###### Portfolio Risk: Analytical Methods

After completing this reading, you should be able to: Define, calculate, and distinguish... **Read More**

**After completing this reading, you should be able to:**

- Explain and describe the mechanics of spot quotes, forward quotes, and future quotes in the foreign exchange market and distinguish between the bid and ask exchange rates.
- Calculate bid-ask spread and explain why bid-ask spread for spot quotes may differ from the bid-ask spread for the forward quotes.
- Compare outright (forward) and swap transactions.
- Define, compare and contrast transaction risk, translation risk, and economic risk
- Describe the examples of the transaction, translation, and economic risk and explain how to hedge these risks.
- Describe the rationale for multi-currency hedging using options.
- Identify and explain the factors that determine the exchange rates.
- Calculate and explain the effect of an appreciation/depreciation of a currency relative to a foreign currency.
- Explain the purchasing power parity theorem and use this theorem to calculate the appreciation or depreciation of a foreign currency.
- Explain how the no-arbitrage assumption in the foreign exchange markets leads to the interest rate parity theorem and use this theorem to calculate forward foreign exchange rates.
- Distinguish between covered and uncovered interest rate parity conditions.

The foreign exchange market is a type of market where the parties involved (collection of hedgers and speculators) exchange one currency for another at the specified rates. The market is also called Forex, Fx, or currency market.

The exchange rate can be defined as the number of units of one currency (the quote currency) that are needed to purchase one unit of another currency (base currency). The exchange market is the world’s largest market, where all forms of exchange transactions are carried out.

Generally, currency quotes always appear as AAA/BBB or AAABBB, where AAA and BBB are different currencies. The currency to the left of the slash is the base currency, while the currency to the right of the slash is the quote currency.

The base currency (in this case, AAA) is **always equal to one unit**, and the quoted currency (in this case, BBB) is **what that one base unit is equivalent to in the other currency**.

The general convention is to quote base currency/ quoted currency. Currency quotes between USD and another currency is the most common exchange rate. Other quotes, for instance, between GPB and CAD, are called **cross-currency**. In most cases, the USD is the base currency, while the other currency is the quote currency. However, when the US dollar is quoted with the British Pound, the Euro, the Newzealand dollar, and the Australian dollar, then the USD becomes the quote currency.

The EUR/USD is quoted as 1.2563. How do we interpret this quote? This quote implies that we need 1.2563 USD to buy one euro.

The **bid price** is the price at which the counterparty is willing to buy one unit of the base currency, expressed in terms of the price currency. On the other hand, the a**sk price** is the price at which a counterparty is willing to sell one unit of the base currency, expressed in terms of price currency. For instance, a dealer might quote EUR/USD exchange rate of 1.3849. This quote implies that the dealer is willing to pay 1.3849 USD to buy 1 Euro. Intuitively, we expect the bid price to be slightly less than the offer price because the dealer’s goal is to make some cash in every transaction. With that in mind, it is easier to single out the bid price or the offer price given a quote.

- The ask price should always be higher than the bid price.
- A market participant requesting the two-sided price quote has the option but not the obligation to transact at either the bid or the ask quoted by the dealer. If a party chooses to transact at the quoted prices, they are said to have “hit the bid” or “paid the offer.”

A spot exchange rate is the prevailing price level in the market used to directly trade one currency for another, which is delivered at the earliest possible time. The standard delivery time for spot currency transactions is no longer than T+2, after which it will be deemed a forward contract. Spot exchange rates are usually quoted with four decimal places. For instance, the spot-bid for EUR/USD could be stated as 1.1745 and the spot-ask as 1.1747.

A forward exchange is a price at which one currency is traded against another at some specified time in the future. The forward exchange rate must respect the arbitrage relationship – the returns from two alternatives, but equivalent investments must be equal (as we will see in covered rate parity).

Forward exchange rates are not quoted with the same base as the spot exchange rates but rather as points that are multiplied by \(\frac {1}{10,000}\) then added to the spot exchange rate.

The following table gives the forward rates as of June 16, 2019. The spot bid and ask rates are 1.1745 and 1.1749, respectively.

$$ \begin{array}{c|c|c} \textbf{Maturity} & \textbf{Bid} & \textbf{Ask} \\ \hline \text{1 month} & {27.12} & {28.60} \\ \hline \text{2 months} & {53.15} & {54.15} \\ \hline \text{3 months} & {81.87} & {83.07} \\ \hline \text{4 months} & {113.59} & {114.99} \\ \hline \text{5 months} & {139.07} & {140.47} \\ \end{array} $$

What is the three-month forward bid and ask quotes?

