ECON3008 Lecture 2 - Arbitrage

Risk-adjusted UIP

For risk-averse investors, no-arbitrage condition is the risk-adjusted version of uncovered interest parity -This equation comes from maximizing the payoffs of (risk-averse) investors who care about both expected returns and return risk: iᵤₖ = iᵤₛ + eᵉ£/$ − e£/$ - RP Where RP > 0 is a currency risk premium. The risk-adjusted UIP equation implies that the foreign currency will be held by risk-averse investors only if it compensates them by paying a risk premium over the risk-free UK return.

Expression for spot rate and what it shows

e£/$ = iᵤₛ - iᵤₖ + eᵉ£/$ -This equation show that the equilibrium spot rate depends on interest rates and the expected future spot rate. -It tells us that a currency's value has two parts: the net interest received, iUS − iUK, and the expected future resale value of foreign currency, eᵉ£/$ . -Hence, UIP predicts that the Pound will appreciate if the Bank of England raises interest rates; if the Fed cuts interest rates; or if the Pound is expected to appreciate in the future.

Linear UIP Equation

It is usually more convenient to work with linear UIP by taking natural logs of 1 + iᵤₖ = (1 + iᵤₛ) (Eᵉ£/$)/(E£/$): iᵤₖ = iᵤₛ + eᵉ£/$ − e£/$ Where we use ln(1 + i) ≈ i for small i, and e£/$ = ln(E£/$) and eᵉ£/$ = ln(Eᵉ£/$ ) Rearrange to give an expression for the spot rate: e£/$ = iᵤₛ - iᵤₖ + eᵉ£/$

UIP Equation

Uncovered Interest Parity 1 + iᵤₖ = (1 + iᵤₛ) (Eᵉ£/$)/(E£/$)

Forward rate under risk aversion

Linear CIP equation: iᵤₖ = iᵤₛ + f£/$ − e£/$ Risk adjusted UIP equation: iᵤₖ = iᵤₛ + eᵉ£/$ − e£/$ - RP iᵤₛ + f£/$ − e£/$ = iᵤₛ + eᵉ£/$ − e£/$ - RP f£/$ = eᵉ£/$ - RP -Forward rate lower under risk aversion! -We see that forward rate is now lower than the expected future spot rate by an amount equal to the risk premium -With f£/$ = eᵉ£/$ - RP only risk-averse purchase forwards

What does the UIP predict

UIP predicts that the Pound will appreciate if the Bank of England raises interest rates; if the Fed cuts interest rates; or if the Pound is expected to appreciate in the future.

Buy and hold example with expected return

Suppose UK investors face two investment opportunities (they have £1 to invest) • Option 1 - HOME INVESTMENT - Invest in UK bank (home) with known interest rate iᵤₖ Return: £(1 + iᵤₖ) • Option 2 - FOREIGN INVESTMENT - Invest in US bank (foreign) with known interest rate iᵤₛ Return: £(1 + iᵤₛ) (Eᵉ£/$)/(E£/$) Where (Eᵉ£/$)/(E£/$) is the investors' expectation of the future spot rate and is fixed and given And iᵤₖ and iᵤₛ are given -Comparing returns on the two investments, we see that the optimal strategy is: Invest in pounds if 1 + iᵤₖ > (1 + iᵤₛ) (Eᵉ£/$)/(E£/$) Invest in dollars if (1 + iᵤₛ) (Eᵉ£/$)/(E£/$) > 1+ iᵤₖ That is, invest in the currency that delivers the highest expected return. For instance, if the foreign currency offers a higher return than home currency, investors will exchange pounds for dollars. The Pound will therefore depreciate, lowering the expected capital gain on foreign currency until expected returns are equal. The FX market is then in equilibrium: all arbitrage opportunities have been taken. The resulting no-arbitrage condition is known as uncovered interest parity (UIP): 1 + iᵤₖ = (1 + iᵤₛ) (Eᵉ£/$)/(E£/$) (UIP equation)

Example: Suppose UK investors face two investment opportunities (they have £1 to invest) • Option 1 - HOME INVESTMENT Return: £(1 + iᵤₖ) • Option 2 - FOREIGN INVESTMENT £(1 + iᵤₛ) (Eᶠ£/$)/(E£/$)

