TermStructure

TermStructure - The
Term
Structure
of
Interest
...

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Unformatted text preview: The
Term
Structure
of
Interest
 Rates
 Term
Structure
of
Interest
Rates
 •  Different
interest
rates
prevail
in
the
market
 for
borrowing
over
different
=me
horizons.
 •  The
term
structure
of
interest
rates
describes
 the
rela=onship
between
the
term
of
 borrowing
and
the
rate
for
borrowing.
 •  It
can
be
expressed
in
many
different
ways.
 Term
Structure
of
Interest
Rates
 •  A
different
term
structure
can
be
derived
for
 each
(type
of)
instrument.
 •  We
will
focus
on
the
market
benchmark
term
 structure
derived
from
the
treasury
market.
 –  Treasury
securi=es
are
default‐free,
so
this
 represents
a
term
structure
that
is
free
from
 considera=ons
of
creditworthiness
 –  The
treasury
market
is
the
most
liquid
market,
so
 there
are
no
illiquidity
concerns
 Types
of
Interest
Rates
 •  •  •  •  Yield‐to‐Maturity
 Spot
Rate
 Forward
Rate
 Short
Rate
 Yield‐to‐Maturity
 •  Widely
used
for
bonds
 •  Given
by
the
constant
interest
rate
that
 equates
the
discounted
value
of
the
future
 cash
flows
under
the
bond
and
its
current
 market
price.
 •  Also
called
the
internal
rate
of
return.
 Yield‐to‐Maturity
 •  For
a
bullet
bond
with
coupon
payment
c,
n coupon
payments,
face
value
F
and
price
P,
the
 yield
(y)
is
the
solu=on
of
the
equa=on:
 1 − (1 + y )− n P = cF + F (1 + y )− n y •  The
yield
curve
gives
the
yield
as
a
func=on
of
 bond
maturity.
 € Yield‐to‐Maturity
 •  OQen
used
to
compare
bonds
with
different
 maturi=es,
issuers,
etc.
 •  Can
be
misleading:

 –  The
yield
on
a
three
year
bond
depends
on
the
 borrowing
rate
at
three
years,
as
well
as
at
all
 previous
coupon
dates.
 –  The
curve
does
not
reveal
year‐by‐year
 informa=on
about
borrowing
costs.
 Spot
Rates
 •  Let
P(t,t+k)
denote
the
price
(at
=me
t)
of
a
 zero
coupon
bond
with
face
value
1,
with
k
 periods
un=l
maturity.
 •  Let
t=0.
The
spot
rate
for
k
periods
to
maturity
 is
the
yield‐to‐maturity
of
a
zero
coupon
bond
 with
k
periods
to
maturity.
It
is
the
solu=on
sk
 of
the
equa=on:
 P (0, k ) = (1 + sk ) −k Spot
rate
curve
for
different
maturi=es
of
Canadian
Treasuries.
 Source:
www.bandofcanada.ca
 Spot
Rates
 •  We
can
also
price
a
bullet
bond
based
on
 observed
spot
rates:
 n P = cF ∑ (1 + sk )− k + F (1 + sn )− n k =1 •  More
generally,
if
an
instrument
pays
the
 (determinis=c)
cash‐flow
Ck
at
the
=me
k
 € periods
in
the
future,
k=1,…,n,
its
price
is:
 n P = ∑ Ck (1 + sk )− k k =1 Forward
Rates
 •  Rates
for
contracts
made
today
for
borrowing
 in
future
periods.
 •  fj,k
is
the
nota=on
for
the
forward
rate
covering
 period
j
to
k
 •  For
example,
f3,5
is
the
interest
rate
for
 agreeing
today
to
borrow
money
3
years
from
 now
and
repay
it
5
years
from
now.
 Forward
Rates
and
Arbitrage
 •  Arbitrage
is
the
opportunity
to
earn
a
riskless
 profit
by
taking
advantage
of
mispricing
in
one
 or
more
markets.
 •  In
financial
theory,
we
generally
assume
that
 the
market
does
not
permit
arbitrage
 opportuni=es.