Recall that forward exchange rates are not quoted with the same base as the spot exchange rates but rather as points that are multiplied by \(\frac {1}{10,000}\) then added to the spot exchange rate.

Since we are given the spot bid rate as 1.1745, then the 3-month forward bid rate is:

$$ 1.1745+\cfrac {1}{10,000}×81.87=1.1745+0.008187=1.182687 $$

Analogously, the 3-month forward ask quote is

$$ 1.1749+\cfrac {1}{10,000}×83.07=1.1749+0.008307=1.183207 $$

The **bid-ask spread** is the amount by which the offer price exceeds the bid price of a currency in a market. Basically, it is the difference between the highest price that the purchaser is willing to pay and the lowest price that a seller is willing to accept.

Continuing with the example above, the bid-ask spread for the 3-month forward rate is calculated as:

$$ 1.183207-1.182687=0.00052 $$

Note that this can also be calculated as:

$$ (1.1749-1.1745)+(0.008307-0.008187)=0.0004+0.00012=0.00052 $$

That is, we could calculate the bid-ask spread by adding the bid-ask spreads of the spot exchange rate and that of the points.

When the forward exchange rate is less than the spot rate, the points are negative. However, it should be apparent that the magnitude of the negative ask price is less than that of the bid price. This makes sure that the ask price is larger than the bid price. Consider the following example.

The following table gives the forward rates as of July 8, 2019. The spot bid and ask rates are 1.3184 and 1.3185 respectively.

$$ \begin{array}{c|c|c} \textbf{Maturity} & \textbf{Bid} & \textbf{Ask} \\ \hline \text{1 Month} & {-9.39} & {-7.67} \\ \hline \text{2 months} & {-18.20} & {-17.29} \\ \hline \text{3 months} & {-32.10} & {-30.91} \\ \hline \text{4 months} & {-40.91} & {-39.42} \\ \hline \text{5 months} & {-45.90} & {-43.95} \\ \end{array} $$

What is the five-month forward bid and ask quotes?

The 5-month forward bid quote is:

$$ 1.3184+\cfrac {1}{10,000} ×-45.90=1.3184-0.004590=1.31381 $$

Analogously, the forward ask quote is:

$$ 1.3185 +\cfrac {1}{10,000} ×-43.95=1.3185-0.004395=1.314105 $$

At this point, we could easily calculate the bid-ask spread:

$$ 1.314105-1.31381=0.000295 $$

Recall that a forward exchange is a price at which two parties agree to trade one currency against another at some specified time in the future. This is termed as an **outright transaction or outright forward transaction**.

On the other hand, a foreign exchange swap is a type of exchange rate transaction where a currency is bought (sold) in a spot market and then sold (bought) in the forward market. From a different angle, an FX swap is a method of funding an asset transacted in foreign currency by paying the interest due in terms of the domestic currency. An example of an FX swap is where a US-based company funds its Chinese investment by borrowing in USD and buying the Chinese Yuan, and after some time, the company exchanges the money back to USD. By doing this, the company can fund its operation in the Chinese Yuan.

Swap transactions are usually profitable if the foreign currency used will be of more value in the forward market, i.e., more of the local currency will be received for less of the foreign currency at the agreed-upon future date.

For instance, if we use the following table:

$$ \begin{array}{c|c|c} \textbf{Maturity} & \textbf{Bid} & \textbf{Ask} \\ \hline \text{1 month} & {27.12} & {28.60} \\ \hline \text{2 months} & {53.15} & {54.15} \\ \hline \text{3 months} & {81.87} & {83.07} \\ \hline \text{4 months} & {113.59} & {114.99} \\ \hline \text{5 months} & {139.07} & {140.47} \\ \end{array} $$

Assume that a US company borrows in USD and buys 1 million EUR today to fund its European operations. At the same time, the company also agrees to sell 1 million EUR for USD within one-month time. In the table above, the points in one month’s time are 27.10. This implies that EUR is valuable in the forward market, and thus, the points reduce the net funding cost in USD since more USD is going to be received in a one-month time relative to the amount that would have been received today.

Currency swaps will be discussed in details in chapter 20

Future quotes are the exchange-traded futures legal contract that stipulates the price in one currency at which another currency can be bought or sold at a future date. A good example is the Chicago Mercantile Exchange (CME) in the US, where diverse futures contracts on exchange rates between USD and other currencies are made.

