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OF THE

MASSACHUSETTS INSTITUTE

OF TECHNOLOGY

WORKING PAPER

ALFRED P. SLOAN SCHOOL OF MANAGEMENT

ON THE INTERACTIONS OF CORPORATE FINANCING

AND INVESTMENT DECISIONS

AND THE WEIGHTED AVERAGE COST OF CAPITAL

S. C. Myers

May 1972

598-72

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

ON THE INTERACTIONS OF CORPORATE FINANCING

AND INVESTMENT DECISIONS

AND THE WEIGHTED AVERAGE COST OF CAPITAL

S. C. Myers

May 1972

598-72

ON THE INTERACTIONS OF CORPORATE FINANCING

AND INVESTMENT DECISIONS AND THE WEIGHTED AVERAGE COST OF CAPITAL

Stewart C. Myers

I. INTkODUCTION

Everyone seems to agree that there are significant interactions

between corporate financing and investment decisions. The most important

argument to the contrary â€” embodied in Modigliani and Miller's (MM's)

famous Proposition I â€” specifically assumes the absence of corporate

income taxes; but their argument implies an interaction when such taxes

are recognized. Interactions may also stem from transaction costs or

other market imperfections.

The purpose of this paper is to present a general approach for

analysis of the interactions of corporate financing and investment de-

cisions, and to derive some of the approach's implications. Perhaps the

most interesting implication is that the weighted average cost of capital

formulas proposed by MM and other authors are not always correct. Except

in certain special cases, a more general "Adjusted Present Value" rule

should, in principle, be used to evaluate investment opportunities.

The paper is organized as follows. Section II presents the

framework for my analysis, which is a mathematical programming formulation

of the problem of financial management. The conditions for the optimum

and the implications for corporate investment decisions are derived.

In Section III, the usual weighted average cost of capital rules are

derived as special cases of the more general analysis. Section IV de-

scribes the errors that can occur if weighted average cost of capital

rules are used in practice. Also, I discuss the Adjusted Present Value

rule as an alternative decisionmaking tool. The last section briefly

describes some topics for further work, notably in the development of

programming models for overall financial planning.

It must be emphasized that this paper is not intended to cata-

logue or deal with all possible interactions of financing and investment

decisions; in other words, there is no attempt to specify the problem of

financial management in full generality. Instead, I present an approach

to analyzing interactions and a specific analyses of the most important

ones.

Although the paper is an exploratory step towards a general model

of financial management, the analysis is nevertheless of immediate interest.

As far as I know this is the first full statement of the assumptions under-

lying cost of capital concepts, and the first explicit calculation of

the errors that can result if the assumptions are false.

II. BASIC FRAMEWORK

Specification of the Problem

We will consider the firm's problem in the following terms. It

begins with a certain initial package of assets and liabilities. For a

brand-new firm, this may be simply money in the bank and stock outstanding.

For a going concern, the package will be much more complicated. Any firm,

however, has the opportunity to change the characteristics of its initial

package by transactions in real or financial assets â€” i.e., by investment

or financing decisions.

The problem is to determine which set of current and planned

future transactions will maximize the current market value of the firm.

Market value is taken to be an adequate proxy for the firm's more basic

2

objective, maximization of current shareholders' wealth.

This type of problem can be approached by (1) specifying the

firm's objective as a function of investment and financing decisions and

(2) capturing interactions of the financing and investment opportunities

by a series of constraints.

Example of the Approach

Before moving on to more general formulations, I will present a

simple numerical example.

Consider a firm which has to decide how much to invest and/or

borrow in the coming year. Let:

X = New investment, in millions of dollars,

y = New borrowing, in millions of dollars.

Also, assume that:

1. Available investment opportunities can absorb $1 million at most.

The investment generates a perpetual stream of after-tax cash

flows. Let the expected average value of these flows be c â€¢ In

this case C = .09y.

2. Assume the market will capitalize the retutms at a rate p_ = .10.

Thus, if all-equity financing is used, these assets generate a net

present value of - $.10 per dollar Invested.

