International Short-Term Financing and Investment 251
(1 + ey ) > (1 + L ) 291
(1+f ) 300
(1+f ) 302
International Long-Term Financing, Capital Structure and the Cost of Capital 284
D = Z D, 283
E = Z 283
k = Z 289
International Long-Term Portfolio Investment 296
Rj,t = b 0,j + b 1, jR m,t + b 2,j$t + e j,t (11.40) 311
dL 319
dm 1 -1'—-1'-2'"2 --v-^"3 '-3 (1149) 319
+ 2ct(S 1, S 4)w 4 +1 1 = 0 319
= 2w3 s 2 (S3 ) + 2s(S 1 , S3 )w 1 + 2s(S2 , S3 )w2 320
+ 2s(S 3 , S4 )w 4 +2 1 = 0 = 2w4 s 2 (S4 ) + 2s(S 1 , S4 )w 1 + 2s(S 2 , S4 )w2 + 2s(S 3 , S4 )w 4 + 2 2 = 0 320
= w 1 + W2 + w 3 -1 = 0 320
w' = [w 1 w 2 w 3 w 4 2 1 2 2] and 331
Foreign Direct Investment 344
i 341
References 341
i 361
Index 363
\n-1 f=l
Figure 4.1 shows the probability distribution or (in the case of historical data) the frequency distribution of the percentage change in the exchange rate under high and low foreign exchange risk. In the case of high foreign exchange risk, the percentage change in the exchange rate takes a wider range of values (extending between C and D) than in the case of low foreign exchange risk (extending between A and B). The total rate of return or the
Probability
or frequency
FIGURE
4.1
The probability (frequency) distribution of the percentage change
in the exchange rate.
TABLE
4.1
Basic
statistics
of quarterly percentage changes (1990-2000).
Exchange
rate
Minimum
Maximum
Mean
Standard
deviation
AUD/USD
-5.24
10.74
0.94
3.96
CAD/USD
-3.18
4.05
0.70
1.85
CHF/USD
-11.94
11.53
0.85
5.82
DKK/USD
-8.64
11.75
1.05
4.95
GBP/USD
-7.10
19.05
0.79
4.68
JPY/USD
-12.54
18.89
-0.26
6.02
NOK/USD
-8.49
15.22
1.19
4.72
NZD/USD
-5.38
12.82
0.90
4.09
SEK/USD
-7.35
28.59
1.55
6.57
value of the asset will have a corresponding probability or frequency distribution. It should be observed that the mean or the expected value (the value on the vertical axis corresponding to the highest probability or frequency) is equal for both distributions. Table 4.1 displays the basic statistics, including the means and standard deviations, of the quarterly changes of the US dollar exchange rates of nine currencies over the period 1990-2000. It can be seen that the Canadian dollar is the most stable currency against the US dollar, whereas the yen is the most volatile (these exchange rates have the lowest and highest standard deviations respectively).
The standard deviation as a measure of risk has been criticised for the arbitrary manner in which deviations from the mean are squared and for treating positive and negative deviations in a similar manner, although negative deviations are naturally more detrimental. To meet these criticisms, one may use the mean absolute deviation (MAD) or the downside semi-variance (DSV), which are respectively given by
1 n _
MAD(S) = -Y St - S (4.10)
n t=1
1 n
DSV (S) = — Y X2 (4.11)
n-1 t= 1 t
where Xt = St - S if St < S and Xt = 0 otherwise.
A more general measure of dispersion is given by
rq ...
D = j_¥ (q - S)a f( S)dS (4.12)
where the parameter a describes the attitude towards risk and q specifies the cut-off between the downside and the upside that the decision maker is and is not concerned about respectively. Many risk measures (including the downside semi-variance) are special cases of, or closely related to, this measure.
4.2 VALUE AT RISK
Value at risk (VAR) is a new approach to risk management that has been accepted by practitioners and regulators as the "right" way to measure risk. For example, the Bank for International Settlements (BIS) has allowed banks to use their own models of VAR to set the capital requirements for market risk. This measure of risk focuses on the tail of the distribution of the rate of return (or the profit/loss or the percentage change in the exchange rate), which means that the emphasis is placed on the worst given percentages of outcomes. In what follows, a brief (and perhaps superficial) treatment of VAR is presented. For more details and extensions, the reader is referred to KPMG- Risk (1997) and Dowd (2002).
Essentially, this approach is used to answer the question "over a given period of time with a given probability, how much money might be lost?". The money lost pertains to the decline in the value of a portfolio, which may consist of a single asset or a large number of assets. The measurement of VAR requires the choice of: (i) a measurement unit, normally the base currency; (ii) a time horizon, which could be a day, a week or longer provided that the composition of the portfolio does not change during this period; and (iii) a probability, which normally ranges between 1% and 5%. Hence VAR is the maximum expected loss over a given holding period at a given level of confidence (that is, with a given probability).
VAR as a measure of risk is illustrated in Figure 4.2 for a position in currency y when the base currency is x. This figure shows a probability distribution (or frequency distribution) for the percentage change in the exchange rate, S, over a chosen holding period. At a particular confidence level (say, c per cent), VAR is calculated by identifying the point on the x-axis that cuts off the top c per cent of observations from the bottom 1 - c per cent tail, which is -a. The maximum possible loss would, therefore, be obtained by applying this VAR factor to the size of the position (VR = aK, where K is the size of the position). If the distribution is that of the size of profit/loss, then VAR can be read directly from the distribution (it would be a).