In the CME setting, USD is always the base currency since investors treat foreign currency as assets value in USD. Assume that the 6-month forward quote for the USD/CAD is 1.2900. The future quote is found by finding the reciprocal of the forward quote. That is:

6-month futures quote is equivalent to the 6-month forward quote is: \(\frac {1}{1.2900}\)=0.7752 USD per CAD

Firms in the foreign exchange market are exposed to risks. They should, therefore, be keen on the extent to which they could accept risk. Once this is known, the firms should decide whether the risk levels are acceptable and if not, and if not, they should apply appropriate hedging strategies.

Three categories of risk are examined, and these include:

- Transaction risk
- Translation risk
- Economic risk

This kind of risk is associated with received and paid capital; it affects the cash flows of a company. Let us look at a simple example:

Assume that an American company imports goods from Kenya, for which it pays in Kenyan shillings. By doing this, the company is faced with USD/KSH risk. That is, if KSH gains strength, then the company will experience little profits if it is required to buy KSH to pay for its services.

In summary, a company buying from a foreign company will be exposed to losses (profits) if: the currency of the foreign company strengthens (loses value), implying that more (less) of the local currency will be needed to purchase one unit of the foreign currency.

Conversely, a company selling to international clients will suffer losses (profits) if the currency of the foreign company weakens (strengthens), implying that more (less) of the received revenue will be needed so as to convert it to the local currency.

Transaction risk is hedged using outright forward transactions and swaps.

In the case of hedging using outright transactions, consider the case of an American company investing in Kenyan Shillings. The company could hedge its position by buying KSH forward, which would hedge the exchange rate paid to Kenyan suppliers while selling USD forward would lock in the FX rate applied to the revenues of USD.

On the other hand, the FX swap is applied when the company owns a foreign company that owns foreign currency that will be used for purchases at a future time to earn interest in its domestic currency. The company would sell the foreign currency in exchange for its domestic currency in the spot market and repurchase it at a stated future time in the forward market.

This type of risk comes up when an institution’s assets and liabilities are in a foreign currency, which must be valued in the institution’s domestic currency when the financial statements are made. Thus, the institution can experience foreign exchange gains or losses.

As compared to transaction risk, translation risk does not affect the cash flows of a company.

A Canadian company has its investments in the US. At the end of the first year, the company netted USD 20 million, and the USD/CAD at that time was 1.3200. At the end of the second year, the company’s net value remained the same as that of the first year. However, the USD/CAD has changed to 1.2700. At the end of the two years, the company wishes to produce its 2-year financial statements. Intuitively, the company experienced a loss of:

$$ (1.32000-1.2700)×20,000,000=\text {CAD } 1,000,000 $$

Foreign gains or losses can also arise from borrowing in foreign currency. To illustrate this, let us look at an example.

Suppose that the Canadian company has a loan of USD 10 million that is supposed to be returned in 10 years (assume that it is paid at par). The interests are paid in USD, implying that the company is prone to transaction risk. Additionally, the company is exposed to translation risk, which is embedded in the repayment of the loan principal.

Given that the USD/CAD at the end of 1st year was 1.3100 and at the end of the 2nd year, USD/CAD had reduced to 1.2800, the value of the loan at the end of the 1st year is:

$$ 10,000,000×1.3100=\text{CAD } 13,100,000 $$

And at the end of the second year:

$$ 10,000,000×1.2800=\text{CAD } 12,800,000 $$

Intuitively, the company earns a foreign gain of

$$ 13,100,000-12,800,000=\text{CAD } 300,000 $$

The above results can be attributed to the fact that the USD has weakened over the CAD. On the other hand, had the USD strengthened, the company would have experienced a loss.

Translation risk is hedged using the forward contracts on the reporting date to decrease the volatility of profits. The forward contract involves a plan by the concerned firm to sell foreign currency assets or retire foreign currency liabilities at a later time.

Another method of hedging translation is by financing the assets in a country with the borrowed funds in that country. By doing this, the gains (losses) on assets are offset by the losses (gains) on the liabilities.

To see this, suppose that in our example above. Suppose the Canadian company has a USD 20 Million worth of investment. If the company analyzes its position and realizes that the translation is imminent, it could finance its investment with a USD 20 million loan.

It is worth noting the difference between transaction risk and translation risk. We can see that transaction risk has direct effects on the cashflows of a company which is not the case for translation risk. However, the effects of translation risk on the reported earnings of a company can be big.

Also, note that it is only reasonable that we hedge translation risks on one future date. Doing otherwise will be overhedging.

Economic risk is the risk that the future cash flows of a firm will be affected by the movements in exchange rates.

Economic risk arises from the exchange rate movements and thus is difficult to quantify. Consider a Canadian sales firm in Brazil. If the BRL (Brazilian real) weakens in value relative to the Canadian dollar, the Brazilian customers will see the firm’s products as expensive. This will decline the demand for the product or prompts the company’s management to reduce the CAD price of its products.

