Mark to Market Value, and Parity Conditions

Economics 14 min readModule 2 of 3Save

What you'll learn

LOS 5.d
Calculate the mark to market value of a forward contract.
LOS 5.e
Explain international parity conditions (covered and uncovered interest rate parity, forward rate parity, purchasing power parity, and the international Fisher effect).
LOS 5.f
Describe relations among the international parity conditions.
LOS 5.g
Evaluate the use of the current spot rate, the forward rate, purchasing power parity, and uncovered interest parity to forecast future spot exchange rates.
LOS 5.h
Explain approaches to assessing the long run fair value of an exchange rate.

If the forward contract price is consistent with covered interest rate parity (discussed later), the value of the contract at initiation is zero to both parties. After initiation, the value of the forward contract will change as forward quotes for the currency pair change in the market.

Mark to Market Value

The value of a forward currency contract prior to expiration is also known as the mark to market value. To compute the value of a forward contract prior to expiration, we take the difference between the forward price we locked in and the current forward price, multiply that by the size of the contract, and then discount for the time period remaining until the contract settlement date.

V_{t} = \frac{( F P _{t} - F P ) \times contract size}{1 + R ( \frac{days}{360} )}

Where:

Symbol	Meaning
$V_{t}$	Value of the forward contract at time $t$ to the party buying the base currency, with $t < T$ , denominated in the price currency
$F P_{t}$	Forward price to sell foreign currency at time $t$ in the market for a new contract maturing at time $T$
$F P$	Forward price specified in the contract at inception, to buy the base currency
days	Number of days remaining to maturity of the forward contract, $T - t$
$R$	Interest rate of the price currency

Worked example: Valuing a forward contract prior to maturity

Yew Mun Yip has entered into a 90 day forward contract long CAD 1 million against AUD at a forward rate of 1.05358 AUD/CAD. Thirty days after initiation, the following AUD/CAD quotes are available:

Maturity	FX rate
Spot	1.0612 / 1.0614
30 days	+4.9 / +5.2
60 days	+8.6 / +9.0
90 days	+14.6 / +16.8
180 days	+42.5 / +48.3

The following information is available (at t = 30) for AUD interest rates:

Maturity	AUD interest rate
30 days	1.12%
60 days	1.16%
90 days	1.20%

What is the mark to market value in AUD of Yip’s forward contract?

Solution

Yip’s contract calls for long CAD (i.e., converting AUD to CAD). To value the contract, we would look to unwind the position. To unwind the position, Yip can take an offsetting position in a new forward contract with the same maturity.

Hence, Yip would be selling CAD in exchange for AUD and, hence, going up the bid (i.e., use the bid price). Note that after 30 days, 60 more days remain in the original contract.

The forward bid price for a new contract expiring in T − t = 60 days is 1.0612 + 8.6/10,000 = 1.06206.

The interest rate to use for discounting the value is also the 60 day AUD interest rate of 1.16%:

V_{t} = \frac{( F P _{t} - F P ) \times contract size}{1 + R ( \frac{days}{360} )} = \frac{( 1.06206 - 1.05358 ) ( 1 , 000 , 000 )}{1 + 0.0116 ( \frac{60}{360} )} = 8, 463.64

Thirty days into the forward contract, Yip’s position has gained (positive value) AUD 8,463.64. This is because Yip’s position is long CAD, which has appreciated relative to AUD since inception of the contract. Yip can close out the contract on that day and receive AUD 8,463.64.

Note: Be sure to use the AUD (price currency) interest rate.

Covered Interest Rate Parity

The word covered in the context of covered interest parity means bound by arbitrage. Covered interest rate parity holds when any forward premium or discount exactly offsets differences in interest rates, so that an investor would earn the same return investing in either currency. If euro interest rates are higher than dollar interest rates, the forward discount on the euro relative to the dollar will just offset the higher euro interest rate.

