Grinold and Kroner Model

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Written By
Dan Buckley
Dan Buckley is an US-based trader, consultant, and part-time writer with a background in macroeconomics and mathematical finance. He trades and writes about a variety of asset classes, including equities, fixed income, commodities, currencies, and interest rates. As a writer, his goal is to explain trading and finance concepts in levels of detail that could appeal to a range of audiences, from novice traders to those with more experienced backgrounds.

The Grinold and Kroner model – sometimes known as the Grinold-Kroner or G-K model – is a financial theory that provides a formula to calculate the expected return on a stock or a stock market index over an asset without credit risk.

The model was developed by Richard C. Grinold and Kenneth F. Kroner and has been a tool for financial analysts and investment managers.


Key takeaways – Grinold and Kroner Model

  • The Grinold and Kroner model is an analytical framework that provides a formula to calculate the expected return on a stock or stock market index.
  • It considers factors such as dividend yield, expected inflation, earnings growth, changes in shares outstanding, and changes in the Price-to-Earnings (P/E) ratio.
  • The model has limitations, including simplistic assumptions, reliance on historical data, inadequate treatment of risk, ignorance of market sentiment and behavioral factors, lack of consideration for some macroeconomic variables, difficulty in estimating inputs, and neglect of market frictions and transaction costs.
  • It has contributed to modern financial theory and has led to the development of other models such as the Fed model, which compares earnings yield to the yield on the 10-year Treasury bond as a measure of market valuation.


The Grinold and Kroner Equation

This model states that the expected return on a stock or a stock market index (E[R]) is the sum of the dividend yield, expected inflation, growth in earnings, changes in shares outstanding, and changes in the Price-to-Earnings (P/E) ratio.

The mathematical representation of the Grinold and Kroner model is as follows:


E[R] = Div/P + i + g – ∆S + ∆(P/E)



  • Div/P represents the dividend yield, which is the expected dividend in the next period divided by the current price.
  • i stands for the expected inflation rate, indicating the rate at which the general level of prices for goods and services is rising.
  • g is the real growth rate in earnings. Adding real growth and inflation essentially results in nominal growth.
  • ∆S represents the changes in shares outstanding, where an increase in shares outstanding decreases expected returns.
  • ∆(P/E) is the changes in the P/E ratio. There is a positive relationship between changes in P/E and expected returns (i.e., multiple expansion).


Capital Market Expectations: Grinold-Kroner Model


Empirical Evidence: Grinold, Kroner, and Siegel (2011)

In a study conducted in 2011, Grinold, Kroner, and Siegel provided an empirical estimation of the inputs to the Grinold and Kroner model.

Their research concluded with a then-current equity risk premium estimate of between 3.5% and 4%.

The equity risk premium is the difference between the expected total return on a capitalization-weighted stock market index and the yield on a risk-free government bond (in this case one with 10 years to maturity).

This means, for example, if a 10-year government bond is yielding 3%, an investor in a certain stock (or the stock market) might expect to require 6.5% to 7% forward return.

The opposite of this yield would be the P/E ratio (1/0.065 = 15.4x, 1/0.07 = 14.3x).


The Fed Model

An interesting offshoot of the Grinold and Kroner model is the controversial Fed model.

This model compares the earnings yield (the inverse of the P/E ratio) to the nominal 10-year Treasury bond yield.

It implies that if the earnings yield is higher than the yield on the 10-year Treasury bond, stocks are undervalued and vice versa.

While it has its critics, the Fed model is often used as a quick and dirty gauge of market valuation.


Limitations of the Grinold Kroner Model

While the model provides a framework for understanding the determinants of equity returns, it also has several limitations.

Here are some of the key limitations of the Grinold-Kroner model:

Simplistic assumptions

The G-K model relies on simplifying assumptions that may not capture the complexity of real-world markets.

For example, it assumes that expected returns are solely determined by dividends, earnings growth, and changes in valuation ratios, neglecting other factors that can influence stock prices.

Reliance on historical data

The model heavily relies on historical data to estimate future returns.

However, past performance may not be a reliable indicator of future performance, especially during periods of significant market changes or structural shifts.

Inadequate treatment of risk

The G-K model assumes that investors are solely concerned with the mean and variance of returns, as measured by the expected return and the volatility of the market.

This assumption does not fully capture the complexity of risk, such as tail risk or non-normal distributions of returns.

Ignores market sentiment and behavioral factors

The model does not consider the influence of market sentiment, trader/investor psychology, or behavioral biases on asset prices.

These factors can have a significant impact on market pricing and may lead to deviations from the model’s predictions.

Lack of consideration for some macroeconomic variables

The G-K model does not explicitly incorporate some macroeconomic variables that can alter asset prices, which can have a substantial influence on equity returns.

Ignoring these factors may limit the model’s ability to accurately predict asset prices.

Lack of global perspective

The G-K model focuses primarily on domestic factors and may not adequately capture the impact of global market dynamics, international trade, or cross-border capital flows on equity returns.

Difficulty in estimating inputs

The G-K model requires accurate estimates of variables such as dividend growth rates and changes in valuation ratios.

However, these estimates can be challenging to determine, and small changes in input assumptions can lead to significant variations in predicted returns.

Neglects market frictions and transaction costs

The model assumes frictionless markets and does not consider transaction costs, market impact, or other trading-related expenses.

In reality, these costs can significantly affect investment returns and may lead to deviations from the model’s predictions.



A big part of finance has to do with the concept of expectations. If I make this decision, what am I likely to get?

The Grinold and Kroner model offers an analytical framework to determine expected returns from a stock or a stock market index.

It integrates a range of factors including dividend yield, expected inflation, real earnings growth, changes in shares outstanding, and changes in P/E ratio.

Although it is a complex model, it offers a comprehensive approach for investors to gauge potential returns.

The model also provides a basis for further financial models and analyses, including the Fed model, contributing significantly to modern financial theory.