40141 – How well does the power utility consumption CAPM perform in UK Stock Returns?

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1 Hansen and Jagannathan (1991) LOP Volatility Bounds

Volatility bounds were first derived by Shiller (1982) to help diagnose and test a particular set of asset pricing models. He found that to price a set of assets, the consumption model must have a high value for the risk aversion coefficient or have a high level of volatility. Hansen and Jagannathan (1991) expanded on Shiller’s paper to show the duality between mean-variance frontiers of asset portfolios and mean-variance frontier of stochastic discount factors. Law of one price volatility bounds are derived by calculating the minimum variance of a stochastic discount factor for a given value of E(m), subject to the law of one price restriction. The law of one price restriction states that E(mR) = 1, which means that the assets with identical payoffs must have the same price. For this constraint to hold, the pricing equation must be true.

Hansen and Jagannathan use an orthogonal decomposition to calculate the set of minimum variance discount factors that will price a set of assets. The equation m = x* + we* + n can be used to calculate discount factors that will price the assets subject to the LOP condition. Once x* and e* are calculated, the minimum variance discount factors that will price the assets can be found by changing the weights, w. Hansen and Jagannathan viewed the volatility bounds as a constraint imposed upon a set of discount factors that will price a set of assets. Therefore, when deriving the volatility bounds, we calculate the minimum variance stochastic discount factors that will price the set of assets. Discount factors that have a lower variance than these values will not price the assets correctly. Furthermore, Hansen and Jagannathan showed that to price a set of assets, we require discount factors with a high volatility and a mean close to 1.

After deriving these bounds, we can use this constraint to test candidate asset pricing models. Models that produce a discount factor with a lower volatility than any discount factor on the LOP volatility can be rejected as they do not produce sufficient volatility. Hansen and Jagannathan find evidence that using LOP volatility bounds, we can reject a number of models such as the consumption model with a power function analysed in papers such as Dunn and Singleton (1986).

2 Methodology

To test whether the power utility CCAPM prices the UK Treasury Bill (Rf) and value weighted market index returns, we first calculate the LOP volatility bounds. The volatility bound is derived by calculating the minimum variance discount factors that correctly price the two assets for given values of E (m). The standard deviations of the stochastic discount factors are then plotted on a graph to give the LOP volatility bound shown in figure one.

Figure 1 here

The CCAPM stochastic discount factors are then calculated for different levels of risk aversion. The mean and standard deviation of these discount factors are then plotted on the graph and compared to the LOP discount factor standard deviations.

Pricing errors can then be calculated and analysed to see whether the assets are priced correctly by the candidate model. To accept the CCAPM model in pricing the assets, we expect the stochastic discount factors variance to be greater than the variance of the LOP volatility bounds. It is also expected that pricing errors and average pricing errors (RMSE) will be close to zero. These results will be analysed more closely in the later questions.

3 Power Utility CCAPM vs LOP Volatility Bounds

In order for the power utility CCAPM to satisfy the Law of One Price volatility bound test at any level of risk aversion, the standard deviation of the CCAPM stochastic discount factor at that level of risk aversion must be above the Law of One Price standard...