Finance: Static Tradeoff Theory

Background

Firms require capital to operate and generate profits for their owners and investors. Typically, firms are financed with a mix of equity and debt (also known as the firm’s capital structure). Each of these financing sources have their advantages and disadvantage making the decision on the optimal mix between the two a challenging problem for the managers of these firms.

In my corporate finance course, I learned that there are tradeoffs between the benefits (tax shield) and harms (financial distress) of debt. Debt, to a certain point, can be beneficial for a firm. However, too much debt can also be harmful—increased risk of default.

Why is debt beneficial for a firm? It’s because of the tax shield. The tax shield is estimated as the debt multiplied by the tax rate. The interest paid on the debt is tax deductible and can lower the earnings before taxes (EBT) or pre-tax income. However, too much debt can cause financial distress due to the risk of default or bankruptcy.

Thus, the fundamental tradeoff is this balance between the benefits and risks of carrying debt to finance the operations of the firm. So, what is the optimal mix between the benefits and risk of debt? That’s what the static tradeoff theory tells us.

Static Tradeoff Theorem

According to the Modigliani-Miller theorem,1 the value of a levered ($V_L$) firm is equal to the value of the unlevered (V_U) firm plus its debt multiplied by the corporate tax rate (T_C):

V_L = V_U + T_C × Debt

 This implies that the value of the firm increases indefinitely as debt increases. However, unrealistic, this provides some theoretical foundation for how firms think about the benefits of the tax shield.

Conversely, a firm without any debt is simply:

V_L = V_U

which implies that the firm’s capital structure (mix of debt and equity) does not impact its value. But this is also unrealistic as firms tend to finance their operations with both debt and equity.

So, how do we incorporate these two disparate ideas?

That’s where the static tradeoff theorem comes in.

The static tradeoff theorem was first documented by Stewart C. Myers in a 1984 paper published in The Journal of Finance.2 In it, Myers describes the challenges of finding the optimal mix between the benefits (tax shield) and risk (default) of debt for a firm. He focused on two approaches: (1) the static tradeoff framework and (2) pecking order framework. I want to focus on the static tradeoff framework in this article (a future article will review the pecking order framework).  

Let’s imagine that there was no risk (financial distress) from carrying debt. That would mean that firms will have only debt as their capital structure. They benefit from the tax shield while carrying no financial risk. We can plot this with value on the y-axis and the debt amount on the x-axis:

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Note that the value of the firm with debt increases in a linear fashion indefinitely with increased debt amount. Additionally, the value of the firm without debt is a constant or horizontal straight line. What’s needed is a way to balance the two extremes. That’s where the static tradeoff theorem comes in.

Let’s reimagine this figure with some additional elements. We need to think about the cost of financial distress; that can’t be ignored. But it’s based on the probability of financial distress happening. This is related to the amount of debt that is carried by the firm. A low amount of debt would be associated with a very low probability of financial risk. However, as the debt amount increases, the probability of financial risk increases exponentially until it’s a certainty.

In the figure below, I added several elements. The red line indicates the actual value of the firm after considering the cost of financial distress alongside the tax shield benefits of debt. This is best represented by the following formula:

V_L = V_U + (T_C × Debt) + (P_F × C_F),

where P_F is the probability of financial distress and C_F is the cost of financial distress. The cost of financial distress represents the decrease in the company value due to the costs of additional expenses or divestment of assets. Having too much debt can force a firm to undergo costly bankruptcy proceedings, selling assets to pay back lenders, and reduce the morale of their workforce. This is the balance against taking on unlimited debt. The green arrow represents the cost of financial distress whereas the blue arrow indicates the tax shield on the debt. The difference between the two will generate the actual value of the firm.

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Applications

Let’s imagine a hypothetical firm is valued at $1000 without any debt (V_U), and that tax rate is 30%. Let’s also assume that the cost of financial distress is $125. We can figure out what the optimal debt amount is using the following formula:

V_L = V_U + (T_C × Debt) + (P_F × C_F)

We’ll estimate the actual value of the firm by increasing the amount of debt from $0 to 500. We will also exponentially increase the probability of financial distress from 0% to 100%. The probability of financial distress increases exponentially with increasing debt.

We can input these values into Excel and plot the results. Based on the table, we can identify when the actual value of the firm is at its maximum value when the debt is $350. The figure also indicates that the maximum actual value of the firm occurs when the debt is $350.

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Conclusions

The static tradeoff theorem provides a simple way for financial analysts to determine the actual value of the firm by incorporating the benefits and risks of taking on debt. Firms with debt can benefit from the tax shield, but there is a price to pay if that debt gets too large. Typical firms that carry large amounts of debt include those with steady cash flow with high tax-deductible expenses. Firms that likely won’t benefit from much debt include fast-growing companies, cyclical companies and those whose values are in their patents or intellectual properties and generally asset-free. There are more nuances associated with the static tradeoff theorem such as the incorporation of the weight average cost of capital (WACC) and market risk, but I’ll save those for a future conversation.

