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**Unformatted text preview: **Time is money
The theory of interest rates Everybody needs money
Good Bad Stock Lots of upside Lots of downside Bond Safety No upside Convertible Bond Lots of upside Limited downside The issuing entity Coupons are paid every 6 months.
The amount represents the total for one year On this date, the issuer pays
$100 to the bond holder
(plus the last coupon payment) Bond prices Unlike stocks
bond prices do not trade in an exchange
prices are determine by independent brokers Clean vs. dirty prices
Bonds, Yields
With annual compounding
Dirty price = X pi (1 + r) ti i = Accrued interest + Clean price Consider a bond that, with a payment of P (t, T ) at time t, p
time T (and has no other intermediate payments). If the interest
assumed constant, then Bonds,
we would Yields
have With annual compounding
Number of days X P (t, T ) = e r(T t) .
Dirty
pi (1 + r) ti
since
theprice
last =
Hence,
coupon payment i
log P (t, T )
= Accrued
price
r = interest + Clean
.
(T t) This is useful since it is P , n
not r that is observed (not quite:
Accrued
interest
=
⇥ Annual Coupon Rate
section). When r is not constant,
365 we simply define the yield rate Consider a bond that, with a payment of
(t,TT)) at time t, pay
log P
P (t,
r(t, T ) = payments). . If the interest r
time T (and has no other intermediate
(T t)
assumed constant, then we would have
Since r determines P , r determines the entire term structure. Canadian Bonds Bonds, Yields
With annual compounding
Bonds,
Yields
X
With annual
compounding
Dirty
price =
pi (1 + r) ti
Bonds,
Yields
X
i
Dirty
price =
pi (1 + r) ti
With annual
compounding
= Accrued
interest + Clean price
i
X
Dirty price =
= Accrued
pi (1 interest
+ r) ti + Clean price
log(P/N )
i
r(T ) =
T )
log(P/N
=r(T
Accrued
interest
+ Clean price
)=
n
T
Accrued
interest =on the
⇥
Annual
Couponconvention.
Rate
The definition of interest rates
is dependent
compounding log(P/N
)
365
r(T=) =n ⇥ Annual Coupon Rate
Accrued interest
T
X
365
r(ti ) ti
For annual compounding, a series of cashflows
P =Xn pisi ediscounted .
r(ti ) ti Coupon Rate
Accrued interest
i pi⇥
P == 365
e Annual
.
X
i
ti
X
P1 =X
pi (1 r(t
+ ir)
.
) ti
= i ppi(1e + r) ti..
PP
1 =
i
If the compounding is n times a year, i
X
i
X
Pn =X
pi (1 + nr ) ttii n .
ti n.
1 = i pp(1
i (1++r r)
PP
.
n =
i
n)
i X
i
X
P1 =X pi er r tit.i n
PnP=yields pii (1
In the limit, instantaneous compounding,
pi +
e nr )ti . .
1 = Compounding •
•
•
•
• i i
Consider a bond that, with a X
payment
of P (t, T ) at time t, pays $1 at
r
a bond
that, intermediate
with
a payment
time
t, paysrate
$1 atr is
P1 =
pi payments).
e oftiP. (t, T ) Ifatthe
time Consider
T (and has
no other
interest
time T (and
has nothen
otherweintermediate
payments). If the interest rate r is
i
assumed
constant,
would have
constant,
then
we with
would
Instantaneousassumed
compounding
has
many
analytical
advantages,
fromt,now
Consider
a bond
that,
a have
payment
) at time
payson,
$1 we
at will use it
r(Toft)P (t, T and
P (t, T ) = e r(T
.
t)
time T (and has no other intermediate
If the interest rate r is
most of the time. P (t, T ) = e payments).
.
assumed constant, then we would have
Hence,
Hence,
log P r(T
(t, T )
=T )log
Pr(t,
=P
e (t, T ) t).
(T t) .
r=
(T t)
Hence,
This is useful since it is P , not r that is observed (not quite: see next
This is useful since it is P , not r log
that
is Tobserved
(not quite: see next
P (t,
)
section). When r is not constant,
we simply .define the yield rate
r
=
section). When r is not constant, we (T
simply
t) define the yield rate
log P (t, T )
log Pis(t,observed
T) .
This is useful since it is P
, not
r that
(not quite: see next
r(t,
T) =
r(t, T ) =
. Zero coupon bonds
Imagine a market where all bonds pay no coupons • Bonds will contain a single cash flow: a single payment of at the time of maturity • A bond will be characterized by three variables: • - The notional, i.e. the payment to occur at maturity. - The price of the bond - The time to maturity
Bond price With these concepts in mind, we define the yield Notional yield
Time to maturity Bonds, Yields
With annual compounding Cashflow
formula
Dirty price = valuation
p (1 + r)
X i ti i = Accrued interest + Clean price
log(P/N
)
The yield curve can then
r(T )be
=used to calculate the price of
T
any series of future cashflows: n
Accrued interest =
⇥ Annual Coupon Rate
365
X
r(ti ) ti
P =
pi e
.
i Consider a bond that, with a payment of P (t, T ) at time t, pays $1
ime T (and has no other intermediate payments). If the interest rate r
ssumed constant, then we would have i = Accrued interest + Clean price Bootstrapping
log(P/N )
T r(T ) = n
Accrued interest =
⇥ Annual Coupon Rate
365
If we return to a world where bonds have couponX
payments, we can still recover the yield curve from those,
avoiding the zero coupon bonds, as follows: P =
pi e r(ti ) ti .
i zero-coupon bonds, therefore • For maturities less than 6 months, all coupons are
X
P1 =
pi (1 + r) ti .
i • Pn = X Valid for 0<T<1 ti n pi (1 + nr ) . For maturities between six months and one year, bonds have a coupon payment within six months, and
i
another payment between six months and a year. X
r ti
P
=
p
e
.
1
i
Known number,
Unknown, can be solved
i
using the previous step
as a one-variable equation P = p1 e r(t1 )·t1 + p2 e r(t2 )·t2 , 0 < t1 < Consider a bond that, with a payment of P (t, T )
From the market
(Dirty
price)T (and has no other intermediate payments). If
time 1
2 < t2 < 1. at time t, pays $1 at
the interest rate r is process can
be extended
to infinity,
thereforehave
allowing us to calculate the yield curve for all maturities,
assumed
constant,
then
we would
• The
assuming coupon bearing bonds for all maturities. P (t, T ) = e Hence, r(T t) . ...

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