Formally, covered interest rate parity requires that (given A/B quote structure):

F = \frac{1 + R _{A} ( \frac{days}{360} )}{1 + R _{B} ( \frac{days}{360} )} S_{0}

Symbol	Meaning
$F$	Forward rate, quoted as A/B
$S_{0}$	Spot rate, quoted as A/B
days	Number of days in the underlying forward contract
$R_{A}$	Interest rate for Currency A
$R_{B}$	Interest rate for Currency B

Recall that:

forward premium (discount) = F - S_{0} = \frac{1 + R _{A} ( \frac{days}{360} )}{1 + R _{B} ( \frac{days}{360} )} - 1 S_{0}

F - S_{0} = S_{0} [\frac{days /360}{1 + R _{B} ( days /360 )}] (R_{A} - R_{B})

Worked example: Covered interest arbitrage

The U.S. dollar MRR is 8%, and the euro MRR is 6%. The spot exchange rate is $1.30 per euro (USD/EUR), and the 1 year forward rate is $1.35 per euro. Determine whether a profitable arbitrage opportunity exists, and illustrate such an arbitrage if it does.

Solution

First, we note that the forward value of the euro is too high. Interest rate parity would require a forward rate of:

1.30 (\frac{1.08}{1.06}) = 1.3245 USD/EUR

Because the market forward rate of $1.35 is higher than that implied by interest rate parity, we should sell euros in the forward market and do the opposite (i.e., buy euros) in the spot market. The steps in the covered interest arbitrage are as follows.

Initially:

Borrow USD 1,000 at 8%.
Purchase $1, 000/1.30 = 769.23$ euros in the spot market.
Invest the euros at 6%.
Enter into a forward contract to sell the expected proceeds at the end of one year, $769.23 \times 1.06 = 815.38$ euros, at USD 1.35 each.

After one year:

Sell the 815.38 euros under the terms of the forward contract at USD 1.35 to get USD 1,100.76.
Repay the USD 1,000 loan at 8%, which requires USD 1,080.
Keep the difference of USD 20.76 as an arbitrage profit.

Uncovered Interest Rate Parity

With covered interest rate parity, arbitrage will force the forward contract exchange rate to a level consistent with the difference between the two country’s nominal interest rates. If forward currency contracts are not available, or if capital flows are restricted so as to prevent arbitrage, the relationship need not hold. Uncovered interest rate parity refers to such a situation; uncovered in this context means not bound by arbitrage.

Consider Country A where the interest rate is 4%, and Country B where the interest rate is 9%. Under uncovered interest rate parity, currency B is expected to depreciate by 5% annually relative to currency A, so that an investor should be indifferent between investing in Country A or B.

Given a quote structure of A/B, the base currency (i.e., currency B) is expected to appreciate by approximately $R_{A}$ − $R_{B}$ . (When $R_{A}$ − $R_{B}$ is negative, currency B is expected to depreciate). Mathematically:

E (%Δ S)_{A / B} = R_{A} - R_{B}

The following example illustrates the use of uncovered interest rate parity to forecast future spot exchange rates using market interest rates.

Worked example: Forecasting spot rates with uncovered interest rate parity

Suppose the spot exchange rate quote is ZAR/EUR = 8.385. The 1 year nominal rate in the eurozone is 10% and the 1 year nominal rate in South Africa is 8%. Calculate the expected percentage change in the exchange rate over the coming year using uncovered interest rate parity.

Solution

The rand interest rate is less than the euro interest rate, so uncovered interest rate parity predicts that the value of the rand will rise (it will take fewer rand to buy one euro) because of higher interest rates in the eurozone. The euro (the base currency) is expected to “appreciate” by approximately $R_{ZAR}$ − $R_{EUR}$ = 8% − 10% = −2%. (Note the negative 2% value.) Thus the euro is expected to depreciate by 2% relative to the rand, leading to a change in exchange rate from 8.385 ZAR/EUR to 8.217 ZAR/EUR over the coming year.

