Interest rate.

which means that the effective rate of return on a short-term investment in a y- denominated asset is approximately equal to the sum of the interest rate on the currency and the percentage change in the exchange rate.

9.5 INTRODUCING THE BID-OFFER SPREADS

If we allow for the bid-offer spreads in exchange and interest rates, the equa­tions used to calculate the effective financing rate and the effective rate of return will be different. Let us start with the effective financing rate. Suppose that an amount, K, of currency y is borrowed at time t at the offer foreign interest rate, i_y,a. The foreign currency amount is converted into the base currency at the bid exchange rate prevailing at time t, S_b,t. Thus, the base currency value of the amount borrowed, L_t, is given by

(9.15)

When the loan matures at time t +1, the y currency amount to be repaid (prin­cipal plus interest), Lt+i, is given by

Lt+1 - KSa,t+ 1(1 + iy,a ) (9.16)

where S_a,t+1 is the offer spot exchange rate prevailing at time t + 1. The effec­tive financing rate is therefore given by

(9.17)

b,t

^KSa,t+1⁽¹ + ⁱy,a ⁾

^Lt ^{- KS}b,t

⁽¹ + ^ey ^{) -} -

KS

or

S

a,t+1

-1

(9.18)

S

b,t

ey- (¹ + iy,a )

Since S_a,t+1 = (1 + m)Sb,t+1, it follows that

-1

(9.19)

^Sb,t+1⁽¹+^m)

^ey ⁽¹ + ⁱy,a ⁾

^Sb,t

or

^ey ^{- (1} + ⁱy,a ⁾⁽¹ + ^Sb ⁾⁽¹ + ^m)-1

-y ' ^ly,a ,

(9.20)

where S_b is the percentage change in the bid exchange rate between t and t + 1. Obviously, the effective financing rate as calculated from equation (9.20) is greater than that calculated from equation (9.11). This is because the bid-offer spread is a transaction cost that is added to the cost of borrowing, making it higher than otherwise. An approximate formula can also be obtained by working out equation (9.20) and ignoring the small terms to obtain

* ^ly ,a + ^Sb + ^m

In this case the choice between foreign currency financing and base currency financing depends on a comparison between the effective financing rate and the base currency offer rate.

(9.21)

Likewise, the formula for the effective rate of return will differ if allowance is made for the bid-offer spreads in interest and exchange rates. The process from which the formula is derived can be modified as follows. Suppose that the foreign currency equivalent of an amount, K, of the domestic currency is available for foreign currency investment at time t at the bid foreign interest rate, iyb The base currency amount is converted into foreign currency at the offer exchange rate prevailing at time t, S_a,t. Thus, the foreign currency value of the amount available for investment is K/S_a,t. When the investment matures at time t + 1, the base currency equivalent of the foreign currency amount received is given by

(9.22)

(9.23)

K

^It+1 = S— ^ + ⁱy,b ^)Sb,t+1

a,t

This is because the amount of the foreign currency realised from the invest­ment is sold at the lower bid rate. The effective rate of return is therefore given by

^(K/Sa,t⁾⁽¹ + Vb ^)Sb,t+1

(1 + ry ) =

K

or

S

V+1

-1

(9.24)

S

a,t

^ry =⁽¹ + Vb ⁾

which means that

S

b,t+1

-1

(9.25)

^Sb,t ⁽¹ + ^m)

^ry =⁽¹ + Vb ⁾

or

(9.26)

-1

1+m

(1 + iy ,b )(1 + Sb)

^ry=-

The effective rate of return calculated from equation (9.25) is lower than that calculated from equation (9.13). This is because the bid-offer spread is a trans­action cost that reduces the net return on investment, making the effective rate of return lower than otherwise. In this case the choice between foreign currency and base currency investment depends on a comparison between the effective rate of return and the domestic bid rate.

9.6 IMPLICATIONS OF CIP AND UIP

Two international parity conditions have some implications for short-term financing decisions: covered interest parity (CIP) and uncovered interest parity (UIP). The implications of the two parity conditions are discussed in turn.