Suppose UK investors face two investment opportunities (they have £1 to invest) • Option 1 - HOME INVESTMENT - Invest in UK bank (home) with known interest rate iᵤₖ Return: £(1 + iᵤₖ) • Option 2 - FOREIGN INVESTMENT - Invest in US bank (foreign) with known interest rate iᵤₛ Return: £(1 + iᵤₛ) (Eᶠ£/$)/(E£/$) Where (Eᶠ£/$)/(E£/$) is the future exchange rate from dollars into pounds and is known And iᵤₖ and iᵤₛ are given -Note that the return on foreign investment has two parts: the capital gain on the exchange rate: Eᶠ£/$)/(E£/$) and the interest earned: (1 + iUS). -The capital gain has a simple interpretation. The investor's pounds are first exchanged for dollars at the current spot exchange rate, E$/£ = 1/(E£/$). The dollars are then held in a US bank until the money is withdrawn and converted back into pounds. The exchange rate which applies in this second conversion is the future spot rate. If (Eᶠ£/$)/(E£/$) > 1 investors make a capital gain because dollars are worth more in terms of pounds. -When comparing home and foreign investment opportunities, investors should look at both interest rates and potential capital gains.

Linear CIP Equation

Taking natural logs of 1 + iᵤₖ = (1 + iᵤₛ) (F£/$)/(E£/$), we find that: iᵤₖ = iᵤₛ + f£/$ − e£/$ Where we use ln(1 + i) ≈ i for small i and f£/$ = ln(F£/$) is the natural log of the forward exchange rate

The current 1-year forward rate is F£/$ = 0.50. An investor has £100,000 to invest and expects the spot rate in a year's time to be E£/$ = 0.60. (b) Suppose now that the investor exchanges the £100,000 for dollars at today's spot rate E£/$ = 1, and deposits the dollars in a US bank account for one year at an interest rate of 25%. The investor enters a forward contract to sell the dollars for pounds at the 1-year forward rate F£/$ = 0.50. Compute the profit or loss on this investment

(b) Now the investor's £100,000 immediately become $100,000 (since E£/$ = 1). 100,000 X 1 = 100,000 iᵤₛ = 0.25 The $100,000 deposited in the US bank becomes 100,000 X (1.25) =$125,000 in a year. The investor is contracted to sell $125,000 for pounds at the rate F£/$ = 0.50, which gives 0.5 × 125, 000 = $62, 500. Use E£/$ = 1 -> $62, 500 X 1 = £62,500 Profit = £62,500 - £100,000 = -£37,500. (=Loss!) Foreign return formula: £(1 + iᵤₛ) (Eᶠ£/$)/(E£/$) Foreign return = FR = £(1 + iᵤₛ) (F£/$)/(E£/$) = 1.25 × 0.5/1 = £0.625 Intuitively, the problem is that £1 gets you only 63p in a year's time!

Risk averse investors

-An important assumption we have made up to this point is that investors are risk-neutral. That is, they care only about the expected return and look at this alone when making investment decisions. -We now consider the case where investors are risk-averse. -Risk aversion implies that investors will take into account riskiness of returns when evaluating investment opportunities. -As we shall see, it is precisely such individuals that may benefit from purchasing forward contracts that 'lock-in' a known future exchange rate.

Arbitrage

-Arbitrage is a trading strategy that takes advantage of profit opportunities arising from price differences. -In its simplest form, arbitrage means buying at a low price and selling at a high price in order to make a profit (Arbitrage is buying low and selling high to make a profit)

Spot rates are: (1) London: E £/$ = 0.66 (2) New York: E NY £/$ = 0.70 Strategy: Buy $s in London and sell in New York for a profit! What happens?

-As traders embark on this strategy, London will be flooded with demand orders for dollars, and New York will be swamped with a supply of dollars to be sold. -There is excess demand for dollars in London, and an excess supply of dollars in New York. -Unable to satisfy the demand for dollars, London dealers will raise the price of a dollar above 66p. In New York, demand for dollars will fall short of supply, so dealers will reduce the price of dollars below 70p. -With the price of dollars rising in London and falling in New York, a new set of exchange rates will be established. -What will the new set of exchange rates be? Both locations will end up with the same exchange rate (see Figure 1). The new exchange rate will lie between the initial exchange rates of 66p and 70p and is determined by the intersection of the supply and demand schedules at E£/$ = 0.68 (Point E). -This is the equilibrium exchange rate. It is established through traders taking arbitrage opportunities until no further arbitrage is possible.

