Modern Portfolio Theory and Capital Asset Pricing Model

Modern Portfolio Theory and Capital Asset Pricing Model Introduction The Capital Asset Pricing Model developed by William Sharpe has significant similarities with Harry Markowitz’s Portfolio theory. In fact, the later is rightly considered as the next logical step from the latter, with both based on similar foundations. There are also differences in how each model/theory is calculated, pertaining to risk considerations. This paper’s main objective is to identify these differences while highlighting the similarities as well to put things into perspective. The report will open with an overview of Markowitz’s portfolio theory and explain it further by means of describing the efficient frontier, the Capital Market Line, risk free asset and the Market Portfolio. The report will then switch its attention to the Capital Asset Pricing Model and explain it further through the Security Market Line. The report will then close by outlining the differences between the two with a view of answering the main objective. What will come through in this report is that Markowitz’s portfolio theory uses standard deviation as its risk measure and takes into account all risk in an efficient portfolio, while the Capital Asset Pricing Model uses the beta co-efficient to measure risk and takes into account both efficient and non-efficient portfolios – further more it measures the risks of individual assets within the portfolio. Modern Portfolio Theory Modern Portfolio Theory (MPT) was introduced by Harry Markowitz, way back in 1952. At a high level it proposes how rational investors use diversification to optimise their investment portfolios and give guidance on pricing risky assets. MPT assumes that investors are risk averse, i.e. given two assets A and B offering the same expected return, investors will opt for asset A if it is less risky. In effect, an investor who expects higher returns would need to accept more risk. The expected trade-off between risk and return depends on the individual’s level of risk aversion. The implication of this is a rational investor (a risk averse investor) will not invest in a portfolio if another one exists offering a better risk-return profile (Fabozzi Markowitz, 2002). For any given level of risk, investors will opt for portfolios with higher expected returns instead of those with lower returns. Another assumption under MPT is that investors are only interested in the expected return and the volatility of an investment, as measured by the mean and standard deviation respectively. Investors do not consider any other characteristics, for example, charges. In effect, based on the assumptions above, investors are concerned about efficient portfolios. To explain portfolio theory further, let us consider the formula for the expected return and risk of a portfolio under MPT. Suppose two assets A and B formed a portfolio in proportion (X) each, the expected return for that portfolio would be: R(p) = X(a)R(a) + X(b)R(b), where: R(p) = expected returns from portfolio R(a) = expected returns from asset A R(b) = expected returns from asset B The standard deviation or risk of that portfolio would be: SD(p) = √(XÂ ²aSDÂ ²a + XÂ ²bSDÂ ²b + 2XaXbRSDaSDb), where: SD(p) = standard deviation of expected returns of portfolio SDa = standard deviation of expected returns of asset A SDb = standard deviation of expected returns of asset B R = correlation coefficient between the expected returns of the two assets The efficient frontier Under MPT, Markowitz examined the efficient frontier curve. The efficient frontier curve gives a graphic presentation of a set of portfolios that offer the maximum rate of return for any given level of risk (McLaney, 2006). According to Markowitz, an efficient investor will opt for an optimum portfolio along the curve, based on their level of risk aversion and their perception of the risk and return relationship (Fabozzi Markowitz, 2002). Figure 1: Efficient Frontier Source: www.riskglossary.com The curve in the diagram above illustrates the efficient frontier. Portfolios on the curve are efficient – i.e. they offer maximum expected returns for any given level of risk and minimum risk for any given level of expected returns. The shaded region represents the acceptable level of investments when risk is compared against returns. For every point on the shaded region, there will be at least one portfolio that can be constructed and has a risk and return corresponding to that point (www.riskglossary.com) As aforementioned, each portfolio on the efficient frontier curve will have a higher rate of return for the same or lower level of risk or lower risk for an equal or better rate of return when compared with portfolios not on the frontier. It is important to note that the efficient frontier is really made up of portfolios rather than individual assets. This is because portfolios could be diversified, i.e. investors can hold assets which are imperfectly correlated (Fabozzi Markowitz, 2002). This will help to ensure that investors can reduce their risks associated with individual asses by holding other assets – a kind of set-off. The Capital Market Line The Capital Market Line (CML) is a set of risk return combinations that are available by combining the market portfolio with risk free borrowing and lending (www.lse.co.uk/financeglossary). The CML defines the relationship between risk and return for efficient portfolios of risky securities. It specifies the efficient set of portfolios can investor can obtain by combining the portfolio (which contains risk) with a risk free asset. The formula for CML is: E (r_c) = r(f) + SD(c)*[E(r_m)-r(f)]/SD(m) Where: E(r_c) = expected return on portfolio c R(f) = risk free rate SD(c ) = standard deviation of portfolio c E (r_m) = expected return on market portfolio SD(m) = standard deviation of market return The CML indicates that the expected return of an efficient portfolio is equal to the risk-free rate plus a risk premium. Both risk and return increase in a linearly along the CML. Figure 2: Capital Market Line Source: www.riskglossary.com In Figure 2 above, the CML is the line touching the efficient frontier curve. It passes through the risk free rate (assumed to be 5%). The point where the CML forms a tangent with the efficient frontier curve is the point called the super-efficient portfolio. The Risk free asset, Sharpe ratio and the Market Portfolio The risk free asset pays a risk free rate and has a zero variance in returns, e.g. government short-term securities. When combined with a portfolio of assets the change in return and risk is