Implementation of VAR analysis
There are at least three approaches to the implementation of VAR analysis, all of which involve the estimation of the statistical distribution of asset returns: these are the parametric (or analytical) approach, the historical approach and the simulation approach. These approaches will be discussed briefly here. For more details, see Dowd (1998, 2002), Hendricks (1996) and KPMG-Risk (1997).
The main assumption of the parametric VAR is that the distribution of asset returns is normal. If the percentage change in the exchange rate is normally distributed, then 95% of the observations will fall within 1.96 standard deviations of the mean and 98% will fall within 2.33 standard deviations of the mean. By taking the latter figure, this means that 98% of all observations fall
Probability
or frequency
FIGURE
4.2
Value at risk.
between S - 2.33ct(S) and S + 2.33ct(S). Thus, VAR with a probability of 1% (a confidence level of 99%) is equal to -K[(S -2.33s(S)] or 2.33ct(S)K if S = 0, where K is the size of the position. Table 4.2 shows the quarterly VARs, with probabilities of 1% and 2.5%, on a position of USD1,000,000 from the perspectives of investors with nine different base currencies. It can be seen that the lowest VAR is found when the base currency is the Canadian dollar, because the CAD/USD rate is the most stable rate, and the highest when the base currency is the Japanese yen.
One problem with the parametric approach is the assumption of normally distributed returns. It has for a long time been established that this is not the case and that the distributions of asset returns have fat tails and tend to be skewed to the left (for example, Mandelbrot (1963) and Fama (1965)). Because of this deviation from the normal distribution, the historical method may be preferred. In this case, VAR (with a probability of 1%) can be calculated by identifying the lowest 1% of the percentage change in the exchange rate, and then applying this value to the size of the position.
The third approach is the simulation approach. Instead of calculating VAR on the basis of the historical rates of change in the exchange rate or by assuming that they are normally distributed, this approach is based on their simulated values.
Pros and cons of the VAR methodology
Value at risk has become a widely used method for measuring financial risk, and justifiably so. The attractiveness of the concept lies in its simplicity, as it represents the market risk of the entire portfolio by one number that is easy to comprehend. It thus conveys a simple message on the risk borne by a firm or
TABLE 4.2 Value at risk on a USD1,000,000 position.
Base currency |
2.5% VAR |
1% VAR |
AUD |
82,868 |
68,216 |
CAD |
36,105 |
29,260 |
CHF |
127,106 |
105,572 |
DKK |
104,835 |
86,520 |
GBP |
101,144 |
83,828 |
JPY |
142,866 |
120,592 |
NOK |
98,076 |
80,612 |
NZD |
86,297 |
71,164 |
SEK |
1 37,581 |
113,272 |
an individual. The concept is also suitable for risk limit setting and for measuring performance based on the correlation between the return earned and the risk assumed. Moreover, it can take account of complex movements such as non-parallel yield curve shifts. In general, it has two important characteristics: (i) it provides a common consistent measure of risk across different positions and risk factors; and (ii) it takes into account the correlations between different risk factors (for example, different currencies).
There are, however, several shortcomings associated with the VAR methodology. First, it can be misleading to the extent of giving rise to unwarranted complacency. And, as we have seen, the VAR is highly sensitive to the assumptions used to calculate it. Jorion (1996) argues that VAR is a number that itself is measured with some error or estimation risk. Thus, the VAR results must be interpreted with reference to the underlying statistical methodology. Moreover, this approach to risk management cannot cope with sudden and sharp changes in market conditions. It neglects the possibility of discrete, large jumps in financial prices (such as exchange rates), which occur quite often. Losses resulting from catastrophic occurrences are overlooked due to dependence on symmetric statistical measures that treat upside and downside risk in a similar manner.
VAR is useful, but it should be handled with care and should be used in conjunction with other measures of risk. For example, it can be complemented by a series of stress tests that account for extremely unfavourable market conditions. It is imperative, however, that VAR should not be viewed as a strict upper bound on the portfolio losses that can occur.
Expected tail loss
The expected tail loss (ETL) is a measure of risk that is also known as expected shortfall, conditional VAR, tail conditional expectation, and worst conditional expectation. The concept is very simple: ETL is the expected value of a loss that is in excess of VAR. It is defined formally as
ETL = E(L | L > VAR) (4.13)
While the VAR tells us the most that can be expected to be lost if a bad event does occur, the ETL tells us what we can expect to lose if a bad event does occur.
Kritzman and Rich (2002) argue that viewing risk in terms of the probability of a given loss or the amount that can be lost with a given probability at the end of the investment horizon is wrong. This view of risk, according to them, considers only the final result, arguing that investors should perceive risk differently because they are affected by risk and exposed to loss throughout the investment period. They suggest that investors consider risk and the possibility of loss throughout the investment horizon; otherwise, their wealth may not survive to the end of the investment horizon. As a result of this way of thinking, Kritzman and Rich suggest two new measures of risk: within- horizon probability of loss and continuous VAR. These new risk measures are then used to demonstrate that the possibility of making loss is substantially greater than what investors normally assume.
Is VAR used in practice?
VAR is widely used by major companies in real life. Microsoft, for example, uses VAR as a management tool to estimate its exposure to market risk, reporting the estimated VAR figures in its annual reports. For the purpose of calculating VAR, Microsoft uses a time horizon of 20 days, which is longer than what is typically used by banks. Another difference is that Microsoft uses a confidence level of 97.5% rather than 99%, which is what is used by banks.
