A Unified Analytical Theory of Production and Capital Structure of Firms

Irrelevance of capital structure in a perfect market

• Modigliani and Miller (1958) • If an imperfection is identified, this type of

imperfection would be gradually reduced over time from competition or regulation. So we might expect capital structures of firms will be less and less relevant and financial decision making becomes simpler and simpler over time.

• Reality is not the case.

Question on the assumption of imperfection

• Miller (1998) on agency cost

• Tax as an imperfection

• But the impacts of taxes can be precisely measured

Empirical investigation on capital structure

• puzzles in corporate finance often result not from “imperfect market” but rather from “imperfect theory” (Molina, 2005).

Relation between production and capital structure of firms

• In early literature production of a firm is independent from financing decisions

• Empirical evidences: firm’s financial decisions are closely related to the operational side of the firm and market environment (Istaitieh and Rodrigues-Fernandez, 2005; Khanna and Tice, 2005).

• A unified theory is desirable.

analytical theory of production and capital structure of firms

• natural extension from an analytical theory of production

• the main result is an analytical formula of variable cost of production as a function of fixed cost and uncertainty

fixed cost and variable cost

• Fixed costs are pre-committed costs

• Variable cost changes with the many environmental factors, including the value of the product.

Extension to capital structure theory

• Problems on capital structure can be naturally incorporated into the theory on production from a simple observation.

• Debt is fixed income for investors and hence fixed cost for issuing firms. The increase of debt increases the fixed cost of firms.

• The decision on capital structure is part of the decision process that determines the level of the fixed cost and variable cost of firms to achieve a high rate of return based on the understanding of current and future market conditions.

Empirical findings

• The new theory, by integrating financial decisions into the general firm decision processes, offers a simple and parsimonious understanding to a broad range of empirical patterns documented in the literature.

Capital structure and the level of uncertainty

• Two projects are producing two different products. Both projects have 1.5 million dollar fixed cost in production, which are financed by equity. Suppose the annual revenue of both products is 1 million. Both production facilities will last for 10 years. The diffusion rate for the first product is 40% per annum and the diffusion rate for the second product is 60% per annum. The discount rate is 5% per annum. What is the variable cost for each product? What are the NPV for each project? The equity owners of the two projects are concerned that their capital structures may not be optimal. They try to determine the optimal fixed cost level for the projects by maximizing NPV value with respect to fixed costs for each project, assuming all other parameters, i.e., diffusion rate, duration of project and discount rate, are the same. What are the optimal fixed costs for both projects? Suppose the extra fixed costs are generated through debts. What are the amounts of debts for both projects? What are the resulting debt equity ratios for both projects?

Solution

S 1 S 1

K 1.5 K 2.5417

R 0.05 R 0.05

T 10 T 10

sigma 0.4 sigma 0.4

c 0.4978 c 0.36

profit 3.5222 profit 3.858

ratio of profit 1.0954

asset 6.3997

equity 5.358

debt/equity ratio 0.1944

Notes on calculation

• Debt level = Optimal fixed cost – operating fixed cost

• 2.5417-1.5

• Asset = (S-C)T

• Equity = asset - debt

S 1 S 1

K 1.5 K 1.7319

R 0.05 R 0.05

T 10 T 10

sigma 0.6 sigma 0.6

c 0.6732 c 0.6487

profit 1.7675 profit 1.7807

ratio of profit 1.0075

asset 3.5126

equity 3.2807

debt/equity ratio 0.0707

Discussion

• Intuition suggests the debt equity ration increases when uncertainty declines.

• The calculation from this example confirms it.

• New industries, high uncertainty, mostly equity financed.

• Mature industry, less uncertainty, more debt financed.

Capital structure and duration of projects

• Two projects are producing two different products. Both projects have 2 million dollar fixed cost in production, which are financed by equity. Suppose the annual output of both products is 1 million. One project will last for 10 years and the other project will last for 15 years. The diffusion rate for both projects is 35% per annum. The discount rate is 5% per annum. What is the variable cost for each product? What are the NPV for two projects? The equity owners of the two projects are concerned that their capital structures may not be optimal. They try to determine the optimal fixed cost level for the projects by maximizing NPV value with respect to fixed costs for each project, assuming all other parameters, i.e., diffusion rate, duration of project, market size and discount rate, are the same. What are the optimal fixed costs for both projects? Suppose the extra fixed costs are generated through debts. What are the debt level of the two projects? What are the debt equity ratios of both projects?

Solution

K 2 K 2.627984

R 0.05 R 0.05

T 10 T 10

sigma 0.35 sigma 0.35

c 0.364565 c 0.289776

market size 10 market size 10

profit 4.354347 profit 4.474257

ratio of profit 1.027538

debt/equity ratio 0.140355

S 1 S 1

K 2 K 3.628384

R 0.05 R 0.05

T 15 T 15

sigma 0.35 sigma 0.35

c 0.51621 c 0.368485

market size 15 market size 15

profit 5.256845 profit 5.844339

ratio of profit 1.111758

debt/equity ratio 0.278626

Discussion

• In general, longer term projects are more leveraged.

• Companies with more tangible assets usually are more leveraged, as tangible assets are long term assets.

• In accounting, long term debts are treated as capital.

• Short term debts are liabilities.

Capital structure and operating leverages

• Two projects are producing two different products. One project has 1 million dollar fixed cost in production and another project has 2 million dollar fixed cost in production, which are financed by equity. Suppose the annual output of both products is 1 million. Both projects will last for 15 years. The diffusion rate for both projects is 40% per annum. The discount rate is 4% per annum. What is the variable cost for each product? What are the NPV for two projects? The equity owners of the two projects are concerned that their capital structures may not be optimal. They try to determine the optimal fixed cost level for the projects by maximizing NPV value with respect to fixed costs for each project, assuming all other parameters, i.e., diffusion rate, duration of project, market size and discount rate, are the same. What are the optimal fixed costs for both projects? Suppose the extra fixed costs are generated through debts. What are the debt equity ratios are both projects?

