"A company must finance its assets with capital. There are two primary sources of capital: debt and equity. That's how you end up with the fundamental accounting equation: $A=L+E$. "$L$" stands for liabilities, which is just another name for debt.
The capital providers collectively own all the assets. The debt holders and equity holders own the assets. They provided the funds to buy the assets.
When the company finances its assets, it has to pay for this financing. The financing rate for debt is called the debt cost of capital, and the financing rate for equity is called the equity cost of capital.
The total risk of financing the company's assets is the same regardless of the capital structure. It doesn't matter if you have a lot of debt or no debt, the total risk is the same. That's because you are not changing the risk of the assets. You're just changing the financing of the assets. The asset cost of capital should not change.
Let me try and explain why the total risk doesn't change based on the financing. Assume you buy a restaurant. You have some risk in owning the restaurant. The expenses could increase if the labor costs go up. Or the revenue could go down if customers don't like the food or a better restaurant opens.
Now assume instead of just you buying the restaurant, we buy the restaurant together. It doesn't have to be 50/50. Maybe you pay for 60%, and I pay for 40%. Also, the risk doesn't have to be shared proportionately. I could just get a fixed rate of return, and you could take on most of the risk and receive most of the upside potential. But regardless of how we share the financing and the risk, the total risk of owning the restaurant doesn't change. The total risk is based on the expenses and revenues of the restaurant. We're just deciding how to allocate that risk between us.
I hope that explains why the total risk of financing the company's assets is not dependent on the capital structure.
The debt cost of capital and the equity cost of capital are not the same. Debt is less risky, because it has a priority claim on the assets. If something bad happens at the company, the debt holders will get paid back first. So the debt cost of capital is less than the equity cost of capital.
Because the asset cost of capital does not change, the weighted average of the debt cost of capital and equity cost of capital must not change. It is constant for the same assets regardless of the financing mix.
Another name for the asset cost of capital is the unlevered cost of capital. That's because the asset cost of capital equals the equity cost of capital if the company has no debt. That would mean all the assets are financed only with equity. So the fundamental accounting equation would be $A=E$.
We've already explained why the asset cost of capital does not change based on the type of financing. The risk is based on the assets. So the unlevered cost of capital will not change based on how the assets are financed.
$$r_U = w_E \cdot r_E + w_D \cdot r_D$$
We often assume the debt cost of capital is constant. But the equity cost of capital increases as you add more debt. That's because the investment gets riskier for the equity holder because equity holders get paid after the debt holders, if there is anything left. So in the above equation, $r_E$ is increasing as $w_E$ is decreasing. That is how the weighted average stays the same.
If the name "unlevered" cost of capital is confusing, just call it the asset cost of capital. That is really more descriptive anyway. The asset cost of capital is the rate required to finance the assets, which is the weighted average of the debt cost of capital and the equity cost of capital.
Using the "unlevered" term can be especially confusing when you're talking about a company that has debt. If a company has debt, why do we need to know the unlevered cost of equity?
Because sometimes you want to compare the cost of capital between companies on an apples to apples basis. So we can compute the unlevered cost of capital for both companies, which represents the cost of capital for both companies assuming they have no debt.
You also need to compute the unlevered cost of capital for a company if you want to determine the equity cost of capital for a different level of debt. Start by computing the unlevered cost of capital with this formula at the companies current level of debt.
$$r_U=w_E \cdot r_E + w_D \cdot r_D$$
Then recognize that $r_U$ will stay constant. So if you change $w_D$, you can solve for $r_E$.
Hopefully this is making sense so far. There is just one more concept to cover.
The asset cost of capital, or unlevered cost of capital, is the expected return investors will earn holding the firm's assets. But this is not the same as the after-tax cost of capital to the firm. Because the firm can deduct interest paid on the debt. So we need another measure to calculate the after-tax cost of capital.
That is the weighted average cost of capital (or WACC). The formula is very similar to the asset cost of capital, but the after-tax cost of debt is used rather than the pre-tax cost of debt. The cost of equity is not adjusted for taxes because firms cannot deduct the dividend payments to equity holders.
$$r_{WACC} = w_E \cdot r_E + w_D \cdot r_D \cdot (1− \tau C)$$
The weighted average cost of capital will not stay constant as the financing mix changes. Companies benefit from using debt because the interest is deductible. So as $w_D$ increases, the WACC will decrease."
Credits to Mike Carmody of CoachingActuaries for this insight.