In 1999, Moosa and Knight (2002) conducted a survey of the practices of Australian public shareholding companies with respect to the use of value at risk analysis. The results of the survey reveal significant unfamiliarity with VAR analysis, as half of the respondents indicated that they were unaware of the existence of this technique. Financial institutions, those involved in international operations and those using derivatives tend to be more familiar with the technique. Moreover, not all of the companies with VAR awareness actually use the technique for measuring risk. The results of the survey produced several findings, including the following: (i) companies that do not use VAR mostly employ scenario analysis; (ii) those not using VAR claim that it is not relevant to their operations; (iii) those using VAR predominantly employ the parametric approach; and (iv) the majority of users employ back testing and stress testing. The results also revealed that financial institutions, companies involved in international operations and those using derivatives are more aware of the existence of VAR analysis and more inclined to use it, and that companies not using VAR but intending to use it are predominantly those involved in international operations and those using derivatives.
According to the Reserve Bank of Australia (2000), the VAR of major Australian banks was 0.02% of their capital base, much lower than the corresponding
4.3 MEASUREMENT OF EXPOSURE TO FOREIGN EXCHANGE RISK 73 TABLE 4.3 VAR of major banks.
Country Foreign
exchange risk (% of Total market risk (% of
capital) capital)
Australia
0.02
0.08
Canada
0.02
0.12
Germany
0.15
0.71
Japan
0.07
0.31
Netherlands
0.02
0.20
UK
0.03
0.16
USA
0.03
0.15
Source:
Reserve Bank of Australia (2000)
figures in other countries. In Germany, for example, the corresponding figure was 0.15%. The 1999 figures are shown in Table 4.3.
4.3 DEFINITION AND MEASUREMENT OF EXPOSURE TO FOREIGN EXCHANGE RISK
Foreign exchange exposure is a measure of the sensitivity of the base currency values of assets, liabilities and cash flows to changes in the exchange rate. For example, we consider asset exposure and operating exposure when respectively assets and operating cash flows are sensitive to changes in exchange rates. Also, we refer to a long exposure to assets and cash inflows, and to a short exposure to liabilities and cash outflows. Exposure may also refer to what is at risk (the foreign currency amount exposed). Whether exposure refers to what is at risk or the sensitivity of the base currency value depends on whether changes in this value and in the exchange rate are measured in absolute or percentage terms.
Let us go back to equation (4.1), which tells us that for each Vy there is a corresponding Vx, given the value of the exchange rate, S. So, as S rises (currency y appreciates), Vx will also rise. However, the underlying assumption here is that changes in the exchange rate do not affect Vy, and this is not necessarily true. Changes in the exchange rate tend to affect Vy, and this effect may be conspicuous or less so. For example, if the underlying asset is a bond, then Vy may be affected by changes in the exchange rate if these changes affect interest rates (via central bank intervention, for example). The same would be true if the underlying asset is a share, as it is plausible to imagine that changes in the exchange rate affect the stock market. In any case, since Vx depends on the exchange rate, exposure is a measure of the sensitivity of Vx with respect to the exchange rate. Hence we may define exposure as
E = (4.14)
dS
and in discrete form as
E = (4.15)
AS
in which case the exposure is measured in terms of currency y (hence, the amount exposed, or at risk). Alternatively, the expression can be written in terms of the (percentage) rates of change, which gives
E = Vr (4.16)
It is obvious from equation (4.16) that exposure in this case is measured without units, more or less like an elasticity. But in all cases, the exposure is the slope of the line (or curve) representing the relationship between Vx and S. Notice, in general, that
Vy = /1(S) (4.17)
and
Vx= f2(S) (4.18)
Therefore
Vx = SVy = Sf1(S) (4.19)
If, for example, Vx = a + ES + e, then the exposure coefficient, E, is calculated as
E = S(Vx , S) (4.20)
s2 (S) ' '
such that
s2 (Vx ) = E2 s2(S) + s2(e) (4.21)
where s 2 ( ) and s( , ) are the variance and covariance respectively, and s 2 (e) captures the residual variability that is independent of exchange rate movements. In what follows, we consider some possibilities for f1(S) andf2(S).
The effect of the exchange rate on the /-denominated value
We will consider four cases involving linear relationships between Vy and S, some of which produce nonlinear relationships between Vx and S. The list of possibilities presented here is not exhaustive. Because the relationship between Vy and S can take several shapes and forms, foreign exchange exposure can be zero, constant or variable. We will now consider the four cases by looking at an asset (that is, a long exposure), but the same description applies to a liability (that is, a short exposure).
Case 1: Vy is independent of the exchange rate
If Vy is constant, assuming the same value at any level of the exchange rate, then
Vy = K (4.22)
Hence
Vx = KS (4.23)
and
dVx = K (4.24)
dS
which means that the exposure does not change with the level of the exchange rate. This case is represented diagrammatically in Figure 4.3. The upper part of the diagram shows that Vy is independent of S, and this is why the relationship is represented by a horizontal line. The bottom part of the diagram shows the derived relationship between Vx and S (equation 4.23). As the exchange rate rises, Vy is unaffected but Vx rises. Thus, the relationship between Vx and S is represented by the line Vx = KS, and the exposure is represented by the slope of this line, which is constant at K. Notice that Vx is represented in the upper part of the diagram by the area under the line Vy = K, whereas in the bottom part it is measured on the vertical axis.
