Debt Instruments and Markets

Professor Carpenter

Convexity 


Concepts
and
Buzzwords

• Dollar
Convexity
 • Convexity
 • Curvature
 • Taylor
series
 • Barbell,
Bullet


Readings

• Veronesi,
Chapter
4
 • Tuckman,
Chapters
5
and
6


Convexity

1

Debt Instruments and Markets

Professor Carpenter

Convexity

• Convexity
is
a
measure
of
the
curvature
of
the
value
of
 a
security
or
portfolio
as
a
function
of
interest
rates.
 • Duration
is
related
to
the
slope,
i.e.,
the
first
derivative.
 • Convexity
is
related
to
the
curvature,
i.e.
the
second
 derivative
of
the
price
function.
 • Using
convexity
together
with
duration
gives
a
better
 approximation
of
the
change
in
value
given
a
change
in
 interest
rates
than
using
duration
alone.


Price‐Rate
Func:on

Example:
Security
with
Positive
Convexity 


Price


Linear
 approximation
of
 price
function


Approximation error

Interest
Rate
(in
decimal)


Convexity

2

Debt Instruments and Markets

Professor Carpenter

Correc:ng
the
Dura:on
Error

• The
price‐rate
function
is
nonlinear.
 • Duration
and
dollar
duration
use
a
linear
 approximation
to
the
price
rate
function
to
measure
 the
change
in
price
given
a
change
in
rates.
 • The
error
in
the
approximation
can
be
substantially
 reduced
by
making
a
convexity
correction.


Taylor
Series

• The
Taylor
Theorem
from
calculus
says
that
the
value
 of
a
function
can
be
approximated
near
a
given
point
 using
its
“Taylor
series”
around
that
point.


 • Using
only
the
first
two
derivatives,
the
Taylor
series
 approximation
is:


1 f ''(x 0 ) × (x − x 0 ) 2 2 1 Or, f (x) − f (x 0 ) ≈ f '(x 0 ) × (x − x 0 ) + f ''(x 0 ) × (x − x 0 ) 2 2 f (x) ≈ f (x 0 ) + f '(x 0 ) × (x − x 0 ) +

€

Convexity

3

Debt Instruments and Markets

Professor Carpenter

Dollar
Convexity

• Think
of
bond
prices,
or
bond
portfolio
values,
as
 functions
of
interest
rates.
 • The
Taylor
Theorem
says
that
if
we
know
the
first
and
 second
derivatives
of
the
price
function
(at
current
rates),
 then
we
can
approximate
the
price
impact
of
a
given
 change
in
rates.


f (x) − f (x 0 ) ≈ f '(x 0 ) × (x − x 0 ) + 0.5 × f ''(x 0 ) × (x − x 0 ) 2
 The
first
derivative
is
minus
dollar
duration.
  Call
the
second
derivative
dollar
convexity.
 • Then
change
in
price
≈
‐$duration
x
change
in
rates

 
 
+
0.5
x
$convexity
x
change
in
rates
squared


€

Dollar
Convexity
of
a
PorBolio

If
we
assume
all
rates
change
by
the
same
amount,
then
 
the
dollar
convexity
of
a
portfolio
is
the
sum
of
the
dollar
 convexities
of
its
securities.
 Sketch
of
proof:


∑ f (x) − f (x ) ≈ (∑ f '(x )) × (x − x ) + 0.5 × (∑ f ''(x )) × (x − x ) i i 0 i 0 0 i 0 0

2

I.e.,
Δ portfolio value ≈ ‐
(sum
of
dollar
durations)
x
Δr
€

+ 0.5 x (sum of dollar convexities) x (Δrates)2 ⇒ Portfolio
dollar
duration
=
sum
of
dollar
durations
 ⇒ Portfolio
dollar
convexity
=
sum
of
dollar
convexities 



Convexity

4

Debt Instruments and Markets

Professor Carpenter

Convexity

• Just
as
dollar
duration
describes
dollar
price
sensitivity,
 dollar
convexity
describes
curvature
in
dollar
performance.
 • To
get
a
scale‐free
curvature
measure,
i.e.,
curvature
per
 dollar
invested,
we
define


convexity =

dollar convexity price

⇒ The
convexity
of
a
portfolio
is
the
average
convexity
of
its
 securities,
weighted
by
present
value:


€ convexity =

∑ price × convexity ∑ price i i

i

= pv wtd average convexity

• Just
like
dollar
duration
and
duration,
dollar
convexities
 add,
convexities
average.


€

Dollar
Formulas
for
$1
Par
of
a
Zero

For
$1
par
of
a
t‐year
zero‐coupon
bond
 price = dt (rt ) = 1 (1+ rt /2) 2t t (1+ rt /2) 2t +1 t 2 + t /2 (1+ rt /2) 2t +2

dollar duration = - dt '(rt ) = dollar convexity = dt ''(rt ) =

For
$N
par,
these
would
be
multiplied
by
N.

