Modigliani-Miller & Unlevered Betas

In 1958 Franco Modigliani and Merton Miller wrote a nobel-prize winning paper describing how leverage effects the value of companies. In this post we’ll discuss some of their results and apply it to the derivation of a well-known formula used to de-lever equity betas.

Suppose there exist two companies that own exactly the same assets and produce exactly the same stream of uncertain earnings. Let one company be completely unlevered, i.e. completely financed with equity, and the other company be partially financed by debt. Denote the annual earnings before interest and tax (or EBIT) of both companies by the random variables  X_1, X_2, X_3, \ldots that are governed by some probability distributions. We assume that the value of each firm is only a function of its stream of after-tax EBIT (or EBIAT). However, since interest payments are tax-deductible, the value of each firm will depend on its leverage.

First suppose that the company is unlevered, i.e. has no debt and is fully financed with equity. In this case, on year i, the company produces after-tax EBIT of X_i(1-\tau) where \tau is the corporate tax rate for this company. The value of the company can be written as

\displaystyle V_U = \mathbb{E}\left[\frac{X_1(1-\tau)}{(1+r_U)} + \frac{X_2(1-\tau)}{(1+r_U)^2} + \cdots \right]

where r_U is some discount factor appropriate for the risk associated with the earnings stream.

Note that we are discounting earnings and not cash. We assume that earnings convert into cash quickly and that management acts in the best interest of the shareholders when reinvesting earnings instead of paying cash dividends to equity holders.

Now consider the case where the company is partially financed with debt. Let D be the market value of the debt and r_d is the interest paid on debt or the cost of debt. On year i, this company will produce earnings after tax and before interest of \displaystyle (X_i-r_dD)(1-\tau)+r_dD=X_i(1-\tau)+r_d\tau D

Notice that we can separate these earnings into earnings identical to the unlevered firm X_i(1-\tau), and fixed earnings r_d\tau D from the tax benefit of debt financing. In fact, we can replicate these earnings exactly by holding a portfolio of the unlevered company and \tau D amount of the debt of the company. Therefore, the value of the levered company must be exactly equal to the value of the portfolio containing the unlevered company and \tau D debt.

V_L = V_U+\tau D

Unlevered Betas

Now we will use the relationship derived above to show how the beta of a company’s equity to the market changes with its leverage. Let E represent the value of the equity of a levered company (share price or market cap), and let M be the level of a market index. By CAPM, we know that

\displaystyle \frac{\Delta E}{E}=\beta_L \frac{\Delta M}{M} + \epsilon

Where \epsilon is a a random with \epsilon \sim \mathcal{N}(0,\sigma^2). Here, \beta_L is said to be the levered beta of the equity. Now suppose that the company had no leverage and denote the hypothetical value of the unlevered firm as V_U. Then we can write

\displaystyle \frac{\Delta V_U}{V_U}=\beta_U \frac{\Delta M}{M} + \epsilon

where \beta_U is the unlevered beta of the company. Solving for \frac{\Delta M}{M} in each and setting those equal to each other, we get

\displaystyle \frac{1}{\beta_L}\left(\frac{\Delta E}{E} -\epsilon\right)=\frac{1}{\beta_U}\left(\frac{\Delta V_U}{V_U}-\epsilon\right)

From above, we know that V_L =D+E=V_U+\tau D so we can replace V_U with E+(1-\tau)D. If we assume D is constant and does not fluctuate with the market, this also implies that \Delta E = \Delta V_U.

\displaystyle \frac{1}{\beta_L}\left(\frac{\Delta E}{E} -\epsilon\right)=\frac{1}{\beta_U}\left(\frac{\Delta E}{E+(1-\tau)D}-\epsilon\right)

Taking expectations

\displaystyle \frac{1}{\beta_L}\left(\frac{\mathbb{E}[\Delta E]}{E} \right)=\frac{1}{\beta_U}\left(\frac{\mathbb{E}[\Delta E]}{E+(1-\tau)D}\right)

\displaystyle \beta_U=\frac{E}{E+(1-\tau)D}\beta_L=\frac{\beta_L}{1+(1-\tau)\frac{D}{E}}

The Time Value of Money

You may have heard the old adage a dollar today is worth more than a dollar tomorrow. But how much more? Would you rather have $100 today or $120 in a month? What about $100 today or $120 in ten years? The study of interest rates (or more precisely risk-free interest rates) tries to answer the question: what is the monetary value of time? How much should you be compensated simply for waiting?

Most discussions of interest rates begin with the assumption that there exists a fictitious bank from which anyone can borrow or deposit any amount of money at a fixed rate r_f

Let’s go back to the hypotheticals posed earlier and assume that r_f is 2% (continuously compounded). Should we choose $100 now or $120 in a month? If we really needed money today (to pay bills, buy something immediately) we might be tempted to choose $100 today. However, using the bank we can simply borrow $100 today and pay back the debt with the $120 we get a month from now. Let’s compare the two options:

Receive $100 Today Borrow $100 Today
Today \$100 \$100
Tomorrow 0 \$120-\$100e^{(.02)(\frac{1}{12})}=\$19.83

Clearly, choosing to receive the $120 in a month is a better option. Now lets consider $100 today or $120 in ten years.

