"If you increase cheap debt, WACC goes down"—where that intuition goes wrong

"The interest rate on debt is 8%, and the cost of equity is 15%. Therefore, if you increase the ratio of cheap debt, the weighted average WACC should go down. In other words, debt is a magic tool for lowering the cost of capital."

—Have you ever heard this explanation? Or perhaps you have calculated it that way yourself.

To be clear on terminology, WACC stands for "Weighted Average Cost of Capital." It is the weighted average of the procurement costs of debt (loans and corporate bonds) and equity (stocks) based on their monetary ratios, which is, so to speak, the price of the company's total capital procurement.

In fact, this initial intuition was proven to be "not valid in principle" over 60 years ago. It was proven by two economists named Franco Modigliani and Merton Miller, and it is known as the "MM Theory." Both of them are Nobel Prize winners in Economics.

However, in textbooks, the MM theory often has a high barrier of mathematical formulas, which tends to discourage readers along the way. "Propositions," "arbitrage," "irrelevance"... just looking at the words lined up makes you want to quietly close the book. I was the same way at first.

So this time, I aim to make this theory "sink in" using almost no mathematical formulas, just 6 diagrams. This content is intended for those who have never studied finance systematically, or those who have calculated WACC in their practical work but have wondered, "Are these numbers really correct?"

To give you the conclusion first, it is this.

By the time you finish reading, you should be able to recall the meaning of this sentence along with the diagrams. Now, let's begin.

1. First, what is the "difference" between debt and equity?

First, let's start with the most fundamental point. Debt (loans/corporate bonds) and equity (stocks) have completely different "rules for receiving money."

Imagine a situation where a company is sold or liquidated and the money is divided up. The rules are simple, and there are only two.

Debt is "first-come, capped." Creditors can receive money before anyone else, but they can only receive up to the amount they lent (300 million yen in the diagram). No matter how successful the company becomes, they will not receive a single yen more than that.

Equity is "residual, unlimited." Shareholders can only receive money last, but everything left over after paying the debt is all for the shareholders. There is no upper limit.

Looking at the graph in Figure 1, you can see that once the enterprise value exceeds 300 million yen, the increased portion flows entirely into the red area (equity).

For example, imagine you are running a shop with two people and you have signed a contract that says, "Person A receives 300 million yen from the sales first. Person B receives everything else." If the sales are 200 million yen, Person A gets 200 million yen and Person B gets zero. If the sales are 1 billion yen, Person A is capped at 300 million yen, and Person B gets 700 million yen. Person A is debt (creditor), and Person B is equity (shareholder).

It is often explained that "the difference between debt and equity is the presence or absence of a repayment obligation," but the perspective that really matters for understanding the theory is this one—"first-come, capped" or "residual, unlimited"—the difference in the rules for one's share. This structure of "300 million yen being the dividing line" is the starting point for everything that follows.

2. Leverage amplifies "fluctuations"

So, what does this rule of division mean for shareholders? Let's look at the same company (enterprise value of 1 billion yen, debt of 300 million yen) in a case where the enterprise value fluctuates by ±10%.

Even if the enterprise value increases from 1 billion yen to 1.1 billion yen (+10%), the debt's share remains fixed at 300 million yen. The additional 100 million yen belongs entirely to equity. Equity goes from 700 million yen to 800 million yen, a change of +14.3%.

The reverse is also true. Even if the enterprise value decreases to 900 million yen (-10%), the debt still takes its 300 million yen. The 100 million yen loss is borne entirely by equity. It is -14.3%.

In other words, while the enterprise value fluctuates by ±10%, the equity fluctuation is amplified to ±14.3%. The amplification factor is "1 billion yen enterprise value ÷ 700 million yen equity = 1.43 times".

