Inflation, Asset Returns and Exchanges Rates in a Monetary Economy with Financial Leverage

Inflation, Asset Retur ns and Exchange Rates in a Monetar y

Economy with Financial Lever age

Weng-Tzong Hsiao

Department of Economics, National Taipei University

Mao-Wei Hung

College of Management, National Taiwan University

Shue-J en Wu

Department of Economics, National Taipei University

ABSTRACT

This paper investigates the effects of financial leverage on asset returns in a monetary economy. We show that monetary policy has real effects on asset prices. But, the real effects of money policy under our model are contrary to those documented in previous studies. Moreover, the structure of financial leverage gives us a new explanation for several important empirical findings such as the risk premium, the correlations of the real rates of return on assets with the rate of inflations , the growth rate of output, and the rate of money growth. We also extend this model to a two-country world and find that stock returns will be affected by exchange rates via financial leverage.

Keywor ds：Financial leverage, Asset returns, Exchange rates, Inflation

Ⅰ. Introduction

The interaction between finance and macroeconomics has been the focus of recent research. In particular, researchers have attempted to understand the relationships between stock returns and marcreconomic variables such as money supply, inflation and output. The goal of this paper is to study the implications of financial leverage on these relationships. In a recent paper, Abel (1999) shows that stock returns are increasing with the leverage ratio. Benning and Protopadakis (1990) resolve the equity premium puzzle using reasonable leverage ratios in the equity market. However, both of these papers are built on a real economy, and hence the type of debt examined is, in both papers, real bonds. In this paper, we consider nominal bonds, which pay a stream of known nominal payoffs.

The model we consider is a real balance monetary economy. The cash-in-advance(CIA) and transaction cost methods are the other two popular alternatives to incorporate money which are mentioned in the literature. Unfortunately, the closed-form solution of general price can not be solved from above. For example, Bohn(1991) and Balduzzi(1996) assume that CIA constraint is binding. And Marshall (1992) uses the model with transaction cost to connect real variables with nominal variables, but the inflation rate can still not be determined.

Although Bakshi and Chen (1996) solve for price level endogeneously, the monetary policy is neutral in their papers . In our paper, we can not only solve for the price level in the model, but we are also able to show that money is not neutral.

The intuition behind our model is easy to understand. The financial risk only introduces production risk in a real economy, but in a nominal economy, it also generates inflation risk. Hence, two effects on stock returns are exhibited in our model: one is the increase in dividends per share, and the other is a redistribution between debtors and creditors resulting from inflation. One should note, however, that in the real economy, the redistribution effect will disappear. Financial leverage is an important factor to connect variables such as the real rate of returns on an asset, the inflation rate, and exchange rates in a model in which the real money balance is presented. When financial leverage is absent, the model will be reduced to the standard one in which monetary policy is neutral. On the contrary, when financial leverage is present, we can show that monetary policy has real effects on asset returns. However, the conclusions with respect to monetary policy may be the opposite of some results in previous papers, such as in Lucas (1978) and Svensson (1985). In these papers, the real rates of return on stocks are increasing with the inflation rate and the growth rate of the money supply in the presence of financial leverage. This finding gives us a new perspective to re-examine some empircal findings in the realm of asset pricing, such as the risk premium, the issue of whether stocks can be employed to hedge against inflation as documented in Fama and Schwert (1977), and whether the exchange rate will affect stock returns.

The fact that money is not neutral has frequently been documented in empirical research. For example, Day (1984), Svensson (1985), Balduzzi (1996), and Basak and Gallmeyer (1999) support that monetary policy has an effect on real asset prices. Most of these papers also support the conclusion that the real rate of return on stock is a decreasing function of the growth rate of the nominal money supply. However, we show that these results may not hold any longer when financial leverage is taken into account formally. In this issue, our theoretical results are consistent with empirical findings in Lastrapes (1998), who finds that real equity prices respond positively to nominal money shocks in France, Holland,

Japan, the UK and the US.

In addition, we study the correlation between the real rates of return on stock and macroeconomic variables such as the rate of inflation, the growth rate of money, and the growth rate of output. Previous studies, such as Fama and Schwert (1977), Fama (1981), Day (1984), Stulz (1986) and Marshall (1992), find that there exists a negative correlation between common stock returns and inflation \footnote{Huybens and Smith(1999) also document this negative relationship in a period of low inflation rate.} This relationship contradicts the traditional view that stocks ought to act as an inflation hedge. Fama (1981) argues that this puzzle occurs in a situation in which there exists a negative relationship between inflation and real activity, which is known as the phenomenon of stagflation. Unlike Fama, our results show

that this puzzle can be resolved via financial leverage. This is an interesting finding regarding the role played by financial leverage in asset markets.

We also extend our model to a two-country world. Chiang and Chiang (1996) finds that, in an international asset pricing model, volatility in international stock returns is mainly influenced by exchange rate volatility, which supports the international capital market integration hypothesis. In our paper, stock returns are shown to decrease with the exchange rate. However, if leverage is absent, then stock

returns are not affected by the exchange rate. We also demonstrate that financial leverage plays an important role in explaining why the correlations of the real rate of return on stock and output growth are different in flexible and fixed exchange rate regimes.