3. New debt is limited to 40 percent of new Investment.

4. The firm has $800,000 in cash available.

5. Any excess cash is paid out in dividends.

6. The additions to debt and equity are expected to be permanent.

In order to specify the objective function in the simplest possi-

bly way, I will assume that MM's view of the world is correct. If so, it

is sufficient to maximize the overall market value of the firm. V is given

V = V + PVTS (1)

where V = the market value of the firm given all-equity financing, and

PVTS = the present value of tax savings due to corporate debt.

Dividends paid are not explicitly included in Eq . (1). Under MM's assump-

tions, dividend policy is irrelevant, given the firm's investment and bor-

rowing decisions.

Therefore, i) , the increase in the market value of the firm, is

-.Ix + .5y. This is to be maximized subject to constraints on the amount

invested (x < 1), the amount of debt issued (y Â£ .4x) and the balance of

sources and uses of funds (x < y + .8).

The solution . â€” This is the linear programming problem depicted

in Figure 1. It is evident from the figure that the solution is at

X = 1, y = .4. The constraints on the amounts invested and borrowed are

each binding at the optimum. The sources/uses constraint is not binding,

however: the firm has $200,000 available for dividends.

Why does the optimal solution cal] for investing in a project

with a negative net present value? The reason is that the project allows

the firm to issue more debt, and the value cf tax savings generated by

New /|N X

Assets

(millions)

1.2

1.0

.6 -

.Ax

. Ix + . 5y = constant

,1

.2

.3

The Problem:

Max:

-.Ix + .5y

Subj

ect

to:

X < 1

y Â°' ^^^

^ (l+P.)

t=0 J

*

In either rule, p. is the "hurdle rate" or minimum acceptable expected

rate of return.

Comparing Eq. (4) to (6) and (7), it is evident that NPV. and

APV. are intended to measure the same thing: the net contribution of j to

the market value, taking account of the interactions of j with other invest-

ment and financing opportunities. There is always some value of p. which

will insure that NPV. = APV., or that

J J

T

(1+p.)'' J t=0 ^ ^^ " J^

t=0 ^

The problem is, how should p. be computed, if not directly from Eq . (8)?

Of the many procedures for calculating p., two are of particular

interest. The first is MM's. They propose

P* = P^jCl - tL), (9)

14

where: p . = The appropriate discount rate assuming all-equity

financing;

T = The corporate tax rate, and

L = The firm's "long-run" or "target" debt ratio.

MM interpret n . as the rate at which investors would capitalize the firm's

expected average after-tax income from currently-held assets, if the firm

were all-equity financed. This would restrict application of the formula

to projects whose acceptance will not change the firm's risk characteristics,

(However, we will see that this is an unnecessarily narrow interpretation

of the MM formula. )

The second proposed formula is:

p* = (1 - t) i-|-+k-^ (10)

where: r = the fimi's current borrowing rate;

k = "the cost of equity capital" â€” that is, the expected

rate of return required by investors who purchase the

firm's stock;

B = market value of currently outstanding debt;

S = market value of currently outstanding stock, and

V = B + S, the total current market value of the firm.

I will refer to Eq. (10) as the " cextbook formula," for lack of a better

name. (The formula, or some variition on the same theme, appears in nearly

all finance texts.) It is not aecessarilv inconsistent with the MM

formula, but it is recommended by many who explicitly disagree with MM's

view of the world.

The task now is to determine what assumptions are necessary to

derive Eqs. (9) and/or (10) from Eq . (A), the general condition for the

optimal investment decision. I will present a set of sufficient condi-

15

tions, and then argue that, in most cases, the conditions are necessary

as well.

Derivation of the MM Cost of Capital Rule

If MM's view of the world is correct, then the value of the

firm will be V , the value of the firm assioming all-equity financing, plus

the present value of tax savings due to debt financing actually employed.

Dividend policy is irrelevant. Assuming this view is correct, the object-

ive function in the mathematical programming formulation is:

ij; = AV + Z y F

^ t=0 '^ "^

where

^t â€” ^^â€” rrr â€¢ (iia)

(1 + r)"^-"^

That is, F is rT, the tax savinp per dollar of debt outstanding in t,

discounted to the present. (It :s assumed that the interest is paid at

t=l.) Eq. (11a) follows from Eq. (1).

