Let's now look at how forward rates like RAs can be priced in theory using what's called a no arbitrage argument.

The idea is simple.

If you can invest from today until some future date in two different ways, those strategies should give you the same return.

Otherwise, an arbitrage opportunity would exist.

In the case of RAs, that means the return from investing from today to a future time T two should be equal to the return from two consecutive steps.

First, investing from today to an intermediate point T one, and also locking in today a forward rate from T one to T two.

This logic gives us a formula that relates spot rates and forward rates.

On screen, you can see that one plus R two times T two equals one plus R one times T one multiplied by one plus F times T one two where R one is the spot rate from today to T one R two is the spot rate from today to T two.

It is the forward rate from T one to T two, and each T is a time period expressed as a day count fraction, for example, actual divided by 360 in years.

This is known as the compounded equivalence approach.

It's fundamental to how we think about forward pricing and forms the basis for valuing a variety of forward starting interest rate contracts.

Now, it's important to note this approach works very well for risk-free forwards like those based on sofa, Sonya, or esra, because the underlying rates are close to default free and there's no credit premium involved, but it does not work as well for eyeball based RAs.

Before the financial crisis of 2007 to 2009, this formula was often close enough in practice because credit and liquidity conditions were relatively stable and there was little or no term premium embedded in eyeball curves.

But since then, the picture has changed.

Market participants now recognize that term eyeball rates contain credit and liquidity premiums, especially over longer periods.

As a result, pricing Eyeball RAs using this pure no arbitrage approach, often produces results that differ significantly from actual market prices.

So while this is the theoretical foundation in the real world, especially for eyeball based instruments, market pricing may reflect other factors including supply and demand funding pressures and credit risk.

Let's now look at a concrete example of FRA pricing.

In practice. We're interested in pricing a three by six FRA, and we have the relevant market inputs.

The trade date is the 1st of April, 2025, and spot is two business days later, so the 3rd of April.

In this case, the three month arrival fixing on the 1st of April is 2.336%, which means the underlying interest rate period runs from 3rd of April to 3rd of July, 2025.

The total of 91 actual days, the six month year arrival fixing is also 2.336% covering the period from 3rd of April to 3rd of October, 2025.

That's 183 actual days.

If we apply the no arbitrage formula we just discussed, we get an implied three by six forward rate of 2.3223%.

That's the theoretical fair value you'd expect under the assumption that term money market rates are risk-free.

But now look at the actual three by six FRA mid-market quote at 11:00 AM CET, just after the arrival fixings were published, it's 2.113%.

That's more than 20 basis points lower than the implied forward rate.

So why the discrepancy? Because as mentioned earlier, this theoretical approach doesn't work for IB RAs anymore.

The no arbitrage logic assumes there's no credit or liquidity risk embedded in the rates.

But that assumption broke down during the financial crisis and hasn't held since.

Term money. Market rates like your IBOR are no longer considered risk-free.

As a result, the way RAs are priced in the real world has had to evolve instead of relying on simple compounded forwards, IB RAs are now priced using short term interest rate futures, interest rate swaps and tenor basis swaps, all of which reflect the actual credit and liquidity conditions in the market.