Download Chapter Thirteen: Term Structure of Interest Rates

Chapter Thirteen:
Term Structure of
Interest Rates
(Econ 512): Economics of Financial Markets
Dr. Reyadh Faras
Econ 512
Dr. Reyadh Faras
Overview
 One of the dimensions relevant for distinguishing
between bonds, the time to maturity is one of the
most important.
 It is on the relationship between each bond’s time
to maturity and its rate of return that analysis of
the term structure of interest rates focuses.
Econ 512
Dr. Reyadh Faras
13.1 Yield Curves
13.1.1 Yield Curves in Principle
 Diversity among bond contracts can be narrowed
because yield curves are almost always
constructed for government bonds, for which the
risk of default is negligible. Thus, their differences
are: time to maturity, n, and coupon rate, c/m.
 Yield curves are constructed for ZC bonds, the
spot yield is plotted against the time to maturity.
 Panel (a) in figure 13.1 depicts an upward sloping
yield curve, which is the conventional and most
observed shape, with bonds of longer maturities
attracting higher yields.
Econ 512
Dr. Reyadh Faras
 The rationale is that bonds with many years to
redemption are riskier than bonds near maturity.
 Hence, L-T bonds command premia, as reflected in
higher yields, relative to S-T bonds. Consequently,
the yield curve is positively sloped.
 Yield curves are not necessarily positively sloped.
 They are, on occasion, observed to slope
downwards, with S-T bond yields exceed L-T ones.
 Yield curves can also be flat, i.e. same y for all n.
 In principle there is no reason why yield curves
should be monotonic throughout.
Econ 512
Dr. Reyadh Faras
 The curve could be negatively sloped for some
maturities and positively sloped for others (panel
(b) in figure 13.1).
 Even though this is uncommon, but when
observed they are usually attributed to specific or
peculiar (unusual) events in bond markets, often
following an abrupt (sudden) reversal or
intensification of monetary policy.
Econ 512
Dr. Reyadh Faras
13.3 Implicit Forward Rates
 Implicit forward rates are indicators of future
interest rates inferred from observed bond prices.
 They provide a way of characterizing the term
structure of interest rates that is equivalent to the
yield curve: for each sequence of spot yields (one for
each maturity) there exists a unique sequence of
implicit forward rates, and vice versa.
 Explicit forward rates are interest rates relevant for
agreements made today on loans that begin and end
at specific future dates.
 Implicit forward rates are forecasts of interest rates
on loans that will begin and end in the future.
Econ 512
Dr. Reyadh Faras
 Being forecasts, there is no guarantee that the
implicit forward rates will be realized when the
future arrives.
 To define an implicit forward rate, consider the
investment of $1 in a ZC bond that matures in 5
years from the present.
 Alternatively, suppose that $1 is invested in a 4
year ZC bond, followed by an investment of the
proceeds in a 1-year bond.
 The 1-year rate that results in the same payoff
after five years for both strategies is the implicit
forward rate between years 4 and 5.
 Note: Markets are assumed to be frictionless.
Econ 512
Dr. Reyadh Faras
 More formally, consider the investment of $1 in a
ZC bond that matures in n years from the present.
 After n years, when the bond matures, the
investment will be worth (1+yn)n, where yn is the
spot yield on an n-period bond purchased today.
 Similarly, $1 invested in a ZC bond that matures
n-1 years from today will accumulate to (1+yn-1)n-1
, after n-1 years.
 The implicit forward rate between n-1 and n years
is the interest rate that equates the payoffs from
the two strategies.
Econ 512
Dr. Reyadh Faras
 In other words, the implicit forward rate of return
that would be received from investing the proceeds
of the (n-1)-year ZC bond, at maturity, for one more
year if the rate of return over n=(n-1)+1 years
exactly equals the yield from holding an n-year ZC
bond to maturity.