Second, assume that

A. = C./p . - I. (lib)

3 J oj J

There are two ways of interpreting Eq. (lib). One is to say that project

j is expected to generate a constant, perpetual stream of cash returns.

16

If C. = C., a constant for t - 1, 2, . . ., Â°Â°, then Eq. (lib) simply

states the project's net present value when discounted at p ., the

"appropriate rate" for j given all-equity financing.

However, MM interpret C. as the expectation of the mean of

the series C ,, C.â€ž, . . ,, C.

where p . is the discount rate specific to the risk characteristics of j's

type. A. may be interpreted as the market value of j if the project

could be divorced from the firm and financed as a separate unleveraged

enterprise.

Eq. (llf) assumes that projects are risk-independent , in the

sense that there are no statistical relationships among projects' returns

18

such that some combinations of projects affect stock price by an amount

different than the sum of their present values considered separately. In

particular, risk- independence implies that there is no advantage to be

gained by corporate diversification.

I have shown elsewhere that risk-independence is a necessary con-

12

dition for equilibrium in perfect security markets.

Eq. (llf) also assumes that projects are "physically independent"

in the sense that there are no causal links between adoption of project j

and the cash returns to other projects â€” that is, it rules out "competi-

13

tive" or "complementary" projects. Such interactions make it impossible

to specify an unique hurdle rate for project j , since the minimum accepta-

ble rate of return on j may depend on whether or not other projects are

accepted. However, I am not concerned with this problem in this paper.

Let us adopt Eq. (llf) and drop Eqs. (lie) and (lie). We can

recalculate the minimum acceptable rate of return on the project.

p* = p â€¢ (1 -TZ./I.) (13)

This has the same form as Eq. (9) but it is not restricted to projects within

a single risk class. In this case it is not plausible to identify Z./I.,

project j's marginal contribution to debt capacity, with L, the firm's over-

all target capitalization ratio. Presumably Z./I. will be more or less

J J

than L, depending on the risk or on other characteristics of the project

in question.

In short, MM's formula can be extended to independent projects

which differ in risk and in their impact on the firm's target debt ratio.

19

What if Investment Projects are not Perpetuities?

So far we have established that Eqs. (11a, b, d and f ) are

sufficient for the generalized MM formula, Eq. (13). Eqs. (11a) and (llf)

are clearly necessary as well. But what about (lib) and (lid) which re-

quire all projects to be perpetuities?

In general, they are necessary: Eq. (13) does not give the cor-

rect "hurdle rate" for projects of limited life. (The question of

whether the resulting errors are serious is taken up in the next section.)

This can be shown by a simple example. Consider a point- input,

point-output project requiring an investment of I. and offering an expected

cash flow of C, in T = 1, and C. =0 for t > 1. Assume p . = p and

jl 3t ^oj ^o

Z., = LI. (and, of course, Z. =0 for t > 1). Then

APV

:ij__ I. +LI. (_J^)

J 1 + P^ J j '1 + r^

The internal rate of return on the project is given by

R, = â€” ^ - 1.

The cost of capital is given by R. when APV. = 0. Thus

*

p. = p - Lrx

1 + r

(14)

Eqs. (13) and (14) are equivalent only in the uninteresting case of p = r.

20

The Textbook Formula

Let us reconsider Eq. (10);

p* = r(l - t)B/V + k(S/V) (10)

The ^ is used temporarily to indicate a proposed value for the cost of

capital, the true value being given by Eq. (8).

Probably it is intuitively clear from the foregoing that p. = p.

only under very restrictive assumptions. First, let us assume that Eqs.

(11a) through (lie) hold. Also, assume that

V = â€” . (15a)

Po

That is V , the current market value of the firm if it were all equity fi-

o

nanced, is found by capitalizing the firm's after-tax operating income at

p . C is, of course, calculated assuming all-equity financing. Also,

Eq. (15a) presumes C = C, t = ] , 2 , . . . , Â«>.

Finally, assume

L. = B/V (15b)

This implies that the firm is already at its target debt ratio.

