Analysis of the choice between domestic and foreign loan facilities

OMEGA The Int. JI of Mgmt Sci.. Vol. 7, No. 4. pp. 333 to 338 ~3 Pergamon Press Ltd 1979. Primed in Great Britain

0306-0483/79/0801-0333502.00/0

Analysis of the Choice Between Domestic and Foreign Loan Facilities NIGEL

MEADE

Imperial College, University of London MALCOLM

J FINNEY

JF Chown & Co Ltd, London

{Receir,',l December 1978: in r,'l i~ed Jbrnl February 1979J

When a corporation borrows from a non-domestic source, such as the Eurobond market, it is obliged to make interest payments in a foreign currency at predetermined periods throughout the life of the loan. By doing this the corporation incurs an exchange risk which may outweigh any advantages of borrowing foreign currencies such as lower interest rates. This note describes a criterion which assists the corporate treasurer in his choice of currency by showing the exchange rates at the end of the life of the loan implied by each choice.

INTRODUCTION W HEN A corporate treasurer wishes to borrow over a long term, he has a choice of a number of currencies with associated interest rates. As a general rule, the more stable currencies offer lower interest rates than the less stable. If the corporation's domestic currency is one of the less stable ones (such as £), the treasurer's choice is likely to be between a high interest loan in his own currency, or a lower interest loan in a more stable foreign currency (such as Swiss Francs). The choice of the lower interest foreign loan implies that the exchange rate between the relevant currencies is not expected to change sufficiently for the effective payments of interest and principal to exceed those of the domestic loan. The choice can be formalised by computing a "breakeven" exchange rate at which the foreign and domestic loans would be equivalent. It should be emphasised that the problem considered is that of making a choice between two or more fixed interest loans over specific

terms, the Eurobond is an example of the type of loan instrument involved. The use of the breakeven rate as a decision criterion for the corporate borrower was criticised by Finney and Meade [2] and an improved criterion was outlined. The purpose of this note is to describe the improved criterion in detail. BREAKEVEN DECISION CRITERION This criterion is nothing more than a straightforward internal rate of return calculation, it was described by Chown [1] in 1973. Consider first a domestic loan over N years with an interest coupon of 100 cA% pa with the interest payable n times pa, the cost of this loan will be the solution x to this equation:

(cA~n)

1 - - i=l (1 + x)i

I (1 + x) N"

0

(1)

Obviously in this case the cost of the loan x is (cMn) domestic money units (dmu) paid n times yearly. 333

Meade, Finney--Analysis of the Choice Between Domestic and Foreign Loan Facilities

334

Secondly consider a loan in a foreign currency over the same period with the same timing of interest payments but with a coupon of lO0 c B % pa. When this loan is taken out the exchange rate is Eo foreign money units (fmu)/ dmu. It is also assumed that the relative depreciation rate between the currencies is rb every l / n of a year. The cost of the foreign loan is the solution y to this equation: N,~ 1

cB/n

.

1

1

i ~ (1 + rb)i (1 + y)i

I

0

(1 "~" rb) Nn (] + y)N.

(2) For the costs of the foreign and domestic loans to be equal: y = x = cA/n

Thus substituting in (2): N.

c./n

I 0

1 -- ~=1 ~" (I + rb)i(l + ca/n) i -- (1 + rb)Nn(l + CA~n)N"

(3) Thus for the cost of the loans to be equal the depreciation rate rb must satisfy (3), that is at the end of N years the exchange rate must be Eo(l + rb) N" fmu/dmu. The solution to (3) is simpler than it at first appears, rb

CB

-

-

SENSITIVITY O F BREAKEVEN EXCHANGE RATES To measure the effect of the irregular fluctuations in exchange rates, a set of random

CA

n+Ca

if rb takes this value, the cost of the foreign loan is equal to that of the domestic loan and the end of term exchange rate Eo(1 + rb) N" is called the breakeven rate. The treasurer bases his choice of loan on his view of the likely exchange rate at the end of the term of the loan in relation to the breakeven rate. Thus the breakeven rate is a simple and easily computed decision criterion. However the assumption of a uniform geometric depreciation of relative currency values is implicit in this calculation. A study of any exchange rate between two currencies soon demonstrates that the rates fluctuate unevenly rather than follow a smooth path. In Fig. l the exchange rate of £ against the Deutschmark (DM) is plotted at 6 month intervals and the uniform depreciation rate over the same period is superimposed. The question that presents itself is to what extent do irregular changes in exchange rates undermine the usefulness of the breakeven rate.

'

8

7

DM I£ 6

5

4

Path of actual exchonge rote at 6 month intervals - - - - U n i f o r m depreciotion rote

0 Jo

I

I

I

I

I

I

I

JU

Jo

Ju

Ja

Ju

Jo

Ju

1972

1973

1974

1975

I

I

I

I

,ka

Ju

Jo

Ju

1976

1977

I 1978

Jonuory and July rotes used

FIG. 1. Depreciation o f £ against the Deutschmark. (Last working day exchange rates)

335

Omega, Vol. 7, No. 4 TABLE I.

