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The term structure of interest rates

Reading

• Luenberger, Chapter 4

• Fabozzi, Chapters 7, 8, 41, 42

Goals

• Understand the term structure of interest rates

• Define forward and spot rates

• Understand expectations dynamics

• Extend the notions of duration and immunization

Kay Giesecke

MS&E 242: Investment Science, The term structure of interest rates 2

Yield curve

• A bond is specified by its face value F , the coupon rate c, the

coupon frequency m and the maturity T

• For a bond (F, c,m, T ) with price P , the YTM λ is the IRR

– In the previous chapter, we fixed a bond with maturity T and

considered the bond price P as a function P (λ) of the yield λ

• Now we consider bonds in a given quality class (e.g. treasury bonds,

AAA corporate bonds) but with different maturities

• The yield curve displays the yield as a function λ(T ) of maturity T

– “Normal curve” is increasing

– “Inverted curve” is decreasing

– Relative pricing information

Kay Giesecke


The term structure

Spot rates

• Focus is on interest rates, not yields

• We consider rates that depend on the length of time for which they

apply–we relax the assumption of a constant ideal bank

• The spot rate st is the annual interest rate for money held from

today (t = 0) until time t; it replaces the time-invariant annual rate

r considered above

• This implicitly assumes a compounding convention, such as annual,

m times a year, or continuous compounding

• The spot rate curve displays st as a function of time t; as the yield

curve, it is increasing most of the time

Kay Giesecke


The term structureDetermining the spot rate from zero bond prices

• Let si be the spot rate for i years with annual compounding

– A dollar deposited at time 0 has value (1 + si)i after i years

– The corresponding discount factor is di = 1(1+si)i

• Consider a zero coupon bond with face value F that matures i years

from now; its price P is given by P = Fdi = F(1+si)i

• For i > 0 we find the corresponding spot rate si via

si =(F

P

) 1i

− 1

• Given the prices of zero bonds with various maturities, we can

construct the spot rate curve

Kay Giesecke


The term structure

Bootstrapping the spot rate from coupon bond prices

• Observe s1 as the one-year rate available today (e.g. one year

Treasury rate)

• Next consider a 2 year bond with annual coupon C and face value F

that has price

P =C

1 + s1+

C + F

(1 + s2)2

This can be solved for s2 given s1 and the terms of the bond

• Next consider 3 year bonds, and so on

Kay Giesecke


Forward rates

• We consider the interest rate that is available for borrowing money

in the future, under terms agreed upon today

• The forward rate ft1,t2 between time t1 ≥ 0 and t2 > t1 is the

annual interest rate for money held over the time period [t1, t2].This rate is agreed upon today. Clearly f0,t = st for all t.

• For a set of spot rates (si) based on annual compounding, the

forward rate fi,j between years i and j > i satisfies

(1 + sj)j = (1 + si)i(1 + fi,j)j−i

so that the forward rate implied by the spot rates is given by

fi,j =(

(1 + sj)j

(1 + si)i

) 1j−i

− 1

Kay Giesecke


Forward rates

Arbitrage argument

• Consider two ways of investing a dollar for j years at the currently

available rates

– Invest in a j year account. A dollar will grow to (1 + sj)j .

– Invest in a i year account for some i < j. At i, take out the

(1 + si)i and invest in a j − i year account that accrues interest

at an annual rate fi,j that you agree upon today. A dollar will

grow to (1 + si)i(1 + fi,j)j−i.

• In the absence of arbitrage opportunities and transaction costs, we

must have

(1 + sj)j = (1 + si)i(1 + fi,j)j−i

Kay Giesecke


Short rates

• The short rate ri at year i is the forward rate fi,i+1

• Short rates are as fundamental as spot rates, since a complete set of

short rates fully specifies the term structure:

(1 + si)i = (1 + r0)(1 + r1) · · · (1 + ri−1)

and also

(1 + fi,j)j−i = (1 + ri)(1 + ri+1) · · · (1 + rj−1)

Kay Giesecke


Compounding conventions

• Above we considered spot rates, forward rates and short rates based

on annual compounding

• All these rates can also be defined based on discrete compounding

several times a year and continuous compounding

• Problem: Express the rates under continuous compounding

Kay Giesecke


Compounding conventions

Solution

• The accumulation factor is estt with st the spot rate for [0, t]

• For t1 ≥ 0 and t2 > t1 the forward rate ft1,t2 satisfies

exp(st2t2) = exp(st1t1) exp(ft1,t2(t2 − t1)) and therefore

ft1,t2 =st2t2 − st1t1t2 − t1

• Assuming that ddtst exists, the short rate rt is given by

rt = limu↓0

ft,t+u = limu↓0

st+u(t+ u)− stt

u= lim

u↓0

st+uu+ t(st+u − st)u

= st + t limu↓0

st+u − st

u= st + t

d

dtst

Kay Giesecke


Why is the spot rate curve almost never flat?