Comparing covered and uncovered interest parity, we see that covered interest rate parity derives the no arbitrage forward rate, while uncovered interest rate parity derives the expected future spot rate (which is not market traded). Covered interest parity is assumed by arbitrage, but this is not the case for uncovered interest rate parity.

Under uncovered interest rate parity, if the foreign interest rate is higher by 2%, the foreign currency is expected to depreciate by 2%, so the investor should be indifferent between investing in the foreign currency or in their own domestic currency. An investor that chooses to invest in the foreign currency without any additional return (the interest rate differential is offset by currency value changes) is not demanding a risk premium for the foreign currency risk. Hence, uncovered interest rate parity assumes that the investor is risk neutral.

If the forward rate is equal to the expected future spot rate, we say that the forward rate is an unbiased predictor of the future spot rate. In such an instance, $F = E (S_{1})$ ; this is called forward rate parity. In this special case, if covered interest parity holds (and it will; by arbitrage) uncovered interest parity would also hold (and vice versa).

Stated differently, if uncovered interest rate parity holds, forward rate parity also holds (i.e., the forward rate is an unbiased predictor of the future spot rate).

There is no reason that uncovered interest rate parity must hold in the short run, and indeed it typically does not. There is evidence that it does generally hold in the long run, so longer term expected future spot rates based on uncovered interest rate parity are often used as forecasts of future exchange rates.

(Domestic) Fisher Relation

Professor Irving Fisher originated the idea that the nominal rate of return is (approximately) the sum of the real rate and the expected rate of inflation.

We can write this relation (known as the Fisher relation) as:

R_{nominal} = R_{real} + E (inflation)

International Fisher Relation

Under real interest rate parity, real interest rates are assumed to converge across different markets. Taking the Fisher relation and real interest rate parity together gives us the international Fisher effect:

R_{nominal, A} - R_{nominal, B} = E (inflation_{A}) - E (inflation_{B})

This tells us that the difference between two countries’ nominal interest rates should be equal to the difference between their expected inflation rates.

The argument for the equality of real interest rates across countries is based on the idea that with free capital flows, funds will move to the country with a higher real rate until real rates are equalized.

Worked example: Calculating the real interest rate

Suppose the nominal South African interest rate is 9.0% and the expected inflation rate is 3.5%. Calculate the real interest rate.

Solution

0.090 = r_{ZAR}^{real} + 0.035

r_{ZAR}^{real} = 0.090 - 0.035 = 0.055 = 5.5%

If we move to a 2 country scenario, we will now have two nominal interest rates and two expected inflation rates. If the real rates for both countries are assumed to be equal, they drop out of the equation, and we are left with the international Fisher relation, as shown in the following example.

Worked example: Using the international Fisher relation

Suppose that the eurozone expected annual inflation rate is 9.0%, and that the expected South African inflation rate is 13.0%. The nominal interest rate is 10.09% in the eurozone. Use the international Fisher relation to estimate the nominal interest rate in South Africa.

Solution

r_{ZAR}^{real} = r_{EUR}^{real} \approx 10.09% - 9.0% = 1.09%

The real rate in each country equals the nominal interest rate in the eurozone less the eurozone expected annual inflation rate.

R_{ZAR} = expected South African inflation + r_{ZAR}^{real} = 13% + 1.09% = 14.09%

Purchasing Power Parity

The law of one price states that identical goods should have the same price in all locations. For instance, a pair of designer jeans should cost the same in Paris as they do in New York, after adjusting for the exchange rate. The potential for arbitrage is the basis for the law of one price: if designer jeans cost less in New York than they do in Paris, an enterprising individual will buy designer jeans in New York and sell them in Paris, until this action causes the price differential to disappear.

Note, however, that the law of one price does not hold in practice, due to the effects of frictions such as tariffs and transportation costs.

