# Transform a 3M FRA Rate to a 6M FRA Rate

I have a question whether it is possible to transform 3M FRA rates to 6M FRA rates without having any spreads available. Let's give an example:

FRA 3M:

FRA 1x4 FRA 2x5 FRA 3x6 FRA 4x7 FRA 5x8 FRA 6x9

And what I need to calculate is:

FRA 6M:

FRA 1x7 FRA 2x8 FRA 3x9 FRA 4x10 FRA 5x11

Is there a possibility doing this?

From a pure mathematical perspective, this is possible. For example, consider dates $t_0 \le t_1 < t_2 < t_3$. Given \begin{align*} L(t_0, t_1, t_2) = \frac{1}{\Delta t_2}\left(\frac{P(t_0, t_1)}{P(t_0, t_2)}-1 \right), \end{align*} and \begin{align*} L(t_0, t_2, t_3) = \frac{1}{\Delta t_3}\left(\frac{P(t_0, t_2)}{P(t_0, t_3)}-1 \right), \end{align*} where $\Delta t_i = t_i-t_{i-1}$, and $P(t, u)$ is the price at time $t$ of a zero-coupon bond with maturity $u$ and unit face value.

Then \begin{align*} L(t_0, t_1, t_3) &= \frac{1}{t_3-t_1}\left(\frac{P(t_0, t_1)}{P(t_0, t_3)}-1 \right)\\ &=\frac{1}{t_3-t_1}\left(\frac{P(t_0, t_1)}{P(t_0, t_2)}\frac{P(t_0, t_2)}{P(t_0, t_2)}-1 \right)\\ &=\frac{1}{t_3-t_1}\Big(\big(1+\Delta t_2 L(t_0, t_1, t_2)\big)\big(1+\Delta t_3 L(t_0, t_2, t_3) \big)-1 \Big). \end{align*}

For example, assuming 30 days a month and the day-count convention is 30/360. If FRA $1\times 4$ is 4% and FRA $4\times 7$ is 5%, then \begin{align*} FRA \,1\times 7 &=\frac{1}{0.5}\Big(\big(1+0.25\times 0.04)\big)\big(1+0.25 \times 0.05 \big)-1 \Big)\\ &=0.04525, \end{align*} that is, 4.525%.

• Thanks a lot. The 3M and 6M are the tenors not the maturity. Or do I misundertand something? Furthermore, do you have a numerical example? – JonDoe Jan 12 '18 at 14:53
• you can consider converting FRA 1x4 and 4x7 to FRA 1x7, assuming 30 days a month and the day-count convention is 30/360. – Gordon Jan 12 '18 at 15:18
• Thanks Gordan, I am a little bit new in the bsuines.Can you show a simple example with values? It does not need to be exact or maybe in QuantLib? – JonDoe Jan 12 '18 at 15:23
• See the updates. – Gordon Jan 12 '18 at 15:27
• Thanks a lot for the update. Just some more silly questions. How die you derive the values 0.5 and 0.25 numerically? – JonDoe Jan 12 '18 at 15:33

An FRA (forward rate agreement) is an interest rate derivative contract with specific documentation. A 3M FRA settles to 3M-IBOR, and a 6M FRA settles to 6M-IBOR, not a 6M rate that is a compounded composite rate of two 3M periods.

Gordon's answer does as much as you can with the given information: it transforms a 3M rate into a 6M 'period' with compositions, but the resulting rate suffers the inherent problem it does not represent 6M-IBOR. My opinion is that the difference (not just spot but also forward) in the 3M-IBOR/6M-IBOR basis is so important in these prices that the specific question stating "without having any [basis] spread available" means you cannot determine the specific prices of 6M FRAs.

In all major currencies, GBP, USD, EUR and JPY the 6M-IBOR basis is different in spot and forward levels leaving scope for a lot of variability. As a trader I would never price a 6M FRA with only the information of 3M FRAs.

An analogy is deriving the yields of French government bonds compared to German government bonds without having the country spread available. You can make some sensible estimations and derive some general conclusions but you can't 'price' them.