OF THE

MASSACHUSETTS INSTITUTE

OF TECHNOLOGY

WORKING PAPER

ALFRED P. SLOAN SCHOOL OF MANAGEMENT

ON THE INTERACTIONS OF CORPORATE FINANCING

AND INVESTMENT DECISIONS

AND THE WEIGHTED AVERAGE COST OF CAPITAL

S. C. Myers

May 1972

598-72

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

ON THE INTERACTIONS OF CORPORATE FINANCING

AND INVESTMENT DECISIONS

AND THE WEIGHTED AVERAGE COST OF CAPITAL

S. C. Myers

May 1972

598-72

ON THE INTERACTIONS OF CORPORATE FINANCING

AND INVESTMENT DECISIONS AND THE WEIGHTED AVERAGE COST OF CAPITAL

Stewart C. Myers

I. INTkODUCTION

Everyone seems to agree that there are significant interactions

between corporate financing and investment decisions. The most important

argument to the contrary â€” embodied in Modigliani and Miller's (MM's)

famous Proposition I â€” specifically assumes the absence of corporate

income taxes; but their argument implies an interaction when such taxes

are recognized. Interactions may also stem from transaction costs or

other market imperfections.

The purpose of this paper is to present a general approach for

analysis of the interactions of corporate financing and investment de-

cisions, and to derive some of the approach's implications. Perhaps the

most interesting implication is that the weighted average cost of capital

formulas proposed by MM and other authors are not always correct. Except

in certain special cases, a more general "Adjusted Present Value" rule

should, in principle, be used to evaluate investment opportunities.

The paper is organized as follows. Section II presents the

framework for my analysis, which is a mathematical programming formulation

of the problem of financial management. The conditions for the optimum

and the implications for corporate investment decisions are derived.

In Section III, the usual weighted average cost of capital rules are

derived as special cases of the more general analysis. Section IV de-

scribes the errors that can occur if weighted average cost of capital

rules are used in practice. Also, I discuss the Adjusted Present Value

rule as an alternative decisionmaking tool. The last section briefly

describes some topics for further work, notably in the development of

programming models for overall financial planning.

It must be emphasized that this paper is not intended to cata-

logue or deal with all possible interactions of financing and investment

decisions; in other words, there is no attempt to specify the problem of

financial management in full generality. Instead, I present an approach

to analyzing interactions and a specific analyses of the most important

ones.

Although the paper is an exploratory step towards a general model

of financial management, the analysis is nevertheless of immediate interest.

As far as I know this is the first full statement of the assumptions under-

lying cost of capital concepts, and the first explicit calculation of

the errors that can result if the assumptions are false.

II. BASIC FRAMEWORK

Specification of the Problem

We will consider the firm's problem in the following terms. It

begins with a certain initial package of assets and liabilities. For a

brand-new firm, this may be simply money in the bank and stock outstanding.

For a going concern, the package will be much more complicated. Any firm,

however, has the opportunity to change the characteristics of its initial

package by transactions in real or financial assets â€” i.e., by investment

or financing decisions.

The problem is to determine which set of current and planned

future transactions will maximize the current market value of the firm.

Market value is taken to be an adequate proxy for the firm's more basic

2

objective, maximization of current shareholders' wealth.

This type of problem can be approached by (1) specifying the

firm's objective as a function of investment and financing decisions and

(2) capturing interactions of the financing and investment opportunities

by a series of constraints.

Example of the Approach

Before moving on to more general formulations, I will present a

simple numerical example.

Consider a firm which has to decide how much to invest and/or

borrow in the coming year. Let:

X = New investment, in millions of dollars,

y = New borrowing, in millions of dollars.

Also, assume that:

1. Available investment opportunities can absorb $1 million at most.

The investment generates a perpetual stream of after-tax cash

flows. Let the expected average value of these flows be c â€¢ In

this case C = .09y.

2. Assume the market will capitalize the retutms at a rate p_ = .10.

Thus, if all-equity financing is used, these assets generate a net

present value of - $.10 per dollar Invested.