 In symbols, the implicit one-year interest rate
beginning n-1 years from today, n-1fn, is defined to
satisfy: (1+yn)n = (1+yn-1)n-1 (1+ n-1fn)
(13.1)
 Rearranging (13.1), the implicit forward rate is:
n-1fn =
 That’s how the implicit forward rate is calculated
from the prices of ZC bond with the 2 relevant
maturities.
Econ 512
Dr. Reyadh Faras
Example
If: P5 =60, p4 = 66, m = 100. Then: 4f5 = (66/60) – 1 = 10%
 If the investor expects the 1-year interest rate 4 years from
now to exceed 10%, he can make a speculative profit by
selling 5-year bonds, and investing the funds in 4-year
bonds. After 4 years, invest the payoff at the 1-year
interest rate.
 If the investor’s expectations turned out true, a profit
would be made after 5 years following the redemption of
the 5-year bonds.
 This investment strategy is risky: there is no guarantee
that the investor’s expectations will be realized.
 Thus, implicit forward rates are important as they enable
inferences about interest rates applicable for loans over
intervals in the future, which are used to determine which
bonds to buy or sell.
Econ 512
Dr. Reyadh Faras
13.4 The Expectations Hypothesis of the Term Structure
 The expectations hypothesis provides a starting point
for all explanations of the term structure.
 It asserts that expectations about future bond yields
determine the shape of the yield curve.
 Consider a world in which 1-year (S-T) and 2-year
(L-T) ZC bonds are traded.
 Suppose that the yields on both are 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
reinvested in one year bonds commencing next year,
thus benefiting from the higher expected yield in the
future.
Econ 512
Dr. Reyadh Faras
 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 the 2year bonds is lower, and the price of the 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 one 1-year bonds.
 Hence, the theory predict that the yield curve has a:
 Positive slope if investors expect interest rates to rise.
 Negative slope if investors expect interest rates to fall.
 Be flat if the 1-year yield is expected to remain at its
current level.
Econ 512
Dr. Reyadh Faras
 Let pn,t denote the price today, t, of a ZC bond with n
years to maturity. Let yn,t denote its spot yield.
 After one year, at t+1, the same bond will have n-1
years to maturity.
 Its price and spot yields are then written as pn-1,t+1
and yn-1,t+1.
 Now compare ZC bonds with 1-year and 2-years to
maturity.
 There are three bonds: (i) 1-year bonds available
today, yielding y1,t; (ii) 1-year bonds issued at t+1,
yielding y1,t+1; and (iii) 2-year bond available today,
yielding y2,t.
Econ 512
Dr. Reyadh Faras
 By assumption all 3 rates are known today.
 Hence, it seems reasonable to claim that in market
equilibrium the payoff from investing $1 in a 2-year
bond must equal that from investing $1 in a 1-year
bond and then reinvesting the proceeds in another 1year bond for the second year:
(1+y2,t)2 = (1+y1,t) (1+ y1,t+1)
(13.8)
 If this equality does not hold, then investors have an
incentive either to issue 2-year bonds and invest the
proceeds in two successive 1-year bonds, or to issue
two successive 1-year bonds and invest the proceeds
in 2-year bonds.
Econ 512
Dr. Reyadh Faras
 (13.8) can be rewritten to exploit the relationship
between bond prices and yields:
y1,t = (p1,t+1 / p2,t) – 1
(13.9)
 In words: the yield on a 1-year bond equals the
expected payoff on a 2-year ZC bond purchased
today and sold after one year.
 The analysis may be extended for n-year ZC bonds.
 In market equilibrium the yield on 1-year bonds
equals the expected return on n-year bonds held for
one year:
y1,t = (pn-1,t+1 / pn,t) – 1 = ((1+yn,t)n / (1+yn-1,t+1)n-1 ) – 1
Econ 512
Dr. Reyadh Faras
13.5 Allowing for risk preferences in the term structure
13.5.1 The liquidity preference theory of term structure
 Hicks was the first to modify the expectations
hypothesis, it is known as the liquidity preference,
liquidity premium or risk premium theory.