Covered interest parity

To assess the implications of CIP for short-term financing decisions, we will again modify the process used to derive an expression for the effective financing rate. This process is modified by assuming that the borrowing firm wishes to avoid foreign exchange risk. To do this, the foreign currency expo­sure can be covered in the forward market. In this case, the foreign currency is bought forward at time t, which means that the amount to be repaid is converted into foreign currency at the forward rate prevailing at time t, Ft. Thus

Lt+ J = KFt (1 + iy ) (9.27)

The effective financing rate is, therefore, given by KF_t (1 + iy )

(1 + e y ) = ' ^y (9.28)

^KSt

which gives

e y= (1 + iy )(1+f) -1 (9.29)

where f is the forward spread. If CIP holds then

1 + i

1+f = (9.30)

¹ + ^ly

By substituting equation (9.30) into equation (9.29) we obtain

^ey = ⁱx ⁽⁹.³¹)

Thus, if CIP holds, the effective financing rate will be equal to the interest rate on the base currency. Hence foreign currency financing will be useless in the sense that it will not be cheaper than domestic currency financing. If, however, CIP is violated such that

1 + i

1+f > (9.32)

¹ + ^ly

which means that the forward spread is larger than the interest rate differen­tial, it follows that

¹ + ⁱx

(9.33)

1 + i

y 0

(1 + e_y ⁾ > (1 + L )

or

e > i

y x

(9.34)

which means that foreign currency financing is not desirable, since it is more expensive than base currency financing. If, on the other hand, CIP is violated such that the forward spread is smaller than the interest rate differential, that is

-, r 1 + i_x

(9.35)

1 + f < z-T- 1 + i_y

then

¹ + ⁱx 1 + i_y

(9.36)

⁽¹ + ^ey ⁾ < ⁽¹ + ⁱy ⁾

or

^ey < ⁱx

(9.37)

which means that foreign currency financing is desirable because it is cheaper than base currency financing. The implications of CIP for short-term financing decisions are listed in Table 9.2.

The same results can be derived by allowing for the bid-offer spreads in interest and exchange rates. Depending on the direction of covered arbitrage, CIP may be written (approximately) as

(9.38)

(1+m)

(1 + i_y ,_b)(l + f )

⁽¹ + ⁱx,a ⁾ = "

TABLE 9.2 Implications of CIP for short-term financing and investment.
The CIP	Direction of	Financing decision	Investment decision
condition	violation
holds	_	Indifference between	Indifference between
		foreign and base	foreign and base
		currency financing	currency investment
Violated	Forward spread is	Base currency	Foreign currency
	larger than interest	financing is more	investment is more
	differential	desirable	desirable
Violated	Forward spread is	Foreign currency	Base currency invest-
	smaller than interest	financing is more	ment is more
	differential	desirable	desirable

(9.39)

or

⁽¹ + ⁱx,b ⁾

⁽¹ +ⁱy,a ⁾

^{y ,a} ' (1+f )(1+m) By combining equations (9.20) and (9.39), we obtain

(9.40)

x,b

1 + S-

b

⁽¹ + ^ey ^{) - (1} + ⁱx,b ⁾

(1+f )

If S_b »f, it follows that ey »i_x,_b . Likewise, if we combine equation (9.26) and (9.38) we obtain

1 + Su

⁽¹ + ^Ty ^{) - (1} + ⁱx,a ⁾

(1+f )

which gives Ty » i_x,_a.

Uncovered interest parity

If UIP holds, then the percentage change in the spot exchange rate will be equal to the interest rate differential. This may be represented by

• 1 + i

1 + S - i+i- (9.42)

¹ + ⁱy

By substituting equation (9.42) into equation (9.11) we again obtain ey = i_x. The implications of UIP for short-term financing and investment are listed in Table 9.3.

9.7 THE PROBABILITY DISTRIBUTION OF THE EFFECTIVE FINANCING RATE AND THE EFFECTIVE RATE OF RETURN

Using the UIP condition as a criterion for foreign currency financing and investment decisions implies that the decision is based on the expected change in the exchange rate such that foreign currency financing will be preferred if the exchange rate is expected to change by less than what is implied by UIP. As we have seen, it is sometimes difficult to obtain a point forecast for the exchange rate. In this case, we use a probability distribution for the expected change in the exchange rate to arrive at a probability distribution for the effective financing rate (the same analysis is valid for the effective rate of return).

(9.41)

Suppose that the expected (percentage) change in the exchange rate assumes the values S1, S2 , ..., S_n with probabilities p\, p2, ..., p_n. In this case the expected value of the (percentage) change in the exchange rate is given by

TABLE 9.3 UIP as a criterion for foreign currency financing and investment.

