1 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
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1
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
MS&E 242: Investment Science, The term structure of interest rates 3
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
MS&E 242: Investment Science, The term structure of interest rates 4
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
MS&E 242: Investment Science, The term structure of interest rates 5
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
MS&E 242: Investment Science, The term structure of interest rates 6
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
MS&E 242: Investment Science, The term structure of interest rates 7
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
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MS&E 242: Investment Science, The term structure of interest rates 8
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