would thus tend to offer better terms in order to persuade lenders to switch over
into the long market . . . . Hicks, Value and Capital, pp. 146–7 (emphasis added).
Speculation thus equilibrates the bonds markets. If long-term bonds are in excess supply, their
prices tend to fall, thus providing incentives for speculators to hold the bonds in the expectation
of returns in excess of the returns from holding short-term bonds. Consequently, bond prices (and
yields) reflect the asymmetry between lenders’ and borrowers’ preferences — the ‘constitutional
weakness on one side’ of the market.
An implication of the liquidity preference theory is that yield curves are positively sloped if
interest rates (i.e. short-period bond yields) are expected to remain constant or to increase in the
future. Only if a fall in interest rates, sufficient to outweigh the term premia, is expected, would the
yield curve display a negative slope.
Hicks’s theory implies that investors, including bond issuers, (a) have diverse risk preferences
with regard to consumption at different future dates, (b) select bond portfolios according to these
preferences, and (c) that the pattern of bond prices adjusts to balance portfolio demands with supplies
in the aggregate.
12
5.3
Preferred Habitat Theory
Preferred habitat (hedging pressure) theory
• Refines liquidity preference theory
• Allow for differing preferences among holders and issuers with regard to bond maturities.
• Prediction: stocks of bonds and investors’ preferences affect bond yields
– not only expectations of the future bond yields.
• Extreme version: Segmented markets hypothesis
– asserts that markets for different bonds are unrelated to one another
– implication: expectations have little impact on observed bond yields
5.4
Applying the Theories in Practice
Empirical factor models
• Goal: to forecast the yield curve at each date – i.e., to assume that bond yields depend on a
number of identifiable explanatory variables (outlined below)
• How? Bonds’ yields assumed to depend on a number of factors.
‘Factors’ are just explanatory variables that theories predict affect bond yields.
• What are these factors? Two approaches:
– Factors are latent (unobserved), reflected in statistical regularities.
Statistical methods, such as ‘principal components analysis’, can be used to infer the
presence of explanatory variables, even though the variables are not observed as separate
variables.
– Factors are observed economic variables (e.g. inflation rate) – the question (largely unanswered) is which variables? With enough effort it is always possible to find explanatory
variables that appear to fit the data, but such empirical models are notoriously bad at predicting ‘out-of-sample’ (with new data) and are of little use without theoretical support
and justification.
• With what results?
– Three latent factors: level, slope, curvature – (i) the overall level of the yield curve,
(ii) the extent to which yields of bonds of different maturity tend to differ (first derivative
of the yield curve); (iii) how those differences differ according to bond maturity (second
derivative of the yield curve).
– Economic variables, could be almost anything – typical approach: long rates function of
short rate and a few macroeonomic variables
A common approach is to assume that the monetary authority (central bank) sets the short
rate of interest, and market forces, via investors’ (unobserved) expectations, determine
rates of higher maturities. Investors’ expectations are then assumed to depend on various
influences in the economy, e.g. business confidence, inflation, ‘quantitative easing’.
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Making the models work
• Latent factor models often obtain close fit with data
But with little a priori motivation
Hence, limited use in predicting responses to policy changes, etc.
• Observed factor models – where the factors are based on observed data, rather than inferred
from statistical regularities in interest rates.
Directly reflect economic forces – a macroeconomic model could be used to predict which
variables, such as inflation or fiscal policy, affect interest rates of various maturities.
But often forecast poorly – probably this stems from inadequacies in macroeconomic theory.
• Arbitrage-free models:
Impose parameter restrictions to satisfy Arbitrage Principle
What this means is that the factor model parameters (loadings) are restricted in such a way that
there are no arbitrage opportunities for trading in bonds of different maturities. The principles
are studied in EC371 – not required for EC372.
Empirical results are mixed – it’s never possible with statistical to know for sure whether the
arbitrage restrictions hold. To the extent that they do hold, little is gained from imposing them.
When the restrictions do not appear to hold in the data several inferences are possible:
1. Perhaps the restrictions do hold but statistical variation (e.g. mismeasurement of variables) masks their validity.
2. Perhaps the results reveal arbritrage opportunities. While possible, this is implausible
inasmuch as specialist investors are likely to have spotted them much sooner and, by exploiting the opportunities, their actions will have moved bond prices (yields) to obliterate
any gains – otherwise, we could all be infinitely rich.
3. Perhaps the results reveal model mis-specification, i.e. that the model is not an accurate
representation of ‘the world’ (e.g. as a consequence of market frictions such as transaction costs). All models are inaccurate to some degree: the issue is how close is the
approximation?
Of the three possible inferences, it seems likely that the third is the most probable.
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Summary
Summary
1. Bonds: contract to make a specified sequence of payments
2. Zero-coupon (ZC): one specified payment at a specified date.
3. Coupon-paying bond: equivalent to a portfolio of ZC bonds
4. Securitised bond (e.g. CDO): backed by other, typically non-marketed, loans
5. Term structure of interest rates (yield curve): spot yields as a function of time to maturity
6. Implicit forward rates: future spot yields implied by ZC prices observed today
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7. Theories of the term structure:
• Pure expectations hypothesis
• Liquidity preference (risk premium) theory
• Preferred habitat (hedging pressure) theory
• Segmented markets hypothesis
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