Investment decisions are the decisions made by the investors through investment analysis which is supported by decision tools. Investors use technical and fundamental analysis of the market to achieve satisfactory return against the risk taken. There have been several theories, models and ideas by the scholars and researchers to analyze the market condition and maximize the portfolio expected return while minimizing the risk level.
One of the most common and used theory is the Modern Portfolio Theory or MPT which help to determine lower risks for an investment. In this theory, there are several mathematical models to analysis different factors in an investment like risk and expected return, diversification, efficient frontier with no risk free asset, two mutual fund theorem, capital allocation line and risk free asset. MPT can also help for asset pricing through systematic risk (market risk or portfolio risk), specific risk measurement and capital asset pricing model etc.
In this paper, we are going to calculate expected return, risk premium, standard deviation, covariance, portfolio return and sharp ration for different fractions through case analysis.
To understand the theories and approaches for investment decisions, we are going to consider the facts of Pioneer Gypsum.
Table 1: Pioneer Gypsum market status
From these facts, we are going to calculate expected outcome in the market for the company.
Modern Portfolio Theory (MPT)
In investment analysis, MPT is a concept of diversification by using mathematical formula with an aim of finding a set of investment assets that has lower risk all together than any individual asset. The concept behind MPT is that the assets in an investment portfolio should not be selected individually. Instead, it is more significant to consider how each asset changes in price regarding to how every other asset in the portfolio changes in price. Normally, assets with greater expected returns are riskier. For a given amount of risk, MPT shows how to select a portfolio with the maximum promising expected return. Contrary, for a given expected return, MPT describes how to select a portfolio with the lowest possible risk (Elton and Gruber, 1997).
In spite of its theoretical significance, there have been some criticisms on MPT about whether it is a perfect investing strategy, the reason for that is this concept of financial market does not reflect the real world in several ways. Efforts to interpret the theoretical basis into a feasible portfolio structure algorithm have been diseased by technical difficulties from the volatility of the key problems with respect to the data in hand. Recent findings have proven that instabilities disappear when a restriction or penalty is included in the optimization process (Brodie, De Mol, Daubechies, Giannone and Loris, 2009).
Many researchers criticize the Modern Portfolio Theory as: it does not reflect the market in practical sense and it does not take its own effect in account for asset prices.
The Modern Portfolio Theory gives several mathematical models regarding market analysis and calculates risk and return.
Expected Return
According to the Modern Portfolio Theory, the mathematical equation of expected return is:
Where Return on the profit
= Return on assed i
wi = Weightingof component asset i
Portfolio return variance:
Here, ?ij = the correlation coefficient between the returns on assets i and j
For two asset portfolio,
The portfolio return:
The portfolio variance:
For Pioneer Gypsum, the expected return would be:
CAPM
CAPM or the Capital Asset Pricing Model was introduced by Jack Treynor in 1961 (French, 2003). The model is used to establish a theoretically approach to determine the required rate of return for an asset, if the asset is to be joined in an well diversified portfolio with the fact that the asset is at non diversifiable risk. The theory takes into account the sensitivity of the asset to non diversifiable risk (i.e. market risk or systematic risk) as well as the expected return of theoretical risk free asset and the expected return of the market. It is often symbolized by the quantity beta (?) in the investment analysis.