3. New debt is limited to 40 percent of new Investment.

4. The firm has $800,000 in cash available.

5. Any excess cash is paid out in dividends.

6. The additions to debt and equity are expected to be permanent.

In order to specify the objective function in the simplest possi-

bly way, I will assume that MM's view of the world is correct. If so, it

is sufficient to maximize the overall market value of the firm. V is given

V = V + PVTS (1)

where V = the market value of the firm given all-equity financing, and

PVTS = the present value of tax savings due to corporate debt.

Dividends paid are not explicitly included in Eq . (1). Under MM's assump-

tions, dividend policy is irrelevant, given the firm's investment and bor-

rowing decisions.

Therefore, i) , the increase in the market value of the firm, is

-.Ix + .5y. This is to be maximized subject to constraints on the amount

invested (x < 1), the amount of debt issued (y Â£ .4x) and the balance of

sources and uses of funds (x < y + .8).

The solution . â€” This is the linear programming problem depicted

in Figure 1. It is evident from the figure that the solution is at

X = 1, y = .4. The constraints on the amounts invested and borrowed are

each binding at the optimum. The sources/uses constraint is not binding,

however: the firm has $200,000 available for dividends.

Why does the optimal solution cal] for investing in a project

with a negative net present value? The reason is that the project allows

the firm to issue more debt, and the value cf tax savings generated by

New /|N X

Assets

(millions)

1.2

1.0

.6 -

.Ax

. Ix + . 5y = constant

,1

.2

.3

The Problem:

Max:

-.Ix + .5y

Subj

ect

to:

X < 1

y Â°' ^^^

^ (l+P.)

t=0 J

*

In either rule, p. is the "hurdle rate" or minimum acceptable expected

rate of return.

Comparing Eq. (4) to (6) and (7), it is evident that NPV. and

APV. are intended to measure the same thing: the net contribution of j to

the market value, taking account of the interactions of j with other invest-

ment and financing opportunities. There is always some value of p. which

will insure that NPV. = APV., or that

J J

T

(1+p.)'' J t=0 ^ ^^ " J^

t=0 ^

The problem is, how should p. be computed, if not directly from Eq . (8)?

Of the many procedures for calculating p., two are of particular

interest. The first is MM's. They propose

P* = P^jCl - tL), (9)

14

where: p . = The appropriate discount rate assuming all-equity

financing;

T = The corporate tax rate, and

L = The firm's "long-run" or "target" debt ratio.

MM interpret n . as the rate at which investors would capitalize the firm's

expected average after-tax income from currently-held assets, if the firm

were all-equity financed. This would restrict application of the formula

to projects whose acceptance will not change the firm's risk characteristics,

(However, we will see that this is an unnecessarily narrow interpretation

of the MM formula. )

The second proposed formula is:

p* = (1 - t) i-|-+k-^ (10)

where: r = the fimi's current borrowing rate;

k = "the cost of equity capital" â€” that is, the expected

rate of return required by investors who purchase the

firm's stock;

B = market value of currently outstanding debt;

S = market value of currently outstanding stock, and

V = B + S, the total current market value of the firm.

I will refer to Eq. (10) as the " cextbook formula," for lack of a better

name. (The formula, or some variition on the same theme, appears in nearly

all finance texts.) It is not aecessarilv inconsistent with the MM

formula, but it is recommended by many who explicitly disagree with MM's

view of the world.

The task now is to determine what assumptions are necessary to

derive Eqs. (9) and/or (10) from Eq . (A), the general condition for the

optimal investment decision. I will present a set of sufficient condi-

15

tions, and then argue that, in most cases, the conditions are necessary

as well.

Derivation of the MM Cost of Capital Rule

If MM's view of the world is correct, then the value of the

firm will be V , the value of the firm assioming all-equity financing, plus

the present value of tax savings due to debt financing actually employed.

Dividend policy is irrelevant. Assuming this view is correct, the object-

ive function in the mathematical programming formulation is:

ij; = AV + Z y F

^ t=0 '^ "^

where

^t â€” ^^â€” rrr â€¢ (iia)

(1 + r)"^-"^

That is, F is rT, the tax savinp per dollar of debt outstanding in t,

discounted to the present. (It :s assumed that the interest is paid at

t=l.) Eq. (11a) follows from Eq. (1).

