## Introduction

The Capital Asset Pricing Model (CAPM) specifies the relationship between risk and the required rate of return on assets when they are held in well-diversified portfolios. It assumes that investors are rational and they choose among alternative portfolios based on each portfolio’s expected return and standard deviation. Furthermore, the model assumes that investors are risk averse and maximize the utility of end of period wealth. Additionally, such investors have homogeneous expectations with regard to asset return. In this case, all assets are marketable and divisible. In other words, it is believed under this model that the capital market is efficient and perfect.

## Literature Review

According to Separability theorem, all investors, regardless of their attitude towards risk, should hold the same risk assets in their portfolios, as shown below

The crucial differences in portfolios held by investors of different psychologies are in the in-between the risky stocks and the non-risk stocks. Different investors will choose different points on the capital market line. In the above figure, point M represents the market portfolio and R_{f }is the rate of return on the riskless asset. All investors combine R_{f }and M, but in different proportions (Oster, 1994). The market portfolio is the âsame mix of risky stocksâ. Investor A for example prefers to play safe with emphasis on riskless assets. Person B has most funds in the market portfolio. Person C could have had leveraged portfolios that were added to his original funds by borrowing at R_{f and} putting all of them into the market portfolio (Levich, 2001).

The interior decorator school of thought suggest that different portfolio of risky assets should be prepared for differing investors to suit their tastes. In other words, points other than M on the all risky portfolio should be taken. This will clearly lead to inefficiencies in the presence of riskless asset if all the assumptions of CAPM hold. If the assumptions do not hold, in particular the borrowing assumption, then the Separability Theorem falls. Separability theorem therefore maintains that investors borrow the money to be invested (Reilly & Brown, 2007).

In the above figure, person A combines the riskless asset with risky portfolio M. Person B selects his own interior decorator policy, while C combines yet another risky portfolio by borrowing at R_{b.} According to the above, it is then possible to say that risk and return are directly proportional. The higher the risk, the higher is the return on investment and vice versa. In this case, for Kevin Murray to increase his expected return, he should increase his portfolio by investing in more assets, which promise a high return. Kevin Murray is a risk taking investor who desires to make more profits or realize more returns. Kevin Murray needs to invest just along the capital Market line to achieve his goal of return maximization as desired (Dempsey, 2013).

The CAPM is given as follows:

R_{i} = R_{F} + [E (R_{M} – R_{F})] Ă

Where R_{i} is required return of security i

R_{F} is the risk free rate of return

E (R_{M}) is the expected market rate of return

Ă denote Beta.

If we graph Ă_{i} and E (R_{i}) then we can observe the following relationship

## Data and Methodology

All assets with correct prices will lie on the security market line. Any security off this line will either be overpriced or underpriced. The security market line therefore shows the pricing of all assets if the market is at equilibrium. It is a measure of the required rate of return if the investor were to undertake a certain amount of risk. The investor has the option of reducing her risk of exposure by going for the less risky assets such as treasury bills and bonds in order to reduce this risk. The data indicates that as long as you need additional returns, there is an additional risk that is associated with it. The investor can decide to take calculated risk by just investing along the security market line. Any portfolio or asset on the security market line is less risky and worthy. Treasury bills and bonds are considered less risky since they have a fixed rate of return and a fixed period of investment and every investor is assured of this return. The risk-averse investors mostly undertake this kind of investment.

## Empirical Results

The holding period return is the amount or return realized from an investment during the period within which the investor holds it. HPR is sometimes referred to as holding period yield (HPY) and can be computed for any asset. Holding period return helps an investor to evaluate what he gets as actual return and what he gets as expected return from an investment. HPR is then computed as follows.

HPR = Income

Historical cost

Or HPR = Income + (current value – initial value)]/Initial value

Expected Dividends = Probability x Dividends = 0.25*4.5 + 0.45*4.0 + 0.25*3.5 + 0.05*2 = 1.125+1.8+0.875+0.1 = 3.9

Therefore, HPR = {[3.9+ (112-110)]/110} = 0.0053

It therefore follows that the investor has received a total return of 0.00% from this investment from the time it was acquired to date. The probability that this investment is viable is 0.00%. Probability distribution like in this case can be skewed. Biasness is a concept, which is commonly used in statistical decision-making. It refers to the degree in which a given frequency curve is deviating away from the normal distribution (Tobin, 1958). There are two types of biasness namely Positive biasness and Negative biasness. Positive biasness is the tendency of a given frequency curve leaning towards the left. In a positively skewed distribution, the long tail extends to the right (Bornholt, 2013).