Searchers have fount quite a few problems in the theory of CAPM. The most notables among them are:
- According to this model, asset returns are normally distributed as random variables and potential shareholders uses a quadratic outline of the utility. However, it is often found that returns in equity and other markets are not circulated in general way. Because of that, large swings (3 to 6 standard deviations from the mean) takes place in the market more than the normal distribution theory would anticipate (Mandelbrot and Hudson, 2004)
- This model also employs that the guesses of potential and active shareholders on possibilities go with the true distribution of returns. Another possibility tells that the expectations of potential and active shareholders are prejudiced, which causes market prices to be unproductive. This factors is studied in behavioral finance, which takes account of psychological assumptions to bring new alternatives for CAPM like the overconfidence based asset pricing theory (Daniel, Hirshleifer, and Subrahmanyam, 2001)
- In theory, a market portfolio should have all types of assets that are held as an investment (i.e. real estate, art works and human capital etc.). In real, such market portfolio is not possible and individuals usually replace the true market portfolio in the place of a stock index. It has been proven that this replacement is not inoffensive and most likely can guide to miss-inferences as to the legality of CAPM. Also, this theory might not be empirically experiment able because of the lack of its potential outcomes in the real market portfolio (Roll, 1977)
The mathematical formula of CAPM depends on two facts, the security market line and its relation with systematic risk and expected return. From this relationship, we obtain the Capital Asset Pricing Model with the following equation:
Here:
= the expected return on the capital asset
= the risk free rate of interest
= the sensitivity of the expected excess asset returns
= the expected return of the market
= the difference between the expected market rate of return and the risk free rate of return
Beta
The term Beta (?) refers to a number is that describes the relation of its returns for a portfolio with those of the financial market all together (Levinson, 2006). If the return of an asset changes autonomously according to the changes in the return of the market, it has a Beta of zero. A positive beta means that the return of the asset follows the return of the market. On contrary, a negative beta means that the return of the asset follows the opposite movement of the returns of the market. The beta coefficient is a key parameter in the CAPM.
Seth Klarman criticized Beta by saying that it fails to consider specific economic developments and business fundaments (Klarman & Williams, 1991).
The mathematical expression of Beta is
Where = beta coefficient
= the rate of return of the asset
= rate of return of the portfolio
= covariance between the rate of the return
Risk Premium
The term Risk Premium refers to the minimum amount of money which must exceed the confirmed return from a risk free asset comparing to the expected return on a risky asset. This is accounted when an individual is holding a risky asset instead of a risk free asset. The premium is the minimum compensation for the risk that the individual is willing to accept.
The mathematical expression for calculating the risk premium is
Where u = concave von Neumann-Morgenstern utility function
= return on the risk free asset
r = random return on the risky asset
x = zero-mean risky component
= hypothetical expected return
Standard Deviation
The standard deviation is used to determine the variation from the average or expected value. A low standard deviation means that the date points are close to the average and a high standard deviation means spread out data points over large array.
Sharpe Ratio
Sharp ratio is used to measure the risk premium per unit of deviation in an investment asset. Mathematically, it is expressed by the equation:
Here, R = asset return
= return on a benchmark asset
= expected value of the excess of the asset return over the benchmark return
= standard deviation
Covariance Matrix
The covariance matrix is the type of matrix where the elements in the i, j position are the covariance between the ith and jth elements of a random vector. Each element of the covariance vector is a random scalar variable, each with a finite number of experiential empirical values or with an infinite or finite number of possible values précised by a theoretical combined probability distribution of all the random variables.
References
Edwin J. Elton and Martin J. Gruber (1997) Modern portfolio theory, 1950 to date. Journal of Banking & Finance 21
Brodie, De Mol, Daubechies, Giannone and Loris (2009). Sparse and stable Markowitz portfolios Proceedings of the National Academy of Science 106 (30)
French, Craig W. (2003). The Treynor Capital Asset Pricing Model, Journal of Investment Management, Vol. 1, No. 2, pp. 60–72
Mandelbrot, B.; Hudson, R. L. (2004). The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin, and Reward. London: Profile Books
Daniel, Kent D.; Hirshleifer, David; Subrahmanyam, Avanidhar (2001). “Overconfidence, Arbitrage, and Equilibrium Asset Pricing”. Journal of Finance 56 (3): 921–965
Roll, R. (1977). “A Critique of the Asset Pricing Theory’s Tests”. Journal of Financial Economics 4: 129–176
Levinson, Mark (2006). Guide to Financial Markets. London: The Economist (Profile Books). pp. 145–6
Klarman, Seth; Williams, Joseph (1991). “Beta”. Journal of Financial Economics 5 (3): 117
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