Receive $100 Today Borrow $100 Today
Today \$100 \$100
Tomorrow 0 \$120-\$100e^{(.02)(10)}=-\$2.14

In this scenario the $120 isn’t enough to pay back our debt, so we should choose $100 today. If we really didn’t mind waiting the 10 years and would rather have more money in the future, we could simply take the $100 today and put it in the bank. In 10 years, your $100 would turn into \$100e^{(.01)(10)}=\$122.14, which is more than the $120 we would have received.

The central idea here is that we can convert money promised in the future into money today, and vice-versa, by using the bank. $100 today is exactly the same as \$100e^{(.02)(1)}=\$102.02 in a year, \$100e^{(.02)(5)}=\$110.52 in five years, and \$100e^{(.01)(10)}=\$122.14 in ten years: We can put $100 in the bank today and withdraw any of those amounts in the future. Similarly, $120 in 10 years is exactly the same as \$120e^{(.02)(-10)} = 98.25 today, \$120e^{(.02)(-9)}=\$100.23 in one year, and \$120e^{(.02)(-5)}=\$108.58: We can borrow these amounts now, in a year, or in five years and pay back our debt with the $120 in ten years.

They are totally and unambiguously interchangeable.

Let’s consider another example. Suppose you want to open a restaurant but are deciding between opening a cheap restaurant and a fancy restauran. For each option you know exactly how much money it will cost to open the restaurant, how much money you’ll make in every year for the next ten years.

Year Cheap Fancy
0 -280,000 -720,000
1 120,000 120,000
2 110,400 120,000
3 101,568 120,000
4 93,442 120,000
5 85,967 120,000
6 79,089 120,000
7 72,762 120,000
8 66,941 120,000
9 61,586 120,000
10 56,659 120,000

When comparing these two options, its convenient to convert future cash flows into today-money. This is called discounting future cash flows to their present value or PV. For the Cheap restaurant, this comes out to be:

\displaystyle -\$280{,}000 + \$120{,}000e^{(.02)(-1)} + \$110{,}400e^{(.02)(-2)}+ \cdots + \$56{,}659e^{(.02)(-10)}=\$655{,}771.12

Thus, running the cheap restaurant for ten years is exactly the same as producing $655,771 today. The PV of the cashflows for the fancy restaurant is:

\displaystyle -\$720{,}000 + \$120{,}000e^{(.02)(-1)} + \$120{,}000e^{(.02)(-2)}+ \cdots + \$120{,}000e^{(.02)(-10)}=\$621{,}744.99

Therefore opening the cheap restaurant is a better investment than opening the fancy restaurant.

Unfortunately, in the real world this fictitious bank doesn’t exist. As an alternative we can deposit our money in U.S. government bonds, or if we’re a creditworthy bank we can deposit money with or borrow money from other creditworthy banks. These deposit and borrow rates typically depend on the length of time you’re depositing or borrowing for, and fluctuate daily. We’ll discuss all of these complexities of real-world risk-free interest rates in a future article.


Hello everyone. I’ve decided to start this blog to share some of the more interesting things I have encountered while working in the credit finance world. This blog will be focusing on explaining the mathematics and intuition behind pricing and evaluating risk on a variety credit instruments.

So what exactly is Credit?

Broadly, credit refers to borrowing or lending activity when there is risk that the borrowing party may not pay back their debt in full. Lending to a home buyer, a public company, or betting with a third party about whether or not public company X will default are all considered to be credit-risky investments. Lending money to the U.S. government, however, would not be considered a credit investment since the U.S. is thought to have no or de minimis default risk.1

Credit instruments include bonds, loans, and a suite of derivatives that are actively traded by banks, mutual funds, and hedge funds.

Why is credit important?

The debt market is actually much larger than the equity market.
In 2010, the total value of all global debt outstanding reached around $158 trillion while the total value of global equities outstanding was $54 trillion.2

Understanding debt is integral to understanding the equity of credit risky companies.
The equity of a company has no intrinsic value if the company is unable to pay off its debts. Therefore any changes in the creditworthiness of a company must impact the value of its equity as well.

The health of the credit market is linked to the health of the overall economy.
When companies can borrow at low interest rates, they can grow more quickly and boost the economy. When too much credit is available to risky borrowers, as it was in the years preceding the 2008 financial crisis, it can lead to economic collapse.

Whom is this blog intended for?

This blog is intended for anyone who is interesting in learning more about the credit markets. I will be starting with posts that include the basics of the various instruments involved before writing about more complex or nuanced topics.

Thanks for reading!

  1. Actually, the default risk of the U.S. government is not really zero and you can trade credit default swaps to bet on the odds that the government defaults in the next five years.
  2. Mapping global capital markets 2011 August 2011