# How to price a forward-rate agreement?

I don't understand how the formula on page 24 of Joshi: Concepts and Practice of MF is derived. Here is the paragraph I don't understand:

A forward-rate agreement is simply an agreement to take some money on deposit, or to borrow some money, at an interest rate fixed today for a fixed period of time starting at a specified future time. For example, a company may be paid for some goods on a known future date, and will buy some other goods on a fixed date after that. The company wishes to make plans on the basis of this and so enters an FRA, thus ensuring that there is no interest-rate risk. How would the bank decide at what rate to offer the FRA? The FRA is easily synthesized by going long and short the appropriate bonds. If the FRA starts at time $$t_0$$, we go short a bond expiring at time $$t_0$$ to bring the money back to the present - thus multiplying the sum deposited by $$(1+r_0)^{-t_0}$$ where $$r_0$$ is the yield of the to bond. To take the money forward to time $$t_1$$, the end of the deposit period, we go long bonds maturing at time $$t_1$$ with yield $$r_1$$ and thus multiply the sum by $$(1 + r_1)^{t_1}$$. In conclusion, in return for the £1 deposit at the start of the FRA, the company receives $$X = (1+r_0)^{-t_0}(1+r_1)^{t_1}$$ at the end. One can then convert this into an equivalent compounding annual interest rate, $$r_2$$, by solving $$(1+r_2)^{t_1-t_0}=X.$$

Here I summarise how the price is synthesised:

1. The company starts with a balance of £1.
2. The company sells a bond for $$£(1+r_0)^{-t_0}$$, the balance is now £$$[1 + (1+r_0)^{-t_0}]$$.
3. Wait until $$t=t_0$$.
4. The company pays the owner of the bond the $$£1 = (1+r_0)^{-t_0}(1+r_0)^{t_0}$$ it owes. The balance is now $$£(1+r_0)^{-t_0}$$.
5. The company buys a bond for $$£(1+r_0)^{-t_0}$$. The balance is now £0.
6. Wait until $$t=t_2$$.
7. The company earns $$X=£(1+r_0)^{-t_0}(1+r_1)^{t_1}$$ for the bond it purchased.

Therefore, the company started with £1 and now has $$X=(1+r_0)^{-t_0}(1+r_1)^{t_1}$$ so we can work out the interest rate $$r_2$$ from $$X$$. Couldn't the company have made more money if instead of shorting it decided to only long the bonds? In other words, it could have done this:

1. The company starts with a balance of £1.
2. The company buys a bond for £1. The balance is now £0.
3. Wait until $$t=t_0$$.
4. The company earns the yield from its bond and the balance is now $$£(1+r_0)^{t_0}$$.
5. The company buys a bond for $$£(1+r_0)^{t_0}$$, the balance is now £0.
6. Wait until $$t=t_1$$.
7. The company earns $$Y = £(1+r_0)^{t_0}(1+r_1)^{t_1}$$ for the bond it purchased.

Where $$Y>X$$? So why wouldn't the company do this instead? Am I misreading how the synthesis works or have I made an error somewhere in my calculations?

• Not he answer you were looking for: in practice, FRAs are not priced, rather their prices are entirely driven by supply and demand. In turn, the supply and demand is driven by market maker's expectation of central bank policy. FRAs would be typically traded up to 1.5 years maturity, the underlying being 6m or 3m IBOR. These IBOr rates in turn practically trade at the Central bank policy rate + fixed spread. As an example, a 6*9 FRA (3m libor, starting in 6 months) would simply reflect the market maker's view of where central banks rates will be in 6 months. No mathematical pricing involved. – Jan Stuller Dec 16 '20 at 20:39