The reminder of the paper proceeds as follows. Section 2 presents our model. Section 3 solves for equilibrium asset prices and the real rate of returns on assets. Section 4 analyzes the link between capital structure and dividends. The second moments are discussed in Section 5 which includes the covariances of stock returns with the inflation rate, the growth rate of the money supply, the growth rate of output and other assets. An open economy model is investigated in Section 6, and the final section presents concluding remarks and some extensions for future study.

Ⅱ. The economy

The economy under our assumptions is made up of representative consumers with rational expectations, and the goods market, money market, and asset market in this economy are all in equilibrium. The key assumptions of this study are as follows.

Each individual allocates her resources among consumption, money balance and assets. The utility function is a function of the consumption of a single perishable good and real money balance. Regarding the functional form of the utility function, the log utility functions, such as those used in Lucas and Stokey (1987), Stulz (1986), Bakshi and Chen (1996), Balduzzi (1996), are considered. In other words, we assume

(

1

)

ln

,

ln

t d t t t

P

M

c

u

=

θ

+

−

θ

where

c

_t is the consumption at time t,

M

_tdis the nominal money demand at time t,

t

P

is the price of consumption good at time t,

M

_td/

P

_t is the real demand for money at time t.

The financial leverage decision is one of the more important financing decisions made by corporations. If the debt one considering is real bonds, then the real dividends are

y

η shown by Abel(1999), where y is the real revenues ( or output level ) and

η

is a constant which is determined by the leverage ratio.

η

=1 indicates that the funds demanded by the firm are financed by equities purely. One should also note that

η

is increasing in leverage ratio. On the other hand, if the debt one considering is nominal bond, the nominal dividends can be

taken as the formula:

( )

η y

P

ND

=

,

where

ND

represents nominal dividends and

P

_y is the nominal revenues. If financial leverage is absent, the nominal dividends will reduce to

P

_y, such as in Bakshi and Chen (1996). Therefore, the real dividend can be represented as follows:

,

1 η η

y

P

RD

=

−

where

RD

represents real dividends. It should be noted that the presence of nominal bonds causes

RD to depend on the price of a consumption good

P

, as the impact of leverage operations would be different at various levels of

P

.

Owing to randomness in the growth rates of output

{ }

x

_t and the growth rates of the nominal money supply

{ }

m

_ts , we have to specify the probability structures of the stochastic processes of

{ }

x

_t and

{ }

m

_ts . Note that money is issued by the government and thus the stochastic properties of

{ }

m

_ts depend on the behavior of the government. However, for simplicity, we do not describe the behavior of the government in detail; we simply assume that

{ }

m

_ts follows some stochastic process. We assume that both

{ }

x

_t and

{ }

m

_ts are independent and identically distributed (i.i.d.). As for the joint distribution of

x

_t and

m

_ts, we assume that





















































+ + 2 2 1 1

,

~

m xm xm x m x s t t

LN

m

x

σ

σ

σ

σ

µ

µ

then

(

2 2

)

1 1 t

~

x m

,

x

2

xm m t

m

LN

x

_{+ +}

µ

+

µ

σ

+

σ

+

σ

It should be noted that

σ

_xm may not be zero, i.e.,

x

_t and

m

_ts may be dependent.

Ⅲ. Asset retur ns

The investor's optimal consumption, money demand and portfolio will be discussed in this section. In our model, the investor's maximization problem can be represented as follows:









=

∑

∞ = t t t

u

E

U

E

0 0 0 0

max

δ

, where

(

1

)

ln

.

ln

t d t t t

P

M

c

u

=

θ

+

−

θ

The budget constrant can be shown as follows:

( )

(

,

)

( )

,

1

,

1 1 t d t t zt t t d t t t t zt

P

M

z

n

p

c

P

M

z

y

P

n

p

η

+

η− η

+

=

+

η

₊

+

+

where

z

_t is the share of asset holdings at time t and

p

zt

( )

n

,

η

is the real price of

an n-period asset at time period t. For example,

p

_zt

( )

∞

,

η

is the price of stock.2 The equilibrium conditions of the goods market, money market, and stock market are

c

_t

=

y

_t,

M

_td

=

M

_ts

=

M

_t and

z

_t

=

1

, respectively.3 In order to obtain a solution, we linearize the closed form solution around

(

δ

,

µ

_i

,

σ

_i2

)

. The optimal solution should satisfy the following first order conditions:

t t

c

η

θ

1

=

, (1)

(

1

)

1

,

1 1 1 + + +

−

=













−

t t t t d t

P

P

M

E

η

δ

η

θ

δ

(2)

(

)

[

_{zt 1}

1

,

_{t 1}1 _{t 1}

]

_{t 1 zt}

( )

,

_t

.

t

p

n

P

y

p

n

E

η

η

η

η

δ

₋

₊

η η

₌

+ + − + + (3)

Equation (1) represents the requirement to obtain optimal consumption, the optimal money demand is represented in equation (2), and the asset demand of the investor is shown in equation (3).