Expectations theory

• While forward rates are known today, the corresponding spot rates

actually realized in the future are random variables

• The expectations hypothesis says that today, the market’s expected

value of the j − i year spot rate available in i years from now is

equal to the forward rate fi,j quoted today

• Example: Let s1 = 0.07 and s2 = 0.08. Then the implied forward

rate is f1,2 = 2·0.08−1·0.072−1 = 0.09. The expectations theory says that

this is the market’s expected value of the 1 year spot rate available

next year.

Kay Giesecke


Why is the spot rate curve almost never flat?Expectations theory

• Since forward rates are implied by a set of current spot rates,

expectations about future spot rates are inherent in current spot

rates available in the market

• The other way around, the expectation of future spot rates

determines current forward rates and thus current spot rates

• The theory argues that the market believes (“expects”) that the

spot rate will be higher in the future (e.g. because of inflation), and

this translates into a rising spot rate curve today

• Caveat: the market expects rates to increase whenever the spot rate

curve is upward sloping, which is almost always the case. Thus

“market expectations” cannot be right even on average.

Kay Giesecke



Liquidity preference theory

• This theory asserts that investors prefer short-term fixed-income

securities over long-term securities

• In other words, investors prefer to stay flexible: they like their funds

to be liquid rather than tied up, and this flexibility costs some yield

• While this is plausible for time deposits, the argument is less obvious

in the case of bonds, which can often be sold in the market

• In that case the argument is that short-term bonds are preferred

over long-term bonds since the former are less sensitive to rate

changes, and thus carry less risk if short term sales are necessary

Kay Giesecke



Market segmentation theory

• This theory asserts that the market for fixed-income securities is

segmented by maturity dates

• Investors desire a specific set of maturities, based on their projected

need for funds or their risk preference

• The demand and supply for a specific maturity is determined by a

specific sect of investors; in the extreme, all points on the spot rate

curve are mutually independent

Kay Giesecke


Expectations dynamicsForecasting future spot rates

• Suppose the expectations about future spot rates implied by current

spot rates will actually be fulfilled

• We can forecast next year’s spot rate curve from the current one,

and this curve implies another set of expectations for the following

year. If these are fulfilled, too, we can predict ahead once again,

generating spot rate curve dynamics

• Let (si) be the current spot rate curve. If expectations will actually

be fulfilled, then the j year spot rate available next year will be

equal to the forward rate f1,1+j implied by (si), given by

f1,1+j =(

(1 + s1+j)1+j

1 + s1

) 1j

− 1, 0 < j ≤ n

Kay Giesecke


Expectations dynamics

Forecasting future spot rates

• Here is an example:

s1 s2 s3 s4 s5 s6 s7

Current 6.00 6.45 6.80 7.10 7.36 7.56 7.77

1yr Forecast

• Calculate the forecast rates using annual compounding

Kay Giesecke


Expectations dynamicsForecasting future spot rates

• Here is an example:

s1 s2 s3 s4 s5 s6 s7

Current 6.00 6.45 6.80 7.10 7.36 7.56 7.77

1yr Forecast 6.90 7.20 7.47 7.70 7.88 8.06

• Since the j year spot rate available next year will be equal to the

forward rate f1,1+j implied by (si), next year’s spot rate forecast is

f1,2 =(1 + s2)2

1 + s1− 1 = 0.069

f1,3 =(

(1 + s3)3

1 + s1

) 12

− 1 = 0.0720

• Note that the forecast curve is shorter by one term

Kay Giesecke


Expectations dynamics

Invariance theorem

• Suppose you have to invest a fixed amount in Treasuries for n years,

without withdrawing funds before n

– Multitude of choices whose values depend on future rates

• Theorem. Suppose interest rates evolve according to expectations

dynamics. Then, with annual compounding, a sum invested in the

interest rate market for n years will grow by a factor of (1 + sn)n

independent of the investment and reinvestment strategy, so long as

all funds are fully invested.

• Interpret this in terms of the short rates, which do not change under

expectations dynamics: every investment earns the relevant short

rates over its duration

Kay Giesecke


Expectations dynamicsInvariance theorem

• Proof for n = 2. You have two choices:

– Invest into a 2 year zero that will have grown to (1 + s2)2 after 2

years

– Invest into a 1 year zero that will have grown to (1 + s1) after a

year, and then reinvest into another 1 year zero at the then

current 1 year spot rate. Under expectations dynamics, this rate

will be equal to today’s forward rate f1,2 for next year (the short

rate r1), and so the investment will have grown after 2 years to

(1 + s1)(1 + f1,2) = (1 + s2)2

by the definition of the forward rate f1,2

• A similar argument applies for any n.

