# Returns of an interest rate swap

I would like to calculate returns for a plain-vanilla (fixed-for-floating) interest rate swap. Consider, that I am long in USD 5-year swap rate, i.e. I'm holding a receiver swap for 5-year swap rate with notional of 100\$. At initiation, the swap should be valued at 0\$, as both the fixed leg and the floating leg of the swap should have the same value.

If I valuate the receiver swap a month later and it would have a value of let's say 2\$, then how should I calculate the return? Is it just (2\$ - 0\$) / 100\$, i.e. change of swap value relative to the notional? I can't quite figure out how I could calculate the return by comparing change of value to previous value (or simply comparing consecutive values), as they can be at initiation 0\\$ and later be negative.

If the correct way is indeed to calculate returns relative to notional, then why is this? After all, I am not investing any capital on the swap (apart from some possible margin payments), so why should I use the notional in return calculations? For me a more natural way would simply be following the value of the swap and calculating the returns with consecutive values, but then I face the problems of negative values and zero values mentioned above.

I think the best concept of return for the interest rate swap is (2-0)/m, where m is the outlay for initial margin. This is rarely calculated at an institution holding a large portfolio of swaps, since the incremental m is unclear (might be a risk reducing trade whose incremental m is zero or negative). But for an individual or a small hedge fund, m is clearly the initial outlay and is the denominator by which returns should be judged.

Let us specify the floating-leg rate reset dates to be $$T_\alpha, T_{\alpha+1}, \ldots, T_{\beta-1}$$, payment dates $$T_{\alpha+1}, \ldots, T_{\beta}$$. Set the day count fraction to $$\tau_i \equiv T_{i+1}-T_i$$.

Let us write down the payoff for the vanilla interest rate swap, assuming that the notional $$N$$ is constant:

$$\pi_t = N \sum_{i=\alpha+1}^{\beta} P(t,T_i) \tau_i \left[ L(t;T_{i-1},T_i)-K \right]$$

Say this swap is entered into at $$t=t_0$$, hence $$\pi_{t_0} = 0$$. Consider two time points in the future, say $$t_1$$ and $$t_2$$, with $$t_1 \leq t_2$$. We can consider a couple of different return metrics.

We could consider the arithmetic difference divided by the notional, call this $$\lambda_1$$:

$$\lambda_1(t_2;t_1) = \frac{\pi_{t_2}-\pi_{t_1}}{N}$$

We could consider the relative difference, call this $$\lambda_2$$:

$$\lambda_2(t_2;t_1) = \frac{\pi_{t_2}}{\pi_{t_1}}-1$$

However, perhaps the most appropriate measure to consider is the PV01, namely, the present value of the swap for a 1 basis point increase in interest rates, call this $$\lambda_3$$:

$$\lambda_3(t) = \pi_{t;r(t)+0.0001} -\pi_{t}$$

The PV01 measure tell us the sensitivity to interest rates, which is the relevant risk factor for the interest rate swap.

You would use $$\lambda_1$$ and $$\lambda_2$$ to track the PnL of the swap for two time points, and $$\lambda_3$$ for risk management purposes. The benefit of using $$\lambda_1$$ (relative to $$\lambda_2$$) is that it can be used from inception.