8/19/2019 10-Duration and Convexity
1/18
comm 324 --- W. SuoSlide 1Slide 1
Duration andConvexity
8/19/2019 10-Duration and Convexity
2/18
comm 324 --- W. SuoSlide 2Slide 2
Interest Rate Sensitivity
8/19/2019 10-Duration and Convexity
3/18
comm 324 --- W. SuoSlide 3Slide 3
Determinants of Bond Price Volatility
8/19/2019 10-Duration and Convexity
4/18
comm 324 --- W. SuoSlide 4Slide 4
Inverse relationship between price and yieldAn increase in a bond’s yield to maturity results in a smaller
price decline than the gain associated with a decrease in yieldLong term bonds tend to be more price sensitive than shortterm bondsAs maturity increases! price sensitivity increases at adecreasing rate
"rice sensitivity is inversely related to a bond’s coupon rate"rice sensitivity is inversely related to the yield to maturity atwhich the bond is selling
#ond "ricing Relationships
8/19/2019 10-Duration and Convexity
5/18
comm 324 --- W. SuoSlide $Slide $
%erivation o& 'ormula 'or (acauley’s%uration
)he slope o& a bond’s price yield relationship measures the bond’s sensitivity to *)(
(odi&ied duration +(od,(acauley duration +(A-,
(acaulay +1.3/, suggested studying a bond’s time structure bye0amining its average term to maturity
Appro0imating the bond price change using duration ise uivalent to moving along the slope o& the bond price yieldcurve
8/19/2019 10-Duration and Convexity
6/18
comm 324 --- W. SuoSlide Slide
0ample -alculation o& (A- and(5%
6iven in&ormationA 71!888 par bond with a *)( o& 189 has three years to
maturity and a $9 coupon rate-urrently sells &or 7/:$; $:
( ) ( ) ( )1 2 37$8 7$8 7$8 71!888
"<1 ;18 1 ;18 1 ;18
74$;4$4 741;322 7::/;//17/:$; $:
+= + +
= + +
=
8/19/2019 10-Duration and Convexity
7/18comm 324 --- W. SuoSlide :Slide :
0ample -alculation o& (A- and
(5%(A- can be calculated using the previous presentvalue calculations
) "< o& -'ach -'s "< as
&raction o& "rice ) weighted by -'
1 4$;4$$ 8;8$1.1 1 = 8;8$1.1 > 8;8$1.1
2 41;322 8;84:1/ 2 = 8;84:1/ > 8;8.43/
3 ://;//1 8;.88. 3 = 8;.88. > 2;:82:8
1;88888 (A- > 2;/4/..
(5% > (A- ÷ +1?y,> 2;/4/.. ÷ 1;18
> 2;$/..
8/19/2019 10-Duration and Convexity
8/18comm 324 --- W. SuoSlide /Slide /
A measure of the effective maturity of a bondThe weighted average of the times until each payment is received, with theweights proportional to the present value of the payment
This interpretation is useful when NPV of a project
Duration is shorter than maturity for all bonds except zero coupon bondsDuration is equal to maturity for zero coupon bondsPrice change is proportional to duration and not to maturity
Or, if we denote D*
= modified duration ,
%uration
1$ /1$ /
8/19/2019 10-Duration and Convexity
9/18comm 324 --- W. SuoSlide .Slide .
0ample
A bond has 189 coupon rate +paid semi annually, and 18 years to maturity!*)( is currently 189
-urrent price 71888! (ac> ;$42:! (od> ;2311
I& there is 8;19 increase in *)(! then its price is 7..3;:.$2
I& there is 8;19 decrease in *)(! then its price is 718.;2$:
)he appro0imation only holds when the change in yield is small
8/19/2019 10-Duration and Convexity
10/18comm 324 --- W. SuoSlide 18Slide 18
@hy is duration a ey conceptB
It’s a simple summary statistic o& the e&&ectiveaverage maturity o& the port&olioC
It measures the sensitivity o& bond price changerelative to the change in yieldIt is an essential tool in immuniDing port&olios &rominterest rate ris CIt is a measure o& interest rate ris o& a port&olio
8/19/2019 10-Duration and Convexity
11/18comm 324 --- W. SuoSlide 11Slide 11
Properties of Duration
For discount bonds, duration = maturityFor coupon-paying bonds, duration < maturity
The higher the coupon, the shorter the durationThe higher the discount rate, the shorter the durationFor bonds selling at par or above, duration increases at adecreasing rate with maturityFor bonds selling at a discount, duration increases withmaturity for a long time, and then declinesBond prices vary proportionately with duration
8/19/2019 10-Duration and Convexity
12/18comm 324 --- W. SuoSlide 12Slide 12
Other commonly used
risk measures%ollar duration +or %elta,
%ollar duration measures the dollar change in a bondEs value to a change in themar et interest rate;
"
8/19/2019 10-Duration and Convexity
13/18comm 324 --- W. SuoSlide 13Slide 13
%uration and -onve0ity
9 "rice -hange
%uration
"ricing error &rom conve0ity
*ield
8/19/2019 10-Duration and Convexity
14/18comm 324 --- W. SuoSlide 14Slide 14
-orrection &or -onve0ity
-onve0ity &or semiannual coupon
Annual coupon
-orrection &or -onve0ity
21 -onve0ity + ,2 P
D y y P
∆= − ∗ ∆ + ⋅ ⋅ ∆
8/19/2019 10-Duration and Convexity
15/18comm 324 --- W. SuoSlide 1$Slide 1$
-onve0ity calculation
/9#ond
)imeSem;
"ayment "< o& -'+189,
@eight t+t?1,0-4
1 48 3/;8.$ ;83.$ ;8:.8
2 48 3 ;2/1 ;83: ;22$:
3
4
48
1848
sum
34;$$3
/$$; 11
. 4;$48
;83$/
;//:1
1;888
;42..
1:;:413
1/;4:$.
8/19/2019 10-Duration and Convexity
16/18comm 324 --- W. SuoSlide 1Slide 1
-onve0ity calculation +cont;,
-onve0ity is computed li e duration! as a weighted
average o& the terms +t2
?t, +rather than t, divided by+1?y, 2
)hus! in the above e0ample! it is e ual to1/;4:$.F1;8$ 2 > 1 ;:$/2
8/19/2019 10-Duration and Convexity
17/18comm 324 --- W. SuoSlide 1:Slide 1:
@hy do people li e conve0ityB
#ond #
8/19/2019 10-Duration and Convexity
18/18324 W SSlid 1/
%uration and -onve0ity o&
-allable #onds
8 Interest Rate
- a l l " r i c e
Region o& positive
conve0ity
Region o& negative conve0ity"rice yield curve is below tangent
$9 189