 –  Otherwise,
investors
would
immediately
invest
in
 arbitrage
opportuni=es
in
huge
amounts.
 •  Supply
and
demand
would
alter
prices
un=l
the
 arbitrage
opportuni=es
no
longer
existed.
 Forward
Rates
and
Spot
Rates
 •  To
prevent
arbitrage,
we
must
have
the
 following
rela=onship
between
forward
and
 spot
rates:
 (1 + f j ,k ) k − j (1 + sk ) k = (1 + s j ) j •  In
terms
of
bond
prices:
 € (1 + f j ,k ) k− j P (0, j ) = P (0, k ) Forward
Rates
and
Spot
Rates
 •  One
period
forward
rates
fk,k+1
are
simply
 denoted
by
fk.
 •  It
is
easy
to
see
that:
 •  Spot
rates
are
geometric
averages
of
forward
 rates.
 €•  There
is
a
one‐to‐one
rela=onship
between
 the
spot
curve
and
the
forward
curve.
 –  Spot
rates
uniquely
determine
forward
rates
and
 vice‐versa
 1 + sk = ((1 + f 0 )(1 + f1 ) (1 + f k −1 ))1 / k Short
Rates
 •  One‐period
interest
rates
that
apply
for
 borrowing
at
future
=mes.
 •  rk
is
the
rate
for
borrowing
between
=me
k
and
 =me
k+1
that
prevails
in
the
market
at
=me
k.
 •  fk
rate
for
borrowing
between
=me
k
and
k+1
 agreed upon at -me zero. •  rk
rate
for
borrowing
between
=me
k
and
k+1
 agreed upon at -me k
(it
is
the
spot
rate
that
 prevails
in
the
future).
 •  Viewed
from
today,
rk
is
random.

 Bootstrapping
 •  The
most
useful
representa=on
of
the
term
 structure
of
interest
rates
is
a
spot‐rate
curve.
 •  We
don’t
observe
spot
rates
directly
in
the
 market.
 –  Bond
prices
(oQen
for
coupon
bearing
bonds)
are
 observed.
 •  The
process
of
inferring
spot
rates
from
 observed
prices
of
bonds
is
called
 bootstrapping.
 Shape
of
the
Term
Structure
 Normal
 sT 
 sT 
 Inverted
 T
 Flat
 sT 
 T
 T
 Theories
of
the
Term
Structure
 •  Expecta=on
Theory
(Forward
rates
represent
 market
future
expecta=ons
of
interest
rates).
 –  Pure/Unbiased
Expecta=ons
Theory
 –  Liquidity
Preference
Theory
 –  Preferred
Habitat
Theory
 •  Market
Segmenta=on
Theory
 Pure/Unbiased
Expecta=ons
Theory
 •  Forward
rates
represent
expected
future
spot
 rates:

 f k = E [ rk ] •  The
slope
of
the
curve
represents
expecta=ons
 of
future
rates:

 –  Upward
sloping
means
‘the
market
expects’
rates
 € to
go
up.
 –  Downward
sloping
means
‘the
market
expects’
 rates
to
go
down.
 Liquidity
Preference
Theory
 •  Investors
favour
liquidity.
 •  Forward
rates
are
expected
future
spot
rates
plus
 a
liquidity
premium.

 •  The
liquidity
premium
Lk
increases
with
k.
 f k = E [ rk ] + Lk , Lk > 0 € –  Reflects
the
fact
that
lenders
prefer
to
lend
for
short
 horizons
(borrowers
prefer
to
borrow
for
long
 horizons).
 •  An
upward
sloping
curve
may
reflect
only
 increasing
liquidity
premiums
(not
necessarily
an
 expected
increase
in
rates).
 Preferred
Habitat
Theory
 •  Same
as
Liquidity
Preference
Theory,
except
that
 it
allows
for
Lk
to
be
posi=ve,
nega=ve,
or
zero.
 f k = E [ rk ] + Lk •  Borrowers
and
investors
have
preferred
maturity
 ranges.
 •  If
supply/demand
for
a
given
maturity
sector
 € does
not
match,
borrowers/investors
may
be
 driven
outside
their
preferred
habitat
if
they
are
 compensated
by
an
appropriate
risk
premium.
 Market
Segmenta=on
Theory
 •  Neither
investors
nor
borrowers
are
willing
to
 shiQ
from
one
maturity
sector
to
the
other
to
 take
advantage
of
opportuni=es
arising
 between
expecta=ons
and
forward
rates.
 •  The
shape
of
the
yield
curve
is
determined
by
 supply
and
demand
for
securi=es
within
each
 maturity
sector,
and
independently
from
other
 maturi=es.
 –  Each
sector
(or
segment)
is
a
separate
market
 ...
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This note was uploaded on 01/13/2011 for the course ACTSC 445 taught by Professor Christianelemieux during the Spring '09 term at Waterloo.

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