Fixed exchange r ate r egime

( ^RR

^it

^η

ⁱ

^{, π}

^t

) ^{= −} ( ^φ

^c

^σ

^Hx

⁺ ( ^{1 − φ}

^c

) ^σ

^Fx

^{+ 2 φ}

^c

( ^{1 − φ}

^c

) ^σ

^HxFx

^{− φ}

^c

^σ

^HxHm

⁻ ( ^{1 − φ}

^c

) ^σ

^FxHm

)

cov

^{2 2 2 2}

( η

i

− ¹ ) ( σ

imHm

− φ

c

σ

Hxim

− ( ¹ − φ

c

) σ

Fxim

) ^.

+

The covariance of the real rate of returns on assets of country I with the rate of money growth of country j and the rate of output of country j are:

( ( ) ^, ) ( ¹ ) ( ¹ ) ^,

cov RR

_it

η

_i

m

_jt+1

= φ

_c

σ

_Hxjm

+ − φ

_c

σ

_Fxjm

+ η

_i

− σ

_imjm (59) and

cov ( RR

it

( ) η

i

, x

jt₊₁

) = φ

c

σ

Hxjx

+ ⁽ 1 − φ

c

⁾ σ

Fxjx

+ ⁽ η

i

− 1 ⁾ σ

jxim (60) where i, j = H, F.

Comparing Proposition 2 and Proposition 6, the correlation of stock returns with the inflation rate, the growth rate of money and the growth rate of output are determined by the interaction of two countries, and in particular, the correlation of the money processes of country H and F will affect them via the leverage ratio.

However, these factors are ignored.

2.2 Fixed exchange r ate r egime

If the exchange rate is fixed at some constant level, from (44) we know that processes in the home money supply follow processes in foreign money policy, such as is mentioned in Lucas (1982), and fixed exchange rate regimes are equivalent to a single currency regime. Therefore the money policy of country H can be presented as the following.

Ft

.

H F

Ht

M

M ε

Ω

= Ω

Hence we can modify (53) directly:

( ^, ) ( ) ^{= 1 1 ,}

i ^{Htc+1 1Ft−+1c Fti−+11}

^.

i

it

n x x m

RR

^{φ φ η}

η

η κ

(61) In (61), the only difference between H asset returns and F asset

returns is the leverage ratio.

Pr oposition 7 In a fixed exchange rate regime, the covariance of the real rate of returns on assets of country i with the exchange rate is:

( )

( ^, ) ^{0 .}

cov RR

_it

η

_i

ε

_t+1

=

(62) The correlation of domestic asset returns with foreign asset returns is determined by real variables and monetary variables:

( ) ( )

( RR

Ht

η

H

, RR

Ft

η

F

) = φ

c

σ

Hx

+ ( 1 − φ

c

) σ

Fx

+ 2 φ

c

( 1 − φ

c

) σ

HxFx

cov

^{2 2 2 2}

+ φ

c

( η

H

+ η

F

− 2 ) σ

HxFm

+ ( 1 − φ

c

)( η

H

+ η

F

− 1 ) σ

FxFm

+ ( η

H

− ¹ )( η

F

− ¹ ) σ

Fm²

^.

In a fixed exchange rate regime, the correlation between stock returns and the exchange rate is undoubtedly zero. We now turn briefly to the correlation between common stocks, H stock and F stock, without considering financial leverage.

( ) ( )

( ^{1 , 1} ) ² ( ¹ ) ( ¹ ) ^.

cov RR

_Ht

RR

_Ft

= φ

_c²

σ

_Hx²

+ φ

_c

− φ

_c

σ

_HxFx

+ − φ

_c ²

σ

_Fx² (63) Not too surprisingly, the correlation is the same as between (57) and (63).

Pr oposition 8The correlation between stock returns and the inflation rate in a fixed exchange rate regime is

( )

( RR

it

η

i

, π

t

) = − φ

c

σ

Hx

+ ( 1 − φ

c

) σ

Fx

+ 2 φ

c

( 1 − φ

c

) σ

HxFx

cov

^{2 2 2 2}

+ φ

c

σ

HxFm

+ ( 1 − φ

c

) σ

FxFm

⁺ ( ^η

ⁱ

^{− 1} ) ( ^σ

^Fm2

^{− φ}

^c

^σ

^HxFm

⁻ ( ^{1 − φ}

^c

) ^σ

^FxFm

) ^.

The correlation of common stock returns with the rate of money growth and the growth rate of output are:

( ( ) ,

+1

) ⁼

cov RR

_it

η

_i

M

_jt

φ

c

σ

HxFm

+ ( ¹ − φ

c

) σ

FxFm

+ ( η

i

− ¹ ) σ

Fm²

and

cov ( RR

_it

( ) η

_i

, x

_jt+1

) ⁼ φ

c

σ

Hxjx

+ ( ¹ − φ

c

) σ

jxFx

+ ( η

i

− ¹ ) σ

jxFm

^.

In order to mantain the constant exchange rate, the money processes of countries H and F must be the same. Hence this world can be treated as a single country. The correlation structure presented in Proposition 8 is similar to Proposition 6, in all aspects except for the money process.