References

1. Modigliani F, Miller MH. The Cost of Capital, Corporation Finance and the Theory of Investment. Am Econ Rev. 1958;48(3):261-297.

2. Myers SC. The Capital Structure Puzzle. J Finance. 1984;39(3):574-592. doi:10.1111/j.1540-6261.1984.tb03646.x

Other helpful resources

“Theories of Capital Structure II – Static Trade-off Theory“ from PrepNuggest website (link)

“M&M Theorem“ from CFI Education website (link)

Michael Knoll’s article, “The Modigliani-Miller Theorem at 60: The Long-Overlooked Legal Applications of Finance’s Foundational Theorem“ (link)

“Exploring the Modigliani-Miller Theorem: A Comprehensive Investor's Overview“ from Investopedia (link)

“Trade-off Theory” from Wall Street Prep website (link)

“The Trade-off Theory (Static Theory)” from LexPlug website (link)

“Financial Distress Costs: Impact, Calculation, and Business Effects” from Investopedia (link)

Excel file

You can download the Excel file from the following my GitHub site (link).

Disclosures and disclaimers

This is a work in progress and updates are expected.

I used Claude Opus 5.5 to help create the example for this article.

This is for educational purposes only.

Finance: Markowitz portfolio variance

Background

I wanted to learn about finance, so I started to take some courses in it. It’s been nearly a year, and I have learned a ton. To help reinforce what I’ve learned (and to share the knowledge), I plan to write about finance every once in a while. For this first article on finance, I wanted to write about the Markowitz portfolio variance.

In finance, one of the most important discoveries was by Harry Markowitz when he figured out how to measure the portfolio variance. It is used in modern portfolio theory to estimate the combination of investments that would reduce idiosyncratic risk, which is inherit in the asset (or a group of assets). Idiosyncratic risk is unrelated to the risk in the market. Rather, it is the risks when assets that you invest in are correlated with each other. The more correlated they are, the more risky they become.

Assets that are part of the same sector or type tend to be correlated with each other. For instance, General Motors is an automobile company, and its stock price will likely be correlated with another automobile company like Ford Motors. If you want to diversify your portfolio, investing in both General Motors and Ford Motors increases your idiosyncratic risk since both investments would be part of the same sector. If one of these firms does poorly, it is likely that the other firm will do poorly.

Markowitz was able to develop an equation that would capture the idiosyncratic risk associated with a combination of assets in an investment. His portfolio management variance calculation uses the standard deviation of one asset (Asset A) and its correlation with another asset (Asset B). In other words, Markowitz’s formula incorporates the correlation between assets. By incorporating the correlation between assets, an investor can estimate the idiosyncratic risk of their investment strategy and avoid stocks in market sectors that would potentially be a greater risk due to their correlation.

Markowitz’s portfolio variance formula for a combination of two assets (i and j) is structured as:

where:

• sigma_p^2 is the portfolio variance

• w_i is the weight of Asset i

• w_j is the weight of Asset j

• sigma_i^2 is the variance of Asset i

• sigma_j^2 is the variance of Asset j

• rho_{i, j} is the correlation between Assets i and j

We can simplify this expression for a two-asset portfolio as:

Motivating Example

Let’s look at an example. Suppose we have two assets (i and j) that we want to diversify our portfolio with. We’re concerned about the idiosyncratic risk between the two. We can use this formula to estimate the idiosyncratic risk. (Note: You can download the Excel exercise from my GitHub repository here.)

Here is the data for the returns for Assets i and j:

We can estimate the average returns for Assets i and j along with their standard deviations.

Then, we can estimate the covariance between the two assets’ returns.

Next, we need to apply weights to the assets. Let’s assume that we want to place more weight to Asset i compared to Asset j. The weight for Asset i will be 0.85; hence, the weight for Asset j is 0.15 (or 1 – 0.85). The sum of the asset weights need to be 1.

Once we have estimated all the parameters, we can now combine them using the Markowitz portfolio variance formula.

The idiosyncratic risk is 0.0043, which is pretty low since the correlation between Asset i and Asset j is very small (rho = -0.000156).

Concluding thoughts

There is more to modern portfolio management, but this is just the first part. As we continue in future lessons, we will build upon this knowledge by learning about how we can use the Markowitz portfolio variance to determine the risk and return tradeoffs.

Disclosure and Disclaimer

Since this is a work in progress, expect updates in the future.

Meanwhile, this is for educational purposes only

References

You can learn more about Harry Markowitz on Wikipedia here.

You can also learn more about portfolio variance on Investopedia here.

You can download the Excel exercise from my GitHub repository here.