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# Merton Model

## What Is the Merton Model?

The Merton model is a mathematical formula that stock analysts and commercial loan officers, among others, can use to judge a corporation's risk of credit default. Named for the economist Robert C. Merton, who proposed it in 1974, the Merton model assesses the structural credit risk of a company by modeling its equity as a call option on its assets.

### Key Takeaways

• In 1974, economist Robert C. Merton proposed a model for assessing the credit risk of a company by modeling its equity as a call option on its assets.
• The Merton model is used today by stock analysts, commercial loan officers, and others.
• Merton's work, and that of fellow economist Myron S. Scholes, earned the Nobel Prize for economics in 1997.

## The Formula for the Merton Model

\begin{aligned} &E=V_tN\left(d_1\right)-Ke^{-r\Delta{T}}N\left(d_2\right)\\ &\textbf{where:}\\ &d_1=\frac{\ln{\frac{V_t}{K}}+\left(r+\frac{\sigma_v^2}{2}\right)\Delta{T}}{\sigma_v\sqrt{\Delta{T}}}\\ &\text{and}\\ &d_2=d_1-\sigma_v\sqrt{\Delta{t}}\\ &\text{E = Theoretical value of a company's equity}\\ &V_t=\text{Value of the company's assets in period t}\\ &\text{K = Value of the company's debt}\\ &\text{t = Current time period}\\ &\text{T = Future time period}\\ &\text{r = Risk-free interest rate}\\ &\text{N = Cumulative standard normal distribution}\\ &\text{e = Exponential term}\left(i.e. \text{ }2.7183...\right)\\ &\sigma=\text{Standard deviation of stock returns}\\ \end{aligned}

## What Does the Merton Model Tell You?

The Merton model allows for easier valuation of a company and also helps analysts determine if it will be able to retain solvency, by analyzing the maturity dates of its debt and its debt totals.

The Merton model calculates the theoretical pricing of European put and call options without considering dividends paid out during the life of the option. The model can, however, be adapted to consider dividends by calculating the ex-dividend date value of underlying stocks.

The Merton model makes the following basic assumptions:

• All options are European options and are exercised only at the time of expiration.
• No dividends are paid out.
• Market movements are unpredictable (efficient markets).
• No commissions are included.
• Underlying stocks' volatility and risk-free rates are constant.
• Returns on underlying stocks are regularly distributed.

Variables that are taken into consideration in the formula include options' strike prices, present underlying prices, risk-free interest rates, and the amount of time before expiration.

## History of the Merton Model

Robert C. Merton is a noted American economist and Nobel Prize laureate, who purchased his first stock at age 10. He earned a bachelor of science in engineering at Columbia University, a master of science in applied mathematics at the California Institute of Technology, and a doctorate in economics at the Massachusetts Institute of Technology, where he later become a professor.

During Merton's time at MIT, he and fellow economists Fischer Black and Myron S. Scholes were all working on problems related to the pricing of options and often aided in each other's work. Black and Scholes published a seminal paper on the topic, "The Pricing of Options and Corporate Liabilities," in 1973. Merton's "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," published early the following year, described what has come to be known as the Merton model.

Merton and Scholes were awarded the Nobel Prize for economics in 1997 (Black had died and was no longer eligible). The prize committee cited them for developing "a pioneering formula for the valuation of stock options. Their methodology has paved the way for economic valuations in many areas. It has also generated new types of financial instruments and facilitated more efficient risk management in society."

Their best known collaboration is often referred to today as the Black-Scholes-Merton model.

## What Is a Call Option?

A call option is a contract that allows the buyer to purchase a stock or other financial asset at a specified price by or on a certain date.

## What Is the Difference Between European and American Options?

European options can be exercised only on their expiration date, while American options can be exercised at any time.

## What Is a Risk-Free Interest Rate?

A risk-free interest rate is the theoretical rate of return on an investment carrying zero risk. It is theoretical because no investment is entirely without risk, although some come closer than others.

## The Bottom Line

The Merton model, developed by economist Robert C. Merton, is a mathematical formula that assesses the structural credit risk of a company by modeling its equity as a call option on its assets. It is often used by stock analysts and commercial loan officers to ascertain a corporation's likely risk of credit default.

Article Sources
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1. MIT Management Sloan School. "Robert C. Merton."

2. The University of Chicago Press Journals. "The Pricing of Options and Corporate Liabilities."

3. The Journal of Finance. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates."

4. Journal of Economic Perspectives. "In Honor of the Nobel Laureates Robert C. Merton and Myron S. Scholes: A Partial Differential Equation That Changed the World."