In this distribution, one should note the following. The mean is usually bigger than the mode and median (Fama, 1970). The median is always found between the mode and the mean. There are more observations below the mean than above the mean. This frequency distribution, as represented in the skewed distribution curve, is a characteristic of the age distributions in the developing countries (Fama & French, 2004).

Negative biasness is an asymmetrical curve in which the long tail extends to the left. These are numerical values, which assist in evaluating the degree of deviation of a frequency distribution from the normal distribution. The following are the commonly used measures of biasness (Das, Markowitz, & Scheid, 2010).

The two coefficients above are also known as Pearson measures of biasness.

## Conclusion

The Pearson coefficients of biasness usually range between negative three and positive three. These are extreme values, that is, negative 3 and positive 3, which therefore indicate that a given frequency is negatively skewed and the amount of biasness is quite high. Similarly, if the coefficient of biasness is negative, it can be concluded that the amount of biasness of deviation from the normal distribution is quite high and the degree of frequency distribution is positively skewed. It therefore follows that the above case is one of the positive skewed distributions as indicated by the figures. A portfolio consisting of one share of index fund and a put option takes a positively skewed distribution since it is very uncertain (Elton, Gruber, & Brown, 2006). It is not possible to tell which direction the share price might take because shares are considered very risky always. Since investment is often a risky venture, many investors take steps to ensure that they do not lose their money during investment. They try as much as possible to minimize such risks or otherwise hedge against such potential risk. Insurance is purchased to guard an investor against loss arising due to price fluctuations in the market. This type of distribution is less skewed since the risk factor has been minimized or taken into consideration by the investor who acquires this investment (Perold, 2004). A normal distribution is usually the ideal condition within which an investor can make a decision since the returns here are uniform and take a particular direction, which is at least known or predictable to the investor. Although in a real investment, there is no ideal condition, thus investors are forced to hedge or reduce their risks. This is why acquisition of insurance is necessary to guard the investor against losing his or her money, as well as investments (Bodie, Kane, & Marcus, 2008).

## References

Bodie, Z Kane, A & Marcus, A 2008, *Investments*, McGraw-Hill Irwin, New York.

Bornholt, G 2013, âThe Failure of the Capital Asset Pricing Model (CAPM): An Update and Discussionâ, *A Journal of Accounting, Finance and Business Studies*, Vol. 49, no. 1 pp 3-10.

Das, S Markowitz, H & Scheid, J 2010, âPortfolio optimization with mental accountsâ *Journal of Financial and Quantitative Analysis, *vol. 45 no. 1, pp 311-334.

Dempsey, M 2013, *The Capital Asset Pricing Model (CAPM): The History of a Failed Revolutionary Idea in Finance?* McGraw-Hill, New York.

Elton, E Gruber, M & Brown, S 2006, *Modern Portfolio Theory and Investment Analysis*, John Wiley, New York.

Fama, E & French, K 2004, âThe capital asset pricing model: Theory and evidenceâ, *Journal of Economic Perspectives*, Vol. 18, no. 3, pp 25-46.

Fama, E 1970, âEfficient capital markets: A review of theory and empirical workâ, *Journal of Finance,* vol. 25 no. 2, pp 383-417.

Levich, M 2001, *International Financial Market*, McGraw-Hill, New York.

Oster, S 1994, *Modern Competitive Analysis*, Oxford University Press, Nueva.

Perold, A 2004, âThe Capital Asset Pricing Modelâ, *Journal of Economic Perspectives*, Vol. 18 no. 3, pp 3-24.

Reilly, K & Brown, C 2007, *Investment Analysis and Portfolio Management*, Southwestern Thomson, New York.

Tobin, J 1958 âLiquidity Preference as Behavior Towards Riskâ, *Review of Economic Studies*, vol. 25, pp 68-85.