Combining equations (1) and (3), we have the relationship for asset returns:

[

_{t t+1}

]

=

1

,

t

RR

S

E

[ ]

1

cov

(

_{[ ]}

,

)

,

1 1 + +

−

=

t t t t t t t

S

E

S

RR

RR

E

(4)

where

RR is the real rate of returns on an asset and

S

is the stochastic discount factor. These two variables are defined as:

( )

η

(

η

_{( )}

)

_η

η η

,

,

1

,

1 1 1 1

n

p

y

P

n

p

n

RR

zt t t zt t + − + +

−

+

=

(5) and

,

,

,

1 1 1 1 1 + + + + +

=

















≡

t t t t t c t t t c t

c

c

P

M

c

u

P

M

c

u

S

δ

δ

(6)

where the second term in the denominator in equation (5) is the real dividend. Equation (5) will reduce to the standard equation in which

η

is equal to one. The expectation of the real rate of returns on an asset cannot be solved directly, unless the covariance between the real rate of returns on an asset and the stochastic discount factor is zero. Therefore, one has to first solve for the prices of an asset and

2

_p

( )

_n

_,

₀

zt denotes the price of a nominal bond.

3 _{The equilibrium condition of the bond market is}

=

₀

t

of a currency in order to derive the real rate of returns on an asset.

The solution for the general price

P

_t can be otained by combining equations (1) and (2), and using the assumption that the growth rates of money and output are i.i.d.. Moreover, that the transversality condition is satisfied is a necessary and sufficient condition for the convergence of the economy. The price can be shown as:

,

t t t

y

M

P

=

Ω

(7) and

,

1

1

1₁ 1 1 − + − +

−

⋅

−

≡

Ω

t t

Em

Em

δ

δ

θ

θ

where the transversality condition is

(

)

0

.

lim

−₊1₁

=

∞ → n t n

δ

Em

Equation (7) shows that the solution for the price of a consumption good is a function of current money stock, the current level of output, and expectations of the money and production processes. Inflation can then be presented as follows.

,

1 t t t

P

P

₊

≡

π

=

m

_t+1

x

_t−₊1₁

.

(8)

The solution for the inflation rate is proportional to the growth rate of the money supply and of output, and this is consistant with other monetary models.

Finally, the closed form solution of the real prices of assets and the real rate of return on assets can be obtained by combining equations (1), (3) and (7).

Proposition 1

In a real balance monetary economy, the price of a canonical asset with financial leverage is a function of money stock, output level, the expected value of the money growth rate, and the variance of the money growth rate.

( ) ( )(

,

,

_t

)

n 1 _t

,

zt

n

n

M

y

p

η

=

κ

η

Ω

− (9)

( )

1

_{( )}

1

,

1 1 + − +

=

t t t

m

x

RR

η

η

χ

η

(10) where

κ ,

( )

n

η

is

(

)

(

( )

(

1

( )

)

)

(

1

( )

)

.

1

δ

1

χ

η

χ

η

χ

η

η

₌

₋

₋

∑

= + n i n i

E

m

t

( )

0

,

η

=

0

κ

means that price of a zero-period asset is equal to zero and

( )

η

E

δ

m

_{t 1}1

δ

exp

{

(

η

1

)

µ

_m

[

(

η

1

)

2

2

]

σ

_m2

}

.

χ

_≡

η−

₌

₋

₊

₋

+

χ

( )

η

<

1

is the boundary

condition.

Pr oof: The proof is given in Appendix A.

return on a canonical asset. As presented in proposition 1, both real and nominal variables will affect the real prices of the asset. This finding is dramatically different from the traditional real balance model which does not involve the leverage strategy,

1

=

η

.

Because the equity does not have a maturity date, its real price can be represented as

( )

,

1

[

( )

(

1

( )

)

]

_t 1 _t

,

zt

M

y

p

∞

η

=

Ω

η−

χ

η

−

χ

η

η− (11)

which is the sum of the discounted value of all future real dividends. The real price of the equity is increasing with output and the money supply, which are both random but still known at time t. The result that real equity prices respond positively to nominal money shocks is consistant with that presented in Lastrapes (1998). The n-period of nominal bonds is an asset that is independent of dividends. The real price of this bond is defined as

( )

,

0

1

[

( )

0

(

1

n

( )

0

)

(

1

( )

0

)

]

_t1 _t

,

zt

n

M

y

p

=

Ω

−

χ

−

χ

−

χ

−

where purchasing power is adversely affected by the active money policy. The price of coupon bonds is

p

_zt

( )

∞

,

0

=

Ω

−1

[

χ

( )

0

(

1

−

χ

( )

0

)

]

M

_t−1

y

_t

.

Next, we want to derive the real rate of returns on an asset. This real rate is a function of the inflation rate which is shown in Appendix A.

( )

,

`

1

,

1 η η

π

η

₌

− ₊ t t t

n

k

x

RR

(12)

where `

k

is a function of the expectations of the growth rate of output and the inflation rate. This result contradicts several early studies, such as Day (1984), Svensson (1985), Balduzzi (1996) and Basak and Gallmeyer (1999), which support a conclusion in which the real rate of return on stock is a decreasing function of the rate of inflation. In fact, in a monetary economy, debtors in general will gain from inflation, which will in turn harm creditors. This means that the real rate of return on stock is increasing with the rate of inflation. Note that the real rate on nominal bonds is a decreasing function of the inflation rate. Furthermore, the closed form solution can be presented as:

( )

_{( )}

1

1 11

.

