# Microeconomic rationale for UIP (uncovered interest rate parity)

The key idea behind UIP is that as for all common financial instruments, the "law of no free lunch" should also hold for currencies. However it differs from traditional replication-based no-arbitrage conditions like CIP in that there is no obvious argument for why the exploitation of opportunities when it does not hold true necessarily depletes all such opportunities, thereby making "free lunches" rare in a world of rational investors.

Concretely, the currency of a country with a relatively higher interest rate needs to gradually depreciate to offset the positive earnings potential. What mechanism drives this gradual depreciation?

I've heard that there are models on this issue but have not been able to locate the literature.

Thanks for all suggestions!

• What's wrong with expected free lunches? – Giskard Jul 29 at 12:12
• To clarify, are you asking for literature or an explanation? – Giskard Jul 29 at 12:12
• @Giskard The fact that UIP (under risk-neutrality) expects there to be no free lunches, but there is no obvious arbitrage argument like for a mispriced forward contract. – Steven Jul 29 at 12:17
• @Giskard Both :) I'd love to read the source material, but I'd also appreciate an explanation if you don't happen to be familiar with the literature I'm searching for – Steven Jul 29 at 12:18

There is an arbitrage argument for UIP because under UIP forward rate must be equal to expected exchange rate so the same arbitrage arguments from CIP can be extended to UIP (under its strict assumptions). You will find it in most international macroeconomics or finance textbooks. For example, according to Nelson Mark's International Macroeconomics and Finance:

If foreign exchange participants are risk neutral, they care only about the mean value of asset returns and do not care at all about the variance of returns. Risk-neutral individuals are also willing to take unboundedly large positions on bets that have a positive expected value. Since $$F_{t} − S_{t+1}$$ is the profit from taking a position in forward foreign exchange, under risk-neutrality expected forward speculation profits are driven to zero and the forward exchange rate must, in equilibrium, be market participant's expected future spot exchange rate

$$F_{t} = E_{t}(S_{t+1}). (1.6)$$

Substituting (1.6) into (1.2) [result given from covered interest parity where instead of expected exchange rate you have forward rate] gives the uncovered interest parity condition $$1 + i_t = (1 + i_t^* )\frac{E_t[S_{t+1}]}{S_t}. (1.7)$$ If (1.7) is violated, a zero-net investment strategy of borrowing in one currency and simultaneously lending uncovered in the other currency has a positive payoff in expectation. We use the uncovered interest parity condition as a first-approximation to characterize international asset market equilibrium, especially in conjunction with the monetary model (chapters 3, 10, and 11). However, as you will see in chapter 6, violations of uncovered interest parity are common and they present an important empirical puzzle for international economists.

The in text square brackets contain my clarifications. So as the above passages show there is an arbitrage argument in UIP. You can find further sources and more detailed explanations in the textbook itself.

• Thanks for the recommendation, I'll pick that book up in the next few days! The part that interests me though is why "expected forward speculation profits are driven to zero". If the exchange rate appreciates more than it should according to (1.7), then an onslaught of rational investors will raise the rate even higher instead of pushing it down to its fair value. In replication-based arbitrage this price increase would lower returns and thereby diminish profit opportunities but this mechanism doesn't apply here because the interest rate is a relative return by definition. – Steven Jul 29 at 13:16
• @Steven the UIP hold only under strict assumptions, including that agents are all risk neutral and homogenous - you cant divorce the UIP from those assumptions because that is how you derive it.Those assumptions are not realistic and that is how its empirical failure is usually excused in the literature – 1muflon1 Jul 29 at 13:21
• Yeah, the most critical empirical issue is definitely the negligence of a risk premium, but thinking strictly inside the bounds model, I don't see why investors flocking to a currency that delivers excess returns violates any of the pertaining assumptions. Could you clarify where my thought experiment overstepped the theoretical boundaries? – Steven Jul 29 at 13:44
• @Steven but if UIP holds the currency cannot offer any excess returns due to the equation 1.7 which is the UIP. The argument there is precisely the same argument as the arbitrage argument in CIP because under UIP expected exchange rate must be exactly equal to the forward rate. Hence if there are any arbitrage opportunities they would instantly disappear. Of course, in real world you can have investors flocking to currency and finding ways how successfully exploit it, but in the real world people have heterogeneous risk appetites, information, transaction costs etc which violate UIP – 1muflon1 Jul 29 at 13:51
• I agree with you on every point, my question is what the microeconomic mechanics of "if there are any arbitrage opportunities they would instantly disappear" are. The argument of no free lunches makes sense, but is not a sufficient explanation for why a single investor is to undertake any action that would mitigate excess earnings potentials (under the assumption that all investors are risk-neutral). – Steven Jul 29 at 14:01

From a real world perspective, “interest rate parity” always holds - so long as you take into account the cross-currency basis swap spread. (Basis swaps are typically ignored in simple academic models, but are important for cross-currency funding). Forward points are set based on the theoretical relationship, with a bid-offer spread. Any market maker that is off market would be mercilessly arbitraged, and would be looking for a new job. There is no debate about this, that’s how market makers operate.

The only room for research questions are the following.

• Are forwards an accurate measure of market participants forecasts for currency values? Given that the basis and interest rate market pricing is determined by fixed income investors who often cannot trade foreign exchange, there is no way of surveying “investors” properly. Cross-currency basis swaps are a funding instrument, and are priced based on funding flows. Meanwhile, the primary driver of short-term rates is rate expectations. As such, currency relative valuation does not show up in the instruments determining forward-spot spreads, other than via the indirect route of central bank reactions to currency movements.
• Are forward points offer “good predictions” (however defined) of the future spot rate? That’s where most of the literature ends up, but I think it is safe to say that currency forwards have little useful information about future currency movements. One thing to note is the realised volatility in currencies is often much larger than the deviation of spot and forward, so it’s equivalent to saying that currencies can go up or down.