linear. The Sharpe Ratio is a measure of the additional return to be obtained about a risk free rate for a given portfolio compared with its corresponding risk. On the efficient frontier the portfolio with the highest Sharpe Ratio is known as the market portfolio. The CML is the result of a comparison between the market portfolio and the risk free asset. The CML surpasses the efficient frontier with the exception of the point of tangency. The Capital Asset Pricing Model While the CML focuses on the risk and return relationship for efficient portfolios, it would be useful to consider the relationship between expected return and risk for individual assets or securities. The Capital Asset Pricing Model (CAPM) would be used for this. CAPM is an extension of Markowitz’s Portfolio Theory or MPT. It introduces the notions of systematic and specific risks. Let us define each: Systematic risk – this is the risk associated with holding the market portfolio of assets Individual assets are affected by market movements Specific risk – this risk is unique to an individual asset and represents that portion of an asset’s return which has no correlation with market movements. CAPM assumes the following (McLaney, 2006, 199): Investors are risk averse and maximise expected utility of wealth The capital market is not dominated by any individual investors Investors are interested in only two features of a security, its expected returns and its variance or standard deviation There exists a risk free rate at which all investors may borrow or lend without limit at the same rate There is an absence of dealing charges, taxes and other imperfections All investors have identical perceptions of each security This lends credence to the assertion that CAPM follows a natural progression from MPT. The assumptions are identical with the main difference being how risks are categorised and treated. This will be explored in detail in a later section. Under CAPM, the market place will compensate an investor for taking a systematic risk but not a specific risk. The rationale for this is that specific risks can be avoided or minimised through diversification. The formula for CAPM is as follows: r = Rf + Beta x (RM-RF), where: r = expected return on an asset Rf = rate of risk free investment RM = return rate of the appropriate asset class Beta is the relative risk contribution of an individual security to the overall market portfolio. It measures the security risk relative to the market portfolio and ignores the specific risk. The beta equation is as follows: Cov (i,M)/(SDm)Â ², where: Cov (i,M) = covariance between market portfolio and security i (SDm)Â ² = variance of the market’s return The betas for all assets are measured in relation to the market portfolio beta which is 1. In effect, if individual beta is greater than 1, then individual asset has a higher risk than the market risk. If individual beta equals 1, then individual asset risk and market risk are the same. If individual beta is less than 1, then the risk of that individual asset is less than the market risk. The value of beta provides an idea of the level or size of the change in an asset’s return when a corresponding change in the returns of an overall portfolio is experienced (McLaney, 2006). Beta has come under criticism from academics and investors who do not appreciate the value of beta as an appropriate risk measure. However, this is somewhat challenged by actual performance of the betas of portfolios and mutual funds. These are regarded as stable and can be used to predict future betas. Security Market Line CAPM can be applied by using the Security Market Line (SML). SML is a graphical representation showing the linear relationship between systematic risk and expected rates of return for individual assets. In the case of the SML, risk is measured by beta. It plots the expected returns on the y axis and the risk as denoted by beta on the x axis. In other words, the SML expresses the linear relationship between the expected returns on a risky asset and its covariance with market returns. Its formula is: Figure 3: CAPM and SML The line in the diagram above is the SML. Differences relating to MPT (CML) and CAPM (SML) To explain the differences, it is useful to consider the relationships between risk and return in the perspective of CML and SML. CML compares the relationship from an MPT perspective, while SML does from a CAPM perspective. The main difference pertaining to MPT’s relationship with CAPM is pertaining to risk. Under Portfolio theory, CML gives an indication of expected returns in comparison with risk. Here the risk is measured in terms of standard deviation of returns. The rationale for this is CML represents the trade-off for efficient portfolios, i.e. the risk is all systematic risk (McLaney, 2006). The SML on the other hand, indicates the risk/return trade-off, using beta as the measure of risk. In this case, only the systematic risk element of the individual asset is taken into consideration. The reason why CML shows no individual security’s risk profile is because all individual securities have an element of specific risk, implying that they are inefficient. CML only looks at efficient portfolios. The table below summarises the main differences between CML and SML Table 1: Tabular difference between CML and SML Summary As has been shown above, CAPM has been developed along the lines of Markowitz’s Portfolio theory. They both use expected returns and risk as the investor’s main determinant of their investment decisions. They both assume that investors are risk averse and do not consider anything else other than risk and returns. However, there are some subtle differences which will now be summarised below: Under Portfolio theory, the CML measures risk by standard deviation or total risk. The SML measures risk by beta or systematic risk under CAPM – it ignores specific risks The CML graph is interested in providing information on efficient portfolios only. The SML graph on the other hand provides insight into both efficient and non-efficient portfolio and securities REFERENCES AND BIBLIOGRAPHY Books Bodie, et al (2006) ‘Investments’ (7th edition), McGraw-Hill/Irwin, London Elton, E et al (2003) ‘Modern Portfolio Theory and Investment Analysis’, Wiley, London Fabozzi, F. Markowitz, H. (2002) ‘Theory and Practice of Investment Management’, Wiley, London McLaney, E. (2006) ‘Business Finance – Theory and Practice’ (7th edition), Prentice Hall, London O’neill, W.J. (2002) ‘How to Make Money in Stocks’, (3rd edition), McGraw-Hill, London Internet Sources www.lse.co.uk www.riskglossary.com www.wikipedia.com