− + +

=

η

η

χ

η

t t t

x

m

RR

(13)

Equation (13) presents the fact that output growth and money growth are good news for debtors. However, if capital structure is neglected, money growth is trivial for stockholders.

In order to study the risk premium clearly, we linearize the result shown above. The expected real rate of return can be shown clearly under the specific distribution:

( )

(

)

.

2

1

1

2

_{x xm x}2 t

ERR

η

=

−

δ

+

µ

+

η

−

σ

+

σ

(14)

Equation (14) is the explicit solution of expected asset returns with financial leverage. There is also the special case that debt financing is absent, and the real rate of stock returns is then

( )

.

2

1

2

1

2 x x t

ERR

=

−

δ

+

µ

+

σ

(15)

The factors of equation (15) are all real, which means that money is neutral. This result is also shown in Bakshi and Chen (1996). The difference between equation (14) and equation (15) known as the leverage premium is

( )

_t

( ) (

1

1

)

_xm

.

t

ERR

ERR

η

−

=

η

−

σ

(16)

If the correlation of output growth rates and the growth rate of the money supply is positive, then this risk premium is positive. Some empirical results, such as in Stulz (1986) and Marshall (1992), support that the correlation of output growth rates and the money supply growth rate is positive. However, this premium is largely ignored in earlier studies. Benning and Protopadakis (1990) and Abel (1999) resolve the "equity premium puzzle" via reasonable leverage ratios in the equity market. On the other hand, this premium is negative when

σ

_xm is negative. The real rate of nominal bonds under the case that

η

is equal to 0 is

( )

,

2

1

2

0

_{x xm x}2 t

ERR

=

−

δ

+

µ

−

σ

+

σ

(17) which is determined by the inflation rate and the stochastic discount factor.

The real bond is an asset which pays a real known payoff, and hence the real rate of return on real bonds is independent of stochastic processes. The explicit solution for the risk free rate can be determined directly from equation (4):

,

1

1 +

=

t f t

ES

RR

.

2

1

2

−

δ

+

µ

_x

−

σ

_x2

=

(18)

For an explanation of the last term which is shown in equation (18), we can refer to Campbell and Cochrane (1999). In this paper, they state: "...As uncertainty increases, consumers are more willing to save, and this willingness drives down the equilibrium risk-free interest rate." They called this effect precautionary savings. Hence, if financial leverage is absent, then the risk premium can be expressed as

( )

1

2

.

x f t t

RR

ERR

−

=

σ

(19)

The risk premium only depends on the risk associated with output in equation (19). However, the total risk premium in this study when considering financial leverage is

( )

_tf 2_x

(

1

)

_xm

.

t

RR

If

σ

_xm≥0, then the risk premium with financial leverage will be higher than that in the traditional model. However, this premium may be negative under

xm x

σ

σ

η

>

1

−

2 if

σ

_xm

<

0

.

This result tells us that this premium is always positive if we do not consider leverage. However, a novel feature is that this premium may be negative if we take leverage into account. This interesting result differs from those in Abel (1999) and Bakshi and Chen (1996). The risk premium is always positive in both studies. Hence, if

σ

_xm

<

0

, then the leverage ratio should not be too high. Otherwise, in equilibrium, no one wishes to hold this risky asset.

The variance of a canonical asset is

( )

(

RR

_t

) (

1

)

2 _m2

2

(

1

)

_{xm x}2

.

Var

η

=

η

−

σ

+

η

−

σ

+

σ

The risk of common stock stems from the risk of output only if leverage does not exist. Yet the money process will influence the variance of stock when leverage exists. The relationship between the leverage ratio and the variance of common stock is shown below:

( )

(

)

₂

₍

₁

₎

₂

_,

var

2 1 1 2 mx m t

R

σ

σ

η

η

η

η

=

−

+

∂

∂

> +

( )

(

)

.

2

var

2 1 2 1 2 m t

R

σ

η

η

η

=

∂

∂

> +

Ⅳ. Capital str ucture

There are two ways for a firm to finance its funds: one way is to sell shares of stock, and the other is to issue bonds. Benning and Protopapadakis (1990) argue that when market returns are adjusted for leverage, the asset pricing model with a time-additive, constant relative risk aversion utility function may resolve the equity premium puzzle. Abel (1999) resolves the equity premium puzzle via reasonable leverage ratios and the effect of catching up with the Joneses perferences. In this article, we apply this ratio to explain some pieces of empirical evidence of financial economics. The connection between leverage ratio and parameter

η

in a monetary economy will be analysed below.

The debts considered here are nominal bonds which will pay a known payoff,

t t

y

P

Θ

, which depends on time and Θ is a constant. The nominal dividends of levered equity can be presented as follows:

{

0

,

}

,

max

_{1 1}

1 t t t t

t

P

y

P

y

where

ND

represents the nominal dividends and Θ≥0.

P

_t+1

y

_t+1 is the total revenue, and

Θ

P

_t

y

_t is the expenditure of interest. Bondholders will receive a payoff

Θ

P

_t

y

_t in a situation in which

t t t t

y

P

y

P

₊1 ₊1

>

Θ

or

.

