Quantitative Analysis for Finance - Essay Example

Quantitative Analysis for Finance The capital asset pricing model (CAPM) is a model that describes the relationship that exists between the risk and the required rate of return on assets held by an investor in a well-diversified portfolio (Pahl 2009, p. 165). A well-diversified portfolio refers to the combination of different assets held by an investor for investments purposes with a goal of minimizing the risks associated with asset. Additionally, according to CAPM, the risk of a portfolio reduces as the number of securities increase. However, at a given point, increase in the number of securities does not reduce risk capital asset pricing model has various basic assumptions. Pandey (1979, p. 98) states that they include; i) That the investors are risk averters ii) That the investors are presumed to be rational in the sense that they select among alternative combinations of assets on the basis of each portfolio’s expected return and the standard variation. iii) That CAPM is a single period model, this means that the investors maximise the utility created at the end of the end of the period. iv) That the capital market is said to be efficient and perfect v) That the presence of a risk-free asset allows the investors to borrow and lend at this rate. vi) That the investors have similar expectations on the returns generated by the assets therefore all investors fall in the same efficient frontier. CAPM is represented as; Re = Rf + β (Rm - Rf) Where Re is the expected return from security “e” Rf is the risk free rate of return Β is the beta coefficient Rm is the return on the market portfolio The four variables that make up the CAPM equation can be examined and measured directly by the use of different formulas. The expected return of denoted by ‘Re’ refers to the weighted average of the expected return of each security held in isolation (Harrington 1987, p.237). The formula used to calculate expected return incorporates various variables that include the risk free rate. The risk free rate is the asset that shows no variability in returns meaning that the returns of such an asset are known with certainty and do not depend on the nature of the economy. In practice, risk free assets are the government treasury bills and bonds. However, the normal returns are constant and the actual returns may be uncertain due to the inflation occurring at any given time (Pahl 2009, p. 198). This rate is considered to have a constant mean which implies that the actual returns are equal to the expected returns. Additionally, they have a zero variance which occasioned by the fact that the actual and the expected returns are equal and therefore no deviations from the mean which enables the investors to invest in this securities due to the fact that they are always sure of the returns to expect from their investment in the treasury bills and bonds. Therefore, the risk free rate (Rf) is determined by the government of a country and it is always bears a fixed rate of return (KüRschner 2008, p. 192). The main disadvantage of the risk free rate is the determination of the risk premiums since the past historical premiums are used and as a result changes in the returns yielded from the stocks in future are not taken into consideration. A strategy to curb this limitation is by measuring this rate with the expected cash flows of a company and this will incorporate the rate of inflation at a particular time. Another variable that helps in the calculation of the expected return is the beta coefficient (β) also known as the non-diversifiable. The beta function refers to a measure of sensitivity of a return on the securities to changes in the market portfolio (Galea, Díaz García, & Vilca 2005, p. 98). Beta risk is caused by factors such as inflation, recession, economic instability and changes in macro-economic variables that affect the firms adversely. Various formulas are applicable while calculating beta. The first formula that we will consider is; β = covariance security and the market portfolio/variance of the market portfolio It is worth mentioning that the formula is only used if the covariance is given. The security market line is another way of calculating beta by the use of the capital asset pricing model equation. Beta is made the subject of the formula; Re = Rf +β (Rm-Rf) β = (Rj - Rf) / (Rm - Rf) Beta coefficient can also be measured by regression analysis. If the beta value computed is equal to one, it implies that the market portfolio has a unit sensitivity to the expected returns (Elton 2010, p. 172). The risk free rate on the other hand has a beta value equal to zero since it has no risk. A security with a beta greater than one implies that the security is more sensitive to changes in the market and therefore the systematic risk is higher than that of the market portfolio. Similarly, a security with beta less than one means that the security is less sensitive to changes in the market portfolio and this could be the risk free asset. One of the major limitation is that the use of the regression model to estimate beta has a little consistency as to what the period of time is. It tends to give different results with respect to the market risk for the investments. Another limitation is that since the true value of beta falls within a given range then the use of only one beta in the model to