This is actually the exact same structure as a home mortgage. Suppose you buy a 50 million yen condo with a 10 million yen down payment and a 40 million yen loan. If the property value drops by 10% (5 million yen), the repayment amount to the bank does not change, so the 5 million yen loss is borne entirely by your own funds. It is calculated that half of your 10 million yen down payment, or -50%, is wiped out. You might think, "But the price only dropped by 10%," but this is the true nature of leverage. Of course, if the price rises by 10%, your own funds increase by +50%. It is amplified by the same factor both up and down.

What is important here is that debt bears none of the fluctuation (this is the callout on the far right of Figure 2). For those who thought, "In Figure 1, wasn't the debt also reduced when the enterprise value fell below 300 million yen?", you are sharp. That is correct; in this example, the enterprise value (900 million to 1.1 billion yen) is significantly higher than the 300 million yen debt, so the setting is that the debt can be considered "mostly in the safe zone." In a world where falling below 300 million yen is a possibility, debt would not remain unscathed, but I will save that discussion for next time along with bankruptcy costs.

Even though the total amount of "fluctuation" in enterprise value does not change, because the portion borne by debt is zero, equity bears all of the fluctuation. This asymmetric distribution will be the foreshadowing for the next chapter.

3. Even so, WACC does not go down

Now, before we get to the main topic, one promise. For a while from here, we will think in a "blank world" with no taxes and no bankruptcy costs. It is a world that does not exist in reality, but I will properly explain why we bother to think in such a world at the end, so please bear with me for a little longer.

The cost of debt is 8%, and the cost of equity is 15%. It looks like "if you increase the cheaper one, the average will go down." However—.

Please look at Figure 3. Even if you increase the debt ratio (D/V) from 0% to 60%, the WACC (dark blue line) remains flat at 15%.

Why? Because as the debt ratio increases, the return rE required by shareholders (red line) increases. For example, if D/V = 30%, rE rises from 15% to 18%. Then,

WACC = 0.3 × 8% + 0.7 × 18% = 15%

The "savings" from mixing in cheap debt are perfectly eaten up by the increase in shareholder costs. Moreover, it is not "roughly offset," but exactly offset. This is what is called the MM second proposition.

The point is that the way rE rises is not based on mood, but is strictly determined by a formula. The callout in Figure 3 shows

rE = 15% + 3/7 ×(15% − 8%)= 18%

This is it, and it takes the form of "where it would have been 15% if there were no debt, a premium is added according to the ratio of debt to equity (3:7)." The higher the debt ratio, the greater the premium, and rE rises indefinitely as shown by the red line in Figure 3. On the other hand, WACC does not move a single step from the starting point of 15%—the cost of capital of the business itself (rA)—.

Debt is certainly cheap. But it is not free. The bill for that cheapness is passed on to shareholder costs.

"Is it really that convenient that shareholders raise their required return?" For those who thought that, that is a good question. The next figure is the answer.

4. Risk does not disappear. It is simply concentrated in equity

This is the point I want you to remember most from this article.

The total amount of a company's business risk is determined by the nature of the business. The risk of a ramen shop is determined by its location, taste, and competition; it does not change by even a gram depending on whether the startup capital was funded by debt or equity. Since the number of customers tomorrow will not change even if you change the method of financing, this is only natural.

In Figure 4, this total amount of risk is represented by "area." If there is no debt, that area is absorbed thinly and broadly by 100% equity. However, when debt is increased, the "receptacle" for equity narrows by the amount of debt, which bears almost no risk. Even though the area (total risk) is the same, the receptacle is narrower. What happens then?

Think of Calpis. If the amount of the undiluted solution (total risk) is the same and you only reduce the water (equity)—it becomes more concentrated, right? In terms of Figure 4, as the thickness of the equity thins from 100% to 70% to 40%, the risk density per yen is concentrated from x1.0 to x1.43 to x2.50.

Shareholders demand a higher return as compensation for taking on this concentrated risk. This is the reason why rE was rising in Figure 3. It is not rising conveniently; it wouldn't be worth it if it didn't rise.