Second, assume that

A. = C./p . - I. (lib)

3 J oj J

There are two ways of interpreting Eq. (lib). One is to say that project

j is expected to generate a constant, perpetual stream of cash returns.

16

If C. = C., a constant for t - 1, 2, . . ., Â°Â°, then Eq. (lib) simply

states the project's net present value when discounted at p ., the

"appropriate rate" for j given all-equity financing.

However, MM interpret C. as the expectation of the mean of

the series C ,, C.â€ž, . . ,, C.

where p . is the discount rate specific to the risk characteristics of j's

type. A. may be interpreted as the market value of j if the project

could be divorced from the firm and financed as a separate unleveraged

enterprise.

Eq. (llf) assumes that projects are risk-independent , in the

sense that there are no statistical relationships among projects' returns

18

such that some combinations of projects affect stock price by an amount

different than the sum of their present values considered separately. In

particular, risk- independence implies that there is no advantage to be

gained by corporate diversification.

I have shown elsewhere that risk-independence is a necessary con-

12

dition for equilibrium in perfect security markets.

Eq. (llf) also assumes that projects are "physically independent"

in the sense that there are no causal links between adoption of project j

and the cash returns to other projects â€” that is, it rules out "competi-

13

tive" or "complementary" projects. Such interactions make it impossible

to specify an unique hurdle rate for project j , since the minimum accepta-

ble rate of return on j may depend on whether or not other projects are

accepted. However, I am not concerned with this problem in this paper.

Let us adopt Eq. (llf) and drop Eqs. (lie) and (lie). We can

recalculate the minimum acceptable rate of return on the project.

p* = p â€¢ (1 -TZ./I.) (13)

This has the same form as Eq. (9) but it is not restricted to projects within

a single risk class. In this case it is not plausible to identify Z./I.,

project j's marginal contribution to debt capacity, with L, the firm's over-

all target capitalization ratio. Presumably Z./I. will be more or less

J J

than L, depending on the risk or on other characteristics of the project

in question.

In short, MM's formula can be extended to independent projects

which differ in risk and in their impact on the firm's target debt ratio.

19

What if Investment Projects are not Perpetuities?

So far we have established that Eqs. (11a, b, d and f ) are

sufficient for the generalized MM formula, Eq. (13). Eqs. (11a) and (llf)

are clearly necessary as well. But what about (lib) and (lid) which re-

quire all projects to be perpetuities?

In general, they are necessary: Eq. (13) does not give the cor-

rect "hurdle rate" for projects of limited life. (The question of

whether the resulting errors are serious is taken up in the next section.)

This can be shown by a simple example. Consider a point- input,

point-output project requiring an investment of I. and offering an expected

cash flow of C, in T = 1, and C. =0 for t > 1. Assume p . = p and

jl 3t ^oj ^o

Z., = LI. (and, of course, Z. =0 for t > 1). Then

APV

:ij__ I. +LI. (_J^)

J 1 + P^ J j '1 + r^

The internal rate of return on the project is given by

R, = â€” ^ - 1.

The cost of capital is given by R. when APV. = 0. Thus

*

p. = p - Lrx

1 + r

(14)

Eqs. (13) and (14) are equivalent only in the uninteresting case of p = r.

20

The Textbook Formula

Let us reconsider Eq. (10);

p* = r(l - t)B/V + k(S/V) (10)

The ^ is used temporarily to indicate a proposed value for the cost of

capital, the true value being given by Eq. (8).

Probably it is intuitively clear from the foregoing that p. = p.

only under very restrictive assumptions. First, let us assume that Eqs.

(11a) through (lie) hold. Also, assume that

V = â€” . (15a)

Po

That is V , the current market value of the firm if it were all equity fi-

o

nanced, is found by capitalizing the firm's after-tax operating income at

p . C is, of course, calculated assuming all-equity financing. Also,

Eq. (15a) presumes C = C, t = ] , 2 , . . . , Â«>.

Finally, assume

L. = B/V (15b)

This implies that the firm is already at its target debt ratio.

Online Library → Stewart C Myers → On the interaction of corporate financing and investment decisions and the weighted average cost of capital → online text (page 1 of 3)