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.


1 Answer 1


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%):

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


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


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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.