Enterprise Risk Management Formula Book
7. Interest rates and bond pricing
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7.1 Spot and forward rates
Suppose 
is the price at
time 0 of a zero-coupon bond that pays 1 at time 
,
is the spot rate
for the period 
, i.e. 0 to ,
and 
is the
instantaneous forward rate at time 0 for time (where
and are both
continuously compounded) Then:
7.2 Duration,
modified duration, (gross) redemption yield (yield to maturity), credit spread,
option-adjusted spread, annualisation conventions
If a bond gives the holder entitlements to cash flows 
at time (and
is assumed not to be subject to default risk) and has a ‘dirty price’, 
,
then its (gross) redemption yield (yield to maturity) is the (sensible) rate of
interest that equates with its present
value, i.e.:
Its duration is then 
and its modified
duration is 
. 
is almost exactly
the same as its PV01, also called DV01.
Its credit spread is the difference between its (gross)
redemption yield and the corresponding yield on a reference security, often a
corresponding government security providing the same cash flows in the event of
non-default. The option-adjusted spread is the corresponding spread taking into
account optionality in the bond in question and/or in the reference bond.
Interest rates may be expressed as
annual rates, 
, semi-annual
rates, 
, quarterly rates, 
, monthly rates, 
or even
continuously compounded rates, 
, where:
The quotation convention of a bond (e.g. ACT/ACT) defines
the amount of accrued interest payable when a bond is bought or sold in between
coupon dates.
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