The UIP condition

Direction of violation

Financing decision Investment decision

Expected change in the spot rate is larger than interest differential Expected change in the spot rate is smaller than interest differential

Indifference between foreign and base currency financing Base currency financing is more desirable

Holds

Violated

Violated

Foreign currency financing is more desirable

Indifference between foreign and base currency investment Foreign currency investment is more desirable

Base currency invest­ment is more desirable

E(S) = X S iPi (9.43)

i=1

The effective financing rate would have a similar probability distribution in the sense that it assumes the values ey 1, ey,2, ..., ey,_n with probabilities px, P2, ..., p_n. The expected value of the effective financing rate is therefore given by

n

E( ey ) = X ey ,iP i (9.44)

i=1

The measure of risk in this case is the variance or the standard deviation of the effective financing rate, both of which represent the variability of the effective financing rate or the dispersion around its expected value. The variance is given by

n

s ²(e) = X Pi [ey,i - E(ey )]² (9.45)

i=1

whereas the standard deviation is measured as the square root of the variance as

s(ey) = ^LPi[eyi - E(ey)]² (9.46)

Alternatively, the expected value of the effective financing rate can be calcu­lated directly from equation (9.11) as

E( ey ) = (1 + iy )[1 + E(S )] -1 (9.47)

Expected values, variances and standard deviations can be calculated if probability distributions are available. If not, one can calculate these statistics

from historical data, in which case the expected value is replaced by the mean or average value. Therefore, if we have a sample of observations on the effec­tive financing rate: ey,1, ey,2,..., ey,_n, the mean or average value, ëy, the variance and the standard deviation are given respectively by

S n _e

-e_y - (9.48)

⁷ n

s2(ey ) -^{- n 1 (} ^^{y )2} (9.49)

⁷ n -1

ⁿ )²

SI i(ey t -

s(ey ) = i ¹ y-1 (⁹.⁵⁰⁾

It is important to remember that financing with the base currency involves no foreign exchange risk, since the cost of financing (which is the interest rate on the base currency) is known with certainty. When foreign currency financing is chosen, the effective financing rate is at best probabilistic, in the sense that it assumes a range of values with some probabilities. This, naturally, implies the presence of risk. The choice between foreign currency financing and domestic currency financing, therefore, depends on the attitude towards risk. The decision maker may in this case want to minimise (maximise) the effective financing rate (the effective rate of return) per unit of risk as measured by the standard deviation. Alternatively, we may assume that the decision maker is a mean-variance expected utility maximiser such that the utility function is given by

E[U(ey )] = E(ey ) -gs ² (ey) (9.51)

where g is the coefficient of risk aversion. The difference between financing and investment decisions is that U'(ey) < 0, whereas U' (ty) > 0, respectively.

Since ey = iy + S, it follows that

s ² (ey) = s ² (iy) + s ² (S) + 2s(iy, S) (9.52)

where s(iy , S) is the covariance of iy and S. Equation (9.52) means that over a long period of time, the variability of the effective financing rate is determined by the variability of the interest rate and that of the percentage change in the exchange rate.

9.8 USING CURRENCY PORTFOLIOS FOR SHORT-TERM FINANCING AND INVESTMENT

There is no reason why a firm cannot choose to finance with (or invest in) a portfolio of currencies, which may or may not include the base currency. In fact, such an operation may be beneficial because it is a very well-known prin­ciple in finance that diversification reduces risk.

Let us look at the matter in general terms, using financing operations for the purpose of illustration. Assume that there are two foreign currencies, y and z. The effective financing rate in currency y assumes the values y with proba­bilities pyi for i = 1, 2, ..., n, such that ey,_n > ey,_n-i > ... > ey,i, which means that the nth value of the effective financing rate is the greatest, whereas the first value is the smallest. Similarly, the effective financing rate of currency z assumes the values e_z,j with probabilities p_z,j for j = 1, 2, ..., m such that ey,_m > ey,m-1 > ... > ey,1. Let us assume that the financing portfolio is formed by assigning weights Wy and w_z to currencies y and z respectively such that Wy +

Wz = 1.

The worst thing that can happen is that the exchange rates of the base currency against the two foreign currencies move in such a way as to produce the highest values of the individual currencies' effective financing rates, ey_n and e_z,_m. In this case the highest effective financing rate of the portfolio, ep,_n,_m, will be given by

^ep,n,m = ^wy^ey,n + ^wz^ez ,m ⁽⁹.5³⁾

Since ey,n is realised with a probability py,n and pz,m is realised with a proba­bility py, m, then ep,n,m is realised with a joint probability calculated as the product of the individual probabilities (that is, py,_np_z,_m). This probability is naturally smaller than any of the two individual probabilities. If the effective financing rates of the individual currencies can assume any value, the effec­tive financing rate of the portfolio will be given by

^ep,i,j= ^Wy ^ey,i + ^Wz ^ez,j ⁽⁹⁵⁴⁾

which materialises with a joint probability py,ip_z,j. Table 9.4 lists all of the possible combinations (that is, for various values of i and j).

