# Calculate interest rate swap curve from Eurodollar futures price

So I was reading Robert McDonald's "Derivatives Markets" and it says Eurodollar futures price can be used to obtain a strip of forward interest rates. We can then use this to obtain the implied forward LIBOR term structure and build the interest rate swap curve. The book also provides a concrete example to illustrate its point but somehow I cannot seem to understand it.  There was no assumption about the terms of the swap, so I was kinda confused. Is it correct to say that the swap rates calculated in the table are based on an interest rate swap that starts in June with quarterly payments?
Moreover, can this method be extended to determine the swap rate on a deferred interest rate swap, i.e., one that instead starts on Sep/Dec with payments being made semi-annually/annually? Thank you!

Consider a fixed-for-floating swap with reset dates $T_0, \ldots, T_{n-1}$ and payment dates $T_1, \ldots, T_n$, where $0<T_0 < \cdots < T_n$. We assume that the swap exchanges the floating rate payments $L(T_{i-1}; T_{i-1}, T_i)\Delta T_i$ and the fixed rate payments $K\Delta T_i$, for $i=1, \ldots, n$, where $\Delta T_i = T_i -T_{i-1}$.

The value of the swap at time $t$, where $0 \leq t \leq T_0$, is given by \begin{align*} \sum_{i=1}^n L(t; T_{i-1}, T_i)\times \Delta T_i\times P(t, T_i) - K \sum_{i=1}^n P(t, T_i)\times \Delta T_i,\tag{1} \end{align*} where $P(t, u)$ is the price (i.e., zero price) of a zero-coupon bond with maturity $u$ and unit notional. The forward swap rate $s$ is the rate $K$ such that the swap value given by $(1)$ is zero, that is, \begin{align*} s = \frac{\sum_{i=1}^n P(t, T_i) \times L(t; T_{i-1}, T_i) \Delta T_i }{\sum_{i=1}^n P(t, T_i) \times \Delta T_i}. \end{align*} For a quarterly resetting frequency, that is, $\Delta T_i=\frac{1}{4}$, then $\frac{1}{4}L(T_{i-1}, T_i)$ is the quarterly interest rate. Moreover, the quarterly swap rate is given by \begin{align*} \frac{s}{4} = \frac{\sum_{i=1}^n P(t, T_i) \times \frac{1}{4}L(t; T_{i-1}, T_i)}{\sum_{i=1}^n P(t, T_i)}.\tag{2} \end{align*}

For the given example, we have that $n=2$, $P(t, T_1) = 0.998566$, $P(t, T_2) = 0.99655$, $\frac{1}{4}L(t; T_0, T_1) = 0.0014358$, and $\frac{1}{4}L(t;T_1, T_2) = 0.0020222$. Based on Formula $(2)$, the quarterly swap rate is given by \begin{align*} \frac{s}{4} = \frac{0.998566 \times 0.0014358 + 0.99655 \times 0.0020222}{0.998566+0.99655} = 0.17287\,\%. \end{align*} The computation for swap rate with semi-annual or annual accrual frequency can be proceeded analogously. Moreover, you can set $T_0$ to any date you like, for example, September or December, as you mentioned.

• Why are we only using two periods to determine K? I understand fixed rates are normally paid semi-annually while floating are paid quarterly, is that the reason? If not wouldn't we want all four periods to compute the swap rate so at the time the agreement is entered into it would be neutral? – HerbN Aug 22 '16 at 20:08
• @HerbN: this is just to put the example from the question in context. In practice, you can have any periods and any frequency as you like. – Gordon Aug 22 '16 at 20:46

To answer the first question directly, the swap in question is a 1 Year swap of a fixed rate vs 3 month Libor. The swap starts in Mid-June (the date of the ED futures expiration) and goes until the next June. There are 4 quarterly payments.

To understand things better, look carefully at Table 8.4 and see how the three columns on the right are computed from information in the Futures Prices column. Note how the ED price for June is used to calculate an implied rate which is then written down in the Sep row of the table. And so forth. (It could be fun to reproduce these calculations in a spreadsheet).

This process can be used to value a swap of any tenor, or a deferred swap. Strictly speaking it should only be used to value swaps with quarterly payments of 3-month Libor. That is what the ED futures are tied to. (You could approximate semi-annual by compounding quarterly interest rates, but that's not recommended because a modern (post 2008) approach says 3-month Libor and 6-month Libor should be taken from different curves and you should not approximate one from the other, they are inherently different in liquidity and risk).