Future fluctuations in exchange rates may result in potential losses, and possible gains, for the parties involved when the dates for the delivery and payment of goods or services do not coincide. The possibility of realising a financial loss as a result of such a transaction is called transaction exchange risk. For example, corporations, institutional investors, and individuals incur transaction exchange risk if they enter into a transaction in which they are required to pay or receive a specific amount of foreign currency at a particular date in the future. Since the future spot exchange rate cannot be known with certainty, and the exchange rate can move in an unfavourable direction, such a transaction could lead to a loss.

To describe the precise nature of the risks associated with these transactions we may want to consider the statistical properties of the exchange rate movements. In particular, the distribution of the exchange rate may provide a range of possible future values for the exchange rate and the likelihood that these will occur in the future will provide insight into the amount of foreign exchange risk that a party to the transaction may encounter. The general principle that we are going to apply is that although we do not know the exact value for future exchange rates, we can quantify the possible changes that may occur by looking at historical values. This would allow us to quantify the amount of risk we may encounter when conducting an international financial transaction.

# 1 Assessing Exchange Rate Uncertainty

The analysis of historical data can be used to derive important insights into what may have happened in the past. In addition, it may also be used as an important proxy for what may happen in the future. Figure 1 presents a histogram of monthly percentage changes in the exchange rate of the U.S. dollar per British pound (i.e. GBPUSD). The figure also includes an appropriate normal distribution curve, with the same mean and standard deviation as the data, which may be used for comparative purposes. Figure 1: Distribution of Monthly Changes in GBPUSD (1971/01 - 2017/07)

The data is measured at a monthly frequency, from January 1971 to July 2016, which provides 547 observations. To compute monthly percentage changes in the exchange rate we calculate

$s_t = \frac{(S_t - S_{t-1} ) }{S_{t-1}}$

where the spot exchange rate at time $$t$$ is denoted $$S_t$$. The horizontal axis describes the percentage changes historically observed for the GBPUSD rate, which range from about -12% to +14.5%. To create the histogram, we create ranges (bins) of equal width that are reflected by the bars in the diagram. The vertical axis represents the percentage frequency of occurrence of the rates of exchange rate change for each bin. The average (mean) monthly percentage change was -0.05% for this exchange rate. Since the distribution is bell shaped, observations near to the mean are more likely to occur. The standard deviation is a measure of the dispersion of possibilities around the mean. For the monthly percentage changes in the exchange rate within this sample, the standard deviation is 3.03%. Exchange rate changes within 1 standard deviation of the mean (between [-0.05% - 3.03%] = -3.08% and [-0.05% + 3.03%] = 2.98%) occur more frequently than changes further away from the mean.

The curved line that has been superimposed on the histogram in Figure 1 represents a normal probability distribution with the same mean and standard deviation as the historical data. If we think that the histogram is a useful guide for the future, we can translate it into a probability distribution for future exchange rate changes. Figure 1 suggests that the assumption of a normal distribution, characterised by its classic bell-shaped curve, is reasonable for the GBPUSD exchange rate. This feature of the data is reasonably common when we consider the behaviour of most major developed world currencies (for monthly rates of change).

However, many emerging market currencies exhibit probability distributions that are distinctively non-normal. An example is Figure 2, which shows the distribution for monthly percentage changes of the South African rand relative to the U.S. dollar. The most prominent feature of this diagram is the large left-hand tail. This indicates that large depreciations of the rand relative to the dollar have occurred with the absence of analogously large appreciations, which allows us to conclude that the historical distribution is asymmetric. In addition, the figure would suggest that this distribution has fat tails, as it displays leptokurtosis. Figure 2: Distribution of Monthly Changes in USDZAR (1980/01 - 2017/07)

Financial managers are also obviously interested in the probability distribution of future spot exchange rates. Given that we observe an exchange rate of $$S_t$$ today, we can find the conditional probability distribution of future exchange rates from the historical probability distribution of the percentage change in the exchange rate. Since the probability distribution of the future exchange rate depends on all the information available at time $$t$$, we say that it is a conditional probability distribution. Consequently, the mean, which is the expected value of this distribution, is also referred to as the conditional mean, or the conditional expectation of the future exchange rate.