Moreover, economic risk alters with the competitive nature of a company. Consider a company in a given country, Kenya, which has no investment operations overseas. Exchange rate movements might be favorable for a foreign investor who sees Kenya as a desirable place for business. The foreigner’s increased business activities could negatively impact the domestic firm’s competitive position.

Compared with translation and transaction risks, economic risk is not easily quantified. However, possible exchange rate movements should be taken into consideration before essential strategic business decisions are made. For instance, a production firm might decide to move production overseas due to favorable exchange rate movements.

Multinational companies are exposed to many different currencies. Just like any other portfolio, multiple exposures to multiple currencies reduce the FX risk due to diversification. That is, volatility from multiple currency exposures is less than exposure to a single currency.

Companies often prefer options to forward contracts since the options provide downside protection against unfavorable exchange rate movement while allowing a firm to benefit from desirable movements. Therefore, hedging using options involves buying options on individual currencies to cover each adverse exchange rate movement.

Alternatively, a firm might buy an option on a portfolio of currencies to which it is exposed in the over-the-counter market. Such options are basket and Asian options.

Like any other financial asset, currency exchange rates cannot be determined with ultimate precision because they are influenced by supply and demand, which are also affected by other factors. Some of the factors affecting the exchange rates include:

- Balance of payments and trade flows
- Monetary policies
- Inflation

Recall that the balance of payments between the two countries is the difference between the value of exports and imports. We shall demonstrate the effect of balance of payments using an example:

Suppose that the exports from country X to country Y increases. If the exporters exchange the foreign currency (which is their revenues) to domestic currency, the demand for country X currency will increase, strengthening its currency relative to country Y’s currency. This causes exports to be more expensive in country Y.

On the other hand, if the imports from country X to Y increase, the currency of country X will weaken relative or that of country Y since the importers will be forced to buy country Y’s currency to pay for the goods they are importing. Consequently, the imported goods to country Y become more expensive, lowering the demand for imported goods.

The central bank’s monetary policy also influences the value of a country’s currency. With all other factors held equal, if Country X, say, raises its money supply by 15% while Country Y keeps its money supply at constant, the value of Country X’s currency would begin to fall by 15% compared to Country Y’s currency. This is due to the fact that 15% more of Country X’s currency is being used to buy the same quantity of goods.

Inflation gives rise to purchasing power parity; a relationship that allows for theoretical arbitrage opportunities, i.e., a trader can buy goods cheaply from a country with a lower inflation rate and sell them at a higher price in a different country with a higher inflation rate since inflation has negative effects on the exchange rates.

This condition reflects the link between the exchange rates and the difference in countries’ inflation rates. The laws of one price state the price of a foreign good x denoted as \({\text P}_{\text f}^{\text x}\) must be the equal price of the similar good in a domestic country, \({\text P}_{\text d}^{\text x}\), using the spot rate \({\text S}_{\frac {\text f}{\text d}}\) (We have used the (\({\frac {\text f}{\text d}}\)) notation for simplicity). Put mathematically,

$$ {\text P}_{\text f}^{\text x}={\text S}_{\frac {\text f}{\text d}}×{\text P}_{\text d}^{\text x} $$

For instance, a product in Canada costs CAD 100. The nominal exchange rate for USD/CAD is 0.76. So, the same product will cost 0.76×100 = USD 76 in the US.

The purchasing power parity amplifies the law of one price to include a broader range of goods and services and not just good x. The law of one price equation transforms into:

$$ {\text P}_{\text f}={\text S}_{\frac {\text f}{\text d}}×{\text P}_{\text d} $$

where:

\({\text P}_{\text f}\)-the price level of the foreign country.

\({\text P}_{\text d}\)-the price level of the domestic country

\({\text S}_{\frac {\text f}{\text d}}\)-nominal exchange rate

Making the \({\text S}_{\frac {\text f}{\text d}}\) the subject of the formula, we get:

$$ {\text S}_{\frac {\text f}{\text d}}=\cfrac {{\text P}_{\text f}}{{\text P}_{\text d}} $$

The inflation rate in the US is 3% per year and 1% per year in Canada. You are also given that the USD/CAD exchange rate is 1.0500. A basket of goods in the US costs USD 100. Assuming that the purchasing power parity holds, what is the new USD/CAD after one year.