And once we reach this point, the reason why "WACC does not move" in Figure 3 can also be explained in a single picture. WACC is the average of the required return for the entire company—that is, the entire area. Since the area does not change, the average does not change either. All that changed was the "way of dividing" the area, and no matter how you change the way of dividing, the total sum remains the same. The operation of increasing debt was not an operation to reduce risk, but merely an operation to change the location of the risk.

5. So, where does the calculation that "WACC has gone down" come from?

Nevertheless, in practice, you sometimes see calculations stating, "WACC went down when we increased borrowing." In acquisition scheme reviews, on the capital policy page of business plans, or in a company's own ROIC (Return on Invested Capital) management materials. Where on earth do those numbers come from?

To reveal the secret, in many cases, the calculation is performed while keeping the shareholder cost rE at the original level even after increasing leverage.

The left side of Figure 5 is that calculation. Even though D/V = 30%, if you use rE as 15%,

WACC = 0.3 × 8% + 0.7 × 15% = 12.9%

It certainly looks like you "saved" 2.1 points from 15%. But as we saw in the previous chapter, if you increase leverage, the shareholders' risk is concentrated, and rE should have risen to 18%. If you recalculate correctly (the right side of Figure 5), WACC is 15.0%—in other words, it hasn't moved by a single millimeter.

The true nature of the apparent "2.1pt savings" was that the compensation for the risk shifted from debt to equity was overlooked in the calculation. It was not savings, but an omission in accounting.

To add for the sake of honor, this is not a story of "the person who calculated it was lazy." Since shareholder cost is not a number that can be directly observed from the market, "keeping it at the current level for now" is a process that is easy to do unconsciously in practice. However, the moment you keep it fixed, that calculation has—without realizing it—made a strong assumption that "moving risk is free." It is worth knowing at least that much.

Note that in actual calculations, there is room for WACC to truly decrease because of the tax-saving effect of interest payments (so it is by no means the case that all practical calculations are wrong). The important thing is not to mix the "real decrease" derived from tax savings with the "apparent decrease" generated from keeping rE fixed. Mixed calculations make the effect of leverage look larger than it actually is.

If you encounter a calculation where "WACC goes down due to leverage" next time, please check it quietly. Ask, "Is that rE adjusted to the level after leverage?" Not to corner them, but as a remark to align the foundation of the discussion.

6. The final blow—investors can create leverage themselves

By now, the logic behind "why WACC doesn't go down" should be clear. Finally, I would like to introduce the most brilliant part of MM theory: the idea that "in the first place, there is no inherent value in a company taking on leverage itself."

Let's compare two investments.

A: Invest 1 million yen in the stock of a leveraged company (D/V=30%).

The company has 300 million yen in debt. The remainder after paying 8% on the debt is your share as a shareholder.

B: Invest in the stock of a debt-free company while borrowing the money "yourself."

The company is debt-free. Instead, you borrow money personally to buy the stock. The way you borrow mimics the company. Since company A had a debt-to-equity ratio of 3 to 7, you also borrow approximately 430,000 yen at an 8% interest rate against your own 1 million yen in capital, and buy a total of approximately 1.43 million yen worth of the debt-free company's stock. You receive the business returns, and the remainder after you pay the 8% interest yourself is your share.

Look at the graph at the bottom of Figure 6. No matter how the business turns out—whether it's +20% or -20%—the net returns for A and B are exactly the same in all scenarios. If the business return is +20%, both net returns are +25.1%. If it's -20%, both are -32.0%. The solid red line (A) and the dashed yellow line (B) overlap so perfectly that they look like a single line.

The reason is simple: in both routes, the same money flows in the same order: "business returns are generated -> first, 8% interest is paid -> you receive the remainder." The only difference is whether that interest payment happens on the company's balance sheet or in your personal household budget. The content is the same, only the place where you borrow is different. The content is the same, only the place where you borrow is different they are.