In the case of the GBPUSD exchange rate, we could make use of a normal distribution to describe events up to period $$t$$. One nice feature of the normal distribution is that the probability of any range of possible future exchange rates is completely summarised by its mean and the standard deviation, which may be regarded as a measure of its volatility. If the expected mean value of the change in the exchange rate is $$\mu$$ and the expected standard deviation of the change in the exchange rate is $$\sigma$$, then we have:

$\text{Conditional mean} = \big[ S_t (1+ \mu) \big]$ $\text{Conditional standard deviation} = \big[ S_t \sigma \big]$

For example, if the current GBPUSD exchange rate is $1.50, and we expect that the pound will appreciate relative to the dollar by 2% over the next 90 days. The conditional expectation of the future spot rate in 90 days is then $$\big[1.50 \times (1 + 0.02)\big] = \1.53$$. If the standard deviation of the rate of appreciation over the next 90 days is expected to be 4%, the standard deviation of the conditional distribution of the expected future spot exchange rate is $$\big[1.50 \times 0.04 \big] = \0.06$$. With the conditional mean and conditional standard deviation of the future exchange rate, we can determine the probability that the future exchange rate will fall within any given range of exchange rates. For example, for the normal distribution, slightly more than two-thirds of the probability distribution is within the range of plus or minus 1 standard deviations of the mean. In our example, this range is from $$\big[$$ 1.53 $$\pm$$ 0.06 $$\big]$$, which spans$1.47 to $1.59. Given a probability distribution of future exchange rates, we can also determine the probability that the exchange rate in the future will be greater or less than a particular future spot rate. For example, suppose we want to know how likely it is that the pound will strengthen over the next 90 days to at least an exchange rate of$1.60. Since $1.60 is greater than the conditional mean of$1.53, by $0.07, and the standard deviation is$0.06, we want to know how likely it is that we will be [0.07/0.06] = 1.167 standard deviations above the mean. For the normal distribution, this probability is 12.16%. Hence, the probability of the exchange rate rising to $1.60 or higher, from a starting value of$1.50, is 12.16%.1

Exchange Rate $$\hspace{3cm}$$ Standard Deviation Forward Market Return
EURUSD 11.17 11.25
GBPUSD 10.57 10.70
USDJPY 11.66 11.81
EURJPY 11.34 11.42
EURGBP 9.25 9.35
GBPJPY 12.37 12.49

Table 1: Standard Deviation of Monthly Exchange Rate Changes and Forward Market Returns

In all of these calculations, you should take note of the importance of the standard deviation (or volatility), which may be used to quantify the degree of risk that is inherent in these transactions. These measures are usually associated with measures of expected return, which is the subject of a later discussion. Examples of reported measures for the standard deviations and the forward market return for various exchange rates are provided in Table 1.

# 2 Volatility Clustering

Most of the techniques that are used to measure the transaction exchange risk rely on a measure of volatility. Hence, an understanding of the evolution of volatility from the changes in the exchange rate is critical. The wider the conditional distribution of future exchange rates, the greater the probability that the exchange rate will take on extreme values. With this in mind, it is important to also consider whether or not volatility has increased (decreased) over time, where the use of a probability distribution that is based on a historical standard deviation may underestimate (or overestimate) the true uncertainty about future exchange rates.