Instead of focusing on individual products, absolute purchasing power parity (absolute PPP) compares the average price of a representative basket of consumption goods between countries using an index such as The United States Consumer Price Index (CPI). Absolute PPP requires only that the law of one price be correct on average, that is, for like baskets of goods in each country.

S (A / B) = \frac{CPI ( A )}{CPI ( B )}

In practice, even if the law of one price held for every good in two economies, absolute PPP might not hold because the weights (consumption patterns) of the various goods in the two economies may not be the same (e.g., people eat more potatoes in Russia and more rice in Japan (日本)).

Relative Purchasing Power Parity

Relative purchasing power parity (relative PPP) states that changes in exchange rates should exactly offset the price effects of any inflation differential between two countries. Simply put, if (over a 1 year period) Country A has a 6% inflation rate and Country B has a 4% inflation rate, then Country A’s currency should depreciate by approximately 2% relative to Country B’s currency over the period.

The equation for relative PPP is as follows:

%Δ S_{A / B} = Inflation_{A} - Inflation_{B}

Where $%Δ S_{A / B}$ is the change in the spot price quoted as A/B.

Relative PPP is based on the idea that even if absolute PPP does not hold, there may still be a relationship between changes in the exchange rate and differences between the inflation rates of the two countries.

Ex Ante Version of PPP

The ex ante version of purchasing power parity is the same as relative purchasing power parity except that it uses expected inflation instead of actual inflation.

The following example illustrates the use of the ex ante version of the PPP relation.

Worked example: Calculating the exchange rate predicted by the ex ante version of PPP

The current spot rate is USD/AUD = 1.00. You expect the annualized Australian inflation rate to be 5%, and the annualized U.S. inflation rate to be 2%. According to the ex ante version of PPP, what is the expected change in the spot rate over the coming year?

Solution

Since the AUD has the higher expected inflation rate, we expect that the AUD will depreciate relative to the USD. To keep the cost of goods and services the same across borders, countries with higher rates of inflation should see their currencies depreciate. The expected change in the spot rate over the coming year is inflation(USD) − inflation(AUD) = 2% − 5% = −3%.

This predicts a new USD/AUD exchange rate of approximately 0.97 USD/AUD.

Because there is no true arbitrage available to force relative PPP to hold, violations of relative PPP in the short run are common. However, because the evidence suggests that the relative form of PPP holds approximately in the long run, it remains a useful method for estimating the relationship between exchange rates and inflation rates.

It is useful to establish how all the parity relations described earlier fit together. Figure 5.1 shows the interrelationships among parity conditions. Though these relationships are not all exact, together they provide an extremely useful framework for thinking about exchange rates.

Figure 5.1: The International Parity Relationships Combined

Several observations can be made from the relationships among the various parity conditions:

Covered interest parity holds by arbitrage. If forward rate parity holds, uncovered interest rate parity also holds (and vice versa).
Interest rate differentials should mirror inflation differentials. This holds true if the international Fisher relation holds. If that is true, we can also use inflation differentials to forecast future exchange rates, which is the premise of the ex ante version of PPP.
If the ex ante version of relative PPP as well as the international Fisher relation both hold, uncovered interest rate parity will also hold.

We can use ex ante PPP, uncovered interest rate parity, or forward rates to forecast future spot rates. As stated earlier, uncovered interest rate parity and PPP are not bound by arbitrage and seldom work over the short and medium terms. Similarly, the forward rate is not an unbiased predictor of future spot rate.

However, PPP holds over reasonably long time horizons. If relative PPP holds at any point in time, the real exchange rate (i.e., the exchange rate adjusted for relative historical inflation between the currency pair) would be constant. However, since relative PPP seldom holds over the short term, the real exchange rate fluctuates around its mean reverting equilibrium value.

The international Fisher effect (and real rate parity) assumes that there are no differences between sovereign risk premia (i.e., all countries are perceived to be equally risky by investors). This is obviously untrue as investors do demand a higher real rate of return (i.e., a risk premium) for investing in emerging market currencies that are perceived to be riskier.

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