# 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 *B*t, assuming continuous compounding, we have

where *C*T, 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

(D.l)

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 *D*i 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

(E.1)

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

(E.2)

will be slightly larger than zero. The term

(E.3)

will be slightly less than zero, but smaller in absolute value than the term in Equation E.2. The term

(E.4)

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.