 The theory asserts that, while investors are influenced
by expectations of future rates of return, they are riskaverse when making decisions about which bonds to
hold (or to issue).
 Risk aversion may imply that investors prefer to hold
short-term, liquid, assets unless a premium is included
in the return expected from long-term assets: bonds
nearing maturity would be preferred to bonds for which
maturity is far off.
Econ 512
Dr. Reyadh Faras
 Consequently, long-term bonds would be held only if the
expected payoff from holding them exceeds that on short
term bonds.
 Thus, the bond price pattern reflects the premia
demanded by investors if the aggregate demand to hold
bonds with different maturities is to match the supply.
 The premia are referred to as risk, liquidity or term
premia.
 Two characterizations of term premia deserve mention:
First: Premia are defined as differences between implicit
forward rates and relevant expected future interest rate:
λ2= 1f2 - Et[y1, t+1]
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Dr. Reyadh Faras
 In words: the term premium equals the implicit forward
rate between 1 and 2 years in the future minus the expected
spot yield on a 1-year bond maturing 2 years from the
present.
 More generally, we can define the term structure relevant for
a 1-year bond maturing n years in the future as:
λn= n-1fn - Et[y1, t+n-1]
Second: another definition of the term premium focuses on
holding yields.
 Consider a 1-year holding period. Let ℓ2 denote the term
premium on a 2-year bond, defined as ℓ2= Et[r2, t+1] – r1, t+1
 That is: the term premium on a 2-year bond equals the
expected return on the bond, if held for one year beginning
today (i.e. Et[r2, t+1]), minus the interest rate (today’s yield on
1-year bond , r1, t+1 , or , equivalently, y1, t).
Econ 512
Dr. Reyadh Faras
 More generally, the term premium on n-year bond is:
ℓn= Et[rn, t+1] – r1, t+1
 The term structure can be expressed as:
Et[rn, t+1] - ℓn = Et[rn-1, t+1] - ℓn-1 = …= Et[r2, t+1] - ℓ2 = r1, t+1
 Hicks argues that the 5-year bond, being riskier than the
1-year bond over the ensuing year, has a higher expected
holding period yield (positive term premium):
Et[r5, t+1] > r1, t+1
 The difference being the term premium, ℓ5.
 Hicks theory predicts that term premia have a positive
relationship with time to maturity: ℓn > ℓn -1 > …> ℓ2> 0.
 There is asymmetry in preferences: lenders prefer to
hold short-term bonds, while borrowers prefer to issue
long-term bonds.
Econ 512
Dr. Reyadh Faras
13.5.2 The preferred habitat theory of the term structure
 The preferred habitat or hedging pressure theory refines
the liquidity preference theory to allow for differing
preferences among lenders and borrowers with respect
to the maturity of the bonds they hold or issue.
 The main implication of this theory is that expectations
of future interest rates are not exclusively responsible
for the pattern of current bond yields: the stocks of
bonds, and investor’s demands to hold them, also
influence the term structure.
 If the differentials in bond prices are large enough then
investors may choose maturities different from those
that they most prefer, but they have to be offered an
incentive to do so.
Econ 512
Dr. Reyadh Faras
 The dispersion (spreading) of risk preferences and
stocks of bonds with different maturities can exert an
impact on the term structure.
 As extreme version of the theory – the segmented
markets hypothesis – asserts that bonds with different
times to maturity can be grouped together, such that
prices of bonds with each group are related to one
another but not to the prices of bonds belonging to other
groups.
 For example, monetary policy (acting on S-T interest
rates) might impact upon bond prices with shorter
maturities only. While actions of pension funds (holding
long-term bonds) might impact prices of longer term
bond but not those with shorter maturities.
Econ 512
Dr. Reyadh Faras