# Interest rate parity

According to interest rate parity the difference between the (risk free) interest rates paid on two currencies should be equal to the differences between the spot and forward rates.

If interest rate parity is violated, then an arbitrage opportunity exists. The simplest example of this is what would happen if the forward rate was the same as the spot rate but the interest rates were different, then investors would:

1. borrow in the currency with the lower rate
2. convert the cash at spot rates
3. enter into a forward contract to convert the cash plus the expected interest at the same rate
4. invest the money at the higher rate
5. convert back through the forward contract
6. repay the principal and the interest, knowing the latter will be less than the interest received.

Therefore, we can expect interest rate parity to apply. However, there is evidence of forward rate bias.

## Covered interest rate parity

Assuming the arbitrage opportunity described above does not exist, then the relationship for US dollars and pounds sterling is:

(1 + r£)/(1+r\$) = (£/\$f)/(£/\$s)

where r£ is the sterling interest rate (till the date of the forward),
r\$ is the dollar interest rate,
£/\$f is the forward sterling to dollar rate,
£/\$s is the spot sterling to dollar rate

Unless interest rates are very high or the period considered is long, this is a very good approximation:

r£ = r\$ + f

where f is the forward premium: (£/\$f)/(£/\$s) -1

The above relationship is derived from assuming that covered interest arbitrage opportunities should not last, and is therefore called covered interest rate parity.

## Uncovered interest rate parity

Assuming uncovered interest arbitrage leads us to a slightly different relationship:

r = r2 + E[ΔS]

where E[ΔS] is the expected change is exchange rates.

This is called uncovered interest rate parity.

As the forward rate will be the market expectation of the change in rates, this is equivalent to covered interest rate parity - unless one is speculating on market expectations being wrong.

The evidence on uncovered interest rate parity is mixed.