Kay Giesecke


Running present value

Present value and spot rates

• Let si be the spot rate for i years with annual compounding

– A dollar deposited at time 0 has value (1 + si)i after i years

– The corresponding discount factor is di = 1(1+si)i

• For a spot rate curve (si), the present value of an investment

(x0, x1, . . . , xn) is given by

PV =n∑

i=0

xidi

Kay Giesecke



• We now start with the final cash flow and work backward to the

present along the cash flow times i

• The running present value at year i of the remaining investment

(xi, xi+1, . . . , xn) is defined by the recursive relation

PV (i) = xi +PV (i+ 1)

1 + ri

where ri is the short rate applying at year i and PV (n) = xn

• Note that the short rates (ri) are known today, so PV (i) is a

deterministic quantity

Kay Giesecke



• Theorem. We have PV (0) = PV .

• Proof. Since r0 = f0,1 = s1, we calculate

PV (0) = x0 +PV (1)1 + s1

= x0 +1

1 + s1

(x1 +

PV (2)1 + f1,2

)= x0 +

x1

1 + s1+

1(1 + s1)(1 + f1,2)

(x2 +

PV (3)1 + f2,3

)= x0 +

x1

1 + s1+

x2

(1 + s2)2+

PV (3)(1 + s3)3

and so on, where we use the definition of the forward rates fi,j

Kay Giesecke


Running present value• Theorem. Suppose that interest rates follow the expectation

dynamics. Then the PV (i) will be equal to the realized present

value of the cash flows at i, where i = 1, 2, . . . , n.

• Proof. Under the expectation dynamics, the short rate ri at time i

available today equals the one year spot rate available at time i.

Then, for i = 1, we have that

PV (1) = x1 +PV (2)1 + r1

= x1 +1

1 + r1

(x2 +

PV (3)1 + r2

)= x1 +

x2

1 + r1+

PV (3)(1 + r1)(1 + r2)

equals the present value of (x1, x2, . . . , xn) at time 1

Kay Giesecke


Floating rate bonds

• A floating rate bond (or note or floater) has a fixed face value,

maturity and coupon dates, but its coupon is tied to the rates when

the coupon is due. Specifically, at each coupon date the coupon rate

for the next period is reset to the then current spot rate for that

period.

• Theorem. At any coupon date before the maturity, the value of a

floating rate bond is equal to its face value (the bond is “at par”).

• Proof. Use a running present value argument.

Kay Giesecke


Duration

• Above, we considered the duration of a bond as a measure for its

sensitivity to yield changes (maturity fixed)

• In the context of the term structure, other measures of sensitivity

can be constructed

• For a given short rate curve (si), we consider a parallel shift in the

curve (si + ∆) for some hypothetical instantaneous change ∆

– Note that the shifted spot rates apply for the same periods as

the original rates

– This generalizes a change in the yield to a non-flat term

structure of spot rates

Kay Giesecke


Duration

• We are interested in the response of the bond price to a parallel shift

• Consider the cash stream (x0, x1, . . . , xn), whose price P (∆) as a

function of the shift ∆ is equal to

P (∆) =n∑

i=0

xi

(1 + si + ∆)i

• The relative price sensitivity is given by the quasi-modified

duration

DQ = − 1P (0)

dP (∆)d∆

∣∣∣∣∆=0

=1PV

n∑i=1

i xi

(1 + si)i+1

which has units of time but is not a weighted average of cash flow

times

Kay Giesecke


Fisher-Weil Duration

• We now consider the case with continuous compounding

• The price P (∆) of the cash stream (x0, x1, . . . , xn) at times

(t0, t1, . . . , tn) ∈ Rn+ is equal to

P (∆) =n∑

i=0

xie−(si+∆)ti

where si is the spot rate applying to [0, ti]

• The relative price sensitivity is given by the Fisher-Weil duration

DFW = − 1P (0)

dP (∆)d∆

∣∣∣∣∆=0

=1PV

n∑i=0

tixie−siti

which has units of time and is a weighted average of cash flow times

Kay Giesecke


Immunization

• The term structure perspective leads to a more robust method of

portfolio immunization, which does not require the selection of

bonds with a common yield

• We construct an immunization portfolio that

– Matches the present value of our obligations

– Matches the quasi-modified or Fisher-Weil duration of the

obligations

• This gives protection against parallel shifts in the spot rate curve;

keep in mind that other shifts are possible as well

Kay Giesecke