Case 2: Vyis inversely proportional to the exchange rate
In this case Vy falls (rises) when the exchange rate rises (falls) such that the change is proportional (equal percentage changes in opposite directions). This means that Vy and S are related as
Vy = K/S (4.25)
which is a rectangular hyperbola indicating that the product of Vy and S (which is the area under the curve) is constant. Hence
Vx = K (4.26)
and
dVx = 0 (4.27)
dS
which means that the exposure is zero (there is nothing at risk or that Vx is insensitive to changes in S). This is because any change in the exchange rate is completely offset by a change (in the opposite direction) in Vy, leaving Vx unchanged. This case is represented by Figure 4.4, where a rise in the exchange rate leads to a proportional fall in Vy, leaving Vx unchanged. Hence the exposure is zero because the slope of the line Vx = K is zero.
y
FIGURE
4.3
Exposure when Vy is independent of the exchange rate.
S
S
(4.28)
Vy = a - bS
where b > 0. Equation (4.28) is represented by a downward-sloping line in the upper part of Figure 4.5. In this case Vx is a nonlinear function of S that can be written as
Vx
Vx = K
5
FIGURE 4.4 Exposure when Vy is Inversely proportional to the exchange rate.
Vx = aS - bS2 (4.29)
as shown in the bottom part of Figure 4.5 (the lowest value on the vertical axis on which Vx is measured is a, not 0) . Starting from a low level, as the exchange rate rises Vx also rises, as represented by a larger area under the line Vy = a - bS. But as the exchange rate rises further, Vx reaches a maximum and starts to decline. This is because as S rises, the product of S and Vy may rise or fall, depending on its initial value.
Vx
=
aS - bS2
FIGURE
4.5
Exposure when Vy is negatively and linearly related to the exchange
rate.
S
Case 4: Vy is positively and linearly related to the exchange rate If Vy is an increasing linear function of the exchange rate then
Vy = a + bS (4.30)
which is represented by an upward-sloping line in the upper part of Figure 4.6. In this case Vx is a nonlinear function of S, which is written as
Vx
x
y
S
Vx
■■ aS + bS2
FIGURE
4.6
Exposure when Vy is positively and linearly related to the exchange
rate.
S
as shown in the bottom part of Figure 4.6 (the lowest value on the vertical axis on which Vx and Vy are measured is a, not 0). As the exchange rate rises, Vx also rises, as represented by a larger area under the line Vy = a + bS. It is important to observe that as the exchange rate rises, Vx rises more proportionately. The curve in the bottom part of Figure 4.6 has an increasing positive slope, implying that exposure increases as the exchange rate rises.
The exposure line
Let us now concentrate on the case when exposure is constant, as in Figure 4.3, expressing the relationship in terms of the changes in Vx and S (AVx and AS) rather than their levels. This relationship is represented diagrammatically in Figure 4.7, where changes in the exchange rate are measured on the horizontal axis and changes in the base currency value of the asset are measured on the vertical axis. The line representing this relationship is called the exposure line. In this case, the exposure line has a positive slope to indicate a positive relationship between changes in the exchange rate and changes in the base currency value of the asset. The equation of the exposure line is
AVx = EAS (4.32)
where the slope of the line, E, is the exposure. In the case of a long exposure, as shown in Figure 4.7, E > 0. Notice that there are two lines: a steep line representing high exposure and a shallow line representing low exposure. Hence zero exposure would be represented by a horizontal line, whereas an infinite exposure would be represented by a vertical line.
We now consider the case of exposure to foreign liabilities (short exposure), as shown in Figure 4.8. In this case, a rise in the exchange rate induces a rise in
AK*
FIGURE
4.7
Long
exposure line (assets).
the base currency value of liabilities, which entails a loss. This is why the exposure line in this case is downward-sloping. It has the same equation as (4.32) except that E <0.
The relationship between risk and exposure can be determined from equation (4.32), which gives
a2 (AVx ) = E2 a2(AS) (4.33)
Equation (4.33) tells us that the variance of changes in the base currency value of foreign assets and liabilities is related to the variance of changes in the exchange rate by a factor that reflects exposure, E2.
Multiple exposure
So far we have dealt with exposure to a single currency, y. In practice, exposure to several currencies is normally the case, as international business firms diversify their investment and financing portfolios. Hence a multiple exposure model may be written as
Vx = a 1 S(x/y i) + a2 S(x/y2 ) + ••• + anSxNn ) (4.34)
where S( x/yi) is the percentage change in the exchange rate between the base currency, x, and currency yi for i = 1, 2, ..., n. The coefficient ai (i = 1, 2, ..., n) measures the exposure to currency y/. In the case of assets, the coefficients are positive, whereas in the case of liabilities they are negative.
There are at least two problems with the empirical model of multiple exposure that is represented by equation (4.34). The first problem is that the model assumes a linear exposure, which may not be the case. A nonlinear exposure may arise because of nonlinearity in the firm's price elasticity of demand. The second problem is that exposure may not be constant over time, whereas this model would produce constant exposures if it is estimated by a conventional estimation method, such as OLS. This problem can be circumvented by resorting to an estimation method that allows the estimated coefficients to vary over time. This can be accomplished by specifying a state space model that can be estimated by utilising the recursive method of the Kalman filter (see Chapter 6).
