Having selected two stocks, LLOY.L and BARC.L, from the FTSE100 Index and downloaded the daily prices, spanning the period January 2008 – December 2010, it is now time to calculate the daily returns and the average returns. The daily returns are calculated by subtracting from the closing share price of the following day the closing share price of the prior day, all divided by the closing share price of the prior day. The daily return of stocks represents the changing of the value of the stock in a short term point of view, so the value in percentage that could be obtained by trading the stock in different days at the midquote(closing) price.
This paper doesn’t take into account dividend in the return equation for simplicity. The average returns are the sum between the daily returns of each stock divided by the number of days: LLOY.L -0,09% and BARC.L 0,05%. As it is an average of the sum of the daily returns, the average return of a stock is like a summary value in a long term point of view. In order to calculate the variance, which is the measure of the variability of measured data from the average value of the set of data , we use the excel prepared formula =VAR and the results are LLOY.L 0,00329681532 and BARC.L 0,00298971628. Variance is also the measure of dispersion of a set of data points around their mean value. The standard deviation of the two stocks is calculated by the prepared formula of excel =DEV.ST or just by calculating the radical square of the variance , LLOY.L 0,05741790071952 and BARC.L 0,054678298027727 and it represents the variability of a distribution. The value which represents the correlation between two variables is the covariance of the two stocks which is 0,00197867681213. Having constructed a portfolio which consists of 50% in each stock it is now time to calculate its return and standard deviation which respectively are -0,00018953911968 and 0,05060604020891: in order to calculate the standard deviation we have to calculate the variance as the standard deviation is the radical square of the variance. The table below shows a set of portfolios of the two stocks with different weights. It is mainly important to notice two factors: the returns and the variance. As the average return of LLOY.L is negative, a rational investor would choose to invest only in BARC.L in order to get a positive return; accordingly to the data of the variance it is significant to notice that it has high values in the extremes of the composition of the weights of the portfolio. High values of variances mean high values of risk. Actually, if an investor does not reduce the exposure to risk by diversification of his portfolio with different assets the risk of losses is higher. The risk is higher because with diversification the investor is able to reduce the specific risk of the firm and will be subjected only to the general market risk. The graph below summarizes a portfolio with different weights of assets. The outcome is what is called the efficient frontier (the upper part) and so the graph shows the correlation between returns and risk (standard deviations). The previous explanation meant to mark the difference between what appears to be the best choice for an investor and so the highest return and what is really the best choice, which is maximizing the utility of the investor by minimizing his exposure to risk.
The minimum exposure to risk is calculated by minimizing the variance and by calculating the exact weights of the assets in the portfolio. This paper has used the excel prepared function solver to get the minimum variance in the portfolio made with those two different stocks.
The minimum variance is 0,25% with weights of 43% LLOY.L and 57% BARC.L as highlighted. In order to achieve a portfolio yielding a return of 25% the weight of each stock should be equal to: LLOY.L -17536% and BARC.L 17636%. This result has been obtained by using the excel function solver and it mirrors the fact that an investor should sell LLOYD.L assets (-17536%) and buy BARC.L assets (17636%) in order to achieve a return of 25%. Question (b): It is now time to define what a “Market Portfolio” is. This paper intends to begin describing the market with risky assets and then deals with a portfolio with risk free assets in order to scrutinize the main differences, because it is just in the composition between these two different set of portfolios that there is the market portfolio. Before dealing particularly with the market portfolio it is important to analyze the officials acting in the financial environment, investors in particular. There are two kinds of investors: risk averse and risk takers. If we consider risk as a measure of uncertainty about both the development of the market and the success of our investment, risk averse investors are those officials who prefer to invest in low risk assets even though they will not achieve a high income. Risk takers instead are those investors whose aim is to bet on the market development and invest in a more risky way in order to get better returns. The main objective of the description of these officials it is to describe the combination of assets of their portfolios.
Risk takers invest on assets which have a high degree of dispersion (the variance and so the standard deviation) in the final income. There is a high chance that the final outcome will not be equal to the expected return. An efficient portfolio for risk takers is made by maximizing the expected returns for a given amount of risk or minimizing risk when the expected return is given. Risk averse investors will invest in assets with a low or almost nonexistent level of risk, even though the expected return is not high. Efficient portfolios are shown in the efficient frontier, which is a curve showing optimal portfolios made by assets of different degrees of risk. The efficient frontier offers the highest return for any level of risk and it is constructed by combining mean and standard deviation, so return and risk. In the graph below the efficient frontier is just AB because for every level of risk (variance) we can get higher expected return in AB instead of in AC: this represents efficiency in the selection of assets for a portfolio. Having analyzed the portfolio made of risky assets it is now time to describe the contrast to a portfolio of risk free assets. A portfolio of risk free assets reduces its risk to zero. The expected return turns into realized return (Expected(R) = Realized(R)). Government bonds are usually the closest example of risk free assets.
The most visible change is that the efficient frontier becomes a straight line. This straight line is called capital market line and its equation is E(Rp)=Rf+[(E(Rm)-Rf)/AÆ’m]xAÆ’p. The expected return on a portfolio depending on a risk free rate of return suffers less risk because the variance of the portfolio is smaller, as the variance of the risk free assets and the covariance are equal to zero. The main advantage given by risk free assets is that investors can borrow or lend any amount of money at the risk free rate of return depending on the weight they have, as the graph below shows. It is important to pay attention to both the risk component and the risk free one in order to find out the Market Portfolio (M, in the graph below) as it is just the tangent point between the efficient frontier and the capital market line. The CML is the line used in the capital asset pricing model to illustrate the rates of return forA efficient portfolios depending on the risk-free rate of return and the level of risk (standardA deviation)A for a particular portfolio.
Markowitz introduced a new goal for investors, which is to maximize their utility. The utility is maximized in the market portfolio, which is the equalization between the efficient frontier and the capital market line. The utility is approximately calculated as the expected return minus the variance of return, which is multiplied to a risk averse variable. If an investor wants to achieve the best utility from the combination between assets of his portfolio, he will try to minimize the variance in order to maximize the expected return. The Market Portfolio is where every investors will want to invest. Actually this portfolio must include all risky assets and as the market is in equilibrium all assets are included in their market value.
Since the Market Portfolio contains all risky assets, it is a completely diversified portfolio, which means that all the unique risk of individual assets (unsystematic risk) is diversified away. In the presence of capital markets, rational risk averse investors select efficient portfolios that lie in the CML with the highest expected Sharpe ratio (risk premium/standard deviation) which means with the highest expected return and the lowest degree of risk. The concept of Market Portfolio is strictly related to the concept of the “Separation Theorem”. James Tobin explained in the Separation Theorem that if an investor holds risky assets and he is able to borrow (buying stocks on margin) or lend (buying risk free assets) at the same rate, then the efficient frontier is a single portfolio of risky assets plus borrowing and lending. Tobin’s Separation Theorem says an investor can separate the problem into first finding that optimal combination of risky assets and so the tangency point (Market Portfolio) and then deciding whether to lend or borrow, depending on his attitude toward risk. If there is only one portfolio plus borrowing and lending, it is got to be the market.
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