APPENDIX D The Relation between Macaulay and Modified Durations
We are going to show, as stated in Section 5.2.3, how the relationship between Macaulay and modified durations are dependent on the type of compounding we choose.
Let us first consider the case of continuous compounding. The modified duration is defined (Equation 5.7) as
If we calculate the bond value Bt, assuming continuous compounding, we have
where CT, here and throughout this appendix, includes the principal payment. Taking the derivative of the above and dividing by the bond value we obtain
which is the definition of Macaulay duration given by Equation 5.8.
Let us now consider the situation in which the yield is not continuously compounded, that is,
If we take the derivative and divide by the bond value we obtain
from which it follows that for noncontinuously compounded yields the relationship between Macaulay duration McD and modified duration is
as given by Equation 5.9.
APPENDIX E The Impact of Discounting on an Asset Swap Spread
In this appendix we show that the impact of discounting on an asset swap spread is limited, as discussed in Section 6.1.2.
Let us for simplicity assume (a fairly weak assumption since it would be a common situation) that the fixed and floating legs of the swap have the same frequency. We are going to discount an asset swap with spread with discount factors Di and then with discount factors and see what the difference, if at all, is with the new asset swap spread . In practice we want to solve
for, since both sides need to be equal to the bond price because the change in discounting was not applied to the bond.
Grouping the terms in E. 1 we obtain
Let us imagine that is slightly greater than , then the term
will be slightly larger than zero. The term
will be slightly less than zero, but smaller in absolute value than the term in Equation E.2. The term
will be slightly less than. Combining the three terms in Equations E.2, E.3, and E.4 we can see how
Of course the same holds if is slightly smaller than , in which case each effect would be opposite in sign.