Asset Returns, Risk and Liquidity

From last week’s homework, what you should realize is that the covered interest parity is really a model to derive the implied forward exchange rate. We take the domestic and the foreign interest rates as a given. Those interest rates reflect the demand and supply of money. And we also take the spot exchange rate as a given. The spot exchange rate reflect the demand and supply of imports and exports. Because interest rates are locked in for a period of time, there is an implied future exchange rate if we decide to borrow in one currency, convert that currency to another currency, and lend it out in that other currency. These three steps gives us the implied forward exchange rate. If the actual quoted forward exchange rate doesn’t satisfy the covered interest parity condition, then, as we all know, there is arbitrage opportunity. This shouldn’t exist because arbitrageurs should restore parity through either appreciation or depreciation of the domestic currency.

However, our ultimate goal in this class is to figure out how to put a price on E or the spot market. What we basically need to know is R, R^*, and F to find E. Domestic and foreign money markets will give you R and R^*. For example, government bond yields at various maturities, certificate of deposits at local banks, etc. But, how do we find F? For now, we will substitute F with another value.

Our starting point is focusing on asset returns to build our exchange rate model. Investors in general care about three things: returns, risk, and liquidity. Returns is simply the percentage increase in the value of an asset over a given period of time. Bank deposits, stocks, and bonds are all a type of assets and produce returns, otherwise we wouldn’t invest in them. However, these assets have different variability in returns. This is the risk dimension of investments. Stocks are riskier than bonds and not all stocks and not all bonds have the same risk within their asset class. Lastly, not all assets have the same level of liquidity. Cash is obviously one of the most liquid because it can be sold and exchanged for many things. Government debt is also fairly liquid because it is an integral part of any diversified portfolio. Hedge funds, pension funds, mutual funds all hold some level of government debt and they may need to rebalance their portfolios often. Something less liquid are small cap stocks. It is in general difficult to find buyers if you are selling and it can be difficult to find sellers if you are buying.

Since our task today is to link R and R^* to find E or the spot exchange rate, we start by assuming that investors only care about returns or expected returns and are indifferent or ignore the risk and liquidity dimensions of assets. In other words, we will be assuming that investors are risk neutral. In that sense, currencies can be viewed as a type of asset that produce returns through depreciation of other currencies. I will explain this idea in the next few slides.

The Asset Return to Currency Deposits

From the previous homework exercises, I think it is clear that everyone in this class knows how interest rates work. They are essentially returns on a type of asset, namely currency deposits. So, in this example, if I take 1 EUR and deposit it in a Euro denominated account at 5% per year, I should receive 1.05 EUR after a year 1+R^* = 1+0.05. Currencies are no different from other assets: currencies can fluctuate in value just like stocks and bonds. They appreciate and depreciate. However, just like stocks and bonds, their change in value or returns are going to be modeled as random. Their exact future value is not known today. We will model currency returns through the expected depreciation of the domestic currency. The expected depreciation rate is in fact the expected return on foreign currencies relative to the domestic currency. We model the expected rate of depreciation of USD over a given time period as E hat e minus E all divided by E. E hat e is the expectation of future exchange rate. If E hat e is greater than E, then investors believe that the USD will depreciate, which means they expect the foreign currency to appreciate or generate expected returns that is positive in terms of the domestic currency.

Subtle Math

For those that are less mathematical, you should just focus on the intuition that E hat e is simply the average exchange rate that investors believe the exchange rate will be in the future. It is simply the mean of the universe of investor beliefs. And so, E hate e minus E all divided by E is the average return on foreign currencies relative to the domestic currency.

For those who want some mathematical rigor, I will go over how we arrive at this expression from the definition of expected returns. Much like stock prices, you can think of E hat as the future price of a stock. This price is unknown and therefore is a random variable. Although we don’t know its value, we often tend to assume we know its distribution or the range of possible values it can take and its likelihood for each value. For example, we can assume a stock price to be normally distributed. To calculate returns is simple: just take the difference between future value and current value and divide the difference by the current value. This entire expression is a random variable. To calculate the expected returns, we take the expected value of this random variable.

What is the expected value? Expected value is just the sum of all possible values weighted by their respective probability. For example, flipping a coin can only have 2 outcomes with 50% probability each. If E hat takes on a value of 1 when the coin is heads and E hat takes on a value of -1 when the coin is tails, then the expected value is 0.

Since the expected value function is linear, we can factor out constants. This allows us to write the expected return on a currency in this following manner.

The Asset Return to Currency Deposits: Expectations

So, the expected return on a foreign currency is just the expected rate of depreciation of our domestic currency and it can be written in the following expression. So, an investor with 1 EUR can expect to hold EUR worth 1 + (E hat e - E)/E USD in a year.

Here is an example. At its introduction the EUR is calculated to be worth 1 USD. If the USD is expected to depreciate from 1 USD/EUR to 1.10 USD/EUR, the expected return on holding 1 EUR in a year is 10 percent (in USD). 1 + (E hat e - E)/E = 1 + (1.1 - 1)/1.