The expected value of the effective financing rate of the portfolio can be obtained by multiplying each entry in the third column by the corresponding entry in the fourth column and then adding up the products. This can be represented by

nm

^E(ep ⁾ = Z Z ^(Wy^ey,i + ^Wz^ez, j ⁾py,ipz,j ⁽⁹.⁵⁵⁾

i = 1 j = 1

where

nm

Z Z p_x,ipy, j = ¹ (⁹.⁵⁶)

i=1 j=1

The variance and the standard deviation of the portfolio's effective financing rate are given by

TABLE 9.4 Possible values for the effective financing rate of a two-currency portfolio.

Effective financing Effective financing Effective financing Joint probability rate of y (ey, j) rate of z (e_z, j) rate of portfolio (p_x jp_y, j)

^(ep, i, j⁾

Py ,1Pz ,1 Py ,1Pz ,2

^py ,1^pz ,m

^ey ,1 ^ey ,1

^ey ,1

^ez ,1 ^ez ,2

e

^wy^ey ,1 + ^wz^ez ,1 ^wy^ey,1 + ^wz^ez ,2

Wy^ey,! + Wz^ez ,m

^ey,2 ^ey,2

^ez,1 ^ez,2

^Py ,2^Pz,1

Py ,2Pz ,2

^wy^ey ,2 + ^wz^ez ,1 ^wy^ey ,2 + ^wz^ez ,2

e

^ez

Py ,2Pz,

y,2

^wy^ey ,2 + ^wz^ez ,m

^ez,1 ^ez,2

wyey ,„ + w₇e

Py ,nPz ,1 Py ,nPz ,2

z^e z ,1

-y^,n

e

^Py ,n ^Pz

-y^,n

^wy^ey,n + ^wz^ez

n m

^{s 2 (e}p ) = Z Z [(^wy ^ey,i + ^wz ^ez,j ) ^{- E(e}p )]² Py,iPz,j (⁹-⁵⁷)

^s(ep ⁾ =

i=1j=1

Z Z ^[(wy^eyi + ^wz^ez,j ^{) -E(e}p ^)]2 Py,iPz,j ⁽⁹-⁵⁸⁾

i=1j=1

Alternatively, the expected value of the portfolio's effective financing rate can be calculated from the effective financing rates of the individual curren­cies as

^E(ep ^{) - w}y ^E(ey ⁾ + ^wz ^E(ez ^{) (9}.⁵⁹⁾

or in terms of historical data

ëp - Wyëy + Wzëz ⁽⁹.⁶⁰⁾

whereas the variance is measured as

s ² (ep ) - w2 s ² (ey ) + w² s² (ez ) + 2WyWz s(ey , ez ) (9.61)

where s( ey , e_z ) is the covariance of the individual effective financing rates. The variance as given by equation (9.61) can also be expressed in terms of the correlation coefficient between the individual effective financing rates,

P^{( e}y ^{, e}z ^),as

FIGURE 9.6 The standard deviation of the effective financing rate of a currency port­folio as a function of the correlation coefficient.

^{o 2 ( e}p ⁾ = ^Wy ^{o 2 ( e}y ⁾ + ^W1 ^{o 2 ( e}z ⁾ + ^2Wy^Wz ^{o( e} y ^{)o( e} z ^{)p e} y ^{, e}z ^{) (9}.⁶²⁾

As we can see from equation (9.62), the correlation coefficient plays an impor­tant role in determining the standard deviation of the effective financing rate of the portfolio. Figure 9.6 shows the relationship between the standard devia­tion and the correlation coefficient as represented by equation (9.62). It is drawn for the special case when o(ey ) = o(e_z ) = 5 and Wy = W_z = 0.5. If p = -1, the standard deviation of the effective financing rate of the portfolio is zero. In this case, risk is eliminated completely by choosing an equally-weighted port­folio. The highest value of the standard deviation of 5 is obtained when p = 1. In this case, there is no reduction in risk. The rate at which risk is reduced increases as p ® -1, and this is why the relationship is nonlinear.

Notice also that correlation between ey and e_z is determined by the correla­tion between the percentage changes in the exchange rates, S(x/y)and S(x/z). If

ey = a + b(e_z ) (9.63) then

iy + S( x/y) = a + b[i_z + S( x/z)] (9.64) which gives

S(x/y) = [a + bi_z -iy] + S(x/z) (9.65)

which shows that the correlation between S( x/y) and S( x/z) determines the correlation between ey and e_z.

CHAPTER 10