Spot rates are: (1) London: E £/$ = 0.66 (2) New York: E NY £/$ = 0.70 Strategy: Buy $s in London and sell in New York for a profit! Summarise what happens

-ER differential presents an arbitrage opportunity -All traders will buy $ in L and sell in NY -As they do, the exchange rate moves: Price Difference ⇒ Arbitrage ⇒ ER Adjustment -The new exchange rate will lie between the initial exchange rates of 66p and 70p and is determined by the intersection of the supply and demand schedules at E£/$ = 0.68 (Point E). -This is the equilibrium exchange rate. It is established through traders taking arbitrage opportunities until no further arbitrage is possible.

Arbitrage with two currencies: No-arbitrage condition

-For any locations A and B: Eᴬ£/$ = Eᴮ£/$ -This is a no-arbitrage condition and the equilibrium -Arbitrage opportunities are taken until none left! =The no-arbitrage condition above states that the exchange rate is the same in both locations. -As we have seen, this is because arbitrage opportunities will be taken by investors, thus eliminating any exchange rate differential.

Investors take on risk when they invest 'uncovered' in a foreign currency. Is there any way to avoid this risk?/Is there any way to guard against currency risk?

-Foreign return is risky - depends on realized capital gain -Yes. Hedge using a forward -Investors can 'lock in' a known return on foreign currency investment by entering into a forward contract. -Forward contracts enable future exchange of currency at an exchange rate decided in advance: the forward rate

CIP Equation

-Forward contracts -The resulting no-arbitrage condition between home investment and riskless foreign currency investment is covered interest parity (CIP): 1 + iᵤₖ = (1 + iᵤₛ) (F£/$)/(E£/$) where F£/$ is the forward exchange rate

Capital gain and expected return - No Arbitrage Condition

-In practice, capital gain is not known with certainty -Investors look at expected return. Compute using expectation Eᵉ£/$ . The no-arbitrage condition is now: 1 + iᵤₖ = (1 + iᵤₛ) (Eᵉ£/$)/(E£/$) (UIP Equation)

Spot rates are: (1) London: E £/$ = 0.66 (2) New York: E NY £/$ = 0.70 Is the above price pair an equilibrium?

-No. Buy $s in London and sell in New York for a profit! -Traders can move funds at zero cost. Sole aim to maximize profit. -A dollar can be bought for 66p in London and sold for 70p in New York. Hence, traders can make a profit of 4p on every dollar bought and sold. -Since they have an incentive to move funds between the two locations - buying dollars in London and selling them in New York - the original price pair is not an equilibrium.

Arbitrage with 3 currencies example: Suppose the currencies are Dollars, Euros and Pounds. -The exchange rates are: E£/$ = 0.50, E£/€ = 0.70, and E€/$ = 1. Is there an arbitrage opportunity here?

-Real-world traders will compare payoffs on multiple currencies. Therefore, we need to compare the Pound-Dollar exchange rate with cross rates for multi-currency trades. Suppose the currencies are Dollars, Euros and Pounds. -The exchange rates are: E£/$ = 0.50, E£/€ = 0.70, and E€/$ = 1. Is there an arbitrage opportunity here? Yes! Buy $s for 50p, sell for 70p! -The payoffs on a single currency exchange and a multiple currency exchange are: • Strategy 1: E£/$ = £0.50 • Strategy 2: E£/€ × E€/$ = No. pounds per Euro X No. Euros per dollar = 1 X £0.70 = value of $1 in ponds = £0.70 -Under Strategy 1, we sell dollars in exchange for pounds and receive 50p per dollar sold. -Under Strategy 2, we first exchange dollars for euros (receiving €1) and then exchange each euro for 70p. -There is a clear arbitrage opportunity here. We can buy dollars at 50p (Strategy 1) and sell them for 70p (Strategy 2), making a profit of 20p on each dollar bought and sold. -Since all traders will follow this strategy, any FX dealer who posts these exchange rates will incur losses and hence be forced out of the market. To avoid losses, dealers will post prices such that the spot rate equals the cross rate: E£/$ = E£/€ × E€/$

Risk-neutral investors' participation in the forward market

-Risk-neutral investors will be on the other side of the market, selling the forward contracts to the risk-averse. -They are happy to bear the risk. -To them, the value of the forward contract is eᵉ£/$, but they have to pay only f£/$ = eᵉ£/$ − RP on each unit of currency they deliver. Hence, they make an expected profit equal to RP.

Introduction to Buy and Hold Investment

-Some time passes between the buy and sell dates, and during this time a return may be earned. When investors buy and hold currencies: (1) Receive interest (2) Capital gains (or losses) -A capital gain occurs when a currency is sold at a higher price than it was initially purchased. -Once time enters into investment decisions, investors will be compensated through the payment of interest, and this will form part of their payoff.