Recall: Covered Interest Parity

Let us review the covered interest parity so that we have something to compare with when I introduce the new model. This again is the covered interest parity equation. I give you two ways to express it. The first equation should be used when you are asked to come up with arbitrage strategies. The second equation is a simplified approximation that we will use for graphical proofs and comparative statics. We can derive the second equation by multiplying out the first equation, canceling out the 1 and assume that R^* multiplied by (F-E)/E is approximately zero.

As shown in the previous lecture and homework assignments, we know that this parity must hold if there is no arbitrage opportunities in the forward exchange markets (futures market). Market participants will follow E and F. Should the covered interest parity fail at any moment, arbitrageurs can reap the profit. You all are arbitrage experts now, so it should be obvious how to make profits anytime this parity is broken. However, let us go over an example, just to make sure.

If (1+R) > (1+R^*) (F/E), then we know that it is more profitable to lend in the domestic currency than in the foreign currency. Suppose the foreign currency is Japanese Yen and the U.S. currency is the domestic currency. Arbitrageurs will borrow Japanese Yen and invest in USD for the higher return. Borrowing Japanese to buy USD will bid up the price of USD (reducing E) until equality is restored.

Uncovered Interest parity (UIP)

The uncovered interest parity is almost identical to the covered interest parity. Let us go over the variables again for this equation. Again, suppose the domestic currency is USD and the foreign currency is EUR. 1+R is the gross return on USD deposit. 1+R^* is the gross return on EUR deposits. E is still the spot exchange rate and E hat e is the expected exchange rate. However, the interpretation is somewhat different now. There are now three assumptions that go into this model: 1) No arbitrage, 2) Investors are risk neutral. This means that they only care about expected or mean returns and don’t care about variance and other measures of risk. This assumptions therefore renders forward contracts meaningless since no one requires risk management. And 3) Perfect information. This last assumption assumes that we everyone shares the same information. Since everyone shares the same information, the market should capture all possible information available and thus prices should reflect all information. This is the Market efficiency hypothesis, which all of you should have learned in your introduction to financial markets class. Since markets are efficient, then we believe that on average, investors’ beliefs are on average correct. In that case, a currency’s expected deprecation or (1 + (E hat e - E)/E) should make up the differences in interest rates.

Uncovered Interest parity (UIP): UIP vs CIP

So, why is the uncovered interest parity called uncovered? And why is the covered interest parity called covered? In finance jargon, cover usually means to reduce risk. In the covered interest parity, we remove future spot exchange rate risk by engaging in a forward contract today. However, in the uncovered interest parity, we are exposed the future spot exchange risk. E hat e is not quoted anywhere and therefore E hat e is not a price to trade on.

However, note that arbitrage argument still applies under the uncovered interest parity. Again, assume the domestic currency is USD and the foreign currency is EUR. In the case in which (1+R) is greater than (1+R^*) times E hat e divided by E, a USD deposit would yield more than a EUR deposit. Arbitrageurs would take similar actions under the covered interest parity. They would sell EUR for USD and bid up the spot rate E until the parity is restored.

Because we have 2 more assumptions, the uncovered interest parity can fail significantly more often.

Uncovered Interest parity (UIP): Assumptions

So, to reiterate, in the case in which domestic deposit rates exceed foreign deposit rates, the spot rate E will be bid up until the uncovered interest parity holds.

In addition to no arbitrage, we are now also assuming that 1) financial markets are efficient and everyone has the same information set. Thus, the market incorporate all possible information into prices. And 2) investors do not care about risk and therefore have no need for forward contracts. Investors only care about expected or mean returns. They are indifferent about variance. Under these assumption, investors beliefs are on average correct. Under these assumptions F must equal E hat e on average.

Uncovered Interest parity (UIP): Formulas

Again, for the uncovered interest parity, we will also work with a simplified version when we do graphical proofs or comparative statics. However, this form should also give you more intuition about the uncovered interest rate. The left hand side R is the net return on the domestic currency. The left hand side is thus the foreign currency net return. The two version are approximately the same for the same reason as we discussed before for the covered interest parity.

So, if R-R^* is greater than 0, this means that the return on the domestic currency is greater than the return of the foreign currency. Investors would buy the domestic currency and appreciate the domestic currency. This will decrease E until the parity is restored.

However, now we can make a different argument since we are dealing with investor beliefs or expectations. In the long-term, investor beliefs are fixed. You should think of E hat e as an anchor. If domestic interest returns are greater than foreign interest returns, then investors believe that there must be a capital loss of holding the domestic currency. In other words, the domestic currency must be devalued in the future. If expectations are fixed, we have to appreciate the domestic currency today.

Interest Parity and Exchange Rate Determination

Let us review. The covered interest parity, as done in your homework, mostly serves to price F, given R, R^*, and E. However, what determines E?

The uncovered interest parity gives us a way. If we have a theory of R, R^* and E hat e, then we know E. We take R and R^* as given.