Furthermore, there is a fundamental rule in the financial world: "the same cash flow can only have the same price" (if the prices were different, you could just buy the cheaper one and sell the more expensive one to make a risk-free profit. Such distortions are immediately eliminated by arbitrage). There is no premium added just because a company does something that an investor can replicate at home. Conversely, even if a company doesn't take on leverage, those who want it can do it themselves.

Therefore, corporate value and WACC are independent of capital structure—this is the first proposition of MM theory, the so-called "irrelevance of capital structure." The story we have built up from Figure 1 has come together here.

Note that in the order of textbooks, this first proposition is the foundation, and the second proposition (the formula for the rise in rE) seen in Figure 3 is the "flip side" derived from it. In this article, I have rearranged them in an order that is easier to grasp. If you are going to relearn from a textbook, just be aware that the order is reversed.

I will answer common points of confusion in advance.

Having read this far, I will touch on two points that some of you may be stuck on.

"But don't leveraged funds and real estate investments actually produce high returns?"

They do. However, as seen in Figure 2, that is because returns and risks are amplified by the same multiplier. Leverage raises expected returns, but at the same time, it deepens the downside by the same amount (you are also buying into a world of -32% net returns). The gain per unit of risk has not increased—that is the MM perspective. You could rephrase it by saying that leverage is an "amplifier," not "alchemy."

"But stock prices actually move on news of capital increases or borrowing, right?"

They do. However, rather than the value of the financing itself, that is explained as the effect of information held by management leaking through the means of procurement (information asymmetry) or changes in the outlook for taxes and bankruptcy risk—in other words, a matter of friction. This is deeply related to the theme of the next article, so I will leave it as foreshadowing for now.

Conclusion—MM theory works precisely because "there is friction in reality"

For those who read this far and thought, "No, no, in reality, corporate value can change with debt." You are correct.

MM theory holds true in a world without taxes, bankruptcy costs, or information asymmetry. In reality, there is a tax-saving effect (tax shield) where interest payments are tax-deductible, excessive borrowing brings the risk and cost of bankruptcy, and it is not easy for individuals to borrow money under the same conditions as a company. Therefore, capital structure in reality has real meaning. In fact, the reason corporate CFOs worry about capital structure is not because they are doing something useless.

However, does that mean MM is a "useless theory"? Quite the opposite. The brilliance of MM lies in the fact that it proved that "capital structure is irrelevant in the absence of friction," thereby identifying that "if capital structure changes firm value, the cause must lie in the friction itself." It is like a "reference value" in a medical checkup. Even if no one matches the reference value exactly, the existence of the reference value allows us to discuss "where and by how much things are off."

Knowing MM allows you to move beyond vague explanations like "WACC goes down because we mixed in cheap debt" and ask a deeper question: "That effect should disappear in the MM world, right? If it remains, it would be due to the tax shield effect, but how much are you estimating that to be?" I believe theory is less of a tool that gives you answers and more of a tool to increase the precision of your questions.

Finally, I will summarize today's six points in one sentence each.

• Figure 1: Debt is a priority claim with a cap; equity is a residual claim with unlimited upside.

• Figure 2: Therefore, leverage amplifies the volatility of equity.

• Figure 3: Even if you increase debt, WACC does not decrease (MM Proposition II).

• Figure 4: Risk does not disappear; it is simply concentrated in the thinned-out equity.

• Figure 5: The claim that "WACC has decreased" arises from forgetting to increase rE.

• Figure 6: Investors can replicate leverage themselves. Therefore, capital structure has no inherent value.

"Risk does not disappear, it just shifts." I would be happy if you could take at least this one sentence home with you.

Next time, we will intentionally introduce "friction" into the MM world. We will explore the trade-off theory: is there an "optimal amount of debt" where the tax shield effect and bankruptcy costs reach a tug-of-war? We will also delve into the question of whether the recently popular claim that "startups should raise funds with cheap, non-dilutive debt" is actually true. Stay tuned.

This article is intended to introduce theory and does not recommend any specific investment or financial decisions.