The inflation rate in the US is 3% per year, implying that the price of a basket of goods increases by 3% each. This is analogous to Canada. So, after one year, the basket of goods in the US is:

$$ {\text P}_{\text d}=1.03×100=\text{USD } 103 $$

The same basket would cost the following in Canada:

$$ {\text P}_{\text f}=1.05×1.01×100=\text{CAD } 106.05 . $$

According to purchasing power parity,

$$ {\text S}_{\frac {\text f}{\text d}}=\cfrac {{\text P}_{\text f}}{{\text P}_{\text d}} =\cfrac {106.05}{103}=1.02961 $$

Therefore, the equilibrium in the exchange rates is determined by the ratio of the national price level of the two countries. However, if the transaction cost is largely coupled with the non-tradable nature of some goods, this condition might not hold.

Moreover, if the transaction costs and other trading difficulties are constant, the deviation of the exchange rate is entirely determined by the difference between the inflation rates of the foreign and domestic countries. Mathematically,

$$ \% \Delta {\text S}_{\frac {\text f}{\text d}} \approx \pi_{\text f}-\pi_{\text d} $$

Where:

\(\% \Delta {\text S}_{\frac {\text f}{\text d}}\)=change in the spot exchange rate

\(\pi_{\text f}\)=foreign inflation rate

\(\pi_{\text d}\)=domestic inflation rate

That is:

$$ \text{Percentage Strengthening of Domestic Spot Rate}=\text{Foreign Inflation Rate}- \text{ Domestic Inflation Rate.} $$

If we go back to our example, the domestic currency weakens by:

$$ \% \Delta {\text S}_{\frac {\text f}{\text d}} \approx \pi_{\text f}-\pi_{\text d}=1\%-3\%=-2\% $$

This implies the domestic spot rate weakened by 2%

To calculate the percentage change, one needs to have a clear understanding of the base currency and the quote currency. Let us take an example of the Chinese Yuan (CNY) and South African Rand (ZAR). Suppose that the exchange rate of ZAR/CNY increased from 1.6459 to 1.8356. Therefore, the percentage of appreciation/depreciation of the Chinese Yuan will be:

$$ \cfrac {1.6459}{1.8356} – 1 = -10.33\% $$

The Chinese Yuan depreciated by 10.33% because it used to take 1.6459 CNY to buy one ZAR, but now it has increased to 1.8356 CNY for one ZAR.

The depreciation of the Chinese Yuan against the South African Rand can also be expressed as an appreciation of the South African Rand against the Chinese Yuan. The appreciation percentage, in this case, will not be equal to the previous depreciation percentage of -10.33%.

To calculate the percentage appreciation of the South African Rand, we have to invert the exchange rate. To invert a currency exchange rate, we have to divide 1 by the exchange rate. For example, if

$$ \text {ZAR/CNY} = 1.6459 $$

Then,

$$ \text {CNY/ZAR} =\cfrac {1}{1.6459} = 0.6076 $$

Therefore, the appreciation percentage of South African Rand if the exchange rate ZAR/CNY increased from 1.6459 to 1.8356. We have to invert this exchange to CNY/ZAR so that the Chinese Yuan is now the base currency and the South African Rand is the quote currency. That is:

$$\frac{\left(\frac{1}{1.6459}\right)}{\left(\frac{1}{1.8356}\right)}-1=\cfrac {1.8356}{1.6459} – 1 = 11.53\%$$

This represents an 11.53 percent appreciation in the South African Rand against the Chinese Yuan because the CNY/ZAR (as a result of inversion) exchange rate is expressed with the Chinese Yuan as the base currency and the South African Rand as the quote currency. In other words, you now need fewer South African Rands to buy one Chinese Yuan.

Thus, we can see that the appreciation percentage of the South African Rand is different from the Chinese Yuan’s depreciation.

A forex trader noticed that the USD/EUR spot rate was 1.2960 and expected to be 1.2863 after one year. Similarly, the CHF/USD spot rate is 0.9587 and is expected to drop to 0.8885. Calculate the euro (EUR) expected appreciation/depreciation against the US dollar over the next year.

We know that we are dealing with USD/EUR quotes. So, we calculate as: $$ \cfrac {1.2960}{1.2863}-1=0.007541=0.7541\% $$

This was expected because clearly, there was a decrease in USD/EUR, indicating that EUR is appreciating.

Nominal interest rates are those rates that are listed in the market and show the return that will be earned on a currency. For instance, 5% per year for a given currency of a country implies that 100 units of a currency are anticipated to grow to 105 in one year.