Tess of the d’Urbervilles :: Literary Analysis, Thomas Hardy

In Thomas Hardy’s classic novel, Tess of the d’Urbervilles, there are many complex and intriguing characters that emerge from it. Two such characters are the two young men who tried to win Tess over, Alec d’Urberville and Angel Clare. These two characters are distinctly different from one another in many ways, but in other, more subtle ways, they possess some similarities. Their physical appearances are noticeably different though it is noted that they are both especially handsome young men. Alec has â€Å"touches of barbarism†(32) in his face while Angel’s has â€Å"grown more thoughtful† (112). They also seem to have the opposite colors of hair. Alec’s hair being â€Å"black† (35) while Angel has the â€Å"palest straw colour† that â€Å"deepens to a warm brown†(112). Furthermore, their mouths have variations as well. The â€Å"badly molded†, but â€Å"smooth†, â€Å"full†(35) lips of Alec are much different than the â€Å"small† and â€Å"delicately lined†(114) ones of Angel. Even more, their eyes are described with opposing adjectives. The difference in the Alec’s â€Å"bold rolling eye† and Angel’s â€Å"fixed, abstracted eyes†(114) are quite present. The variety in their faces are most definitely there. The descriptions of their general features relate to their actions. Alec is a womanizer who jumps from woman to woman without regard to their feelings, carelessly flinging around their hearts and virginity. Barbaric. His hair, the darkest possible color, can be associated to a dark personality related to sin and other negative aspects of life. His mouth, which is shaped roughly, like his actions and character, can sometimes be overlooked because of the smoothness of them. Like the smooth trait of his lips, Alec can use trickery to disguise his real motives with people, which include his us of sly words. He certainly has a bold personality, like his eyes, having the audacity to steal a â€Å"cursory kiss†(70) from Tess when he can. The rolling part of his eyes can be taken literally. Because he cannot keep his eyes still, his eyes are forever roaming. It shows his unwillingness to commit to one woman because they are always wondering to others. Angel who is described with reserved, quieter adjectives, is in fact more so Alec. His reservedness often leads many women to fall for his as well because humans, for the most part, naturally want what they cannot have. And Angel who is â€Å"too much taken up with’ his own thoughts to notice girls†(113) fills these requirements.