Suppose (1 + iᵤₛ) (Eᶠ£/$)/(E£/$) > 1+ iᵤₖ at the current spot rate, so that investors want to invest all their money in US banks. Is this situation sustainable?

-The answer is no. -We are back in an arbitrage scenario: investors get 1 + iᵤₖ by investing in the UK, but this is beaten by investing abroad. -Investors will therefore put their money in US banks by exchanging pounds for dollars, implying excess demand for dollars in the FX market. -The spot rate E£/$ will thus increase: a Pound depreciation (= Dollar appreciation). As this happens, the capital gain on the exchange rate falls. The spot rate keeps rising until the capital gain, (Eᶠ£/$)/(E£/$), has fallen enough to make the return on foreign investment equal the unchanged domestic return. Arbitrage opportunities are then eliminated. Once returns are equalised, the FX market is in equilibrium. The no-arbitrage condition is therefore: 1 + iᵤₖ = (1 + iᵤₛ) (Eᶠ£/$)/(E£/$)

Is E£/$ = E£/€ × E€/$ an accurate prediction for real-world exchange rates?

-The answer is yes: the product of the cross rates is approximately equal to the direct exchange rate (see Table 1 in the ebook) -The small discrepancies between E£/$ and E£/e × Ee/$ in the final column arise because Equation (2) holds exactly only if bid-ask spreads are zero.

Uncovered Interest Parity

-The investment said to be 'uncovered' because the return on foreign investment is risky. -In particular, although investment decisions are based on the expected return on foreign investment, the actual return received will depend on whether the foreign currency appreciates or depreciates. -There is always the risk, for instance, that the foreign currency will depreciate sharply, 'wiping out' any return for the investor. Home currency investment avoids this risk.

Interpreting f£/$ = eᵉ£/$

-The result that f = eᵉ has an interesting interpretation. -It implies that the expected future spot rate can be inferred from the forward rate - an observable market price. -This is useful from a policy perspective because it gives central banks a way of assessing investor beliefs about the prospects of their currency. This is certainly more convenient than gathering data on expectations through surveys, though it is not without problems. -An interesting application is when a country fixes its exchange rate at some target e STAR. If investors think the fixed exchange rate is credible, then their expectation should be that eᵉ = e STAR. In this case, we should observe that f = e STAR as well. Hence, if the forward rate deviates from the target exchange rate (i.e. f ≠ e STAR), this may signal that investors doubt the credibility of the peg and will soon bet against it.

No-arbitrage condition for three currencies

-To avoid losses, dealers will post prices such that the spot rate equals the cross rate: E£/$ = E£/€ × E€/$ -It tells us that currencies are priced so that payoffs on spot exchanges are equalized. -Like the two-currency no-arbitrage condition, it applies in any locations in which currencies are freely traded.

Example: Optimal strategy - Buy and hold investment - home vs foreign investment

-When comparing home and foreign investment opportunities, investors should look at both interest rates and potential capital gains. -Comparing returns on the two investments, we see that the optimal strategy is: Invest in pounds if 1 + iᵤₖ > (1 + iᵤₛ) (Eᶠ£/$)/(E£/$) Invest in dollars if (1 + iᵤₛ) (Eᶠ£/$)/(E£/$) > 1+ iᵤₖ That is, invest in whichever currency delivers the highest return.

No-arbitrage condition: Buy and hold investment - home vs foreign investment

1 + iᵤₖ = (1 + iᵤₛ) (Eᶠ£/$)/(E£/$) -When investors see the arbitrage opportunity, all invest in $ -Pound depreciates against Dollar: E£/$ ↑ -Process stops when: 1 + iᵤₖ = (1 + iᵤₛ) (Eᶠ£/$)/(E£/$)

Forward rate - risk neutral investor

Linear UIP equation: iᵤₖ = iᵤₛ + eᵉ£/$ − e£/$ Linear CIP equation: iᵤₖ = iᵤₛ + f£/$ − e£/$ If both UIP and CIP hold, the forward exchange rate equals the future expected spot rate: f£/$ = eᵉ£/$ -What is the intuition for this result? For investors to enter into a forward contract, the investment must offer a return at least as large as the expected return on risky (uncovered) investment of the same maturity, i.e. f ≥ eᵉ. But those on the other side of the forward contract, such as speculators, would never offer a contract with f > eᵉ , because this would imply them delivering more pounds for the investor's dollars than they expect to earn themselves! Therefore, the only contract acceptable to both parties is f = eᵉ .