If the exchange rate is managed floating, we can easily show that

( ) ( ) ( ) ( ) ^,

, 1

, 1

^φ ₁ ^1φ₁ ^η ₁^{1 1κ η} ₁^{1 κ}

η

η = κ

_Ht^c_{+ Ft}⁻₊^c _Ht^H₊^{− −} _Ft^F₊⁻

H H

Ht

n x x m m

RR

where

κ

is the control coefficient with 0≤

κ

≤1. Note that

κ

=0 implies a flexible exchange rate regime and

κ = 1

implies a fixed exchange rate. In this regime, the results stated above including the returns on assets and the correlations of asset returns with money growth, inflation, output growth, etc., are the results of a linear combination of both a fixed and a flexible regime.

We have shown the role of capital structure on asset pricing in a two-country world. The conclusions of this setting can be summed up as follows. First, stock return is decreasing with the exchange rate, however, if leverage is absent, the exchange rate will not affect stock returns. At the same time, if the leverage ratio is 0, a portfolio of stocks H and F cannot diversify away risk. Second, the returns of stocks H and F would be the same without financial leverage. Finally, the

correlations of the real rate of return on stock i with output growth and the real rate of return on stock j differ in the flexible and fixed exchange rate regimes if there exists financial leverage.

Ⅶ. Conclusion

In this paper we consider the effect of financial leverage on the relationships between stock returns and macroeconomics variables in a real balance monetary economy. Financial leverage is a commonly used financial tool, but, there are few theoretical studies on its effects. To our knowledge, this paper is the first paper which attempts to employ financial leverage as a vehicle to analyze the correlations of stock with other economic variables such as money supply, the inflation rate, and economic growth, in a theoretical manner.

Several interesting results are obtained. First, the real rate of return on common stock is a function of the output growth rate and the inflation rate. This result greatly differs from those documented in traditional monetary models. Second, the correlation of the real rate of returns on common stock with the rate of inflation may be positive without the help of the phenomenon of stagflation. The covariance of stock returns with the growth rate of output is influenced by the interaction of the money process with the growth rate of output. Third, we analyze asset pricing in an open economy, where we find that asset returns are equal for both domestic and foreign assets if we neglect the effect of financial leverage. However, the foreign stock return is affected by the exchange rate if there exists financial leverage, and the portfolio of domestic and foreign stocks will be able to diversify away this risk.

The effect of financial leverage on asset pricing has been investigated in this paper. However, this model is a simple one; further extensions may be possible. For example, the financial leverage ratio can be built into the CIA model in order to analyze asset returns. It has been proved that the inflation rate is a negative factor for real rate of stock with CIA model, however, this may not be true when we take leverage ratio into account. This ratio may also apply to an incomplete market, such as an economy with taxation or transaction costs.

Appendix A: Pr oof of Pr oposition 1

Proof: The real rate of canonical asset returns is:

( ) ( )

( ) ^η

η η

^{η η}

, ,

, 1

¹

1 1 1

n p

y P n

n p RR

zt

t t zt

t

− + +

+

− +

=

, (A1)

where n is term period and

η = 1

means that without financial leverage,

η = 0

^is the asset of nominal bonds, where

η > 1

means there exists financial leverage.

According to the optimization program, we have

( )

[

¹

^{1 , η}

^{11 11 1 11}

] ^{( ) , η ,}

δ E

_t

p

_zt+

n − x

_t⁻₊

+ P

_t^η₊⁻

y

^η_t+

x

_t⁻₊

= p

_zt

n

(A2)

One of assumptions in this study is that the growth rate follows i.i.d., and therefore we take sign t out of the expectation. The prices of assets can be solved for with the recursive method.

( ) ^, η κ

_t^{η 1 η}_t

^,

zt

n P y

p =

⁻ (A3) where k is a function of the expectation of economic growth rate and inflation rate.

We can then solve for the real rate of returns on assets from (A1) and (A3):

( ) ^,

1

^,

The real rate of return on stock is an increasing function of the inflation rate, shown in (A4). Putting (10) into (A3), the price of an asset can be solved for as follows:

( ) ( )( ^{, ,} )

¹ t

^,

condition is

χ < 1

. Furthermore, we can rewrite (A2) as:

( ) ( ) ( ^, η χ η κ ^{1 ,} η )

1

χ ( ) η ^. κ n = n − + a

_n−

Using the recursive technique, we have the following relation of asset prices:

( ) ( ) ( ( ) )

Placing (A5) into the definition of the real rate of return on an asset, the closed form solution is shown below.

The solution for the real rate on assets follows.

( ) ( )

By (A6), we know that this solution is independent of term period. The special case, without financial leverage, is that this closed form solution is independent of the money process, presented as follows:

( ) ^{, 1 = 1} ( ) ¹

^t+1

^,

t

n x

RR χ

The real rate on nominal bonds is:

( ) ^{0 = 1} ( ) ⁰

^{t−+11 t+1}

^.

t

m x

RR χ

In fact, this real rate is a function of the inflation rate.

Appendix B: Der ivation of equation (27)

Proof: The source of debt financing in this study is the nominal bond. Its real rate of return is:

According to the first order conditions of optimal processes, we have:

( ) ( ( ) )

Using the recursive method, the closed form solution is presented below.

( ) ( )

t closed form solution of the real rate of asset returns is

( ) ^{, 0 1 1} ( ^{1 1} ) ^{( ) 0 ,}

From (B4), we know that this depends on term period. This stems from the fact that dividends are adjusted every period but always lag slightly behind the current state.

The rate of this asset is thus random but known. The expectation of the real rate of an asset over one period is:

( ) ^,

which is equal to (17). The real rate of returns on assets which have paid a payoff but never paid back principal is:

( ) ^{, 0 1} ( ) ^{0 ,}

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