why swap rate not dependent on valuation date?

When I review my course on swaps, I read the following sentence:

the value of the swap rate is independent of the valuation date(even though the PV's of the individual legs of the swap are clearly not)

However, I don't quite get this. We know $$P_0^{fix}(t) = \sum \alpha_j C P_0(t, T_j^c)$$ where $\alpha_j$ are the day count fractions on the fixed leg and $T_j^c$ is the coupon date.

$$P_0^{float}(t) = \sum \delta_j L_j P_0(t,T_j^f)$$ where $L_j$ is the LIBOR forward rate, $\delta_j$ is the day count fraction applying to the floating leg.

We know that valuation date t belongs to $[0,T_0]$ where $T_0$ is spot date. So swap rate can be obtained as the following formula: $$C^* = \frac{P_0^{float}(t)}{A_0(t)} = \frac{\sum \delta_j L_j P_0(t,T_j^f)}{\sum \alpha_j P_0(t, T_j^c)}$$

It seems there is no way to get $\frac{P_0(t,T_j^f)}{P_0(t, T_j^c)}$ to get rid of $t$.

So how come the value of swap rate is not dependent on t??

• I think you are correct. I'm not sure what the book is referring to. The dependence of C on t is somewhat second order- for example, if the Libor forwards are all the same (flat yield curve ) then C does not depend on t.
– dm63
May 3 '17 at 5:31
• The thing about the value of a swap rate is it reflects how you give up the interest rate from 1 side and take up the interest rate from the other. Maybe it's saying that rate valuation is independent of trade date. Can you post the whole paragraph so we get the context? May 3 '17 at 9:09
• what it probably means is that the swap rate only depends on the forward discount curve on sector $[\min(T^c_0, T^j_0), \max(T^c_{N^c}, T^f_{N^f})]$. But it is still the forward discount curve viewed from $t$. May 3 '17 at 11:24

Hi guys I have one answer now which is quite simple. You just need multiply $P_0(0,t)$ to get rid of $t$, meaning that it has nothing to do with valuation date.