# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Forward rates

Consider the following situation. A businesswoman wants to borrow money for 1 year, but she wants to start the loan at the end of 1 year, i.e. borrowing is planned in the future and not today. While she knows today’s term structure, so she is aware of today’s spots rates $r_1$, $r_2$ etc, she is exposed to the risk of not knowing what the rates would be a year from now.

Well, it turns out that using the bond/bank market she can still manage something which will lock-in her effective rate exactly, that is there would be no risk attached to it, and she can get on with her business. Let’s see how she can work this out.

If she can use the banks today, she will borrow money for 2 years and lend (deposit) all of that money to the bank for 1 year. And what has she achieved in the process? What she has got is that she has managed to get  cash inflow at the end of 1 year (money ‘coming back’ from one year lending) and cash outflow at the end of two years (original money to be returned to the bank at the end of two years). With the loan money coming from the bank and that money being put in the bank for shorter maturity means there is no money left over ‘today’ – which is exactly what she wanted.

So, how do the cash flows look like at the end of years 1 and 2. Given a million rupee loan, for example, her due to the bank in 2 years time is exactly the future value of that, i.e.

$\displaystyle FV_2 = (1 + r_2) ^ 2$

And inflow at the end of 1 year from the million rupee invested in the bank would be

$\displaystyle FV_1 = (1 + r_1)$

That is, she is in a way artificially created a one year loan for herself with the money $FV_1$ coming in at the end of the first year, and the amount $FV_2$ to be paid out at the end of two years – a one year loan starting at the end of one year. And what is the effective interest rate? Well, that is easily found as:

\begin{aligned} \mbox{Effective rate} &= \frac{FV_2}{FV_1} - 1 \\&= \frac{(1 + r_2)^2}{1 + r_1} - 1 \end{aligned}

It turns out that there is a specific name for this effective rate, and it is called the forward rate for the period 1 to 2, and we write it as $f_{12}$. So with this, we have the following result:

$\boxed{1 + f_{12} =\frac{(1 + r_2)^2}{1 + r_1}}$

or alternatively:

$(1 + r_2)^2 = (1 + r_1)(1 + f_{12})$

which can be pictorially represented is:

It turns out that forward rate as described above is only one example of a class of instruments called Forwards and Futures, and we’ll talk about such things at a fair bit of length separately. For now, we come back to the term structure discussion and use the idea of forward rates to understand the shape of the term structure.

Written by Vineet

October 7, 2016 at 8:46 pm