_{1}

^{*}

This paper has used the Arbitrage Theorem (Gordan Theorem) to show that first, all securities are derivatives for each other, and they are priced by the same risk neutral probability measure. Second, after the firm changes its debt-equity ratio, the equityholders can always combine the new equity with other existing securities to create a home-made equity which will give exactly the same time-1 payoff of the old equity. That is, we have a capital structure irrelevancy proposition: changes in firms’ debt-equity ratios will not affect equityholders’ wealth (welfare), and equityholders’ preferences toward variance are irrelevant. Third, when the firm moves from a more certain project to a more uncertain one, the time-0 price of equity will increase, but (because the time-1 payoff of common bond has an upper bound) the time-0 price of common bond will decrease. Fourth, different labor contractual arrangements will not affect the time-0 price of labor input. When the firm moves from a more certain project to a more uncertain one, the time-0 price of labor input will increase if it is under the share or the mixed contract.

The seminal work of Black and Scholes (1973) has inspired many researches on pricing and hedging different financial contracts [

The remainder of this paper is organized as follows. Section 2 derives the Arbitrage Theorem and uses the theorem and several examples to show that securities are derivatives for each other; and that after the firm changes its debt-equity ratio, the equityholders can always combine the new equity with other existing securities to create a home-made equity which will give exactly the same time-1 payoff of the old equity. Concluding remarks appear in Section 3.

Theorem 1:

Let S be a nonempty, closed convex set in R^{n} and. Then, there exists a unique point with minimum distance from y. Also, is the minimizing point if and only if for all.

Proof We first show the existence of a minimum point. Since S is not empty, there exists a point. Define and hence,

.

Since means finding the minimum of a continuous function over a nonempty, compact set, by Weierstrass’ theorem, there exists a minimizing point in S that is closest to the point y.

To show the uniqueness of minimum point, suppose that there is another which is also a minimum point, i.e.,. By convexity of S,. By Schwartz inequality, we get

If strict inequality holds, we have a contradiction to being the closest point to y. Therefore, equality holds, and we must have for some λ. Since,. If λ = –1, then

which contradicts the assumption. Thus, λ = 1, and we have.

Suppose for all. Then,

Since and by assumption, for all, i.e., is the minimizing point. Conversely, assume that is the minimizing point. Let and 0 ≤ α ≤ 1. We have

and. Therefore, from

we can get for all 0 ≤ α ≤ 1. Dividing this inequality by any such α > 0and letting α → 0^{+}, we have for all.

Theorem 2 (Separating Hyperplane Theorem):

Let S be a nonempty, closed convex set in R^{n} and. Then, there exists a nonzero vector and a scalar α such that and for each.

Proof From Theorem 1 we know that because the set S is a nonempty, closed convex set in R^{n} and, there exists a unique minimizing point that

for all.

Letting and, we have z^{t}(x –)≥ 0 and hence, for each. Also, z^{t}y – α = z^{t}(y –) = –(y –)^{t}(y –) < 0 or.

Theorem 3 (Farkas Theorem):

Let A be an m × n matrix and be a vector. Then, exactly one of the following systems has a solution:

System 1: Ax ≥ 0 and c^{t}x < 0 for some

System 2: A^{t}y = c and y ≥ 0 for some

Proof 1) Suppose that System 2 has a solution; that is, there exists a and y ≥ 0 such that A^{t}y = c. Then, if for any such that Ax ≥ 0, then; that is, System 1 has no solution.

2) Suppose System 2 has no solution. Form the set. Note that the set S is a closed convex set: Let and. Then there must exist such that x_{1} = A^{t}y_{1} and x_{2} = A^{t}y_{2}. Also,

where.

Since, by Theorem 2, there exists a nonzero vector and a scalar α such that z^{t}c < α and for each. Because,. for each y ≥ 0. Since y can be made arbitrarily large and α is a fixed number, we must have Az ≥ 0. We have therefore constructed a vector such that Az ≥ 0 and z^{t}c < 0, i.e., System 1 has a solution.

Theorem 4 (Gordan Theorem or Arbitrage Theorem):

Let A be an m × n matrix. Then, exactly one of the following systems has a solution:

System 1: Ax > 0 for some

System 2: A^{t}p = 0 for some, p ≥ 0, e^{t}p = 1 where

.

Proof 1) Suppose that System 1 has a solution: Ax > 0 for some. Then, we can construct a negative scalar δ < 0 and a vector

such that Ax +δe ≥ 0, or and

.

Define, , and. We have, and; that is, System 1 of Theorem 3 has a solution.

2) With the same definitions, System 2 of Theorem 3 can be interpreted as: There exists a vector p ≥ 0 and such that. That is,

or A^{t}p = 0 and e^{t}p = 1 (i.e.,).

In System 2 of the Arbitrage Theorem, the vector p is usually termed as the risk neutral probability measure, and p_{i}, i = 1, ···, m, can be interpreted as the current price of one dollar received at the end of period if state i occurs. If the matrix A has rank m (i.e., the matrix has m independent rows), the risk neutral probability measure p will be unique. We now use the Arbitrage Theorem to clarify some ambiguous (and erroneous) arguments in the literature.