The model represented by equation (4.34) can be used to formulate general multicurrency exposure relationships. If we measure value in terms of currency y1, then we have
Vyr = b 1 S(y 1 /x) + b 2 S(y 1 /y 2 ) + •••+ bn S(y 1 /Vn ) (4.35)
Vy2 = c 1S(V2 /x) + c2 %2 /V 1) + •••+ cnS(V2 /Vn) (4.36)
If, for example, x is the US dollar and y1 is the pound, then equation (4.34) relates changes in the US dollar value of the (US) company to changes in various exchange rates. If this US company has British shareholders, whose base currency is the pound, then equation (4.35) relates the pound value of the company (which is what matters for British shareholders) to changes in the exchange rates of the other currencies against the pound. The same interpretation can be given to equation (4.36) if y2 is, for example, the Japanese yen. Adler and Jorion (1992) have shown that the exposure coefficients of equation (4.34) are related to those of equation (4.35) as follows
(b2, b3, ..., bn) = (a2, a3, ..., an) (4.37)
and
b1 = 1 - (a2 + a3 + ... + an) (4.38)
4.4 TRANSACTION EXPOSURE
Transaction exposure to foreign exchange risk arises if payables and receivables (cash inflows and outflows) are denominated in a currency that is different from the base currency. It measures the sensitivity of the base currency value of contractual cash flows to changes in the exchange rate, which means that it can be determined from accounting statements. It is, therefore, a cash flow exposure that may be associated with trade flows (resulting from exports and imports) and capital flows (for example, dividends and interest payments). This kind of exposure arises, for example, from (i) a foreign currency asset or a liability that is already recorded on the balance sheet; and (ii) a contract or an agreement involving a future foreign currency cash flow. Thus, transaction exposure is specific to transactions where the risk arises from the future value of the exchange rate. It therefore corresponds to Case 1 in the previous section, where the y-value of the cash flow is known precisely but its x-value is not known in advance because of changes in the exchange rate.
Measurement of the transaction exposure of a multinational firm with subsidiaries requires a calculation of the consolidated net amount in currency inflows and outflows for all subsidiaries. For example, suppose that there are a long position and a short position on currency y amounting to Ki and K2 respectively. The exposure in this case is a measure of the sensitivity of the net position to changes in the exchange rate. Hence
E = dS S K1 - K 2 )] (4.39)
which obviously shows that if Ki = K2, then E = 0.
Two points are worthy of consideration here. The first is the degree of variability of each exchange rate. An exposure to a foreign currency that fluctuates sharply against the base currency is a source of more concern than an exposure to a currency that is relatively stable. Second, attention should be paid to the correlation coefficients of the underlying exchange rates. If the exchange rates between the base currency and other currencies are strongly and positively correlated, then the foreign currencies will all depreciate or appreciate against the base currency more or less proportionately. If they are positively but weakly correlated then these currencies will tend to move in the same direction but in different proportions. Negative correlation implies that other currencies move against the base currency in different directions, thus providing some sort of natural hedge.
Perfectly positive correlation leads to a perfect hedge when there are a short position on one currency and an equivalent long position on another currency. Consider a long position on currency y and a short position on currency z. The x-currency value of the two positions are
Vx,t ( y) = Vy/tSt ( x/y) (4.40)
VXti ( z) = Vz A (x/z) (4.41)
where Vx,t(y) is the x-currency value of the y position at time t and Vx,t(z) is the same for the z position. Suppose that Vy and Vz are not affected by changes in exchange rates, which gives
Vx,t+ i(y) = VytSt ( x/y)[1 + S( x/y)] (4.42)
(4.43)
(4.44)
If S(x/y) and S(x/z) are perfectly correlated, it follows that S(x/y) = S(x/z), which gives Vx = 0. Thus, any profit (loss) on the long position will be exactly offset by the loss (profit) on the short position, so that the change in the net base currency value of the position would be zero.
4.5 ECONOMIC AND OPERATING EXPOSURE
Economic exposure arises because changes in exchange rates affect the firm's domestic and foreign cash flows. This may sound like transaction exposure, but the difference lies in the fact that we are in this case concerned with noncontractual or unplanned future cash flows. These cash flows pertain to sales in foreign and domestic markets, as well as input costs, whether these inputs are domestic or foreign. While transaction exposure arises from transactions that are planned, currently in progress or have already been completed, economic exposure refers to changes in future earning power as a result of changes in exchange rates.
Economic exposure arises because of the effect of changes in exchange rates on a firm's cash flows (revenues and costs) and its equity value. If the reference is to the firm's operating cash flows, then the exposure is called operating exposure. If, on the other hand, we are concerned with the firm's equity value, then it is equity exposure. We will deal with equity exposure in a subsequent chapter, but in this section we concentrate on operating exposure.
Economic exposure is sometimes portrayed as consisting of transaction exposure and operating exposure. Operating exposure can be defined as the extent to which the firm's operating cash flows would be affected by random changes in exchange rates. Lessard and Lightstone (1986) argue that business executives are more familiar with transaction exposure than with operating exposure. They also argue that operating exposure has become increasingly important because exchange rates have become more volatile and business has become more international.
4.5 ECONOMIC AND OPERATING EXPOSURE 85 TABLE 4.4 Questions revealing the extent of operating exposure.
Question
Where does the firm sell its products?
Who are the competitors?
How sensitive is demand to price?
Where are the production facilities located?
What are the sources of input? How are the inputs or outputs priced?
Implication
Higher exposure from sales in foreign markets
Higher exposure results if main competitors are foreign
The higher the elasticity of demand the higher is exposure
Foreign production facilities result in higher exposure
Sourcing inputs from abroad implies higher exposure
Foreign currency pricing leads to higher exposure
that produces, sells its products and sources its inputs domestically is immune from exposure. Exposure may result from foreign competitors entering the domestic market for exchange rate-related reasons, and from the exposure of other (domestic) firms that the purely domestic firm deals with. Thus, even purely domestic firms that have no cross-border operations may be subject to operating exposure because changes in exchange rates are likely to change its competitive position in the domestic market, affecting its market share, revenue and profit.
