Credit Risk Spreads in Local and Foreign Currencies
Dan Galai
and
Zvi Wiener
School of Business Administration
The Hebrew University of Jerusalem
Jerusalem 91905
Israel
Preliminary draft, version of 28-Sep-07
Keywords: credit spread, foreign debt, Merton’s model
JEL codes: G12; G13; G15
We wish to thank Zvi Bodie, Michel Crouhy, Rick Levich, Haim Keidar-Levy, Benzi Schreiber and Tony Saunders for
helpful discussions of the issues and to participants of the seminars on Risk Management in Moscow, and the IMF. We
acknowledge financial support from the Zagagi Center and the Krueger Center at the Hebrew University.
Credit Risk Spreads
Abstract
We study in this paper the considerations that should be taken into account in the decision
of currency decomposition of debt.
The problem we are addressing in this paper is what should be the differential credit spread
for a given firm, whether it issues bonds in local currency or in a foreign currency. For a
firm which wants to raise a given amount in debt, and which can either float bonds
denominated in local currency, say dollar, or in foreign currency, say euro, what will be the
resulting credit spread in each case Can we argue that since the leverage in both cases is
the same (by construction) than also the risk premium should be the same? Given the
investments of the firm we can find a financial decision which minimizes the probability of
default for a given leverage.
Introduction.
Today's world is characterized by growing globalization of enterprises, banks and financial
markets. In the past, a local firm was able to raise money only in local financial market. It
was rare for an American company to raise debt in Europe for example. Currently,
companies operating internationally, buying and selling in global markets, and dealing with
many currencies, have opportunities to finance their operations in different markets. Being
able to raise money abroad means that a local company should exploit such an opportunity.
We study in this paper the considerations that should be taken into account in the decision
of currency decomposition of debt.
The problem we are addressing in this note is what should be the differential credit
spread for a given firm, whether it issues bonds in local currency or in a foreign currency.
For a firm which wants to raise a given amount in debt, denoted by B, and which can either
float bonds denominated in local currency, say dollar, or in foreign currency, say euro, what
will be the resulting credit spread in each case. Credit spread is defined as the yield to
maturity of the bond, minus the default-free rate, in the relevant currency. Can we argue
that since the leverage in both cases is the same (by construction) than also the risk
premium should be the same?
2
Credit Risk Spreads
The finance literature dealt with hedging activities by multinational firms and the rationale
behind them. However, the literature has ignored the other side of the same coin, namely,
how financing decisions can be affected by the nature of the assets of the corporation. In
particular, the relationship between the composition of the assets of the corporation and the
currency of the debt is studied in this paper.
The Model.
We propose to analyze the differential credit spreads by using the economic model of
Merton (1974). An American firm, with assets' value of V is floating pure discount debt with
face value F, and market value of B, to be paid at time T, when the risk-free rate is a
constant r. All the parameters above are in dollar terms.
It can be shown, under the appropriate set of assumptions, needed for the Black-Scholes
and Merton model, that the yield to maturity for such a loan, y, is given by
y
1 B
1  V

ln    r  ln  rT N (d1 )  N (d 2 ) 
T F
T  Fe

(1)
where
 
T

 2 , d  d  T
2
1
 T
 V
ln  rT
Fe
d1  
2
And where  is the standard deviation of the rate of return on the assets of the firm.
Following the standard Merton’s model we can express the value of the bond as
B  Fe  rT  Put (strike  F )  Fe  rT  V  N (d1 )  Fe  rT  N (d 2 )
Or after a simple transformation we have
B  V  N (d1 )  Fe  rT  N (d 2 )
(2)
Now assume that the same firm is considering issuing debt denominated in euros,
for the same present value of debt, i.e., x0BE = B where subscript E denotes the euro
currency, and x0 is the current exchange rate (assumed to be 1:1 for simplicity of
presentation). We have to find simultaneously the appropriate face value of the eurodenominated bond, FE and the yield to maturity of the bond, RE, such that x0BE = B. We
further assume that the time to maturity, T, is unchanged, and the risk-free Eurodenominated interest rate is rE.
3
Credit Risk Spreads
Since we introduce another stochastic variable beyond V, namely, the exchange rate
between the dollar and the euro, we have to add an assumption that the exchange rate also
follows Wiener process with a constant x, which stands for the standard deviation of the
rate of change of the exchange rate, and a constant correlation, , with the rate of return on
the assets of the firm.
In order to find the face value of the foreign debt, FE, such that the current value of
the debt in dollars is equal to B, we need to solve the following equation:


B  V  N (d1 )  Fe  rT  N (d 2 )  x0  VE N (d1E )  FE e  rET  N (d 2 E )  x0 BE
(3)
Where subscript E denotes values in euros and x0 is the current exchange rate. For
example E is the volatility of assets of the firm in euro. One can easily show that
 E2   2   X2  2 X  . The only unknown in equation (3) is FE and the right hand side is
monotonic in this variable, thus there exists always a unique solution (which cannot be
found analytically, since d1E and d2E are functions of FE.
The spread for a debt denominated in foreign currency is given by
sE  yE  rE  

1  VE

ln 
N
(

d
)

N
(
d
)
1E
2E 
T  FE e rET

(4)
and
d1E
 VE
ln 
FE e rET


E
  E2
 
T
2

, d 2 E  d1E   E T
T
(5)
Numerical Examples and Illustrations.
Let us illustrate this approach with numerical examples. Assume an American company
with assets value of V=$100, issuing a pure discount (zero coupon) bond in US dollars with
a current value of B=$70, maturing in T=5 years. The standard deviation of the rate of
return of the assets (in dollars) is =20%. We further assume that the riskless interest rate
in the US is r=5%. Given these parameters, it can be shown that the face value of the bond
must be F=$98.27 in order to get a current value of $70.
From equation (1) for the bond spread, it can be shown that the yield to maturity on
the bond is 6.78%, or a spread of 178bp. The risk-neutral probability of default (RNP) as
measured by N(-d2) is 35.4%. We can also calculate the present value of the credit risk of
4
Credit Risk Spreads
the bond (see Crouhy, Galai, and Mark (1999)), by the value of the 5-year put option on V
with exercise price F.

N (  d1 ) 

