6. Duration and convexity

8/7/2019 6. Duration and convexity

1/14

6. Duration and convexity

1

1. Duration as an elasticity2. Duration as a measure of time3. Modified duration

4. Convexity


2/14

Duration as an elasticity

2

o Duration measures interest rate risk betterthan maturity

10 year Treasury definitely has more interest rate

risk than a 1 month T-bill BUT 10 year Treasury has more interest rate riskthan a 10 year mortgage

Mortgages are fully amortizing so investors receive mostprincipal before T

o Duration is mainly first derivative of bondprice function with respect to discount rate

Elasticity = [ % P ] / [ % R ] NoteR is change in discount factor (e.g.,1.04)


3/14

Maturity to proxy elasticity

3

o After high inflation of 1970s regulatorsrequired banks to report the maturity profile oftheir loans

oMaturity buckets for less than 1 year, 1-3,3-5, 5-10, 10 and over Banks with high fraction of assets in high maturitybuckets pressured to reduce exposure to rate

changesoWhat if all the 10 year loans are floating rate?oWhat if all the 10 year loans are fullyamortizing and the 5 year loans are zeroes?


4/14

Duration and time

4

o Duration is the weighted average of the timesthat cash flows arrive to the investor

Zero coupon bond has duration equal to T

Most debt has cash flows before T Weighted average includes times before T so weight on Tless than 100% Duration is usually less than T

oWeight = fraction of total present value of

cash flow (bond price) arriving at period t

oD = 7nt=1[Ct t/(1+r)t]/ 7nt=1 [Ct/(1+r)

t]


5/14

D = 7nt=1[Ct t/(1+r)t]/ 7nt=1 [Ct/(1+r)

t]

Where

D = duration

t = time period that has a cash flow

Ct = cash flow to be delivered in t periodsn= term-to-maturity & r = yield to maturity

Duration


6/14

y Or, to rewrite it

D = 7nt=1[PV(Ct) t]/ Bond price

y Or,

D = 7nt=1[PV(Ct)/Bond price] t

D = 7nt=1wt t

If calculate wt can check that weights sum to 1

6

Duration


7/14

y Calculate D of a 1-year 10% semi-annual coupon bond whenYTM=10%:

P = 5 + (100 + 5) = 100(1.05) (1.05)2

PV of CF1 = $4.76 PV of CF2 =$95.24

Weights = .0476 and .9524

D = (.0476*.5) + (.9524 * 1) = .9762

Duration example


8/14

Duration decreases whenever cash f lows comeearly:

Shorter maturity Higher coupon Amortization of principal

Duration decreases as interest rates rise Higher discount factor means late cash flows dont count

as much When interest rates are zero last cash flow has highest

possible weight Bank regulators rely less on maturity buckets and

more on duration of portfolio

Factors affecting duration


9/14

Modified duration

9

o Duration is elasticity = -[ % P ] / [ % R ] Much more interested in R than % R What happens when Fed raises rates by 25 bp?

(assume all rates go up 25 bp) Not, what happens when rates go up 100% (from25 bp to 50 bp or 100 bp to 200 bp)?

More useful to work with modified duration (MD) than D

o Modified duration = D/(1+r)

if bond has annual coupons (almost never)oModified duration = D/(1+.5r)

if bond has semi-annual coupons (almost always)


10/14

Modified duration

10

o Use MD to get to % P faster than with D

oD = -[ % P ] / [ % R ]

Rewrite when looking for %

P % P = -D * % R % P = -MD * R

oR is often same as Fed Funds rate change


11/14

Modified duration example

11

o Suppose a bond has MD of 8 What happens when Fed raises rates by 25 bp?(assume all rates go up 25 bp)

%P = MD *

R % P = 8 years* R

% P = 8 years* 25 bp/year % P = 200 bp % P = 2%

I.e., a par bond will start trading at 98A portfolio of $200m with MD=8 will lose $4 m. ifrates rise 25 bp (new value = $196 m.)A bank with a MD on loans of 8 will lose moremoney on its investments than one a loan MD of 4


12/14

Modified duration

12

o Duration and modified duration approximatethe price change ( D = [ % P ] / [ % R ] )

First derivative of bond price function:

Linear approximation of nonlinear functionWorks well for small changesMakes some assumptions about rate changesalong entire yield curve only first derivative insome circumstances that may not be realistic

oBetter measure of price change if use secondderivative too (Taylor series)


13/14

Convexity

13

oConvexity adds in second derivative effect CX factor is always positive Price estimated using duration or MD is

always too low First derivative is tangent line (slope =-MD) The tangent line only equals curve at one spotand is off at all others

Off more if function is curvierTwo bonds with same MD can have different CXfactors one is more sensitive to rate changes.


14/14

Convexity

14

Documents

6. Duration and convexity