Building on the previous post, if we imagine a firm with the following beginning characteristics:

$500 million Debt

$500 million Market Capitalization

$25 million Annual Interest Expense

then, if we view equity as an option on the enterprise (basically the Merton Model), we have a call option with a Strike Price of $500 million, an underlying asset value of $1 billion, and an option premium of $25 million per year (in the form of interest).

Now, imagine the market outlook for this equity rises, and the Enterprise Value increases to $1.5 billion. In this Merton-like way of looking at it, we are now paying a $25 million premium for a call $1 billion in-the-money, compared to the original position which was paying a $25 million premium for a call only $500 million in-the-money. From this perspective, the premium values have gone up. (On options, the premium usually declines as the strike price moves away from the underlying asset price. This change in relative rates means that the "strike price" of equity shares is more "in the money", but is still paying the same premium.)

There could be several causes of an increase in value:

1) Decreased Equity Risk Premium

2) Decreased Debt Interest Rate

3) Increased Expected Earnings

Or, an example in the opposite direction (shown in the above graph):

Imagine that interest rates change so that interest expense is $50 million, but equity risk premiums remain stable. Enterprise value will decrease, because the higher premium will mean that equity holders bid the Enterprise value down so that they are less in-the-money than they were when the premium was lower.

Here is a graph of payouts to equities, presented as call options. Just as call options limit the loss of the buyer to the strike price, equities limit the loss to the firms' owners to the level of equity. In effect, debt holders are selling call options on the enterprise to the equity holders and charging them a premium. The difference in premiums is barely noticeable in this graph because the premiums are very low compared to the enterprise value. There are two reasons. (1) Equities are generally very "in-the-money". There is usually a low chance of bankruptcy, which would be the equivalent of having the options expire out of the money. (2) Most call options expire within a few months or years. Equities are call options with no expiration date. I am expressing the premium here in annual terms, and the annual premium will be very small compared to the perpetual value of the equity. On an actual call option, the premium is paid up front and sort of amortizes away over time. With debt, interest is accrued and paid periodically. But, this doesn't really change the analogy.

The next graph is a view of the annual proportional returns to equity in the same 3 scenarios. Again this looks very much like a graph of returns on a set of call option contracts with different strike prices or different levels of implied volatility. There is a tradeoff between a lower breakeven level (where the line crosses the y-axis) and the slope of the payoff line as a percentage of the price of the options.

A lower equity risk premium (by increasing the market capitalization) or a lower interest rate (by lowering the "premium" on the equity option) both have the effect of leading to a new equilibrium enterprise value that lowers the slope of the proportional payout and pulls the y-axis breakeven point up toward zero. And vice versa.

So, holding earning potential stable, the value of the firm is a product of the relationship between debt interest rates, equity risk premiums, and leverage. And, note that, when the relative required returns of equity and debt holders diverge, the scenarios above point to higher leverage when debt is relatively more expensive! (In hindsight, I'm getting ahead of myself. I don't think this last sentence is supported by the simple analysis above, but it is by the analysis below.)

**What about leverage?**

Conventional wisdom is that firms leverage up when interest rates are low. But, some financial analysis gives the opposite intuition. The Modigliani-Miller Theorem posits that we should be indifferent to debt vs. equity, as, in a market without asymmetrical frictions, the risk premiums should adjust with leverage so that Enterprise value is unaffected. From this starting point, debt is favored due to preferential tax treatment. As leverage increases, the cost of both debt and equity increase, so there is usually some optimal leverage level where enterprise value is maximized. As with the scenarios above, counterintuitively, since higher relative debt expenses create more tax savings, this model suggests that the higher the relative cost of debt, the more debt a firm should utilize.

If we price equity as a call option, using Black-Scholes, this is indeed the outcome we get. Here is the graph of the relative value per share of a single firm in various interest rate contexts. All operational expectations and tax rates are constant here. All of these changes are a product solely of interest rates.

As a reminder, here is the Capital Asset Pricing Model:

The discount rate applied to the future cash flows of a firm, for the purposes of valuation, is a combination of the risk free rate, the relative market-correlated volatility of the firm's equity (Beta), and the Equity Risk Premium (expected market return minus the risk free rate).

So, the following graph displays the relative value of a share of stock in a company whose equity is valued with a Black-Scholes model using Enterprise Value (equity + debt) as the underlying asset on the option. Expected volatility, tax rate, revenue, and earnings are stable. The variables that change are the risk free interest rate (RFR), the unlevered Equity Risk Premium (ERP), and the amount of debt the firm uses. The required return on equity is RFR+ERP. Each line shows the share value of the firm with a given RFR/ERP combination, as the debt level increases.

Note that, as we should expect, valuations rise for the unleveraged firm (no debt, all equity) as the composite interest rate declines (see labels on left scale). But, because of the tax advantage of debt, the other relationships between rates and valuations are counterintuitive.

If ERP is stable, but RFR falls (the red arrows), the cost to the firm for debt and equity would both fall. Intuition would suggest that firms might leverage up in response to cheaper debt, but because of the counterintuitive value of debt, the optimal firm would deleverage.

If RFR is stable, but ERP falls (the purple arrows), the cost of debt would remain stable, but the cost to the firm of equity would fall. Intuition says this should cause firms to offer more equity, because investors will demand fewer earnings for the same amount of capital. Again, the optimal firm does the opposite, and leverages up with debt, even as enterprise value rises.

Finally, if RFR rises while ERP falls by an equal amount (the green arrows), the cost of equity remains stable while the cost of debt rises. Surprisingly, even though the discount rate on equity capital has not changed, (so that the value of the unlevered firm would not change at all) the optimal firm can increase its Share Price and Enterprise Value by leveraging up, and trading equity for debt as the debt becomes more expensive!

For instance, look at the scenarios where the cost of equity capital is 5%. For the scenario where RFR=1% and ERP=4%, the firm's Share Price is relatively unresponsive to leverage. But moving up to the scenario where RFR=4% and ERP=1%, now the firm can leverage up with the more expensive debt and increase its Share Price by nearly 20%, just by adjusting its capital base...to the now more expensive debt!

**PS**

If this seems like it can't be true, keep in mind that the beta of the firm's equity is increasing with leverage. So, if RFR=2% and ERP=3%, then the unlevered firm's equity will require 5% returns. But, when the firm replaces 50% of the equity with debt, the firm's earnings per share will be twice as volatile, so equity will now require 2+(2*3)=8% returns. So, for firms that have required returns to equity of 5% when they have no debt, the required rate of returns will rise more for the firms with higher ERPs as debt (and beta) rises. So when we compare these firms that have equal unlevered valuations as they utilize debt, earnings per share will be higher when debt is cheaper, but risk adjusted value to equity in these cheap debt scenarios will be lower.

**An Aside**

I will dig into the implications of this in upcoming posts, but as an aside, this analysis shows one of the many ways that taxes on capital damage an economy. The advantage of leverage is greatly increased by the presence of corporate taxes. If we didn't tax corporate profits, firms would tend to be much less leveraged. Ownership would be more widely spread, and the economy would be less vulnerable to panics, crashes, defaults, and bankruptcies.

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