Example 1. All Securities Are Derivatives.

Assume a one-period, two states of nature world with no transaction costs. There are a money market (Security 1) which provides 1 + 0.25 dollars at time one if one dollar is invested at time 0 (i.e., the interest rate is r = 0.25), and two other securities (Security 2 and Security 3) with current prices 4 and 500 dollars, respectively, which provide:

Note that the two securities are not governed by the same risk neutral probability measure (i.e., System 2 of the Arbitrage Theorem has no solution):

i.e., we cannot find a vector, 0 ≤ π ≤ 1such that

By System 1 of the Arbitrage Theorem, arbitrage exists: e.g., at time 0, we can short sell one share of Security 3, buy 60 shares of Security 2 and invest 260 (= 500 – 4 × 60) dollars in the money market, and at time 1 we can get net profit:

Hence, in equilibrium (with no arbitrage), the time-0 prices of Security 2 and Security 3 will change so that they are priced by the same risk neutral probability messure, say,

and

Suppose that two European call options are based on Security 2 (with strike price 4 dollars) and Security 3 (with strike price 650 dollars), respectively:

Since all the securities are governed (priced) by the same, unique risk neutral probability measure

we will have:

Also, at time 0, by buying n shares of the underlying asset and selling one call to construct a portfolio which gives certain time-1 payoff, the prices of the two European calls can be derived from Security 2 or Security 3:

For:

or

For:

or

The time-0 price of Security 2 can be derived from Security 3 or the options, and the time-0 price of Security 3 can be derived from Security 2 or the options:

For:

or

For:

or

Thus, we can conclude that all securities are derivatives for each other, and all securities are underlying assets for each other. This result refutes Cox, Ross and Rubinstein’s (1979) [^{1}

Example 2. Home-made Securities.

In Example 1, assume that Security 3 is a firm and is the sum of five shares of equity:

Suppose that the fourth and the fifth shares of equity (and) of the firm are changed into riskless debts (and):

The market value of the firm (Security 3) at time 0 is still 500 dollars; that is, the market value of firm is independent of its debt-equity ratio. This is just a restatement of Modigliani-Miller’s first proposition.^{2}

Comparing Equation (1) with Equation (2), it is found that more debt means higher variance of equity’s time-1 payoff:

But after the firm changes its debt-equity ratio, the equityholders can always buy only 0.6 shares of the new equity () and invest 40 (= 100 – 0.6 × 100) dollars in the money market to recreate the time-1 payoff of the old equity ():

or

Suppose that in Equation (2), debts are risky:

Again, after the firm changes its debt-equity ratio, the equityholders can always buy 2/3 shares of the new equity () and invest dollars in the money market to recreate the time-1 payoff of the old equity ():

That is, after the firm changes its debt-equity ratio, the equityholders can always combine the new equity with other securities (e.g., money market) to create a “home-made equity” which will give exactly the same time-1 payoff of the old equity.^{3} We now have a capital structure irrelevancy proposition: changes in firms’ debt-equity ratio will not affect equityholders’ wealth (welfare), and equityholders’ preferences toward variance are irrelevant.^{4} This result refutes the claims in the literature that “the use of debt rather than equity funds to finance a given venture may well increase the expected return to the owners, but only at the cost of increased dispersion of the outcomes” (Modigliani and Miller, 1958 [^{5}

Example 3. Pricing Debt and Equity Contracts.

In Example 2, assume Equation (4) where Security 3 is a levered firm:

Suppose that the firm moves to a more uncertain project, and its time-1 payoff is rather than.

Then, Equation (4’) becomes:

It is found that when the firm moves from a more certain project (its time-1 payoff is either $750 or $250) to a more uncertain one (its time-1 payoff is either $900 or $100), the variance of the time-1 payoff of the firm (and the variance of the time-1 payoff of the equity) increases, the time-0 price of equity increases, but the time-0 price of debt decreases.^{6} Also, this redistribution effect of wealth between debtholders and equityholders has nothing to do with their attitudes toward risk. These results and the results of Example 1 refute the claims that “there is a fundamental distinction between holding an option on an underlying asset and holding the underlying asset. If investors in the marketplace are risk-averse, a rise in the variability of the stock will decrease its market value” (Ross, Westerfield and Jaffe, 2010 [

Example 4. Pricing Convertible Bonds.

In Example 2, assume Equation (2) where Security 3 is a levered firm. Assume that one of the firm’s riskless debts is changed into a convertible bond:

Adding this convertible bond dilutes the time-0 value of the equity (which decreases from 100 dollars to 93.75 dollars). The time-0 price of the option (the right) of converting the bond into a share of equity is 18.75 (= 118.75 – 100) dollars.

Example 5. Pricing Different Contracts.

