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'.

• If you differentiate your first equation with respect to $c$, you get $A(T)$ right? In other words, CV01 is the annuity factor. – Helin Jun 23 '18 at 9:23