20 - bond duration and convexity
TRANSCRIPT
Notes
Page 1
Economics of Capital Markets
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Bond Duration and ConvexityBond Duration and Convexity
Major Topics:
Economics of Capital Markets
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IntroductionIntroduction
Notes
Page 2
Economics of Capital Markets
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Bond Duration and Convexity IntroductionBond Duration and Convexity Introduction
We already know the relationship between bond prices and yields (or required rates of return)– Some others may be new
Economics of Capital Markets
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1. The value of a bond is inversely related to changes in the investor’s required rate of return.
– The higher the required rate of return, the lower the bond value.
Bond Duration and Convexity Introduction (Continued)
Bond Duration and Convexity Introduction (Continued)
Notes
Page 3
Economics of Capital Markets
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$0
$50
$100
$150
$200
0% 2% 4% 6% 8% 10% 12% 14% 16%
Yield
Pric
e
( ) ( )
++
+−= nn
B0 y1
Fy1
11yCP
1. Prices and returns on bonds are inversely related
Bond Duration and Convexity Introduction (Continued)
Bond Duration and Convexity Introduction (Continued)
Economics of Capital Markets
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2 The market value of a bond will be less than the par value if the investor’s required rate is above the coupon interest rate
Bond Duration and Convexity Introduction (Continued)
Bond Duration and Convexity Introduction (Continued)
Notes
Page 4
Economics of Capital Markets
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Premium bonds
Par bonds
Discount bonds
Bond Duration and Convexity Introduction (Continued)
Bond Duration and Convexity Introduction (Continued)
Economics of Capital Markets
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3. Long-term bonds have greater interest rate risk than short term bonds.
$0
$50
$100
$150
$200
$250
0% 2% 4% 6% 8% 10% 12% 14% 16%Yield
Pric
e
10 Year20 Year5 Year
20
10
5
51020
Bond Duration and Convexity Introduction (Continued)
Bond Duration and Convexity Introduction (Continued)
Notes
Page 5
Economics of Capital Markets
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3. Corollary: Low coupon bonds have greater interest rate sensitivity than high coupon bonds.
$0$50
$100$150$200
1% 3% 5% 7% 9% 11%
13%
15%
Yield
Pric
e
CouponZero
Coupon
Zero
Zero
Coupon
Bond Duration and Convexity Introduction (Continued)
Bond Duration and Convexity Introduction (Continued)
Economics of Capital Markets
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The sensitivity of a bond’s value to changing interest rates depends on both the length of time to maturity and on the pattern of cashflows provided by the bond
Bond Duration and Convexity Introduction (Continued)
Bond Duration and Convexity Introduction (Continued)
Notes
Page 6
Economics of Capital Markets
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DurationDuration
Economics of Capital Markets
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We want to know how the price of a bond changes as the yield changes
Bond Duration and Convexity DurationBond Duration and Convexity Duration
Notes
Page 7
Economics of Capital Markets
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To derive the elasticity, use
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
∑= +
++
=n
1tnt
B0 y)(1
F y)(1
C P
Economics of Capital Markets
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Take the first derivative with respect to y to get
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
+
++
+++
+++
=
++
+++
++
+= ++
nn21
1n1n32
B0
y)(1nF
y)(1nC ...
y)(12C
y)(11C
y11-
y)(1(-n)F
y)(1(-n)C ...
y)(1(-2)C
y)(1(-1)C
dydP
Notes
Page 8
Economics of Capital Markets
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Multiplying through by
where
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
Dy1
1-
P1
y)(1nF
y)(1nC ...
y)(12C
y)(11C
y11-
P1
dydP
B0
nn21B0
B0
+=
∗
+
++
+++
+++
=
B0P1
0 P1
y)(1nF
y)(1nC ...