€

Convexity

5

Debt Instruments and Markets

Professor Carpenter

Percent
Formulas
for
Any
Amount
of
a
Zero
 duration = dollar duration N × t /(1+ rt /2) 2t +1 t = = 2t price N ×1/(1+ rt /2) 1+ rt /2 dollar convexity N × (t 2 + t /2) /(1+ rt /2) 2t +2 t 2 + t /2 = = price N ×1/(1+ rt /2) 2t (1+ rt /2) 2

convexity =

€

• These
formulas
hold
for
any
par
amount
of
the
zero
–
they
are
 scale‐free.
 • The
duration
of
the
t‐year
zero
is
approximately
t.
 • The
convexity
of
the
t‐year
zero
is
approximately
t2.
 • (If
we
defined
price
as
dt
=
e‐rt,
and
differentiated
w.r.t.
this
r,
 then
the
duration
of
the
t‐year
zero
would
be
exactly
t
and
the
 convexity
of
the
t‐year
zero
would
be
exactly
t2.)


Class
Problems

Calculate
the
price,
dollar
duration,
and
dollar
 convexity
of
$1
par
of
the
20‐year
zero
if
r20
=
6.50%.


Convexity

6

Debt Instruments and Markets

Professor Carpenter

Class
Problems

Suppose
r20
rises
to
7.50%.
 1) Approximate
the
price
change
of
$1,000,000
par
using
 only
dollar
duration.



2)
Approximate
the
price
change
of
$1,000,000
using
 both
dollar
duration
and
dollar
convexity.





3)
What
is
the
exact
price
change?


Class
Problems

Suppose
r20
falls
to
5.50%.
 1) Approximate
the
price
change
of
$1,000,000
par
using
 only
dollar
duration.



2)
Approximate
the
price
change
of
$1,000,000
using
 both
dollar
duration
and
dollar
convexity.





3)
What
is
the
exact
price
change?


Convexity

7

Debt Instruments and Markets

Professor Carpenter

Sample
Risk
Measures

Duration
and
convexity
for
$1
par
of
a

 10‐year,
20‐year,
and
30‐year
zero.

Maturity 
 10 
 20 
 30 
 Rate 
 6.00% 
 6.50% 
 6.40% 
 Price 
 0.553676 
 0.278226 
 0.151084 
 $Dura/on 
 5.375493 
 5.389364 
 4.391974 
 Dura/on 
 9.70874 
 19.37046 
 29.06977 
 $Convexity 
 54.7987 
 107.0043 
 129.8015 
 Convexity 
 98.9726 
 384.5951 
 859.1356 


For
zeroes,

 • 
duration
is
roughly
equal
to
maturity,
 • 
convexity
is
roughly
equal
to
maturity
squared.


Dollar
Convexity
of
a
PorBolio
of
Zeroes

• Consider
a
portfolio
with
fixed
cash
flows
at
different
 points
in
time
(K1,
K2,
…
,
Kn
at
times
t1,
t2,
…,
tn).
 • Just
as
with
dollar
duration,
the
dollar
convexity
of
the
 portfolio
is
the
sum
of
the
dollar
convexities
of
the
 component
zeroes.
 • The
dollar
convexity
of
the
portfolio
gives
the
correction
 to
make
to
the
duration
approximation
of
the
change
in
 portfolio
value
given
a
change
in
zero
rates,
assuming
all
 zero
rates
change
by
the
same
amount.
 n portfolio dollar convexity =

∑K j=1 j

×

t 2 + t j /2 j (1+ rt j /2)
2t j +2

€
Convexity 8

Debt Instruments and Markets

Professor Carpenter

Class
Problem:

 Dollar
Convexity
of
a
PorBolio
of
Zeroes

Consider
a
portfolio
consisting
of

 • $25,174
par
value
of
the
10‐year
zero,
 • $91,898
par
value
of
the
30‐year
zero.

Maturity 
 10 
 20 
 30 
 Rate 
 6.00% 
 6.50% 
 6.40% 
 Price 
 0.553676 
 0.278226 
 0.151084 
 $Dura/on 
 5.375493 
 5.389364 
 4.391974 
 Dura/on 
 9.70874 
 19.37046 
 29.06977 
 $Convexity 
 54.7987 
 107.0043 
 129.8015 
 Convexity 
 98.9726 
 384.5951 
 859.1356 


What
is
the
dollar
convexity
of
the
portfolio?


Convexity
of
a
PorBolio
of
Zeroes
 n $convexity convexity = = present value n ∑K j=1 j

×

t 2 + t j /2 j (1+ rt j /2) Kj tj 2t j +2

n

∑ (1+ r j=1 /2)

2t j

=

∑ (1+ r j=1 Kj /2)
2t j

×

t 2 + t j /2 j (1+ rt j /2) 2
2t j

tj n

∑ (1+ r j=1 Kj tj /2)

= average convexity weighted by present value ≈ average maturity2 weighted by present value

€
Convexity 9

Debt Instruments and Markets

Professor Carpenter

Class
Problem

Consider
the
portfolio
of
10‐
and
30‐year
zeroes.
 • The
10‐year
zeroes
have
market
value

 
$25,174
x
0.553676
=
$13,938.
 • The
30‐year
zeroes
have
market
value

 
$91,898
x
0.151084
=
$13,884.
 • The
market
value
of
the
portfolio
is
$27,822.
 • What
is
the
convexity
of
the
portfolio?