P  V  N (d1 )  Fe rT N (d 2 )  N (d 2 ) Fe rT  V
N
(

d
)
2 

(6)
Where N(-d2) is the RNP of default and the value in the bracket is the present value of the
potential shortfall of the corporate bond from the promised value of F at time T (also called
LGD – loss given default).
In our numerical example the present value of the credit risk is $6.53, or 6.53% of the
assets of the firm.
Now let us add that the current exchange rate x0=1, the risk-free rate in euro is also
5%, and the standard deviation of the exchange rate is X=6%. In Table 1 we show the
results for the bond, denominated in euros, with current value of $70, for different
correlations between the value of the assets and the exchange rate
5
Credit Risk Spreads
Table 1. Euro-debt spread, face value, PD, and the cost of credit risk for different
correlation values for a given current value of debt ($70). (Both dollar and euro risk-free
interest rates are 5%.)
Correlation
Face value (in €)
Spread
PD (risk neutral)
Put value
-0.4
102.6
265 bp
43.0%
9.92
-0.3
101.8
249 bp
41.7%
9.29
-0.2
101.0
233 bp
40.4%
8.66
-0.1
100.2
217 bp
39.1%
8.04
0
99.4
202 bp
37.7%
7.43
0.1
98.6
186 bp
36.2%
6.83
0.2
97.9
171 bp
34.6%
6.24
0.3
97.1
155 bp
33.0%
5.66
0.4
96.4
140 bp
31.2%
5.09
In Table 1, it is shown that for a correlation factor of 20%, for a face value of the debt €97.9
(euro), the present value of the bond is $70 (dollar). In such a case the spread over the
euro-denominated risk free rate is 171 bp (compared to 178 bp for the dollar denominated
bond), and the RNP of default is 34.6% (compared to 35.4%). The present cost of the
credit risk is $6.24 (compared to $6.53). In other words, issuing euro-denominated bond for
the same present value in dollars can lead to lower credit risk and lower probability of
default.
The last result is even more apparent if the correlation parameter is 40%. In this case
the spread declines to 140 bp only, and the value of the credit risk is $5.09. As can be
expected the results change for low abd negative correlations. For example, for =-20% the
face value of debt should be €101 and the credit spread should be 233 bp.
The intuition behind these results is clear, once it is considered. By issuing debt in a
foreign currency (the euro), part of the asset risk is being offset by the currency risk, when
there is a high correlation between the rate of return on assets, measured in local currency
(the dollar) , and the exchange rate. It is like a “natural hedge” between the assets and the
liabilities of the firm. When the correlation factor is zero (or negative), introducing a foreign
currency liability adds another risk factor to the firm, and makes it riskier.
Figure 1 describes the spread for the FX bonds as a function of the correlation rate
between the rate of return of the firm in local currency and the rate of change of the
6
Credit Risk Spreads
exchange rate, for different levels of the volatility of the exchange rate. The base case, as
described in Table 1, is for X=6%.
In Figure 1 we also show the spreads for X=3% and X=9%. At first glance it looks
odd that all three graphs intersect at high correlation rates, though all spreads decline with
. To understand the logic behind the intersection we should remember that
 E2   2   X2  2 X 
(7)
2
This function, for the volatility of the firm in euros shows clearly why  E is a declining
function of , and also that the product of  and X (for a given ) affects the speed of
2
2
decline of  E . The credit spread is an increasing function of  E (and 2) so when  and
X increase, the credit spreads decrease.
400
300
200
100
-1
-0.5
0.5
1
Figure 1.
The horizontal line is the spread for a local debt, the other three lines show spreads in
foreign currency for X=3%, 6%, and 9% with V=100, r=5%, =20%, T=5y, rE=3%.
The flat line at spread of 178bps (on the vertical axis) is the case where the volatility of the
exchange rate is zero, X=0%, and all the business risk in dollars (or euros) stems from the
dollar volatility of assets. The region below the flat line depicts the spreads in euros that are
below 178bps, or, in other words, the FX spreads that are smaller than the local spreads.
Implications and Conclusions.
Our analysis clearly shows that the credit spreads for a company, with a given leverage in
market value terms, are not constant and depend on the currency decomposition of the
7
Credit Risk Spreads
debt. The higher the correlation between the debt currency the lower is the resulting credit
spread.
Our results have interesting implications, beyond the determination of the credit
spreads and the probability of default, for the practical case of corporate taxation. Higher
spreads mean higher interest expenses and hence higher tax deductions. Therefore, if we
start with a given V and B, the tax effect may increase the value of the firm to its
shareholders (and bondholders) for higher credit spreads.
References.
Crouhy M., D. Galai and R. Mark "Credit Risk Revisted," Risk (Credit Risk Supplement),
(March, 1998), pp. 40–44, also reprinted in Credit Risk: Models and Management (ed. By I.
Shimko), Risk Book (1999).
Cornell, B. and A. C. Shapiro. "Managing Foreign Exchange Risks." Midland Corporate
Finance Journal, Fall 1983, pp. 16-31.
Crouhy M., D. Galai and R. Mark Risk Management, McGraw-Hill, May, 2000.
Galai, D., 1988, Corporate Income Taxes and the Valuation of Claims on the Corporation.
Research in Finance Vol. 7: 75-90.
Levich,R.M. International Financial Markets, 2nd edition, McGraw-Hill 2001
Litterman, R. and T. Iben, 1991. Corporate Bond Valuation and the Term Structure of Credit
Spreads, Financial Analysts Journal (Spring), 52-64.
Merton, R. C. 1974. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.
Journal of Finance 29: 449-70.
Merton, R. C., 1977. On the Pricing of Contingent Claims and the Modigliani-Miller
Theorem, Journal of Financial Economics 5, 241-249.
8
Credit Risk Spreads
Appendix A. Determination of the face value of debt in the foreign currency.
By using the equation (3) we can implicitly solve for the relationship between the face value
of the debt in local and in the foreign currencies. Figure A1 shows the relationship between
FE and F for various levels of X for a given total value of assets V=100. Each point on the
graph corresponds to a different present value of debt. In this case B increases with F (it
should be noted that in Table 1 we keep B constant).
X 6%
FF
X 20 %
300
X 30 %
250
200
150
100
50
80
100
120
140
160
180
200
F dom
Figure A1. FF as a function of F for
V=100, r=5%, =20%, T=5y, rE=3%, X=6%, 20%, 30%, =10%.
As can be expected, FE is increasing with F and the rate of increase is a function of the
volatility of the exchange rate. The sensitivity of FE(F) increases with X. Figure A2 depicts
the relationship between FE and F for various levels of the correlation  and fixed X=6%.
20 %
FF
0%
180
160
20 %
140
120
100
80
80
100
120
140
160
180
200
F dom
Figure A2. FF as a function of F for
V=100, r=5%, =20%, T=5y, rE=5%, X=6%, = -20%, 0, +20%.
9
Credit Risk Spreads
Appendix B. Calculation of Betas.
Table 2. Betas of stock and bond for various correlations. V=$100, the current value of the
debt is $70. Both dollar and euro risk-free interest rates are 5%. In the case of domestic
debt beta of equity is 2.6479, and beta of debt is 0.2938.
correlation
Beta of Equity
Beta of Debt
PD (risk neutral)
-0.4
2.5188
0.3491
43.0%
-0.3
2.5397
0.3401
41.7%
-0.2
2.5617
0.3307
40.4%
-0.1
2.5847
0.3208
39.1%
0
2.6090
0.3104
37.7%
0.1
2.6346
0.2995
36.2%
0.2
2.6616
0.2879
34.6%
0.3
2.6903
0.2756
33.0%
0.4
2.7207
0.2625
31.2%
V
 N (d1E )  V
S
V

 N (d1E )  V
x0 BE
 SE 
 BE
2.5
2
1.5
1
0.5
-0.75
-0.5
-0.25
0.25
0.5
0.75
Figure B1. Beta of equity and debt as a function of the correlation.
10