Formulation of the model and presentation of relevant theory

This thesis will not go into developing a single discount rate, but simply look at the connection between the EU‟s largest stock markets and the price of carbon futures in order to create a Beta coefficient ( ). The appropriate risk premium and discount rate for the carbon futures can then be calculated from a model such as the Capital Asset Pricing Model (CAPM)140 or the Arbitrage Pricing Theory (APT)141. The beta measures the extent to which returns on an asset and the market move together. Formally, CAPM Beta for an asset i is defined as:

where is the return of asset i, the return of the market portfolio M and the variance of M over a specified period of time. Covariance measures how much the return on two risky assets move in tandem. A positive covariance means that the asset returns move together, while a negative covariance means that the returns move inversely. Variance and standard deviation is a measure of volatility and riskiness of an asset or portfolio. In this analysis,

139

Brooks, Chris. (2002): Introductory econometrics for finance. (Page 367-368). Cambridge University Press

140

CAPM is a model that gives a precise prediction of the relationship that we should observe between the risk of an asset and its expected return: Eri = rf + βi[E(rm) – rf]. Assumptions: All investors aim to maximize utilities, are rational risk-

averse, are price takers and cannot influence prices, can lend and borrow unlimited under the risk free rate of interest. Securities are highly divisable into small parcels. No transaction or taxation costs incurs. Source: Bodie, Zvi et al. (2005): Investments. 6th Ed. (In Chapter 9: The Capital Asset Pricing Model, p281-316). McGraw-Hill

141

APT is an asset pricing theory that is derived from a factor model, using diversification and arbitrage arguments. The theory describes the relationship between expected returns on securities, given that there are no opportunities to create wealth through risk-free arbitrage investments. Source: Bodie, Zvi et al. (2005): Investments. 6th Ed. (In Chapter 11: Arbitrage Pricing Theory, p343-363). McGraw-Hill

the two risky assets are carbon emission futures and the stock market portfolio. If is greater than 1, the carbon emission futures‟ rate of return is more sensitive to changes in the level of the overall market than is the average asset. The CAPM is an estimation of the discount rate given that the asset is added to a well-diversified portfolio. There are heavy assumptions behind this model, see reference 140, and it is thus not believed to be perfectly applicable in the real world. CAPM is however widely used and it will be the theoretical basis of this part of the analysis. When using CAPM, the model only takes into account the market risk of the asset, i.e. the tendency of an asset‟s returns to responds to swings in the market, since it assumes all investors are well-diversified and not affected by individual risks in each asset (nonsystematic risk). In this case, the nonsystematic risk of EUA futures will be the main price drivers discussed in Chapter 4.1 such as weather and ETS policies.142

Beta for financial use can be calculated directly from two time-series in a spread sheet or by using regression analysis. In practice it is common to use between two and five years of monthly returns in order to estimate Beta. Computing Beta with very short horizon returns (such as daily returns) may lead to problems related to non-synchronous trading. This

will be explained and become evident in the first test. Regression analysis is one of the most

important tools used in econometrics and is concerned with describing and evaluating the relationships between a given variable and one or more other variables. While dealing with relationships between variables, it is important to be aware of the difference between regression and correlation. The correlation between two variables measures the degree of linear association between them and it is not implied that changes in X cause changes in Y or vice versa. The correlations used in this analysis will also be explained in a significance perspective depicted from the P-value. The P-value of a test is the probability of observing a test statistic at least as extreme as the one computed given that the null hypothesis is true.143 In a simple regression, the dependent variable (y) and the independent variable (x) are treated very differently. The y variable is assumed to be random and have a probability distribution while the x variable is assumed to be non-stochastic and have fixed values in repeated samples. Regression as a tool is thus more flexible and more powerful than correlation. Note, however, that as a general rule we cannot determine the value of y for a

142

Bodie, Zvi et al. (2005): Investments. 6th Ed. (In Chapter 9: The Capital Asset Pricing Model, p281-316). McGraw-Hill

value of x that is far outside the range of the sample values of x.144 Finding a relationship between two variables and proving that this is a significant one is thus two different questions. It is important to assess how well the linear model fits the data. If the fit is poor, we should discard the linear model, draw the conclusions needed and perhaps seek another one.

The following null-hypothesis has been developed:

 H0: No linear relationship exists between returns on carbon emission futures and the

rate of return on the overall stock market.

The following alternative hypothesis has been developed:

 H1 : A linear relationship exists between returns on carbon emission futures and the

rate of return on the overall stock market.

The null hypothesis specifies that there is no linear relationship, which means that the linear slope projected by the regression is 0, i.e. . We perform a two-tail test to determine whether there is sufficient evidence to infer that a linear relationship exists, ≠ 0. The alternative hypothesis test this.

A linear regression approximates a straight line t between the two data points in

question. The mathematic description of the linear regression line in the following analysis is

Coefficient α is a constant and coefficient β is the Beta already introduced. ε is a random disturbance term which is not be calculated, but added to the model to make it more realistic since all of the data points cannot lay exactly on a straight line.145

144

Brooks, Chris. (2002): Introductory econometrics for finance. (In Chapter 3: p42-43). Cambridge University Press

Type I error occurs when we reject a true null hypothesis, and a Type II error is defined as not rejecting a false null hypothesis. A significance level of 5% is set in order to reject the null hypothesis, 2.5% on each side. Hence, a P-value under 0.025 on the constant will be needed to reject. The test statistic used is the „Student t distribution‟ because of the limited amount of data and knowledge about the mean of the following samples:

where sy is the standard error of y which is calculated by a formula with v = n – 2 degrees of

freedom, where n is the number of observations. The critical value of t will thus be found separately in each test, and have to be smaller than the observed value in the analysis in order to reject the null hypothesis.146

The null hypothesis does not initially say anything about the significance of the relationship. Because of this, the tests run will be interpreted in two ways: First, a Beta to be used in a CAPM of carbon emissions futures will be computed the best way possible. Second, the null hypothesis will be tested with a two-way significance level of 5%. This approach is in line with the formulation of the theoretical problem in 4.3.1. A regression analysis gives numerous outputs, and the test of Beta addresses only the question of whether there is enough evidence to infer that a linear relationship exists. But it will also be useful to measure another strength of that linear relationship: This regression analysis will also focus on the coefficient of determination, which is denoted . In financial analysis, the slope coefficient β is a measure of the asset‟s market-related or systematic risk because it measures the volatility of the asset price that is related to the overall market volatility. An example: If for asset i we will interpret this to mean that for each 1% increase in the market portfolio M, the average increase in asset i‟s return is 0.70%. measures the proportion of the total risk that is systematic, and if , 45% of asset i‟s total risk is market related and 55% associated with events specific to asset i rather than the market. This is called

146

Keller, G., Warrack, B. (2003). Statistics for Management and Economics. (In Chapter 8: Continuous Probability Distributions, p252-258). Brooks/Cole – Thomson Learning

nonsystematic risk.147 In this case, as earlier mentioned, the nonsystematic risk of EUA futures will be the main price drivers discussed in Chapter 4.1 such as weather and ETS policies.