Unlike transaction exposure, operating exposure is determined by (i) the structure of the markets in which the firm sources its inputs and sells its products, and (ii) the firm's ability to mitigate the effects of exchange rate changes by adjusting its markets, product mix and sourcing. Generally speaking, a firm is subject to a high degree of operating exposure when either its costs or its revenues are sensitive to exchange rate changes. On the other hand, when both costs and revenues are sensitive or insensitive to exchange rate changes, the firm has no major operating exposure (zero exposure if they are equally sensitive). Given the market structure, the extent to which a firm is subject to operating exposure depends on its ability to stabilise cash flows in the face of exchange rate changes. One has to remember that financial costs, such as interest on debt, are not relevant in determining operating cash flows. However, we will find out later that interest payments are a determining factor of net cash flows and hence net cash flow exposure.
Consider Figure 4.9, which shows four different possibilities for the effect of changes in the exchange rate on operating cash flows in terms of the base currency, x. Specifically, this figure shows the effect of the exchange rate on revenues, Rx, costs, Cx, and profit, px . In Figure 4.9(a), revenues and costs are equally sensitive to changes in the exchange rate, so profit is the same at any
p
--
0
CX
RX
5
(d)
Costs are more sensitive than revenues
FIGURE 4.9 The effect of exchange rate changes on base currency operating cash flows.
level of the exchange rate. In this case, there is no operating exposure, as (net) operating cash flows are not sensitive to changes in the exchange rate. In Figure 4.9(b) costs are not affected by changes in the exchange rate, but revenues rise as the exchange rate rises. This would be the case if the firm sources its inputs domestically and sells its products abroad. Thus, as the exchange rate rises profit rises. Figure 4.9(c) shows the case when both costs and revenues rise with the level of the exchange rate, but revenues rise faster. In this case there is a positive operating exposure, as profit rises with the exchange rate. Finally, Figure 4.9(d) shows the case when costs are more sensitive then revenues. In this case profit declines as the exchange rate rises, reaching zero at a certain level of the exchange rate and turning negative if it rises further.
Measurement of operating exposure
5
(c)
Revenues are more sensitive than costs
R
X
C
X
(b)
Costs
are insensitive whereas revenues are sensitive
5
R
X
C
X
5
(a)
Revenues and
costs are equally sensitive
exchange rate. For the purpose of the following discussion, the level of profit is taken to be the net operating cash flow. If this level changes from p xt to p x t+1 as the exchange rate changes from St to St+1, we get
(4.45)
E _ pPx_ _ (p x,t+ i ^lx,t )
p _ S " (Sf+ i/Sf )-i
Equation (4.45) can be rearranged to obtain
S
t+1
1 + EP
-1
(4.46)
S
or
!t±1
St
(4.47)
-1
p
-1 _ Ep
x ,t
which means that the percentage change in the base currency cash flow is equal to the operating exposure times the percentage change in the exchange rate.
It is obvious from equation (4.45) that the operating exposure may assume any value: positive, negative or zero. If, for example, p x = 0 when S ^ 0, then Ep = 0. And if p x = S, then Ep = 1. Finally, if p x < 0 when S > 0 then Ep < 1. Figure 4.10 shows these possibilities in terms of the slope of the operating exposure line.
Components of the operating exposure
Consider the following relationships for an exporter whose base currency is x:
p x_ Rx -Cx (4.48)
Rx_ Px Q (4.49)
Q _ f (Py ), f< 0 (4.50)
P
Py _ ^ (4.51)
Cx_ F(S), F > 0 (4.52)
where Q is the quantity sold. Equation (4.48) defines profit, or net operating cash flows, in terms of currency x; equation (4.49) defines revenues in terms of currency x; equation (4.50) is a demand function; equation (4.51) relates the y price to the x price via the exchange rate; and equation (4.52) is a cost function. Changes in S, therefore, affect px via several channels, and this is why there are many components of operating exposure. These components will be discussed in turn.
p
x
FIGURE
4.10
Possible
operating
cash
flow
exposures.
Revenue exposure and cost exposure
Revenue exposure and cost exposure can be defined respectively as:
(4.53)
EC-
^ -
C
s
(4.54)
E - Rx-
(St+ ^ ) -1
(C x,t+ 1/Cx,t )-1
(st+ iS ) -2
p
x,t
+ C x
R
x
,t
-x,t
Rx - p x
R
x,t
Multiplying both sides of equation (4.55) by RX/t and rearranging, we obtain
pX pX,t = RXRX,t -CXCx,t (4.56)
Dividing by St Xt, we obtain
R
C
R
x
C
x
P
x
x,t
x,t
è
p
x ,t
Since Cx t/px t = (Rx t/p x t)-land (Rx t/px t) = 1/m, it follows that
(4.58)
Ep
-
m
m-1
Again, Er and Eq may assume various values. For example, Er = 1 if the y- operating revenues are not affected by changes in the exchange rate, such that the x-operating revenues change by the same percentage change as the exchange rate. Eq, on the other hand, depends on the geographical distribution of the operations and suppliers. Thus Eq may be zero for a purely domestic firm with no imports of inputs such as raw materials. We say "may be" because Eq ^ 0 if the domestic suppliers have operating cost exposures (this is indirect exposure, as we are going to see later). On the other hand, Eq = 1 for a firm with a subsidiary producing entirely abroad with inputs that are not imported, and neither are their prices affected by exchange rate changes. In general, cost exposure faced by an importing firm is the opposite of the exporter's operating revenue exposure (if Er = 1 for an exporter, then Eq = 0 for the importer).