1

>

Θ

+ t

m

Otherwise, they receive nothing. By first order conditions, we have the relationship that

(

)

{

}

[

~

n

−

1

,

Θ

+

max

0

,

1

−

Θ

m

−₊1₁

]

=

~

( )

n

,

Θ

,

E

_t

χ

_t

χ

δ

where

χ

~ n

( )

,

Θ

is a function to be determined. However, for simplicity, we assume that

n

=1, and

χ

( )

0

,

Θ

=

0

. So that

( ) ( )

1

,

~

1

,

,

~

t zt

y

P

Θ

=

χ

Θ

where

~

( )

1

,

max

(

0

,

1

11

)

− +

Θ

−

=

Θ

E

m

t

χ

.

The coefficient of the variation of these dividends is shown below:

{

}

[

]

{

}

[

]

(

)

2 1 1 1 1

,

0

max

,

0

max

t t t t t t t t

y

P

y

P

E

y

P

y

P

Var

Θ

−

Θ

−

+ + + + ₌

[

{

}

]

{

}

[

]

(

)

2 1 1

,

0

max

,

0

max

Θ

−

Θ

−

+ + t t

m

E

m

Var

If the bond is default free, then the coefficient of the variation of these dividends can be rewritten as

[ ]

[ ]

[ ]

(

)

.

1

1

2 1 1 2 1 + + +

⋅









₋

Θ

t t t

m

E

m

Var

m

E

(20)

Using the assumption that the growth rate of money is lognormal and i.i.d., then the variance of the growth rate of money can be presented as

[ ]

m

t+1

=

exp

(

2

m

+

m2

) ( )

(

exp

m2

−

1

)

Var

µ

σ

σ

(21)

and

[ ]

(

E

m

_t+1

)

2

=

exp

(

2

µ

_m

+

σ

_m2

)

.

(22) Combining equations (21) and (22), equation (20) can be then rearranged as

{

}

[

]

{

}

[

]

(

−

Θ

)

≅

Θ

−

+ + + + 2 1 1 1 1

,

0

max

,

0

max

t t t t t t t t

y

P

y

P

E

y

P

y

P

Var

[ ]

.

1

1

2 2 1 m t

m

E

σ

⋅









₋

Θ

+ (23)

The nominal dividend defined in the previous section is

(

_{+1 +1}

)

η

.

≡

_{t t}

t

P

y

ND

. (24)

Plugging equation (17) into equation (23), the closed form solution for the dividend is

(

)

η t t t

m

M

ND

+1

=

Ω

+1 (25)

and hence, the coefficient of variance of this dividend is

(

)

[

P

_{t 1}

y

_{t 1}

]

exp

(

2

_m 2 _m2

) (

(

exp

2 _m2

)

1

)

2 _m2

.

Var

_{+ +} η

=

ηµ

+

η

σ

η

σ

−

≅

η

σ

(26)

Comparing equation (23) to equation (25), the relationship of financial leverage and

η

can be presented as

[ ]

,

1

1

1 +

Θ

−

=

t

m

E

η

where

Θ

E

[ ]

m

is the leverage ratio.

The coefficient

η

is a function of the leverage ratio, and

η

is equal to 1 when financial leverage does not exist, and greater than 1 if leverage strategy is considered. Note that

η

is infinity as the leverage ratio,

Θ

E

[ ]

m

, approaches 1. This result means that the funds of a firm come almost all from issuing bonds.4

The real rate on bonds corresponding to this levered equity is (the derivation is in Appendix B):

( )

₍

₎

_{( )}

,

0

1

1

1

1

0

,

~

1 1 1 1 1 + − + + −

−

−

+

−

−

≡

n_{n t n t t} t

n

x

m

x

R

R

χ

ξ

ξ

ξ

ξ

(27)

What the bondholders know is the nominal payoff

Θ

P

_t

y

_t that should be paid in the next period, but not any future payoffs. This property leads to the conclusion that the real rate of a bond depends on the period. The real rate of return on this bond should however be the same as the nominal bond analyzed in the previous section, otherwise there exists the possibility of arbitrage. The real rate of this bond in one period can be shown as follows:

( )

1

,

0

[

1

( )

0

]

,

~

1 1 1 + − +

=

_{t t} t

m

x

R

R

χ

The expectation of this rate is

( )

1

,

0

2

[ ]

1

2

,

~

2 x xm x t

R

R

E

=

−

δ

+

µ

−

σ

+

σ

which is the same as that for nominal bonds in equation (21).

Ⅴ. An analysis of covar iances of asset retur ns with economic var iables

The correlations between the real return of equity and other economic variables, such as the inflation rate, money growth rate, output growth rate, and the rate of

4 _{For more details, please see Abel (1999). In this article, we do not simulate the expected} asset returns of levered equity and canonical assets.

bonds, are the focus of many previous studies. The results for these correlations in our model are given in Proposition 2 . Note that the signs of all these covariances depend on the financial leverage ratio.