Real interest rate is those rates that are adjusted to accommodate the effects of inflation. The real interest is given by:

$$ {\text r}_{\text{real}}=\cfrac {1+{\text r}_{\text{nominal}}}{1+{\text r}_{\text {inflation}}}-1 $$

where

\({\text r}_{\text{real}}\)=real interest rate

\({\text r}_{\text{nominal}}\)=nominal interest rate

\({\text r}_{\text {inflation}}\)=rate of inflation

The above equation is usually approximated as:

$$ {\text r}_{\text{real}} \approx {\text r}_{\text{nominal}}-{\text r}_{\text {inflation}} $$

For instance, if the nominal interest rate is 5% and the inflation rate is assumed to be 2%, then the real interest rate is approximated as:

$$ {\text r}_{\text{real}} \approx 5\%-2\%=3\% $$

Or

$$ {\text r}_{\text{real}} =\cfrac {1.05}{1.02}-1=0.02941 \approx 2.941\% $$

Note that this is pretty close to the approximation.

Note also that the real and nominal interest rates can both be negative.

This is a no-arbitrage condition which states that an investment in a foreign market that is entirely hedged against exchange rate risk should give the same return as a similar investment in a domestic market.

Consider an investor who wishes to start with 1 unit of domestic currency so that he ends up with an amount in foreign currency terms. To achieve this, there are two ways to accomplish this:

- The trader can invest in the funds in the risk-free foreign rate of interest \(({\text i}_{\text f})\) so that the funds grows to \((1+{\text i}_{\text f} )^{\text T}\) at time T. At time
*T*, the trader can enter into a forward contract to exchange \((1+{\text i}_{\text f} )^{\text T}\) for foreign currency at a foreign forward rate of exchange \({\text F}_{\frac {\text f}{\text d}}\) to get: $$ (1+{\text i}_{\text f} )^{\text T} {\text F}_{\frac {\text f}{\text d}} $$ - The trader immediately exchange the proceeds to USD and invest in risk-free domestic rate of interest to get \({\text S}_{\frac {\text f}{\text d}} (1+{\text i}_{\text d} )^{\text T} \)

Since we are assuming there is a no-arbitrage condition, then these two investments should give the same result. That is:

$$ (1+{\text i}_{\text f} )^{\text T} {\text F}_{\frac {\text f}{\text d}}={\text S}_{\frac {\text f}{\text d}} (1+{\text i}_{\text d} )^{\text T} $$

Rearranging we get:

$$ {\text F}_{\frac {\text f}{\text d}} ={\text S}_{\frac {\text f}{\text d}} \left( \cfrac { (1+{\text i}_{\text d} )^{\text T}}{ (1+{\text i}_{\text f} )^{\text T} } \right) $$

Where

\({\text i}_{\text d} \) =The interest rate in the domestic currency or the quoted currency

\({\text i}_{\text f} \)=interest rate in the foreign currency or the base currency

\({\text S}_{\frac {\text f}{\text d}}\)=current spot exchange rate

\({\text F}_{\frac {\text f}{\text d}}\)=forward foreign exchange rate

In other words, the forward exchange rate should give the same rate involving the spot exchange rate, domestic and foreign risk-free yields either in domestic market instruments or currency-hedged foreign market instruments within the same investment horizon.

For this condition to hold, it is assumed that:

- There is no transaction cost and
- The domestic and foreign market instruments should be identical in liquidity, time to maturity, and the default risk.

Note that we have to use the notation f/d, where f stands for the foreign currency and d for the domestic currency. So, f/d can be anything else such as CAD/USD, in which USD is the domestic currency.

Generally given the exchange rate XXXYYY or XXX/YYY then:

$$ {\text F}_{\frac {\text{XXX}}{\text{YYY}}}={\text S}_{\frac {\text{XXX}}{\text{YYY}}} \left(\cfrac { (1+{\text i}_{\text{YYY}} )^{\text T}}{(1+{\text i}_{\text {XXX}} )^{\text T} } \right) $$

Where the variables are defined as above.

The following are interest rates as listed in an interbank market:

$$ \begin{array}{c|c|c|c} \textbf{Currency} & \textbf{Libor (annualized)} & \textbf{Currency} & \textbf{Spot Rate} \\ {} & {} & \textbf{Combinations} & {} \\ \hline \text{USD} & {0.30\%} & \text{USD/EUR} & {1.6975} \\ \hline \text{EUR} & {5.00\%} & \text{JPY/EUR} & {0.0085} \\ \hline \text{JPY} & {0.30\%} & \text{JPY/USD} & {82.25} \\ \end{array} $$

A Japanese company investment manager wants to estimate all-in investment returns on a hedged EUR currency exposure if the covered interest parity holds.

In this question, we do not require any calculations because it is just a matter of intuition. If the covered interest rate parity holds, then the all-in investment for the Japanese company is the fully-hedged EUR Libor, which is equal to a one-year JPY Libor of 0.30%. So, the investment return is simply 0.30% since, according to covered interest rate parity, an investment in a foreign market that is entirely hedged against exchange rate risk should give the same return as a similar investment in a domestic market.