Conversion exposure
When cash flows in terms of y are not affected by changes in the exchange rate, we have on our hands a pure conversion exposure, which is obtained when Er = 1 and Eq = 1. A pure conversion exposure arises if changes in the exchange rate do not affect Py, but affect Px via the equation Px = SPy. This may seem like transaction exposure, but it is not. Under transaction exposure, the cash flows are contractual and the conversion takes place. Under operating exposure the cash flows are not contractual and are unknown in advance, and conversion may or may not take place.
Price exposure
In general, Px/S measures a combination of conversion and price exposures. Let us for this purpose consider the following cases for an exporter to examine the effect of changes in the exchange rate on operating revenues in base currency, Rx.
The selling price is set in terms of currency x, so that Px is unaffected by changes in the exchange rate.
The selling price is set in terms of currency x, but it is changed to offset changes in S partially.
We will consider what happens when the exchange rate declines (currency y depreciates) under the three possibilities. These are shown in Figures 4.11, 4.12 and 4.13 respectively. Each of these figures consists of four related parts. The upper right-hand part shows the relationship between Px and Py expressed as Py = (1/S)Px. The slope of the line is 1/S, which means that a decline in the exchange rate is represented by a movement of the line to the left (and vice versa). This is because as S declines the line becomes steeper. The upper left-hand part is a demand function relating the quantity sold, Q, to Py. The lower right-hand part is a 45° line (Px = Px) used to rotate the axes, so that Px would appear on the vertical axis. Finally, the lower left-hand part shows combinations of Px and Q, so that total revenue is represented by the area of the rectangle defined by Q and Px, because Rx = PxQ. As S changes, Rx will change as a result of changes in Px and/or Q.
Consider Figure 4.11 first, which represents the first case. As the exchange rate declines, the exporter reacts by reducing Px proportionately so that Py would remain unchanged. Since there is no change in Py, Q will not change. Revenue in terms of x will decline only because of the decline in Px. The decline in Rx, DRx, is represented by the shaded area in the lower left-hand part of the diagram.
Consider now Figure 4.12 representing the second case in which the exporter sets the selling price in terms of x. As the exchange rate declines, without a change in Px, Py will rise, leading to a fall in Q. Revenue in terms of x will decline only because of the decline in Q. Again, the resulting change in revenue, DRx, is represented by the shaded area.
The third case is shown in Figure 4.13. As the exchange rate declines, Py rises. The exporter reacts by reducing Px less than proportionately, so that we have on our hands a change in both Px and Py. Revenue in terms of x will decline because of the decline in both Q and Px, and this is why there are two shaded areas in Figure 4.13: DRx (Q) is the decline in revenue resulting from lower Q, whereas DRxPx results from lower Px. Whether DRx(Q) > DRx(Px) or DRx(Q) < DRx(Px) depends on the change in Px relative to Py and also on the elasticity of demand (the slope of the demand function).
Demand exposure
Changes in the exchange rate lead to changes in demand and hence revenue in terms of x, given that Rx = PxQ, Q = f(Py) and Py = (1/S)Px. As the exchange rate changes Py changes, leading to a change in Q and hence a change in Rx. Even if Py is kept unchanged by changing Px, changes in S may lead to changes in other factors that may affect demand.
Q
FIGURE
4.11
The effect of a decline in the exchange rate on the revenue of an
exporter when the price is set in terms of y.
P
P
y
P
y
X
Px
Competitive exposure
Demand and price exposures may be the result of a firm's competitive position. If currency y appreciates (the exchange rate rises), more companies with a base currency x would be inclined to compete for business in the country whose currency is y. Thus, the demand for existing firms' products in this
Q
FIGURE
4.12
The effect of a
decline
in the exchange rate on the revenue of an exporter when the price is
set in terms of x.
P
P
y
y
P
X
P
X
country will decline, as represented by a downward shift in the demand curve, and this will affect revenues, as shown in Figure 4.15. Notice that without a competitive exposure, the rise in the exchange rate leads to a decline in Py and a rise in base currency revenue for the existing firm.
Indirect exposure
A firm that has no direct foreign exchange exposure due to conversion, price, demand or competitive effects could still have an indirect exposure if it is a supplier to firms that have direct exposures. If changes in the exchange rate lead to a fall in the profitability of these firms, the supplier will suffer a reduction in the demand for its products. Again, this is represented by an inward shift in the demand for its products, as shown in Figure 4.15.
Multimarket exposure
A firm that derives its revenue by selling in various markets is said to have a multimarket exposure. If this firm operates in a foreign market and the
domestic market, it will have revenue exposure only to the extent of the revenue derived from the foreign market.
4.6 A FORMAL TREATMENT OF OPERATING EXPOSURE
Q
FIGURE
4.13
The effect of a decline in the exchange rate on the revenue of an
exporter when the decline is offset partially.
P
P
y
y
P
X
P
X
Px
FIGURE
4.14
The effect of a decline in the exchange rate under elastic and
inelastic demand.