Proposition 2

In the setting of the logarithm function of utility, the covariance of the real rate of equity with the inflation rate is

( )

(

,

)

(

)

(

1

)

cov

RR

_t

η

π

_t

=

−

σ

_x2

−

σ

_xm

+

η

−

(

σ

_m2

−

σ

_xm

)

.

(28)

The covariances of the real rate of equity with the rate of money growth and the growth rate of output are:

( )

(

,

) (

1

)

cov

RR

t

η

m

t+1

=

η

−

.

2 xm m

σ

σ

+

(29)

( )

(

,

₊1

)

=

cov

RR

_t

η

x

_t

σ

2_x

+

(

η

−

1

)

σ

_xm

.

(30)

Pr oof: The proof is given in Appendix C.

There is much evidence that stock returns and inflation have been negatively related during the post-1953 period. This result is puzzling given the previously accepted understanding that common stock should be a hedge against inflation. Fama and Schwert (1977), Fama (1981), Day (1984), Stulz(1986), and Marshall (1992) all document that the correlation between real equity returns and inflation is negative in Postwar U.S. data. Fama and Schwert (1977) states "...Although the results are less consistent, common stock returns also seem to be negatively related to the unexpected inflation rate and to changes in the expected inflation rate. Thus, contrary to long-held beliefs, but in line with accumulating empirical evidence, common stocks are rather perverse as hedges against inflation..."

Here we provide a possible reason for this puzzle. In equation (28), if we neglect financial leverage, the covariance of real rates of returns on common stock and the rate of inflation is

( )

(

1

,

₊1

)

=

cov

RR

_t

π

_t

−

(

σ

_x2

−

σ

_xm

)

.

(31)

Therefore, the condition that stock cannot hedge against inflation is

σ

_x2

>

σ

_xm. However, the covariance of stock returns and inflation corresponds to the covariance of inflation and real activity as shown here.

(

₊1

,

₊1

)

=

cov

π

_t

x

_t

−

(

σ

_x2

−

σ

_xm

)

.

(32)

If common stock cannot hedge against inflation, then the correlation between the inflation rate and real activity is positive, which is the phenomenon of stagflation. Fama (1981) explains that the negative stock return-inflation relationship is caused by the negative relationship between inflation and real activity. Here, if capital structure is considered, then there will exist another term to explain the negative relationship between stock returns and inflation. The conditions that common stock may not hedge against inflation are

xm m xm x

σ

σ

σ

σ

η

−

−

+

<

1

₂2 if 2

,

xm m

σ

σ

<

and xm m xm x

σ

σ

σ

σ

η

−

−

+

>

1

₂2 if

σ

_m2

>

σ

_xm

,

Therefore we conclude that the existence of a negative correlation between common stock and inflation does not depend upon stagflation. This explanation differs from that in Fama (1981). In a word, we may state that this is puzzle due to capital structure rather than stagflation. On the contrary, this puzzle may not occur if

xm m xm x

σ

σ

σ

σ

η

−

−

+

>

1

₂2 if

σ

_m2

>

σ

_xm

,

and xm m xm x

σ

σ

σ

σ

η

−

−

+

<

1

₂2 if

σ

_m2

<

σ

_xm

,

Next, the correlation of the real rate of return on an asset with the growth rate of money and the rate of output are shown in equations (29) and (30). If leverage does not exist, then the covariance of the real rate of equity with the money growth rate is decided by

σ

_xm only, and the covariance of stock return with the output growth rate is always positive. However, when considering capital structure, the former is affected by the variance of the growth of money and the latter may not always be positive. The condition of a positive value for the correlation between stock returns and money growth is

xm

σ

>0, or 2

1

m xm

σ

σ

η

>

−

if

σ

_xm

<

0

.

Although

σ

_xm is negative, the correlation between stock returns and money growth may still be positive in the case that financial leverage is high enough.

Stulz (1986) proves the same result that expected real returns on common stocks are negatively related to money growth. This result is supported by Proposition 2 in which capital structure is absent. However, Proposition 2 also presents the other interesting fact that the correlation between the real rate of returns on stock and the rate of output growth is not always positive if financial leverage exists. That condition is

xm x

σ

σ

η

>

1

−

2 if

σ

_xm

<

0

.

Marshall (1992) uses the transaction cost in a monetary economy and finds that the correlation of the real rate of return on assets with the growth rate of money is positive.

Next we derive the relationship between the real rate of stocks and nominal bonds. The result is stated in Proposition 3.

Proposition 3

The covariance of the rate of common stock with that of nominal bonds is:

( )

( )

(

,

0

)

cov

RR

_t

η

RR

_t =

σ

_x2

−

σ

_xm

−

(

η

−

1

)

(

σ

_xm

−

σ

_m2

)

.

(33)

The pr oof: The proof is given in Appendix D.

Miller and Schulman (1999), using U.S. data from 1952 to 1992, find that the correlation between stock returns and bonds is positive for almost the whole period. They think that this fact results from money illusions.

The special case in Proposition 3 is

( )

( )

(

1

,

0

)

cov

RR

_t

RR

_t =

σ

_x2

−

σ

_xm

.

(34)

At the same time, let us recall equation (32), the covariance between the inflation rate and the growth rate is

(

,

₊1

)

=

cov

π

_t

x

_t

−

(

σ

_x2

−

σ

_xm

)

,

=

−

cov

(

RR

_t

( )

1

,

RR

_t

( )

0

)

.

