Covered interest rate parity

The following common approximation is valid when S is not too volatile :


An example
In short, assume that

This would imply that one dollar invested in the US < one dollar converted into a foreign currency and invested abroad. Such an imbalance would give rise to an arbitrage opportunity, where in one could borrow at the lower effective interest rate in US, convert to the foreign currency and invest abroad.

The following rudimentary example demonstrates covered interest rate arbitrage (CIA). Consider the interest rate parity (IRP) equation,


Assume:

the 12-month interest rate in US is 5%, per annum
the 12-month interest rate in UK is 8%, per annum
the current spot exchange rate is 1.5 $/£
the forward exchange rate implied by a forward contract maturing 12 months in the future is 1.5 $/£.
Clearly, the UK has a higher interest rate than the US. Thus the basic idea of covered interest arbitrage is to borrow in the country with lower interest rate and invest in the country with higher interest rate. All else being equal this would help you make money riskless. Thus,

Per the LHS of the interest rate parity equation above, a dollar invested in the US at the end of the 12-month period will be,
$1 · (1 + 5%) = $1.05
Per the RHS of the interest rate parity equation above, a dollar invested in the UK (after conversion into £ and back into $ at the end of 12-months) at the end of the 12-month period will be,
$1 · (1.5/1.5)(1 + 8%) = $1.08
Thus one could carry out a covered interest rate (CIA) arbitrage as follows,

1.Borrow $1 from the US bank at 5% interest rate.
2.Convert $ into £ at current spot rate of 1.5$/£ giving 0.67£
3.Invest the 0.67£ in the UK for the 12 month period
4.Purchase a forward contract on the 1.5$/£ (i.e. cover your position against exchange rate fluctuations)
At the end of 12-months

1.0.67£ becomes 0.67£(1 + 8%) = 0.72£
2.Convert the 0.72£ back to $ at 1.5$/£, giving $1.08
3.Pay off the initially borrowed amount of $1 to the US bank with 5% interest, i.e $1.05
The resulting arbitrage profit is $1.08 − $1.05 = $0.03 or 3 cents per dollar.

Obviously, arbitrage opportunities of this magnitude would vanish very quickly.

In the above example, some combination of the following would occur to reestablish Covered Interest Parity and extinguish the arbitrage opportunity:

US interest rates will go up
Forward exchange rates will go down
Spot exchange rates will go up
UK interest rates will go down