For now, we will consider E hat e as the longer-term anchor. It is the widely shared exchange rate expectations. Then the following equation must hold or approximately hold. Rewriting the first equation into the second should give us even more intuition about our new model. On the right hand side, we have R^* minus R. If this difference is positive, this tells us that foreign interest rates is greater than the domestic interest rate. This is sustainable only if E - E hat e is positive. If this difference is positive, then investors believe there will be appreciation of the domestic spot price.

Uncovered Interest Parity and the Spot Exchange Rate

Under the uncovered interest parity, how should the spot rate evolve overtime. Let us look at the evolution of the spot rate over, say, 90 days. The x axis is time and the y axis is the nominal exchange rate or E. Suppose in the first case E > E hat e. At time zero, the spot rate is at E_0. The vertical difference between E_0 and E hat e is equal to (R^* - R) E_0. If R^* - R is greater than 0, then the domestic currency must appreciate over time (E decreases).

Now let us consider the case in which E < E hat e. Again, at time zero, the spot rate is E_0. The vertical difference between E_0 and E hat e is equal to (R^* - R) E_0. If R^* - R is less than 0, the domestic currency must depreciate (E increases) over time.

The point of this exercise is to help you think about how to make arguments or claims about E given changes in R and R^*.

Spot Exchange Rate and Expected Currency Return

Now, let us try to visually understand how the expected foreign currency return is related to the spot exchange rate E. The x axis is the expected currency return. It is equal to R^* + (E hat e - E)/E. The y axis is the spot exchange rate E. So, holding E hat e and R^* constant, we should see that the spot exchange rate and the expected currency return should have a negative relationship. Increasing E or depreciating the domestic currency will lower the expected currency return of holding foreign currencies. However, decreasing E or appreciating the domestic currency will increase the expected currency return since the denominator is getting smaller.

Uncovered Interest Parity

Now let us add in the interest return on the domestic currency. This will be just a vertical line since R is just a constant with respect to the spot exchange E. When R is equal to R^* plus domestic currency depreciation, the uncovered interest parity is satisfied. At there intersection, we can pin down E_0. We have therefore graphically solved for E. So now let us do some comparative statics.

Increase of USD Interest Rate

We take R and R^* as given. Suppose we have an initial R and therefore an initial E_0. Suppose the domestic interest rate increases to R’. The R vertical line shifts to the right and we arrive at a new intersection. This gives us E’, which is lower than E_0. This means that the domestic currency appreciated. So let us go through the argument.

1. We assume that expectations are fixed in the future since on average investor beliefs are accurate and correct.
2. The domestic interest rate is now greater for the domestic currency. Foreigners will only accept this differential if there is an equivalent capital loss on the domestic currency. This means that the dollar will depreciate in the future. So, the domestic currency must appreciate today so that the expectation of the future dollar wipes out the current appreciation.

Increase of Foreign Currency’s Interest Rate

Now, suppose there is an increase of foreign currency’s interest rate. R is fixed. Since we are not changing E, we are not moving along the curve. So the entire curve must shift up if foreign currency interest rate increases. Remember, (E hat e - E)/E is interpreted as the expected depreciation of the dollar, which is akin to capital gains or losses. Shifting the entire curve upward gives us a higher level of E. This means that the domestic currency will depreciate. Let us go over the argument.

1. There is now a greater interest returns on the foreign currency.
2. This is only acceptable market equilibrium if those interests will be wiped out. That is, interest gains of the foreign currency must equal the capital gains on the domestic currency.
3. Since the domestic currency must appreciate in the future, the domestic currency must depreciate today.

Increase in Expected Future Exchange Rate

What happens if there is a change in future expectations? Upward shift, same logic as last slide.

Empirical Results on Interest Parity

CIP is confirmed in most empirical studies. However, UIP often fails empirical. These failures can be due to several reasons that violate the efficient markets hypothesis and the risk neutral investor assumption. Here are some explanations:

1. If there is a risk premium in currency returns beyond interest rates, then the uncovered interest rate will be violated. This is because currency holders are compensated for the additional risk associated to holding currencies beyond interest rate differentials.
2. If markets do not process information efficiently, then expectations can be wrong. Then the uncovered interest parity does not necessarily have to hold.
3. If the spot exchange rate follows a random walk, then the spot exchange has a time varying mean. Expected future exchange rate isn’t fixed.
4. Again, expectations could be wrong if expectations do not account for infrequent events.
5. Lastly, there could be fundamental changes in institutions

How Efficient are Currency Markets?

I want to conclude on a final quote by Alan Greenspan in 2003. Alan Greenspan before the financial crisis was a true believer in the efficient market hypothesis. He believe that the market knows all and processes all information accurately and therefore investor beliefs are on average correct. However, this sharply contrasts with the performance of many funds who try to exploit violations in the uncovered interest parity. Through carry trades, they are essentially betting that the currency won’t appreciate or appreciate enough to offset the interest rate differentials between nations. Deutsche Bank for example, has a fund that has shown noteworthy returns.