If the common stock returns are co-movement with the bond returns, we know that there must exist stagflation. However, when presented with financial leverage, then the conditions for positive correlation between two assets are

2 2

1

m xm xm x

σ

σ

σ

σ

η

−

−

+

<

if

σ

_xm

>

σ

_m2

,

or 2 2

1

m xm xm x

σ

σ

σ

σ

η

−

−

+

>

if

σ

_xm

<

σ

_m2

,

which may not result from stagflation or money illusion.

Ⅵ. Asset pr icing in a two-countr y wor ld

1. Basic analysis

In this section, we would like to consider asset pricing in an open economy. For simplicity, we assume that there are only two countries (Home and Foreign) in the world. The preferences of people in both countries are indifferent to nationality. The utility function is a logarithm and each agent will maximize life time utility which is restricted by intertemporal budget constraints. The differences are the stochastic processes of output and of money. The markets in this open economy are the commomdity market, the money market, and the asset market. There are five variables : the consumption prices of H and F, the asset prices of H and F, and the exchange rate, which are all determined endogenously.

This two-country model is based on Lucas (1982) in which each agent is assumed to randomly receive a lump-sum transfer at the beginning of time t. Agents in each country consume goods including both H goods and F goods. The output processes of H and F are distributed randomly, and each country only produces its goods. The way to get these goods is to hold stocks of both countries. The stockholders have the right to receive dividends. The utility of the representative consumer is:

(

1

)

ln

,

ln

C

_t

θ

W

_t

θ

+

−

where

C

_t is a composite good with a Cobb-Douglas function shown as follows

,

1 c c Ft Ht t

c

c

C

=

φ −φ

where

c

_Ht is the consumption of H goods,

c

_Ft is the consumption of F goods and

1

0

<

φ

_c

<

.

Money demand includes both H currency and F currency. The real balance can be presented as follows:

,

1 * q q Ft t d Ft t Ht d Ht t

P

M

P

M

W

φ φ

ε

ε

−

















=

where

M

_Htd

( )

M

_Ftd is the demand of H(F) currency,

P

_Htis the price of H goods and

( )

* Ft

P

is the price of F goods denominated in F currency, and

ε

_t is the exchange rate,

0

<

φ

_q

<

1

. The intertemporal budget constraint facing the agent is

(

) (

)

(

)

(

(

)

(

)

)

d Ft t d Ht Ft Ft Ft t F Fzt t Ht Ht Ht H Hzt

n

P

y

z

P

n

P

y

z

M

M

P

η

+

ηH

+

ε

η

+

ε

ηF

+

+

ε

+1 * *

,

,

(

,

)

(

,

)

1 1

,

* 1 * 1 d Ft t d Ht Ft Ft t Ht Ht Ft F Fzt t Ht H Hzt

n

z

P

n

z

P

c

P

c

M

M

P

₊

+

₊

+

+

+

₊

+

₊

=

η

ε

η

ε

ε

where

y

_it is the output of country i and

P

_zit

(

n

,

η

_i

)

is the price of a period n asset zi with

η

_i, i=H,F . If PPP conditions are satisfied, then

P

_Ft

=

ε

_t

P

_Ft*. Since we assume that each country only produces one good, PPP will be satisfied naturally. In the two-country model, the individual has to determine the demands for consumption, money and the holding of assets H and F. The first order condition can be shown as follows :

,

2

Ht t Ht c

P

y

λ

θφ

⋅

=

(35)

(

1

)

2

_{t Ft}

,

Ft c

P

y

λ

φ

θ

−

⋅

=

(36)

(

1

)

2

₁

,

1 t t Ht q

E

M

E

δ

λ

λ

φ

θ

δ

−

+

₊

=

+ (37)

(

1

)

(

1

)

2

[

_{1 1}

]

,

1 t t t t Ft q

E

M

E

δ

ε

λ

ε

λ

φ

θ

δ

−

−

+

_{+ +}

=

+ (38)

(

) (

)

(

)

[

P

_{zHt 1}

n

1

,

_H

P

_{Ht 1}

y

_{Ht 1 t 1}

]

P

_zHt

(

n

,

_H

)

_t

,

E

η

H

λ

η

λ

δ

₋

₊

η

₌

+ + + + (39)

(

)

(

)

(

)

[

P

_{zFt 1}

n

1

,

_{F t 1}

P

_Ft* ₁

y

_{Ft 1 t 1}

]

P

_zFt

(

n

,

_F

)

_t

,

E

F

λ

η

λ

ε

η

δ

₊

−

+

_{+ + +} η ₊

=

(40)

where the equilibrium conditions are:

2 2 , 2 it s it d it it it M M M y c = = = and 2 1 = it z .

Under this setup, the prices of consumption, money and assets can be solved endogenously. Note that there are two regimes of exchange rates, flexible and fixed, which will be discussed later. Foreign inflation will affect the home country under the fixed exchange rate regime. Under the flexible exchange rate regime, inflation risk coming from abroad can be avoided, but it is not completely free from exchange rate risk.

