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.

Communicating data effectively with data visualizations - Part 5 (Colors)

Background

As we decide the type of chart we want to visualize our data with, we should also think about our color scheme. Color should represent the data value. We should be able to see a chart, identify the color that interests us and associate a value with it. However, it is easy to misidentify the data that the color is supposed to represent and it is also easy to confuse the magnitude of the difference between groups.

In this tutorial, we will discuss some basic elements of color theory that will help you to select the best color palette for your graphical presentations. We will use the following simple data and color schemes to illustrate some of these lessons (Figure 1). The X column denotes some arbitrary category that uses the alphabet and the Y column denotes the values associated with each category.

Figure 1. Example data used to illustrate the different color schemes.

Figure 2. Color schemes used in this example (single-hue color and rainbow color schemes).

 

Bias-Precision Tradeoff

There is always a tradeoff between bias and precision. Bias is associated with the ability to accurately identify the data value associated with the color. Precision is associated with the ability to correctly identify the data with little variation. Figure 3 illustrates the differences between the two.

Figure 3. Bias and Precision tradeoff.

For example, if you had a single-hue progression color scheme (blue), it is easy to identify the color that contains the higher value versus the color that contains the lower value. In Figure 4A, the darker shade of blue is associated with a higher value compared to the lighter shade of blue. However, it may not be easy when you have a rainbow color scheme because the values can be arbitrarily associated with a different color (Figure 4B). In other words, Figure 4A helps to reduce the bias while Figure 4B can generate a lot of bias.

Figure 4. Comparison between single hue color scheme and rainbow color scheme.

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Additionally, Figure 4B is easier to distinguish between the groups. You can easily identify Group F compared to Group E because the colors are distinctly differentiable. However, in Figure 4A, it is very difficult to distinguish between Group F and Group E because the shades of blue associated with both groups appear similar. Therefore, the ability to precisely detect the differences in the groups is limited by the single-hue color scheme (Figure 4A) compared to the rainbow color scheme (Figure 4B).

Another way to look at these principles is to think about the concurrence between the true value and the estimated value. Suppose you were given these color schemes and asked to estimate the correct value. How do you think you’ll do? Would it be easier to estimate the true value from the single-hue color scheme or the rainbow color scheme?

Figure 5. Performance of color schemes according to estimated and true values.

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I generated data in Figure 5 to illustrate the principles in this tutorial. Figure 5 compares performance between the single-hue color scheme and rainbow color scheme. Figure 5A has low bias but some scatter (moderate precision) across the gray line, which denotes the 1:1 accuracy between true and estimated values. However, Figure 5B has high bias but very little scatter (high precision) due to the easy identification of the groups. Whereas, there was some uncertainty in correctly identifying the true value within each group.

Summary

When deciding on the color scheme for your data, take into consideration what is more important. Is it critical that your audience precisely distinguish one group from another? Or is it more important to have them visually identify the correct value associated with the color? Ideally, you want to be able to use a color scheme that has low bias and high precision, but in reality, you will need to make a tradeoff between the two.

References

I used the following references to develop this tutorial.

Liu Y, Heer J. Somewhere over the rainbow: An empirical assessment of quantitative colormaps. ACM Human Factors in Computing Systems (CHI) 2018, April 21-26, 2018, Montreal, QC, Canada. http://dx.doi.org/10.1145/3173574.3174172

Cromwell W. Colour schemes in data visualisation: Bias and Precision. Presented at the useR! 2016 international R User conference, June 15, 2017. URL: https://channel9.msdn.com/Events/useR-international-R-User-conference/useR2016/Colour-schemes-in-data-visualisation-Bias-and-Precision