P
X
Case 1: The exporter in terms of currency x
Assume that the firm sells all of its output in the country whose currency is y. Total revenue in terms of x, Rx, is
Rx = SPyQ (4.59)
If the unit cost is cx, the total cost of production is
Cx = CxQ (4.60)
For profit maximisation, marginal cost must be equal to marginal revenue, which gives
dRx dCx
—- = —- (4.61)
dQ dQ v '
FIGURE
4.15
Competitive exposure when the exchange rate declines.
p
p
y
y
P
X
P
X
or
^
+ QdPy'
.
py
dQ
0
dPy
SPy
+ SQ —t = SPy
y
dQ y
(4.62)
=
cx
(4.63)
1
SPy I1 - h 1 = cx
where
(4.64)
h
=
is the elasticity of demand. Therefore, the profit maximising price in terms of y is
P = cX (4 65)
y S[1 -(1/h)] '
which says that the exporter should set the price in terms of y according to the unit cost of production (in terms of x), the exchange rate, and the elasticity of demand. Assuming that the unit cost and the elasticity of demand are constant, and differentiating equation (4.65) with respect to S gives us
dPy r Py
-b = - = -JL < 0 (4.66)
dS S2 [1 -(1/h)] S v ;
which follows by substituting the value of cx from equation (4.63). Equation (4.66) means that an appreciation of y (a rise in S) lowers the profit-maximising price in that currency. What happens to total revenue can be gleaned from the equation
dR dQ dPy dPy dPy
— = PyQ + SPy —L + SQ —L = PyQ + SQ(1 -h) — (4.67)
dR Py
dRx
=
PyQ
- SQ(1 - h)
By substituting the value of dPy/dS from equation (4.66), we obtain
= PyQ -SQ(1 -h) = hPyQ > 0 (4.68)
which means that an increase in S leads to an increase in sales revenue in terms of currency x.
The effect on cost in terms of currency x is also calculated by differentiating Cx with respect to S, which gives
dCx=Cx dQ = Cx ^Q =-Cx dQpy s (4.69)
or
dCx
=
hCxQ
dS
S
> 0 (4.70)
Now, consider the effect on profit, p x , measured as the difference between revenue and cost. Since
p x= Rx -Cx (4.71) it follows that
dPx = dRx -dCx = hQPy -1 > 0 (4.72)
dS dS dS è y S 0
where the factor in parentheses is the mark-up per unit in currency y. If this is positive, then a rise in S would lead to a rise in profit in terms of x.
Case 2: The exporter in terms of currency y In this case, revenue and costs are given by
Ry = PyQ (4.73)
Cy=
^ (4.74)
For profit maximisation, dRy/dQ = dCy/dQ, which gives
Py = ^ (4.75)
palgrave 1
palgrave 2
i 19
i 23
CHAPTER 3 44
Other Kinds of Arbitrage and Some Extensions 46
3.1 COMMODITY ARBITRAGE 46
Py ,t+ 1/Py ,t 55
3.2 ARBITRAGE UNDER THE GOLD STANDARD 61
3.3 ARBITRAGE BETWEEN EUROCURRENCY AND DOMESTIC INTEREST RATES 63
3.4 EUROCURRENCY-EUROBOND ARBITRAGE 74
3.5 ARBITRAGE BETWEEN CURRENCY FUTURES AND FORWARD CONTRACTS 52
3.6 REAL INTEREST ARBITRAGE 54
3.7 UNCOVERED ARBITRAGE WHEN THE CROSS RATES ARE STABLE 55
3.8 UNCOVERED INTEREST ARBITRAGE WHEN THE BASE CURRENCY IS PEGGED TO A BASKET 57
^ ^ 1=1 »j ^ j=1 and 57
E1= S 0 - S j (3.40) 57
3.9 MISCONCEPTIONS ABOUT ARBITRAGE 83
CHAPTER 4 65
Hedging Exposure to Foreign Exchange Risk: The Basic Concepts 65
4.1 DEFINITION AND MEASUREMENT OF FOREIGN EXCHANGE RISK 65
4.2 VALUE AT RISK 69
4.3 DEFINITION AND MEASUREMENT OF EXPOSURE TO FOREIGN EXCHANGE RISK 74
4.4 TRANSACTION EXPOSURE 82
4.5 ECONOMIC AND OPERATING EXPOSURE 84
4.6 A FORMAL TREATMENT OF OPERATING EXPOSURE 95
dQ_py_ dPy Q 102
4.7 TRANSLATION EXPOSURE 123
Financial and Operational Hedging of Exposure to Foreign Exchange Risk 105
Fb,t - Sb,t 123
p - K(Fa,t - Sb,t+ 1 ) 133
Measuring the Hedge Ratio 158
, s(DPu,t, DPA,t If t-1) ,t t . 164
h = ^ ^ r" = P(X 1,t , X 2 ,t ) 164
s 2(DP A,t|f t-1 ) 164
DP u,t = t-1 + Z ai DP U,t-i +Z bi dPa ,t-i +x 1,t (6.33) 186
DP A,t =-bft-1 + Z ci DP U,t-i +Z d i DP A,t - i +X 2 ,t (6.34) 186
s(DPU,t, DPA,t If t-1, DPU,t-i, DPA,t-i) 186
s2 (APA,t If t-1' APU,t-i ' APA,t-i ) (6 35) 186
I,t ' X 2,t ) / s(X i,t ) ^ 186
= P(X 1't ' X 2 ,t ) 186
_ P(X 1,t , X 2 ,t)s(X 1,t)s(X 2 ,t) -2 (f t-11 AP U,t-i, APA,t-i) 187
s2 (X2 ,t ) + b2 s 2 (f t-11 APU,t-i, APA,t-i ) 187
Speculation in the Spot and Currency Derivative Markets 184
Sb,t 198
Sb,t 1 + m 198
Sb,t 1 + E(mt+1) 199
E( Sb,t+1 ), S 200
if the position in x is available. If the funds have to be borrowed, the condition changes to 201
Speculation: Generating Buy and 209
Sell Signals 209
St = f (Xt) 257
where the vector of variables varies according to the underlying fundamental model. The current level of the exchange rate may deviate from the equilibrium level because of the effect of random shocks that tend to have a temporary effect. Hence the current level of the exchange rate may be represented by 258