Lecture6_DurationTermStructure
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Transcript of Lecture6_DurationTermStructure
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7/28/2019 Lecture6_DurationTermStructure
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Fall 2011 Duration and Term Structure Prof. Page
BUSM 411: Derivatives and Fixed Income
6. Duration and Convexity
6.1. Interest rate sensitivity
Bond prices and yields are inversely related: as yields increase, bond prices fall and
vice versa
An increase in a bonds yield to maturity results in a smaller price change than a equal
decrease in yield (convexity)
Prices of long-term bonds tend to be more sensitive to interest rate changes than prices
of short-term bonds
The sensitivity of bond prices to changes in yields increases at a decreasing rate asmaturity increases
Interest rate risk is inversely related to the bonds coupon rate: high-coupon bonds are
less sensitive to changes in interest rates than low-coupon bonds
The sensitivity of a bonds price to yield changes is inversely related to the yield at
which the bond currently sells
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Fall 2011 Duration and Term Structure Prof. Page
6.2. Duration
Maturity is the key determinant of a bonds interest rate risk
Even the relation between coupon rates and interest rate sensitivity really boils down
to maturity:
Bonds with higher coupons have more of their cash flows occurring at shorter
horizons relative to low or zero coupon bonds
Hence, high coupon bonds have a lower effective maturity
Duration is a measure of the effective maturity of a bond or portfolio of bonds
Duration combines the effects of coupon rates and actual maturity into a summary
measure of interest rate risk
Macaulay duration is simply a weighted average of the horizons of each of the bonds
cash flows, weighted by their present value:
D =T
t=1
t
CFt/(1 + y)
t
Bond price
where CFt is the cash flow in period t.
Duration tells us how much a bonds price changes for a given change in (gross) yields:
B
B= D
(1 + y)
(1 + y)
Practitioners often express this in a simpler, more intuitive form by defining modified
duration, D = D/(1 + y), so that
B
B= Dy
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Fall 2011 Duration and Term Structure Prof. Page
6.3. Properties of duration
The duration of a zero coupon bond equals its time to maturity
Holding maturity constant,a bonds duration is lower when the coupon rate is higher
Holding coupon rate constant, duration increases with maturity
Holding other factors constant, duration of a coupon bond is higher when yield to
maturity is lower
Duration of perpetual bond is 1+yy
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Fall 2011 Duration and Term Structure Prof. Page
6.4. Duration and convexity
Duration is only a local, linear approximation of the relationship between yield changes
and bond price changes
We can improve our approximation of this relationship by accounting for convexity
The formula for the convexity of a bond with maturity of T years and annual couponpayments is
Convexity =1
P(1 + y)2
Tt=1
CFt
(1 + y)t(t2 + t)
We can then incorporate convexity into our expression for price changes as a function
of yield changes:P
P= Dy +
1
2
Convexity(y)2
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Fall 2011 Duration and Term Structure Prof. Page
Example:
30-year bond with 8% coupon, selling at initial yield to maturity of 8%
Macaulay duration:
D =3
0t=1t
CFt/(1 + y)
t
$1000
= 12.16
Modified duration:
D = D/(1 + .08) = 11.26
Convexity:
Convexity =1
$1000(1.08)2
30
t=1 CFt
(1.08)t(t2 + t) = 212.4
Suppose the yield increases from 8% to 10%:
Price =$80
.10
1
1
(1.10)30
+
$1000
(1.10)30= $811.46
a decline of 18.85%
The linear duration rules predicts a price change of
PP
= Dy = 11.26 .02 = .2252 or 22.52%
Accounting for convexity gives
P
P= Dy+(
1
2)Convexity(y)2 = 11.26.02+(
1
2)212.4(.02)2 = .1827 or 18.2
a much more accurate prediction.
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Fall 2011 Duration and Term Structure Prof. Page
6.5. The Term-Structure of Interest Rates
6.5.1 The yield curve
Interest rates can and often do differ for cash flows of different maturities.
The relationship between yield and maturity is known as the term-structure of
interest rates, and the graphical representation of this relationship is called the yield
curve.
Examples:
Figure 1: Treasury yield curves
Yield curves are typically constructed from the yields on Treasury securities, in order
to isolate the relationship between yield and maturity (no default risk)
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Fall 2011 Duration and Term Structure Prof. Page
6.5.2 The yield curve and future interest rates
Where does the shape of the yield come from?
Interest rate certainty (we know what the future path of interest rates will be):
Suppose the yield curve is upward sloping as in the example above: the 1-year
yield is 5% and the 2-year yield is 6%.
Consider two strategies: (i) buy a 2-year zero-coupon bond, or (ii) buy a 1-year
zero coupon bond and roll it over next year into another 1-year bond.
If we invest the same initial amount (say $100) in both strategies, they must offer
the same return (or else wed have an arbitrage opportunity, since neither strategy
involves any risk):
Buy & hold 2-year zero = Roll over 1-year bonds
$100 (1.06)2 = $100 (1.05) (1 + r2)
Solve for r2:
r2 =(1.06)2
(1.05) 1 = .0701 or 7.01% > 5%
Upward sloping yield curve means interest rates will rise!
Spot rate: the current yield on a zero coupon bond of a given maturity. In the
example above, the 1-year spot rate is 5% and the 2-year spot rate is 6%.
Short rate: the yield for a given time interval (say a year) at different points in
time. In the example above, todays short rate is 5% and next years short rate
is 7.01%.
More generally, we can find the short rate for n periods ahead using the formula:
(1 + rn) =(1 + yn)
n
(1 + yn1)n1
In reality, we dont know future interest rates with certainty, so we refer to the interest
rate backed out in this manner the forward rate. The forward rate need not equal
the actual future short rate or even the expected future short rate if investors require
some sort of liquidity premium.
Forward rates and interest rate risk:
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Investors with short horizons prefer to lock in an interest rate by investing in
short-term bonds, rather than long-term bonds to be sold for an uncertain price
in the future.
Short-term investors would require a liquidity premium to invest longer-term
bonds forward rate is higher than expected future short rate
Long-term investors would prefer to lock in long-term interest rates, rather than
subjecting themselves to interest rate risk by rolling over
Forward rate is lower than expected short rate!
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Fall 2011 Duration and Term Structure Prof. Page
6.5.3 Theories of the term structure
Expectations hypothesis
Slope of the yield curve is due to expectations of changes in short-term interest
rates
Under this hypothesis, forward rate equals the market consensus expectation of
future short rate
This can lead to either upward or downward sloping yield curves depending on
what the expectations of future short rates are
Liquidity preference
People prefer liquidity (matching maturity to investment horizons)
Short-term investors dominate the market, so long-term bonds must offer a liq-
uidity premium in order to get individuals to invest in them.
The liquidity premium on longer-term bonds leads to an upward sloping yield
curve (which is what we usually observe)
The two theories arent mutually exclusive.
Expectations of future short rates interact with required liquidity premia to produce
various shapes of the yield curve:
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6.5.4 Interpreting the term structure
If term structure reflects market expectations of future interest rates, we can use the
term structure to infer the markets expectations
This expectation can serve as a benchmark for our own analysis and help guide ourinvestment decisions
Problem: we cannot tell how much an upward sloping yield curve is due to expectations
of interest rate increases and how much is dues to a liquidity premium
fn = E[rn] + Liquidity premium
Still, very steep yield curves are typically taken as an indicator of interest rate increases
We can more safely interpret a downward sloping yield curve as evidence that interest
rates are expected to decline
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