Capital Asset Pricing for UK Stocks under the Multivariate Skew-Normal Distribution
The standard theories of modern ﬁnance are based on the assumption that returns on risky ﬁnancial assets follow a multivariate normal distribution. The best known example of the explicit assumption of normality is perhaps the option pricing formula developed by Black and Scholes (1973), and Merton (1973). In other areas of ﬁnance theory, the assumption is made implicitly. For example, the theory of mean-variance portfolio selection developed by Markowitz (1952) does not require normality of returns per se, but does assume that asset returns are characterized by their ﬁrst and second moments and that, furthermore, an investor’s utility function will be a function only of the mean and variance of portfolio returns. Implicit normality is also a feature of the CAPM, the capital asset pricing model, of Sharpe (1964), Lintner (1965), and Mossin (1966). The CAPM, which is derived using Markowitz’ eﬃcient frontier in conjunction with various assumptions about investor behavior, is a linear model that relates the expected return on an asset, i say, over a given investment period to the expected return on the market portfolio, m say, over the same investment period. This model states that
µi = Rf + βi(µm −Rf ), (11.1) where Rf is the return on a risk-free instrument, such as a T-bill or LIBOR in the UK. The parameters µi and µm are, respectively, the expected return on asset i and the market portfolio, and βi is the covariance between returns on asset i and the market portfolio, divided by the variance of returns on the market portfolio. In the derivation of the CAPM, the market portfolio is a portfolio of “all assets.” For the purposes of this article, it may be thought of as a stock market index, such as the FTSE100 or the S&P500.