Assume that a US investor starts with 1 unit of USD to end up with Canadian dollars (CAD). So, the spot rate is quoted as CAD/USD. If the covered rate parity holds then:

$$ \begin{align*}{\text F}_{\frac {\text{CAD}}{\text{USD}}}&={\text S}_{\frac {\text{CAD}}{\text{USD}}} \left( \cfrac {\left(1+{\text i}_{\text{USD}} \right)^{\text T}}{ \left(1+{\text i}_{\text{CAD}} \right)^{\text T}} \right)\\ \end{align*} $$

If

$$ \begin{align*} {\text F}_{\frac {\text{CAD}}{\text{USD}}} &< {\text S}_{\frac {\text{CAD}}{\text{USD}}} \left( \cfrac {\left(1+{\text i}_{\text{USD}} \right)^{\text T}}{ \left(1+{\text i}_{\text{CAD}} \right)^{\text T}} \right) \\ \end{align*} $$

Then:

$$ \begin{align*}{\text F}_{\frac {\text{CAD}}{\text{USD}}} \left(1+{\text i}_{\text{CAD}} \right)^{\text T} &< {\text S}_{\frac {\text{CAD}}{\text{USD}}} \left(1+{\text i}_{\text{USD}} \right)^{\text T}\\ \Rightarrow \cfrac { {\text S}_{\frac {\text{CAD}}{\text{USD}}} \left(1+{\text i}_{\text{USD}} \right)^{\text T} }{ {\text F}_{\frac {\text{CAD}}{\text{USD}}} } &>\left(1+{\text i}_{\text{CAD}} \right)^{\text T}\\ \end{align*} $$

This implies that the investor has more CAD than required to pay the borrowed amount; hence he can make a riskless profit.

On the other hand, if

$$ \begin{align*}{\text F}_{\frac {\text{CAD}}{\text{USD}}}&>{\text S}_{\frac {\text{CAD}}{\text{USD}}} \left( \cfrac {\left(1+{\text i}_{\text{USD}} \right)^{\text T}}{ \left(1+{\text i}_{\text{CAD}} \right)^{\text T}} \right) \\ \end{align*} $$

Then:

$$ \begin{align*}{\text F}_{\frac {\text{CAD}}{\text{USD}}} \left(1+{\text i}_{\text{CAD}} \right)^{\text T} &> {\text S}_{\frac {\text{CAD}}{\text{USD}}} {\left(1+{\text i}_{\text{USD}} \right)^{\text T}} \\ \end{align*}$$

Therefore, the investor would have more of USD than the borrowed amount in USD and make a riskless profit.

Suppose that the risk-free rate of interest in USD and EUR are 3% and 6% per year, respectively. Given that the USD/EUR spot rate is 0.90. What is the 6-month USD/EUR forward rate?

Using the formula:

$$ \begin{align*} {\text F}_{\frac {\text{USD}}{\text{EUR}}} & ={\text S}_{\frac {\text{USD}}{\text{EUR}}} \left( \cfrac {\left(1+{\text i}_{\text{EUR}} \right)^{\text T}}{ \left(1+{\text i}_{\text{USD}} \right)^{\text T}} \right) \\ & =0.90× \cfrac {1.06^{0.5}}{1.03^{0.5}}=0.9130 \\ \end{align*} $$

Note that the spot rate (0.90) is less than the forward rate (0.9130), implying that USD is (theoretically) stronger than the EUR.

When T<1 $$ {\text F}_{\frac {\text f}{\text d}}={\text S}_{\frac {\text f}{\text d}} \left( \cfrac { (1+{\text i}_{\text d} )^{\text T}}{(1+{\text i}_{\text f} )^{\text T} } \right) $$

For simplicity let \({\text F}_{\frac {\text f}{\text d}}=\text F\) and \({\text S}_{\frac {\text f}{\text d}}=\text S\)

The above equation can be approximated as:

$$ \cfrac {\text F}{\text S}=\cfrac {1+{\text i}_{\text d} {\text T}}{1+{\text i}_{\text f} {\text T}} $$

If we subtract 1 from both sides,

$$ \cfrac {\text F}{\text S} -1 =\cfrac {1+{\text i}_{\text d} {\text T}}{1+{\text i}_{\text f} {\text T}} -1 $$

We get:

$$ \cfrac {{\text F} – {\text S} }{\text S}=\cfrac {1+{\text i}_{\text d} {\text T}-1-{\text i}_{\text f} {\text T}}{1+{\text i}_{\text f} {\text T}}=\cfrac {{\text i}_{\text d} {\text T}-{\text i}_{\text f} {\text T}}{1+{\text i}_{\text f} {\text T}} $$