Maturity 
 10 
 20 
 30 


Rate 
 6.00% 
 6.50% 
 6.40% 


Price 
 0.553676 
 0.278226 
 0.151084 


$Dura/on 
 5.375493 
 5.389364 
 4.391974 


Dura/on 
 9.70874 
 19.37046 
 29.06977 


$Convexity 
 54.7987 
 107.0043 
 129.8015 


Convexity 
 98.9726 
 384.5951 
 859.1356 


Barbells
and
Bullets

• Consider
two
portfolios
with
the
same
duration:
 • A
barbell
consisting
of
a
long‐term
zero
and
a
 short‐term
zero
 • A
bullet
consisting
of
an
intermediate‐term
zero
 • The
barbell
will
have
more
convexity.


Convexity

10

Debt Instruments and Markets

Professor Carpenter

Example

• Bullet
portfolio:
$100,000
par
of
20‐year
zeroes
 
market
value
=
$100,000
x
0.27822
=
27,822
 
duration
=
19.37
 • Barbell
portfolio:
from
previous
example

 
$25,174
par
value
of
the
10‐year
zero
 
$91,898
par
value
of
the
30‐year
zero.
 
market
value
=
27,822
 duration = (13,938 × 9.70874) + (13,884 × 29.06977) = 19.37 13,938 + 13,884

• The
convexity
of
the
bullet
is
385.
 • The
convexity
of
the
barbell
is
478.

€

Securi:es
with
Fixed
Cash
Flows:
 More
disperse
cash
flows,
more
convexity

• In
the
previous
example,

 the
duration
of
the
bullet
is
about
20
and

 the
convexity
of
the
bullet
is
about
202=400.
 • The
duration
of
the
barbell
is
about

 0.5
x
10
+
0.5
x
30
=
20

 
but
the
convexity
is
about
 
0.5
x
102
+
0.5
x
302
=
500
>
400
=
(0.5
x
10
+
0.5
x
30)2
 • I.e.,
the
average
squared
maturity
is
greater
than
the
average
 maturity
squared.
 • Indeed,
recall
Var(X)
=
E(X2)-(E(X))2
 • Think
of
cash
flow
maturity
t
as
the
variable
and
pv
weights
as
 probabilities.

Duration
is
like
E(t)
and
convexity
is
like
E(t2). • So
convexity
≈
duration2
+
dispersion
(variance)
of
maturity.


Convexity

11

Debt Instruments and Markets

Professor Carpenter

Value
of
Barbell
and
Bullet

Barbell : V2 (s) = 25,174 91,898 + (1+ (0.06 + Δr) /2) 20 (1+ (0.064 + Δr) /2) 60 100,000 (1+ (0.065 + Δr) /2) 40

Bullet : V1 (s) =

€

At
current
rates,
 they
have
the
same
value

 and
the
same
slope
(duration).
 But
the
barbell
has
more
curvature
(convexity).


Δr

Does
the
Barbell
Always
Outperform
the
 Bullet?

• If
there
is
an
immediate
parallel
shift
in
interest
rates,
 either
up
or
down,
then
the
barbell
will
outperform
the
 bullet.
 • If
the
shift
is
not
parallel,
anything
could
happen.
 • If
the
rates
on
the
bonds
stay
exactly
the
same,
then
as
 time
passes
the
bullet
will
actually
outperform
the
 barbell:
 • the
bullet
will
return
6.5%
 • the
barbell
will
return
about
6.2%,
the
market
value‐ weighted
average
of
the
6%
and
6.4%
on
the
10‐
and
 30‐year
zeroes.


Convexity

12

Debt Instruments and Markets

Professor Carpenter

Value
of
Barbell
and
Bullet:
 One
Year
Later

If
there
is
a
large
enough
 parallel
yield
curve
shift,
 The
barbell
will
be
worth
 more.
 If
the
rates
don’t

 change,
the
bullet
will
 be
worth
more.


Convexity
and
the
Shape
of
the
Yield
Curve?

• If
the
yield
curve
were
flat
and
made
parallel
shifts,
more
convex
 portfolios
would
always
outperform
less
convex
portfolios,
and
 there
would
be
arbitrage.
 • So
to
the
extent
that
market
movement
is
described
by
parallel
 shifts,
bullets
must
have
higher
yield
to
start
with,
to
 compensate
for
lower
convexity.
 • This
would
explain
why
the
term
structure
is
often
hump‐ shaped,
dipping
down
at
very
long
maturities
where
convexity
is
 greatest
relative
to
duration—investors
may
give
yield
to
buy
 convexity.
 • Some
evidence
suggests
that
the
yield
curve
is
more
curved
 when
volatility
is
higher
and
convexity
is
worth
more.


Convexity

13