Many financial researchers have spent considerable time examining exchange rate data and they have come to the conclusion that exchange rate volatility is certainly not constant over time. In fact, as is true for the returns on many other financial assets, percentage changes in exchange rates show a pattern known as volatility clustering. When volatility is high, it tends to remain high for a period of time, while periods of low volatility are similarly persistent. Hence, asset markets in general, and the foreign exchange market in particular, appear to go through periods of tranquillity and periods of turbulence. To illustrate this pattern, we use monthly data on the USDZAR exchange rate in Figure 3, which plots the monthly changes in the exchange rate along with the value of the series squared (which is a measure of volatility). Figure 3: Monthly Changes and Volatility in USDZAR

The graph clearly reveals relatively long periods of low volatility and a number of highly turbulent episodes. For example, there is a spike in volatility during the crises in Asia  and Russia . In addition, the South African currency crisis in  and the global financial crisis [2008-2009] have also given rise to periods of increased volatility. The most volatile period of arose in October 2008, when volatility in both equity and foreign exchange markets reached unprecedented heights during the crisis.

A number of models have been developed to fit the observed pattern of volatility clustering in such data. One of the most popular models that has been developed for this purpose is the Generalized Auto-Regressive Conditional Heteroskedasticity (GARCH) model that is described in Bollerslev (1986). This model for the conditional variance could be expressed as,

$\begin{eqnarray*} y_t &=& \mu + \sigma_t \varepsilon_t \\ \sigma^2_t &=& \alpha_0 + \sum^K_{k=1} \alpha_k (y_t - \mu)^2_{t-k} + \sum^J_{j=1} \beta_j \sigma^2_{t-j} \end{eqnarray*}$

where $$y_t$$ represents the exchange rate returns, and $$\mu$$ is used to capture the mean in this series. The error term is assumed to take the properties $$\varepsilon_t \sim \mathsf{i.i.d.} \mathcal{N}[0,1]$$, while the coefficients $$\alpha_0$$, $$\alpha_k$$, and $$\beta_j$$ are parameters that can be estimated from the data. The $$\beta_j$$ coefficients reflect the sensitivity of the current conditional variance to the past conditional variance, $$\alpha_k$$ reflects its sensitivity to current news, and $$\alpha_0$$ is the minimum variance we would predict even if the past volatility and news terms are zero. Depending on the frequency of the data and where it is assume that $$J=K=1$$, the $$\beta_1$$ coefficent is usually between 0.85 and 0.95, and $$\alpha_1$$ is much lower (for example, between 0.05 and 0.15).

This model may be used to describe the persistence in volatility. If the conditional variance is high today, it is likely to be high tomorrow (and similarly so for periods of low volatility). This persistence in $$\sigma^2_t$$ can generate the patterns of volatility clustering we see in the data. If we are in a period of low volatility, but the exchange rate suddenly and unexpectedly moves in either direction, volatility immediately shifts to a higher level for a while through the $$\alpha_k$$ terms. This shift will tend to persist because of the feedback that is incorporated through the $$\beta_j \sigma^2_{t-j}$$ terms. That is when $$\sigma^2_t$$ take on a large value, we would expect that $$\sigma^2_{t+1}$$ will be higher as well, since the $$\beta_j$$ terms are always expected to be positive.

# 3 Conclusion

Given the above statistical properties of most exchange rates, it could be suggested that this data would resemble most other financial data and should be treated as an asset price. Indeed Bekaert and Hodrick (2012) pose the question:

Is it reasonable to expect exchange rate forecasts to be characterized with large variability? We think the answer is yes because exchange rates are the relative prices of currencies and currencies are assets. Thus, exchange rates are asset prices, such as stock prices, and we should expect exchange rates to behave very much like other asset prices, such as stock prices, which are also very difficult to predict. If exchange rates were easy to predict, lots of easy money would be made betting that one currency would strengthen relative to another.

Hence, as we do with other financial assets, making use of probability distributions allows us to quantify the probability of realising various future exchange rate values. This facilitates the procedure for quantify transaction exchange risk, after considering the properties of historical data. Of course, this is not a perfect science as the measurement of risk that we have used relies on the estimate for the standard deviation, which may vary over time.

1. You can either make use of statistical tables, the R function pnorm(z,0,1), or the Excel function NORM.S.DIST(z,TRUE). In this case the value for $$z$$ is -1.167, since we are looking into the tail of the distribution.↩︎