So,

$$ \cfrac {{\text F} – {\text S} }{\text S}=\cfrac {{\text i}_{\text d} {\text T}-{\text i}_{\text f} {\text T}}{1+{\text i}_{\text f} {\text T}} $$

This can be approximated as:

$$ \cfrac {{\text F} – {\text S} }{\text S} \approx({{\text i}_{\text d} {\text T}-{\text i}_{\text f} {\text T}}){\text T} $$

In the last expression, F-S expressed as a percentage of the spot rate is equivalent to the number of points divided by 10,000 and it is an approximate value of the interest rate differential at time T.

Suppose that the risk-free rate of interest in USD and EUR are 3% and 6% per year, respectively. Given that the USD/EUR spot rate is 0.90. What is the 6-month USD/EUR forward rate expressed as points?

In the previous calculation, we had calculated the forward rate as 0.9130. So, the 6-month forward points are:

$$ (0.9130-0.9000)×10,000=130 $$

If we express in terms of percentage of spot rate, we have:

$$ \cfrac {0.9130-0.9000}{0.9000}=0.014444=1.444\% $$

While covered interest parity is concerned with forward rates and depends on arbitrage arguments, Uncovered interest parity is concerned with exchange rates themselves.

The uncovered interest parity condition postulates that the expected yield from a risky foreign investment must be equal to that of an equivalent domestic currency investment.

It states that the change in spot rate over the investment period should be averagely equal to the difference between the interest rates in two different countries, or simply, the expected appreciation or depreciation should approximately offset the difference in the interest rates.

While using the (f/d) notation (domestic (d) currency as the base currency), assume that an investor has a choice of venturing in one-year domestic market investment and a risky (unhedged) foreign market investment. The uncovered parity condition compels the investor to weigh between the **certain** return from domestic investment and expected return from the risky foreign investment (in terms of foreign currency).

The foreign investment return in domestic currency will be given by:

$$ (1+{\text i}_{\text f} )\left(1-\%\Delta {\text S}_{\frac {f}{d}} \right)-1 $$

This can also be represented as:

$$ \approx {\text i}_{\text f}-\%\Delta {\text S}_{\frac {f}{d}} $$

Also, the uncovered interest rate parity implies that the anticipated change in the spot rate over the investment period should show the difference between the foreign and domestic interest rates. This is mathematically represented as:

$$ \%\Delta {\text S}_{\frac {f}{d}}^{\text e}={\text i}_{\text f}-{\text i}_{\text d} $$

Where \(\Delta {\text S}^{\text e}\) is the future change in the spot rate.

Currencies A (domestic) and B (foreign) have risk-free rates of interest of 2% and 5% respectively. Assuming that the uncovered interest rate parity holds, what percentage would B weaken (strengthen) relative to A?

According to uncovered interest rate parity

$$ \%\Delta {\text S}_{\frac {f}{d}}^{\text e}={\text i}_{\text f}-{\text i}_{\text d}=5\%-2\%=3\% $$

So, we would expect currency B to weaken by 3% relative to the value of currency A.

Therefore, the assumption brought forward by the uncovered interest rate is that when a country has higher interest rates, its currency will depreciate, which offsets the high yields, bringing the return of the two investments to the same level.

## Question

The following are interest rates as listed in an interbank market:

$$ \begin{array}{c|c|c|c} \textbf{Currency} & \textbf{Libor (annualized)} & \textbf{Currency} & \textbf{Spot Rate} \\ {} & {} & \textbf{Combinations} & {} \\ \hline \text{USD} & {0.70\%} & \text{EUR/USD} & {1.8975} \\ \hline \text{EUR} & {7.00\%} & \text{EUR/USD} & {0.0075} \\ \hline \text{JPY} & {0.50\%} & \text{USD/JPY} & {82.25} \\ \end{array} $$

Assuming that the Uncovered Interest Parity holds, by how much is the JPY currency expected to change relative to USD over one year?

A. -0.2%

B. 0.2%

C. 0%

D. 0.4%

## Solution

The correct answer is

A.According to uncovered interest rate parity, the expected change in a spot exchange rate is equivalent to the difference between the interest rates corresponding to each currency (Libors). That is,

$$ \%\Delta {\text S}_{\frac {f}{d}}^{\text e}={\text i}_{\text f}-{\text i}_{\text d}=(0.5-0.7)\%=-0.2\% $$ Therefore, JPY currency has decreased by 0.2% relative to the USD currency.