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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