Elie Wiesel Quotes Essay

The opposite of love is not hate, it’s indifference. The opposite of art is not ugliness, it’s indifference. The opposite of faith is not heresy, it’s indifference. And the opposite of life is not death, it’s indifference.† ― Elie Wiesel â€Å"There may be times when we are powerless to prevent injustice, but there must never be a time when we fail to protest.† ― Elie Wiesel â€Å"Friendship marks a life even more deeply than love. Love risks degenerating into obsession, friendship is never anything but sharing.† ― Elie Wiesel â€Å"We must always take sides. Neutrality helps the oppressor, never the victim. Silence encourages the tormentor, never the tormented.† ― Elie Wiesel â€Å"To forget the dead would be akin to killing them a second time.† ― Elie Wiesel, Night â€Å"When a person doesn’t have gratitude, something is missing in his or her humanity. A person can almost be def ined by his or her attitude toward gratitude.† ― Elie Wiesel â€Å"There is divine beauty in learning†¦ To learn means to accept the postulate that life did not begin at my birth. Others have been here before me, and I walk in their footsteps. The books I have read were composed by generations of fathers and sons, mothers and daughters, teachers and disciples. I am the sum total of their experiences, their quests. And so are you.† ― Elie Wiesel â€Å"Human suffering anywhere concerns men and women everywhere.† ― Elie Wiesel, Night â€Å"For the dead and the living, we must bear witness.† ― Elie Wiesel â€Å"Never shall I forget that night, the first night in camp, which has turned my life into one long night, seven times cursed and seven times sealed†¦.Never shall I forget those moments which murdered my God and my soul and turned my dreams to dust. Never shall I forget these things, even if I am condemned to live as long as God Himself. Never.† ― Elie Wiesel, Night â€Å"One person of integrity can make a difference.† ― Elie Wiesel â€Å"If the only prayer you say throughout your life is â€Å"Thank You,† then that will be enough.† ― Elie Wiesel â€Å"Only the guilty are guilty. Their children are not.† ― Elie Wiesel â€Å"Write only if you cannot live without writing. Write only what you alone can write.† ― Elie Wiesel â€Å"Which is worse? Killing with hate or killing without hate?† ― Elie Wiesel â€Å"I pray to the God within me that He will give me the strength to ask Him the right questions.† ― Elie Wiesel, Night â€Å"I swore never to be silent whenever and wherever human beings endure suffering and humiliation. We must always take sides.† ― Elie Wiesel â€Å"Wherever men and women are persecuted because of their race, religion, or political views, that place must — at that moment — become the center of the universe.† ― Elie Wiesel â€Å"No human race is superior; no religious faith is inferior. All collective judgments are wrong. Only raci sts make them† ― Elie Wiesel â€Å"Then came the march past the victims. The two men were no longer alive. Their tongues were hanging out, swollen and bluish. But the third rope was still moving: the child, too light, was still breathing†¦ And so he remained for more than half an hour, lingering between life and death, writhing before our eyes. And we were forced to look at him at close range. He was still alive when I passed him. His tongue was still red, his eyes not yet extinguished. Behind me, I heard the same man asking: â€Å"For God’s sake, where is God?† And from within me, I heard a voice answer: â€Å"Where He is? This is where–hanging here from this gallows†¦Ã¢â‚¬  That night, the soup tasted of corpses.† ― Elie Wiesel, Night â€Å"For in the end, it is all about memory, its sources and its magnitude, and, of course, its consequences.† ― Elie Wiesel, Night â€Å"Think higher, feel deeper.† ― Elie Wiesel â€Å"There are victories of the soul and spirit. Sometimes, even if you lose, you win. † ― Elie Wiesel â€Å"We must take sides. Neutrality helps the oppressor, never the victim. Silence encourages the tormentor, never the tormented. Sometimes we must interfere. When human lives are endangered, when human dignity is in jeopardy, national borders and sensitivities become irrelevant. Wherever men and women are persecuted because of their race, religion, or political views, that place must – at that moment – become the center of the universe.† ― Elie Wiesel, The Night Trilogy: Night/Dawn/The Accident â€Å"I shall always remember that smile. From what world did it come from?† ― Elie Wiesel, Night â€Å"For the survivor who chooses to testify, it is clear: his duty is to bear witness for the dead and for the living. He has no right to deprive future generations of a past that belongs to our collective memory. To forget would be no t only dangerous but offensive; to forget the dead would be akin to killing them a second time.† ― Elie Wiesel, Night â€Å"I don’t want my past to become anyone else’s future.† ― Elie Wiesel â€Å"We must not see any person as an abstraction. Instead, we must see in every person a universe with its own secrets, with its own treasures, with its own sources of anguish, and with some measure of triumph.† ― Elie Wiesel â€Å"Most people think that shadows follow, precede or surround beings or objects. The truth is that they also surround words, ideas, desires, deeds, impulses and memories.† ― Elie Wiesel â€Å"Night is purer than day; it is better for thinking and loving and dreaming. At night everything is more intense, more true. The echo of words that have been spoken during the day takes on a new and deeper meaning. The tragedy of man is that he doesn’t know how to distinguish between day and night. He says things at night that should only be said by day.† ― Elie Wiesel, Dawn

Smoking Is A Well Known Habit - 1354 Words

Smoking is a well known habit. Many people smoke despite its obvious and widely advertised deleterious effects on health. Smoking is not only bad for the immediate users, but also the people surrounding them. It is an addiction that can effect others without their consent. Not only is it unhealthy physically but also a very expensive habit. It costs millions a year, and that is just to buy enough cigarettes to satisfy your addiction, not counting the cost of medical care needed, renovations of homes that smoking has ruined, and more problems branching from smoking. Having restrictions on smoking would not only benefit the individual smoker, but also people around them. Some may argue that its just a harmless habit and that putting a†¦show more content†¦Some