Period of loan (years) Interest rates % p.a. (payable twice yearly) £ DM

I

2

3

4

5

10

15

8.50 3.25

8.75 4.00

9.50 4.75

10.00 5.25

10.25 5.50

11.00 5.50

15.00 5.75

paths to the breakeven rate are generated and the effective costs of the foreign loans are calculated. DM loans (foreign) were compared with £ loans (domestic) over the terms and using the rates shown in Table 1. In order to generate the random paths from the initial rate to the breakeven rate, a mathematical model of the exchange rate movements must be hypothesised. No such models were found in the literature, however Granger and Morgenstern [3] demonstrated that a random walk model was appropriate for security prices on the New York Stock Exchange. As both security prices and exchange rates are decided by similar market forces the suitability of this model was tested for a set of seven major exchange rates. The results of this test provided no evidence to dispute the hypothesis of a random walk (copies of these results can be obtained from Meade). The random walk model is IogEi=logEi_l

+Ei

where

log(l + rl) ~ N ( # , a 2)

where # = log(1 + rb)

and a 2 is estimated from data. Using the Bank of England data illustrated in Fig. 1, the value of a 2 for depreciation of £ against DM over 6 month intervals (to coincide with the loan requirements) from January 1972 to July 1978 was found. This value, a = 0.085, was used to simulate exchange rate changes over the term of the loan, starting at the same initial rate and finishing at the appropriate breakeven rate. For each simulation run the effective interest cost was calculated, that is, the value of y in equation (2). The mean effective costs for each term of loan and 80% confidence intervals are shown in Fig. 2. One would expect the mean effective cost to be close to that of the domestic loan since each realisation of the cost of the foreign loan assumes the final breakeven rate is realised. This is the case, but the variation in the effective cost of the foreign loan is significant and should not be ignored. Thus

E(¢~) = 0

and

16

--

15

--

14

--

cov(ai,El_,)= 0 for s4:O,

and Ei is the exchange rate at time i. The period to period depreciation is ri where

d o. 6 c

t3

12

1 + rl = EdEi_l

Since the exchange rate at the end of the period is fixed at the breakeven rate, this determines the mean value for ri in the simulation of the paths of the exchange rate. In the simulation Normality of Ei is also assumed, this can be justified by the Central Limit theorem in that the change in the exchange rate considered is a sum of many smaller fluctuations. Thus the intermediate exchange rates were generated using

~,o

a

I

I

1

I

I

1

I

1

2

3

4

5

10

15

Term FIG.

of l o o n

(yeor's)

2. Domestic cost of borrowing

(£)

and

80~o

confidence

intervals for effective cost of a D.Mark loan given end of term breakeven rate.

~36

Meade, F i n n e y - - A n a l y s i s o f the Choice Between Domestic and Foreign Loan Facilities

E

.I I

....---_g~ \

E

\,~

1t

/ -/

""7---" "

] r

I I

rL

T e r m of loan

~'1

more extreme paths, that is those that depart most from the uniform depreciation. The purpose of the rest of this section is to describe a convenient form of extreme path which can be used to generate the probability distribution of breakeven exchange rates. To define the extreme outcome, an upper or lower confidence interval for the change in exchange rate over time is used, bearing in mind that the starting point Eo and the finishing point EN, are both defined. Continuing the assumption that the change in the exchange rate is lognormally distributed, the upper confidence limit on the cumulative exchange rate at time t is as follows:

FIG. 3. Uniform breakeven rate and two depreciation rates

lenin 0 to equivalent costs of borrowing.

the case for the use of the breakeven rate as a sole decision criterion is seriously weakened by the variability of exchange rate changes and it is clear that consideration of the variation of cost should be included in a modified decision criterion.

log(I + r i l > t p + k v / t a

Pr

= l-¢,(k)

As the exchange rate changes are constrained to terminate at E s , , another upper confidence limit on the cumulative exchange rate change is: log(1 + rl) > tl~ + k v'~Nn - tkr

Pr \i=

I

= l -- 'l~k)

MODIFICATION OF THE DECISION CRITERION The variations in the depreciation rate mean that the breakeven exchange rate will also vary. That is, the cost of the foreign loan will be equal to that of the domestic loan with a final exchange rate other than the uniform breakeven exchange rate. Thus in order to improve the decision criterion it should be possible to compute the probability distribution of the breakeven exchange rate using the information about the variations in the exchange rate. The diagram in Fig. 3 illustrates the way in which different paths can lead to different breakeven exchange rates. Using uniform breakeven rate as the standard for comparison; for curve A, the currency depreciates faster than the uniform rate, thus interest rates are initially cheaper and for the effective cost of the loan to equal that of the domestic loan the resulting payment of the principal must be higher than that of a loan following the uniform depreciation rate. For curve B the reverse is the case, comparatively high interest payments must be offset by a lower repayment ot principal. The breakeven exchange rates further from the uniform breakeven rate are the results of

fort>0

\i=l

for t < Nn

Similar expressions exist for the lower confidence limits. The extreme path chosen which results in equality of cost of foreign and domestic loans is defined by these confidence limits. The final exchange rate is EN, and the intermediate rates are determined by Ei = Eoexp[ilog(l

+ r) + k a \

min(i, N n - i)]

(4l

where (1 + r) ~'"E o = E~.

