Relating Swap Annuity Factors to Coupon DV01

Summary

The document asks why a swap’s annuity factor should be proportional to its coupon value change for a one basis point move, called CV01. It starts from the par swap pricing equation, expressing the fixed coupon and discount-factor sum alongside the maturity discount factor. It then compares the annuity factor with a formula for DV01 when the coupon equals the yield, finding that the annuity factor is one hundred times that DV01 under the stated setup.

The author questions whether this establishes the claimed relationship, since saying DV01 and CV01 are similar may assume the conclusion rather than explain it. The document provides an algebraic comparison but no resolution or proof connecting the measures. Its formulas assume a unit par amount, semiannual payments, and the stated yield-based setup; the result should not be generalized to other conventions without checking definitions and assumptions.

Key ideas

• The par swap equation relates the coupon to discounted payment factors and the maturity discount factor.
• The annuity factor is defined as the sum of discount factors across semiannual payment dates.
• Under the stated yield setup, the document derives that the annuity factor equals one hundred times DV01.
• The author raises an unresolved question about why this DV01 comparison establishes proportionality to CV01.

Tags

# Why is the annuity factor proportional to the CV01?


# Why is the annuity factor proportional to the CV01?












For an asset with par amount of one unit (with a semiannual payment regime) we have

$$\frac{C(T)}{2}\sum_{t=1}^{2T}d\Big(\frac{t}{2}\Big) + d(T) = 1$$ $$\implies\frac{C(T)}{2}A(T) + d(T) = 1,$$

where $$A(T) = \sum_{t=1}^{2T}d\Big(\frac{t}{2}\Big).$$

The CV01 is defined as the change in swap value for a 1bp decline in the coupon rate. The above equation for the annuity factor supposedly implies that the annuity factor to a swaps maturity is proportional to the CV01 of the swap.

My problem

I can't see how that final statement holds. I've attempted the following justification:

For a par swap, we know that the DV01 is $$DV01_{c=y}=\frac{1}{100y}\Big(1 - \frac{1}{(1 + \frac{y}{2})^{2T}}\Big),$$ and we also know that the annuity factor can be written as $$A(T) = \frac{1}{y}\Big(1 - \frac{1}{(1 + \frac{y}{2})^{2T}}\Big)$$ $$\implies A(T) = 100\times DV01_{c=y}.$$

The DV01 and the CV01 are pretty similar, meaning that $A(T) \propto CV01.$

However, I am not content with this explanation because I feel that "DV01 similar to CV01" is more of a product of $A(T)\propto CV01$, as opposed to the other way around. This other way around (i.e. the way I explain it) results in the statement "DV01 similar to CV01" being somewhat 'hand-wavy'.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.