y)(12C
y)(11C D B
0nn21 >∗
+
++
+++
++
=
Economics of Capital Markets
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Now we can get
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
0 D- P
y1 dy
dP η B0
B0 <=
+=
Notes
Page 9
Economics of Capital Markets
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Definition– Duration measures the responsiveness of a
bond’s price to interest rate changes:
B0
n
B0
n
1tt
Py)(1
nF
P
y)(1Ct
D Duration +++⋅
==∑=
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
Economics of Capital Markets
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Despite the name, the duration is not the same as the time to maturity
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
Notes
Page 10
Economics of Capital Markets
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Rewrite D as
B
n
nnt
B
nn
1tB
0
tt
Py)(1
n
γ,F γ tw
Py)(1
nF
P
y)(1Ct
tD
+=+=
++
+⋅
=
∑
∑=
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
Economics of Capital Markets
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The duration shows the “actual” weighted length of time needed to recover the current cost of the bond,
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
B0P
Notes
Page 11
Economics of Capital Markets
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For a zero coupon bond, duration equals the maturity
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
+
=
++
+⋅
= ∑=
B0
n
B
nn
1tB
0
tt
P1
y)(1Fn
Py)(1
nF
P
y)(1Ct
tD
0
Economics of Capital Markets
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But
soD = n
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
nB
0 y)(1FP+
=
Notes
Page 12
Economics of Capital Markets
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For non-zero bonds, D < n– Generally, the higher C and the higher y, the
shorter is durationAlso, the longer the maturity, the greater the duration so the greater is the price change for a small yield change
Bond Duration and Convexity Duration (Continued)
Bond Duration and Convexity Duration (Continued)
Economics of Capital Markets
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ConvexityConvexity
Notes
Page 13
Economics of Capital Markets
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We noticed earlier that the price-yield curve is convex to the origin
Bond Duration and Convexity ConvexityBond Duration and Convexity Convexity
Economics of Capital Markets
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Consider this situation
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
y
PB
PB*
y*
Tangent Line
Notes
Page 14
Economics of Capital Markets
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The tangent line shows the arte of change of PB to a change in y at the tangency point
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
Economics of Capital Markets
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Notice that, starting at y*, if we move away from y*, the price change will always be underestimated by the tangent
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
y
PB
PB*
y*
Underestimate
Underestimate
Notes
Page 15
Economics of Capital Markets
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The accuracy of using the tangent depends on the convexity of the price-yield curve
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
Economics of Capital Markets
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Expand the price equation using a Taylor Series Expansion– Write the first two terms plus the error as
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
( ) error dydy
Pd21 dy
dydp dP 2
2
B0
2B0 ++=
Notes
Page 16
Economics of Capital Markets
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Dividing by , we get
– The first term is the duration for a small yield change
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
( )B0
B0
2
2
B0
2
B0
B0
B0
B0
Perror
Pdy
dyPd
21
Pdy
dydP
PdP
++=
B0P
Duration
Economics of Capital Markets
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The second term corrects for the convexity– The second derivative is called the dollar
convexity of a bond
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
2
B0
2
dyPd $Convexity =
Notes
Page 17
Economics of Capital Markets
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It is easy to show that the second derivative of the price-yield relationship is
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
( )( )
( )( )∑
=++ +
++
++
=n
1t2n2t2
B0
2
y1F1nn
y1C1tt
dyPd
Economics of Capital Markets
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The duration and the convexity adjustment term can be summed to get the estimated price change due to duration and convexity
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
Notes
Page 18
Economics of Capital Markets
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Example: 25-year, 6% coupon selling to yield 9%– Duration = 10.62– Convexity = 182.92– Assume the yield rises 200 basis points to 11%
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
Economics of Capital Markets
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Example (Continued)
Percentage change in price due to duration:= -duration*dy= -(10.62)*(0.02)= -21.24%
Percentage change in price due to convexity:= 0.5*convexity*(dy)2
= 0.5*182.92*0.022
= 3.66%
Percentage change in price = -21.24% + 3.66% = -17.58%
Bond Duration and Convexity Convexity (Continued)
Bond Duration and Convexity Convexity (Continued)
Notes
Page 19
Economics of Capital Markets
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Consider two coupon bonds, A and B– B has greater convexity
Bond Duration and Convexity Application of Duration and ConvexityBond Duration and Convexity Application of Duration and Convexity
y
PB
PB*
y*
B
A
Same Duration
Economics of Capital Markets
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Consider two coupon bonds (Continued)
– Bond B will have a greater price no matter what happens to the yield
– Also,
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Bond Duration and Convexity Application of Duration and Convexity (Continued)
More is Bon Gain Falls YieldLess is Bon Loss Rises Yield
⇒⇒
Notes
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Economics of Capital Markets
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The market takes B’s greater convexity into account when pricing B– The market prices the convexity since the
convexity adds a risk element
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Economics of Capital Markets
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Pricing depends on expectations– If investors expect very small yield changes, the
advantages of holding B are small – both A and Boffer the same approximate price (i.e., return)
» There is little paid to the convexity
– If a big change is expected, B will be priced higher than A
» B has more risk
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Notes
Page 21
Economics of Capital Markets
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There are three properties of bonds due to convexity
1. As the required yield increases (decreases), the convexity of a bond decreases (increases).» This is referred to as positive convexity
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Economics of Capital Markets
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There are three properties (Continued)
1. As the required yield increases: Implication– As the yield rises, the price declines but is “slowed”
by the decline in the duration– As the yield falls, the price rises but is “enhanced”
by the increase in the duration
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Notes
Page 22
Economics of Capital Markets
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There are three properties (Continued)
1. As the required yield increases: Graphically
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Bond Duration and Convexity Application of Duration and Convexity (Continued)
y
PB
↑↓
↓↑
Duration ,y AsDuration ,y As
Economics of Capital Markets
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There are three properties (Continued)
2. For a given yield and maturity, the lower the coupon, the greater the convexity of a bond• A zero coupon bond has the highest convexity
3. For a given yield and duration, the lower the coupon, the smaller the convexity• A zero coupon bond has the lowest convexity
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Notes
Page 23
Economics of Capital Markets
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There are three properties (Continued)
Bond Duration and Convexity Application of Duration and Convexity (Continued)
Bond Duration and Convexity Application of Duration and Convexity (Continued)
583.780%/25-year25.180%/5-year182.926%/25-year20.856%/5-year160.729%/25-year19.459%/5-year
ConvexityBond(Coupon/Maturity)