Upper and lower extreme paths like curves A and B in figure 3 are defined by taking positive and negative values of k. Equation (4) defines the shape of the path but the final rate Eu. is the solution to this equation: N.

I + I

cn/n I + (1 + ca/n)i(Ed'Eo) (1 + ca/n)~C"(Em,/Eo)

0

(5)

where Ei is defined in (4). Equation (5) is similar to equation (3) but the uniform depreciation rate is replaced by the extreme path depreciation rate. For a given pair of currencies the value of k determines the probability that a sequence of exchange rates starting at Eo and finishing

Omega, Vol. 7, No. 4

337

ing ~ = Q(k, Nn), where 20t is the probability of a breakeven occurring outside the limits. For a given value of 0t, the value of k is determined from (7) and the values of the upper and lower limits form (4) and (5).

O70

0,65

EXAMPLE AND C O N C L U S I O N S

u 060

I I

055

I

I

I

I

I

2 4 5 10 Term of loan (years)(twice yearly repayments) (2) (4) (6) (8) (10) (20) (Nn)

t5 (30)

FIG. 4. The function of Z(Nn) at EN, will result in a foreign loan costing more than a domestic one. That is, if F~ is a realisation of the exchange rate at time i, constrained only by Fo = E0 and FN, = EN,, then the effective cost of a foreign loan is C where C is the solution of (6): cs/n -

I +

i

1

, ( I - + - C/n)i(Fi/Eo)

+

(1 +

C/n)lV"(EN,,/Eo)

- 0 (6)

The probability of the event C > cA is a function of k and the conditions of the loan ca and Nn,

The recommendation resulting from this analysis is that the single point breakeven exchange rate is replaced by a band of exchange rates where breakeven is possible. The point is best made by the example shown in Fig. 5 for a treasurer whose domestic currency is Sterling considering a Deutschmark borrowing using the rates shown in Table 1. If the end of term rate falls in region A, then the DM loan will be cheaper with a probability of at least 90% (0t = 10%) in region C the £ loan will be cheaper with a probability of at least 90%. In region B above 'the' breakeven rate the DM loan will be cheaper with a probability between 50% and 90%; below it £ loans will be cheaper with probability between 50% and 90%. The same procedure can be carried out for any value of ~t, so that the probability distribution of the breakeven rates can be defined as precisely as required. By using this modified criterion the treasurer will have a clearer representation of the effects associated with the end of term exchange rates than the somewhat deceptive picture given by the single breakeven rate.

i.e. Prob (C > cA) = Q(k, cs, Nn) where Q is a function whose structure is to be determined. Values of C were found for randomly generated paths for F~, for different values of N n and ca. N o significant effect on Q(.) was found due to cB, but the number of payments of interest, Nn, did have an effect. The function Q(k, Nn) was found to be adequately described in terms of the Normal distribution and an empirical function Z ( N n ) as follows Q(k, Nn)

--~ 1 -

~

.

35

:" ~ . ~ ' ~ " "

REGION A

_~U~

_ ~ The breakeven rate

f_

i

limit

~. ~ ' ~

REGIOCN

REGION C

"" "

REGIONB

'

(7)

The form of Z ( N n ) was found by experimentation and is shown in Fig. 4. Thus an upper and lower confidence limit on the end of term breakeven exchange rates can be found using the value of tr and specify-

10

o.e

I

I

I

I

I

I

I

1

2

3

4

5

10

t5

Term of loan (yeors)- (interest paid tw/ce yearly)

FIG, 5. 80% limits on breakeven rates for D.Mark versus £,

338

Meade, Finney--Analysis of the Choice Between Domestic and Foreign Loan Facilities

REFERENCES 1. CrlOWN JF (1973) Financial Times Tax Newsletter (April). 2. Flr~NEV M & MEADE N (1978) A practical approach to corporate borrowing and exchange risk. Euromoney (October), 191-197.

3. GRANGERCWJ • MORGENSTERNO (1970) Predictability of Stock Market Prices. Heath Lexington, Massachusetts, USA. FOR CORRESPONDENCE: Dr. Niyel Meade, Department of Manayement Science, Imperial College of Science & Technoloqy, Exhibition Road, London SW7 2BX, UK.

ADDRESS