In Example 4, assume Equation (6) where Security 3 is a levered firm. Suppose that the firm’s hiring an additional labor (a manager) can increase its time-1 payoff from to:

The time-0 value of the whole firm will be 570 dollars, and the time-0 value of the labor input will be 70 dollars:

Note that different labor contractual arrangements will not affect the time-0 prices of the labor input and the whole capital input (which includes equity, debt and convertible bond):^{7}

Fixed-wage contract:

Share contract (where the labor’s share:; the capital providers’ share:):

Mixed contract (where the labor obtains 50 dollars and has share:, and capital providers’ share is:):

Suppose the firm moves to a more uncertain projectand its time-1 payoff is rather thanand assume that the labor is the first to get payment, the common bondholder is the second to get payment, the convertible bondholder is the third to get payment, and the equityholder obtains the residual:

Fixed-wage contract:

Share contract (where labor share:; capital providers share:):

Mixed contract (where the labor obtains 50 dollars and has share:, and capital providers’ share is:

):

The time-0 prices of the whole firm, the equity and the convertible bond will increase. The time-0 price of the common bond will decrease. The time-0 price of the labor input will decrease if it is under the fixed-wage contract. The time-0 price of the labor input will increase if it is under the share or the mixed contracts.

This paper has used the Arbitrage Theorem (Gordan Theorem) to show that first, all securities are derivatives for each other, and they are priced by the same risk neutral probability measure. Second, after the firm changes its debt-equity ratio, the equityholders can always combine the new equity with other existing securities to create a home-made equity which will give exactly the same time-1 payoff of the old equity. That is, we have a capital structure irrelevancy proposition: changes in firms’ debtequity ratios will not affect equityholders’ wealth (welfare), and equityholders’ preferences toward variance are irrelevant. Third, when the firm moves from a more certain project to a more uncertain one, the time-0 price of equity will increase, but (because the time-1 payoff of common bond has an upper bound) the time-0 price of common bond will decrease. Fourth, different labor contractual arrangements will not affect the time-0 price of labor input. When the firm moves from a more certain project to a more uncertain one, the time-0 price of labor input will increase if it is under the share or the mixed contract.

In incomplete markets, securities may not be derivatives for each other, but with no arbitrage (System 2 of Theorem 4 has a solution), they will still be priced by the same (which may not be unique) risk neutral probability measure. For example, assume that only two securities (one of them is a money market with interest rate r = 0.25) exist in a no-arbitrage, one-period, five states of nature world:

Suppose that there is a new security: Security 3 whose time-1 payoff is: .

Because c lies in the subspace spanned by

and

(i.e.,), the time-1 payoff of Security 3 can be derived (replicated) by those of Securities 1 and 2:

The time-0 price of Security 3 is: , and with no arbitrage, the three securities are priced by the same risk probability measure:

Suppose that the time-1 payoff of Security 3 is:

.

Because, the time-1 payoff of Security 3 cannot be replicated by those of Securities 1 and 2. But with no arbitrage (System 2 of Theorem 4 has a solution), all the three securities will be priced by the same risk probability measure:

where p_{1}, p_{2}, p_{3}, p_{4}, p_{5} may not be unique.

In incomplete markets, after the firm changes its debtequity ratio, the equityholders may not be able to create a home-made equity to replicate the time-1 payoff of the old equity. For example, assume that only two securities (one of them is a money market with interest rate r = 0.25) exist in a no-arbitrage, one-period, five states of nature world:

where the risk neutral probability can be

or.

Assume that Security 2 is an all equity firm and it plans to issue a riskless debt (debtholder pays 0.8 dollar at time 0, and obtains 1 dollar at time 1):

By,

or, by,

That is, recapitalization through issuing riskless debt does not change the market value of the firm (i.e., the time-0 price of Security 2 is always 4 dollars), and the time-0 prices of equity and debt are independent of the risk neutral probability measure used. Also, after the firm issues riskless debt, the equityholder can always create a home-made equity by combining the new equity (or) with investing 0.8 dollar in the money market, which will give exactly the same time-1 payoff of the old equity:

.

Suppose that the time-1 payoff of the debt is uncertain:

and the time-1 payoff of the equity is:

.

Since where

andb cannot be replicated by a and e_{1}. That is, the equityholder cannot combine the new equity e_{1} with the money market to create a home-made equity to replicate the time-1 payoff of the old equity:

.

Also, with different risk neutral probability measures:

By,

By,

The time-0 price of Security 2 can be the sum of and (which is equal to 4.36444 > 4), and with no arbitrage,

where the risk neutral probability can be

or.

If this is the case, then with no arbitrage, the time-0 price of Security 2 in Equation (B1) will be adjusted to 4.36444 dollars in the first place (see also Litzenberger and Sosin, 1977 [

where p_{1}, p_{2}, p_{3}, p_{4}, p_{5} may not be unique.

In some cases, the time-0 price of a firm may decrease when the firm moves to a more uncertain project. For example, assume a firm exists in a no-arbitrage, oneperiod, two states of nature world (where the market interest rate is: r = 0.25):

That is, the unique risk neutral probability for this world is:

and

Suppose the firm moves to a more uncertain project, and its time-1 payoff is

instead of .

Then, the time-0 prices of the whole firm, the equity and the debt decrease:

Since no one (especially, the equityholder) benefits, the firm will not move to this more uncertain project, and Equation (C2) does not exist.