Douglas Breeden and Robert Lucas, a Nobel laureate in economics, provided the foundation of the consumption capital asset pricing model (CCAPM) in 1979 and 1978, respectively. Their model is an extension of the traditional capital asset pricing model (CAPM). It's best used as a theoretical model, but it can help to make sense of variation in financial asset returns over time, and in some cases, its results can be more relevant than those achieved through the CAPM model. Read on to discover how this model works and what it can tell you.

## What Is CCAPM?

While the CAPM relies on the market portfolio's return in order to understand and predict future asset prices, the CCAPM relies on the aggregate consumption. In the CAPM, risky assets create uncertainty in an investor's wealth, which is determined by the market portfolio (e.g., the S&P 500). In the CCAPM, on the other hand, risky assets create uncertainty in consumption—what an investor will spend becomes uncertain because his or her wealth, (i.e., income and property) is uncertain as a result of a decision to invest in risky assets.

In the CAPM, the risk premium on the market portfolio measures the price of risk, while the beta states the quantity of risk. In the CCAPM, on the other hand, the quantity of market risk is measured by the movements of the risk premium with consumption growth. Thus, the CCAPM explains how much the entire stock market changes relative to the consumption growth.

## Is CCAPM Useful?

While the CCAPM rarely is used empirically, it is highly relevant in theoretical terms. Indeed, the CCAPM is not used, as was the standard CAPM, in the real world. Therefore, a firm evaluating a project or the cost of capital is more likely to use the CAPM than the CCAPM. The major reason for this is that the CCAPM tends to perform poorly on empirical grounds. This may be because a proportion of consumers do not actively take part in the stock market and, therefore, the basic link between consumption and stock returns assumed by the CCAPM cannot hold. For this reason, the CCAPM may perform better than the CAPM for people who hold stocks.

From an academic point of view, the CCAPM is more widely used than the CAPM. This is because it incorporates many forms of wealth beyond stock market wealth and provides a framework for understanding variation in financial asset returns over many time periods. This provides an extension of the CAPM, which only takes into account one-period asset returns. The CCAPM also provides a fundamental understanding of the relation between wealth and consumption and an investor's risk aversion.

## Calculating CCAPM

A simplified version of the CCAPM can take a linear representation between a risky asset (a stock, for example) and the market risk premium. However, the difference is the definition of the so-called implied risk-free rate, implied market return and the consumption beta. Therefore, the formula for CCAPM is as follows:

$\begin{aligned} &r_a = r_f + \beta_c ( r_m - r_f ) \\ &\textbf{where:} \\ &r_a = \text{expected returns on risky asset (e.g. a stock)} \\ &r_f = \text{implied risk-free rate (e.g. 3-month Treasury bill)} \\ &r_m = \text{implied expected market return} \\ &r_m - r_f = \text{implied market risk premium} \\ &\beta_c = \text{consumption beta of the asset} \\ \end{aligned}$

The implied returns and risk premium are determined by the investors' consumption growth and risk aversion. Moreover, the risk premium defines the compensation that investors require for buying a risky asset. As in the standard CAPM, the model links the returns of a risky asset to its systematic risk (market risk). The systematic risk is provided by the consumption beta.

## Consumption Beta

The consumption beta is defined as:

$\begin{aligned} &\beta_c = \frac { \text{Covariance between } r_a \text{ and consumption growth} }{ \text{Covariance between } r_m \text{ and consumption growth}} \\ \end{aligned}$

As shown in Figure 1, a higher consumption beta implies a higher expected return on the risk asset.

Figure 1: Expected return vs. consumption beta.

In the CCAPM, an asset is riskier if it pays less when consumption is low (savings are high). The consumption beta is 1 if the risky assets move perfectly with the consumption growth. A consumption beta of 2 would increase an asset's returns by 2% if the market rose by 1%, and would fall by 2% if the market fell by 1%.

The consumption beta can be determined by statistical methods. An empirical study, "Risk and Return: Consumption Beta Versus Market Beta" (1986), by Gregory Mankiw and Matthew Shapiro tested the movements of the United States' consumption and stock returns on the New York Stock Exchange and on the S&P 500 Index between 1959 and 1982. The study suggests that the CCAPM implies a higher risk-free rate than the CAPM, while the CAPM provides a higher market risk (beta), as shown in Figure 2.

Measures | CAPM | CCAPM |

Risk-Free Rate | 0.35% | 5.66% |

Beta | 5.97 | 1.85 |

Figure 2: Test of the CAPM and CCAPM. *Source: "Risk and Return: Consumption Beta Versus Market Beta"*

The question is, how much would the return on a risky asset be at the risk-free rate and beta in Table 1? Figure 3 illustrates an experiment on the required returns of a risky asset at different market returns (column 1). The required returns are calculated by using the CAPM and CCAPM formulas.

For example, if the market return is 3%, the market risk premium is -2.66 multiplied by the consumption beta 1.85 plus the risk-free rate (5.66%). This yields a required return of 0.74%. By contrast, the CAPM implies that the required return should be 16.17% when the market return is 3%.

Market Return | Stock Return - CAPM | Stock Return - CCAPM |

1.00% | 4.23% | -2.96% |

2.00% | 10.20% | -1.11% |

3.00% | 16.17% | 0.74% |

4.00% | 22.14% | 2.59% |

5.00% | 28.11% | 4.44% |

6.00% | 34.08% | 6.29% |

Figure 3: Experiment on returns of a risky asset

The two cases of market return at 1% and 2% do not necessarily imply that investing in a risky asset is rewarded with a positive return. This, however, contradicts the fundamental aspects of risk-return requirements.

## CCAPM Isn't Perfect

The CCAPM, like the CAPM, has been criticized because it relies on only one parameter. Because many different variables are known to empirically affect the pricing of assets, several models with multifactors, such as the arbitrage pricing theory, were created.

Another problem specific to the CCAPM is that it has led to two puzzles: the equity premium puzzle and the risk-free rate puzzle (RFRP). The EPP shows that investors have to be extremely risk-averse in order to imply the existence of a market risk premium. The RFRP says that investors save in Treasury bills despite the low rate of return, which has been documented with data from most industrialized countries in the world.

## The Bottom Line

The CCAPM remedies some of the weaknesses of the CAPM. Moreover, it directly bridges macro-economy and financial markets, provides an understanding of investors' risk aversion and links the investment decision with wealth and consumption.