The price of goods H can be solved from equations (35) and (37):

,

2

Ht Ht H q c Ht

y

M

P

=

Ω

φ

φ

(41) where the transversality condition is satisfied:

(

)

0

,

lim

1 1

=

− + ∞ → n Ht n

δ

Em

and

.

1

1

1 ₁ 1 1 − + − +

−

⋅

−

≡

Ω

Ht Ht H

Em

Em

δ

δ

θ

θ

In the logarithm utility, the price of H goods is determined by the output process and the money process of country H. The foreign country does not affect the price of H good. This result is similar to that in the closed economy model. The price of F goods can be solved from equations (36) and (37):

,

1

2

Ft Ht H q c Ft

y

M

P

=

−

Ω

φ

φ

(42)

From equation (42), we may note that the price of F goods is a function of the money process of the home country and the output process of the foreign country. The output level of the foreign country will decrease the price of this good, however the home money will cause it to increase. Hence, combining equations (41) and (42), the relative price of these two goods can be shown as:

.

1

* Ft Ht c c Ht Ft t ft

y

y

P

P

P

φ

φ

ε

₌

−

=

This relative price is a function of relative output of the home and the foreign country; it does not depend on the money process. The exchange rate can be derived from equations (35) and (38) :

(

)

.

1

Ft Ht q F q H t

M

M

φ

φ

ε

Ω

−

Ω

=

(43)

Equation (43) shows that the exchange rate is determined only by the currencies of the two countries. If the exchange rate regime is fixed, then

ε

_t is a constant,

ε

. The money supply in country H will follow the following process:

(

)

.

1

_q Ft H q F Ht

M

M

ε

φ

φ

−

Ω

Ω

=

(44)

In order to maintain a fixed exchange rate, country H must adjust the money supply to match its foreign policy completely, which makes it seem like a single money world. Although there is no exchange risk, there is monetary uncertainty coming from abroad. The assumption of the logarithm utility prevents any influence stemming from disturbances in the foreign currency. The difference between the influence of the flexible exchange rate regime and the fixed exchange rate regime is that monetary uncertainty will affect the performance of the home country directly under the fixed exchange rate regime. The results shown above are consistent with those in previous studies. Next, the people who live in each country consume a composite of goods which consists of

y

_H and

y

_F. Hence, the general price has the following form (see Obstfeld and Rogoff (1996)):

.

Ft Ft Ht Ht t t

C

P

c

P

c

P

=

+

(45)

where

P

_t is the general price in the home country. Substituting the function of composite goods, and the price of consumption goods, equations (41) and (42), into equation (45), we can obtain

























−

+









−

=

− − − c c c c c c c c Ft Ht t

P

P

P

φ φ φ φ

φ

φ

φ

φ

1

1

1 1 ₁

.

c c Ft Ht Ht q H

y

y

M

φ φ

φ

−

Ω

=

(46)

The general price is determined by the money stock of country H and the output of both H and F. The inflation rate can be shown as:

.

1 1 1 1 + − + − +

=

Ht Ft Ht t

x

x

m

c c φ φ

π

(47)

Under the two-country model, the inflation rate is affected by the foreign output growth rate in the same direction.

2. The implications of asset pr icing

After showing the basic analysis for the two contrary model, the asset pricing implication will be analyzed below. There are two common stocks in this model and agents have to hold stock in order to consume the products of country H or F. The real rate of return on stock H is defined follows:

( )

1

(

1 1

)

/

1

,

zHt t Hy Ht zHt H Ht

P

P

y

P

P

RR

H + + + +

+

=

η

η

(48)

where

P

_zHt

≡

p

_zHt

P

_t

,

p

_zHt is the real price of asset zH, and the real dividend of stock H is

(

1 1

)

/

1

.

1 + + + +

≡

Ht Ht t Ht

P

y

P

RD

ηH

Next, we want to analyze the relationship between the real rate of return on stock H and financial leverage. An interesting finding is that the effect of the exchange rate on stock H valued in H currency is financial leverage. The process of this result is presented below. The real rate of return on stock F is

( )

(

1

)

/

1

,

* 1 1 1 zFt t Ft Ft t zFt F Ft

p

P

y

P

p

RR

F + + + + +

+

=

η

ε

η

(49)

where the real dividends of stock F valued in H currency is defined by

(

)

1 1 * 1 1 1 + + + + +

≡

t Ft Ft t Ft

P

y

P

RD

F η

ε

which can be rewritten as

(

)

.

1 1 1 1 1 1 + + + − + +

≡

t Ft Ft t Ft

P

y

P

RD

F F η η

ε

(50)

Equation (50) shows the fact that the dividend of stock F will not be affected by the exchange rate if financial leverage is absent. This implies that the two-country world will reduce to a one-country world which produces two goods, H and F, at the same time. Moreover, if the exchange rate rises, the real dividend of stock F will decrease. Plugging equation (50) into equation (49), the real rate of return on stock F can be rearranged as:

( )

(

1 1

)

/

1

.

1 1 1 zFt t Ft Ft t zFt F Ft

p

P

y

P

p

RR

F F + + + − + +

+

=

η η

ε

η

Next, we will show the characteristics of these two stocks in different exchange rate regimes.