# Is this an inconsistency between Swap and LIBOR?

I'm a little confused by what I see as an inconsistency between quoted £ swap rates and £ LIBOR.

From the FT on 25/4/14:
1-year Swap (semi-annual): Bid - $0.63\%$; Ask - $0.66\%$
LIBOR: 6-month - $R_{0.5}=0.63006\%$; 1-year - $R_{1}=0.92438\%$

Based on the LIBOR rates, I calculated a 1-year swap rate as: \begin{equation} R_{swap}=2\times\frac{1-\left(1+R_1\right)^{-1}}{\left(1+R_{0.5}\right)^{-0.5}+\left(1+R_{1}\right)^{-1}}=0.9216\% \end{equation} Although I'm not sure whether 6-month LIBOR is quoted as annual or semi-annual compounding, for such small rates the impact is negligible.

If anybody could tell me where I've gone wrong (and the basis for the 6-month LIBOR quote), I'd greatly appreciate it.

Firstly, understand that the 1y Libor is not useful here; the swap is 2 6-month periods, which will each fix on 6m Libor. These days, the *ibor fixings at different tenors are essentially separate, and 0x6 & 6x12 do not compound up to 0x12.

So we have 6m fixing at 0.63006%, and a 1y swap at 0.645% mid. To do this properly, we would need a discounting curve based on SONIA (the GBP OIS curve) as a standard 6m Libor GBP IRS will be daily margined and OIS accrued these days.

Dates, for precision:

Fixing date 25/4/14
GBP spot 25/4/14 (t+0 ccy)
6m 27/10/14 (185 days)
1y 27/4/15 (182 days)


Cash flow at 6m, for an NPA of £1m (GBP IRS rates are quoted Actual/365):

$$C_{0.5} = (0.0063006 \times (185/365)) \cdot 1,000,000 = £3,193.45$$

Cash flow at 1y:

$$C_1 = (L_{0.5} (182/365)) \cdot 1,000,000$$

Against this, we have the fixed leg. Now GBP 6m IRS are usually quoted Semi-annual as well, so we have 2 payments again:

$$K_{0.5} = (0.00645\times (185/365)) \cdot 1,000,000 = £3,269.18$$ $$K_1 = (0.00645\times (182/365)) \cdot 1,000,000 = £3,216.16$$

We need to discount these back to Spot to find out how much the unknown payment needs to be to net out ($D(t)$ is discount factor at $t$)

$$\text{PV}(\text{float}) = C_{0.5} D_{0.5} + C_1 D_1$$ $$\text{PV}(\text{fixed}) = K_{0.5} D_{0.5} + K_1 D_1$$

Note that most IRS are not quoted Semi-annual vs 6; often it is Annual vs 6, so the fixed and float will not line up like this.

Since this is a Par swap, $\text{PV}(\text{float}) = \text{PV}(\text{fixed})$, so:

$$C_{0.5} D_{0.5} + C_1 D_1 = K_{0.5} D_{0.5} + K_1 D_1$$ $$C_1 D_1 = (K_{0.5} - C_{0.5}) D_{0.5} + K_1 D_1$$ $$C_1 = (K_{0.5} - C_{0.5}) \frac{D_{0.5}}{D_1} + K_1$$

Choose $D_{0.5}=0.9987$ and $D_1=0.9975$ for simplicity (~0.25%):

$$C_1 = (3,269.18 - 3,193.45) \frac{0.9987}{0.9975} + 3,216.16\\ = 75.73 \times 1.00120 + 3,216.16 = £3,291.98$$

But

$$C_1 = (L_{0.5} (182/365)) 1,000,000$$

Rearranging:

$$L_{0.5} = \frac{C_1}{ 1,000,000} . \frac{365}{182} \\ = 0.660205 \%$$

Thus the implied 6m Libor fixing on 27/10/14 is $0.660205\%$. Sanity check: $(0.63 + 0.66)/2 = 0.645$, which matches our 1y rate. By all means redo that separately for Bid and Ask to get a 2-sided value.