estimate the risk can lead to errors when the investors make investment decisions (Francisco, 1987). Additionally, beta coefficient is regarded to be backward looking and do not incorporate the effects of risks of a company at different times. This implies that the use of beta in the model can lead to errors in calculating the expected return (Re). An alternative strategy to the beta is the option pricing technique which is used to compute the beta coefficient that is progressive. The main advantage of the option pricing is the fact that the computations involved are very straightforward and are readily available free of charge. Another variable used in computing the expected return is the return on the market portfolio usually denoted as Rm. The market portfolio is considered to be the most efficient portfolio and it consists of securities of the stock market. Also known as the optimal portfolio, it is identified by super imposing the investors indifference curves on the efficient frontier and the indifference curves that lies at the point of tangency between the investors indifference curves and the efficient frontier represents the optimal or market portfolio (Koch 2009, p. 192). Market portfolio can also be measure by the use of the capital market line which is a lie that shows the trade-off between risk and return of the market portfolio and the risk free asset. The market return is at the point where the securities in a risky market portfolio and risk free investments are joined by the capital market line (CML). The return on market portfolio obtained from plotting the investor’s indifference curve proves to be a constraint because the plotting process is considered to be a theoretical exercise and also predicting the returns and the correlation to obtain the return on market portfolio is quite difficult in practice (Pahl 2009, p. 173). To curb this vice, separability theorem applies and it states that the financing and investment decisions are separate and distinct. This implies that the investor will invest their finances in the same securities irrespective of how capital was obtained, the investors tend to imitate the market portfolio by buying securities constituting the market portfolio therefore all the investors will lie at the point where there maximise their returns at a given level of risk. After measuring the three constraints, the expected return can be computed by incorporating them as follows; Re = Rf +β (Rm-Rf) which is the ultimate capital asset pricing model. However, this model has several limitations. These include; i. The model uses historical data. ii. It assumes a well- diversified portfolio is in existence. iii. This model is a single index model and uses only one beta. iv. The model only focuses on the systematic risk or beta risk and disregards the other aspects of risk. v. The model is a single period model which implies that this model looks at the year-end returns only. In order to overcome these limitations, the arbitrage pricing theory (APT) is used instead. This model tries to explain the relationship between the risk and expected rate of return using several betas and it is also a multiple period model unlike the capital asset pricing model. Arbitrage pricing theory assumes than, in equilibrium, the return on the arbitrage portfolio is zero, if the return is positive then the process is eliminated immediately through the process of arbitrage trading so as to improve the expected returns. This model was formulated by Stephen Ross (1976) and it is written as follows; Re = Rf + β1 (Rm - Rf) + β2 (Rm - Rf) +----------+βn (Rm - Rf) Despite the limitations of capital asset pricing model, there exists several advantages. These include the provision of a market based model for determining the risk return relationship. It combines risk and return into a single index and a can be used for evaluation of portfolio performance (Pahl 2009, p. 172). Additionally, CAPM provides a basis of establishing the risk adjusted discounting rate (RADR) Reference list PAHL, N. (2009). Principles of the Capital Asset Pricing Model and the Importance in Firm Valuation. München, GRIN Verlag GmbH. http://nbn-resolving.de/urn:nbn:de:101:1-2010090222477. PANDEY, I. M. (1979). Financial management. [Delhi], Vikas Publishing House. HARRINGTON, D. R. (1987). Modern portfolio theory, the capital asset pricing model, and arbitrage pricing theory: a users guide. Englewood Cliffs, NJ, Prentice-Hall. PAHL, N. (2009). Principles of the Capital Asset Pricing Model and the Importance in Firm Valuation. München, GRIN Verlag GmbH. http://nbn-resolving.de/urn:nbn:de:101:1-2010090222477. KÜRSCHNER, M. (2008). Limitations of the Capital Asset Pricing Model (CAPM) Criticism and New Developments. München, GRIN Verlag GmbH. http://nbn-resolving.de/urn:nbn:de:101:1-2010082913342. GALEA, M., DÍAZ GARCÍA, J. A., & VILCA, F. (2005). Influence diagnostics in the capital asset pricing model under elliptical distributions. Campinas, UNICAMP. ELTON, E. J. (2010). Modern portfolio theory and investment analysis. Hoboken, NJ, J. Wiley & Sons. FRANCISCO, C. R. (1987). The capital asset pricing model with high non-homogeneous expectations: theory and evidence on systematic risks to the beta. [Manila, Philippines], The Institute. KOCH, C. (2009). The Arbitrage Pricing Theory as an Approach to Capital Asset Valuation. München, GRIN Verlag. Read More