S 1 1 1

K 3.349835 2 1

R 0.04 0.04 0.04

T 15 15 15

sigma 0.4 0.4 0.4

c 0.425279 0.540881 0.685679

profit 5.270984 4.886782 3.714818

profit ratio 1.078621 1.418908

debt/equity ratio 0.276222 0.632557

Discussion

• Projects with low operating leverages often have high financial leverages.

Dividend payout policy• Two projects are producing two different products. Both projects

have 2 .5 million dollar fixed cost in production. Suppose the annual output of both products is 1 million. Both projects will last for 10 years. The discount rate is 8% per annum. The diffusion rate for one projects is 50% per annum and for another project is 60% per annum. What is the variable cost for each product? What are the NPV for two projects? The equity owners of the two projects are concerned that their capital structures may not be optimal. They try to determine the optimal fixed cost level for the projects by maximizing NPV value with respect to fixed costs for each project, assuming all other parameters are the same. What are the optimal fixed costs for both projects? Suppose the excess fixed costs can be reduced through additional dividend payout. What are the amounts of additional dividend payout you recommend for two projects?

Solution

S 1 S 1

K 2.5 K 2.1326

R 0.08 R 0.08

T 10 T 10

sigma 0.5 sigma 0.5

c 0.5457 c 0.5799

profit 2.0434 profit 2.0681

ratio of profit 1.0121

additional dividend payout 0.3674

Notes on solution

• Fixed cost on right half of the slide is the level of fixed cost that maximize profit.

• Additional dividend payout is the difference between two fixed costs.

S 1 S 1

K 2.5 K 1.5665

R 0.08 R 0.08

T 10 T 10

sigma 0.6 sigma 0.6

c 0.637 c 0.7149

profit 1.1296 profit 1.2845

ratio of profit 1.1371

additional dividend payout 0.9335

Discussion

• When the diffusion rate is high, the dividend payout is high.

• For declining businesses with high diffusion rate, dividend payout should be speeded so capital can be utilized in higher return projects.

Life cycle of financing

• Initially, uncertainty is high, mainly equity financing.

• Later, with lower uncertainty, debt finance kicks in.

• As firms grow and develop more tangible assets, debt/equity ratio can increase further.

• In a declining stage, cash flows should be distributed to more promising areas.

Example

• Berkshire Hathaway

• When Warren Buffett controlled Berkshire Hathaway, it had large amount of cash flows due to past success. But its share price was very low due to its dim future prospect.

• Warren Buffett diverted its cash flows to higher return investments.

Life cycle of firms and human beings

• There are three basic principles in corporate finance: the investment principle, the financing principle and the dividend principle. We will discuss human life from these three principles.

• A young firm often tries different kinds of businesses before settle on one. Similarly, a small child often tries different things. One year, she may learn skating, next year, skiing, then, soccer, swimming and many other things. You may find your favorite sport eventually and stick to one or several you do best. When a firm is young, it may try out many different things before settling down on one or several main businesses. When a firm is young, a lot of investments are needed. When a firm matures, little investment is needed. It turns into a cash cow. Similarly, when a person is young, she needs to attend school and receive a lot of different trainings. When she is at a mature age, she takes very little training but earns a high income.

• The ability to learn new things changes with age, both with humans and with firms. A child can pick up a new language easily. An adult almost impossible. A child can learn new skills, such as skating, with great ease. For an adult, it often takes great courage to step on ice for the first time. The same is with firms. Young firms can change direction very easily. But established firms can rarely invest in new types of business with ease. This is because small fixed cost systems are more flexible than high fixed cost systems.

• A firm can be financed by either equity or debt. A small child is mainly financed by her parents, that is, financed by equities. When she grows up and earns an income, she has more abilities to get debt financing. For example, she can apply a mortgage or obtain a credit card easily if she has steady income. When she gets older, she may have paid off her mortgages and other debts. At this stage, she mainly finances her activities with her own internally generated funds, that is, her own incomes.

• You will find similar financing patterns for firms. It is almost impossible to get debt financing for start up firms. They generally are funded by owners’ own money, or equities. When a firm generates more steady income and accumulate tangible and intangible assets, they can get debt financing easier to expand. At mature stage, firms rarely need external financing. Internally generated funds are often more than enough to cover investment needs.

• A firm’s value is total sum of dividends discounted over time.  A small child apparently doesn’t pay out dividend, just like a young firm rarely pays out much dividend.  When a person grows up, her ability to help others grows. The most substantial dividend for a person must be her children. If we look at patterns of child bearing over the years, the ages that people start to have children become older and older. This pattern is similarly reflected in corporate dividend payouts. The time that a firm starts to pay out dividend has increased over time and the payout ratio has declined over time because the cost of investment has increased over time.

• Similarly, people start to have children later because of the longer years of school education and longer period of training in works. Is it a good trend? Canadian birthrate has dropped to 1.5 per woman, which is unsustainable. The high tech firms, which pay out scanty dividends, have dropped sharply in their value since 2000. In general, higher dividend growth is equal to high earning growth (Arnott and Asness, 2003).  In any business, common sense is our most important asset. By offering a unified understanding of human life and corporate life, we can extend our everyday experiences directly to financial matters.

Conclusion

• Firm’s capital structure is an integrated part of corporate strategy.

• “market imperfection” is not needed in understanding empirical patterns.