Risk Management - Part D

Manolis Anastopoulos @ University of Leicester - Risk Management

3. Capital Asset Pricing Model

William Sharpe in 1964, and others, took the previous work further, by formalizing the Capital Asset Pricing Model (CAPM), which showed how the market must price individual securities in relation to the market portfolio. More specifically, their model – the CAPM – as stated in (Friedman 1987), ‘describes the relationship between the expected risk and expected return’. This model – the CAPM, can be used both as apricing model that predicts an asset’s required rate of return, and as a technique in the cost of equity analysis[1], as well.

The formula that generally describes the aforementioned relationship-the CAPM, is the following:

Rk = Rf + (Rm – Rf ) x beta

Rk = The required (expected) rate of return

Rf = The rate of a risk-free investment, i.e. cash

Rm = The market return

Beta =The price of the beta coefficient


Assuming we have a stock with a beta of 0.8, we would expect a rate of return (Rk) of 12.6% if the rate of the risk-free investment (Rf) is 7% and the market return (Rm) is 14%. It is worth noticing that because the stock’s beta is 0.8 (less than zero) the stock’s expected rate of return (Rk) is 12.6%, which is less compared to the market’s return (Rm = 14%). Conversely, if the beta is greater than 1, say 1.4, which determines a more risky stock than the market (beta =1), then the Rk would be 16.8%, which is higher compared to the market’s return (Rm = 14%).

The above mentioned higher rate compared to the market rate, is what such an investor actually requires because of the extra risk that he undertakes when choosing a stock with a beta greater than 1.

3.1 Beta Coefficient

The beta coefficient, as stated in their book (Gitman & Madura 2001), ‘Is a relative measure of non-diversifiable risk’. The beta is the covariance of a stock’s return in relation to the whole stock market’s return and it is measured by the R-square. As it is known, the R-square measures the degree of response between the independent and dependant variable, e.g. between the whole market return and a company’s share return.

When we consider ‘the whole stock market’s return’, we can refer, for example, to the General Index of the Athens Stock Exchange (with a beta coefficient of 1), which is a good indicator of the whole stock market’s behavior.

Stocks with beta higher than 1, are more volatile than the market and, consequently, are more risky. Conversely, stocks with beta lower than 1, are less volatile than the market and less risky, as well.

As a result, it can be argued that high-beta shares, due to their potential of higher-than-market returns, usually ‘attract’ investors willing to risk. Unfortunately, the expected higher-than-market return would be worked against the investor, i.e. a- less- than market return, should the stock market decline.

In conclusion, beta can be used to predict the volatility of a stock’s price related to the market risk, i.e. the systematic or the non-diversifiable risk. However, we should not forget that in order to find the beta coefficient, we have actually relied on past comparisons between company’s share return and the market return, which does not necessarily  predict accurately the future behavior of the company’s share.

4. Usefulness and Limitations of the Portfolio Theory and CAPM

Undeniably, the Portfolio Theory that later evolved into Modern Portfolio Theory introduced significant tenets of modern finance. The concepts below mentioned are still regarded as dominant tools for the current portfolio managers.

  1. The concept of diversification clarified to potential investors that by increasing the number of stocks or assets in their portfolios, they are actually reducing the non-systematic risk that is associated with each individual asset or, in other words, they are reducing their uncertainty. As a result, the overall risk of their portfolio can approach more closely the systematic risk, i.e. the market’s risk.
  2. Diversification is attainable, not only by increasing the number of shares, but also by including different assets in portfolios. In such a case, the result would be the same, i.e. to change in the investor’s favor the relationship between return to risk.
  3. Another important rule that followed the introduction of diversification was the concept of correlation, which enables prospective fluctuations of individual securities to cancel each other to a satisfactory extent.
  4. The Portfolio Theory by using statistical tools - standard deviation or coefficient of variation – managed to measure the volatility. Thus, both the risk-averse investors and those willing to risk were able to trace, through quantitative data, those portfolios that matched their desire for the undertaken risk in respect to the expected return.
  5. As a result, the efficient portfolio that succeeds the highest average rate of return for a given level of risk or, in other words, portfolio that ‘optimally’ balance risk and return, had been born. Thus, investors, for any given value of volatility (risk) can choose an optimal portfolio that gives them the greatest possible rate of return.
  6. Finally, through the analysis of the efficient frontier and the capital market line, the ‘target’ of all investors was determined as the point where the capital line touches the ‘original’ efficient frontier, which represents an identical portfolio - the market portfolio.

Undoubtedly, Markowitz was the first who actually quantified the risk. William Sharpe, Lintner, Mossin, and Famma in 1964 took the Markowit’z work further, by formalizing the Capital Asset Pricing Model (CAPM), which showed the relationship between individual securities in relation to the market portfolio.

More specifically, the CAPM contributes to the following:

  1. The introduction of the CAPM separated the non-systematic risk- the risk incorporated in each individual security, and the systematic risk, or market risk- the risk that affects all equities.
  2. According to CAPM, all investors should hold the market portfolio, leveraged or deleveraged with positions in the risk-free assets.
  3. The formula that describes the CAPM introduced the beta coefficient which actually is the covariance of a stock’s return in relation to the whole stock market’s return.
  4. Since it had already been proved that a number of about 20 shares were sufficient to significantly reduce the risk that was possible to eliminate, i.e. the non-systematic risk, leaving only the systematic risk to affect the diversified portfolio.
  5. Finally, the CAPM offers to investors a ‘quantified’ relationship between risk and return and thus enables them to ‘require’ an expected rate of return for any given level of risk, also offering companies a useful ‘tool in the cost of equity analysis.

Despite the usefulness of the aforementioned concepts, there have been many necessary assumptions in order to conclude to the aforesaid results.

More specifically, we can refer to the following assumptions:

  1. The theory in question, as well the associated models, are based on the hypothesis of an efficient market, which, among others, prerequisite:
    • The existence of many small investors, that is to say, ‘price takers’.
    • All investors have the same level of economic knowledge. They also all have free access to all information, which is not the case.
    • There is no restriction on investments, no taxes, and no transaction cost, which certainly does not respond to the real business environment.
    • All investors make their decisions under the principle of the ‘rational’ behavior or, in other words, they are all behaving as risk-averse investors who prefer higher return and lower risk. However, this is contentious. To give an example, we can refer to those people who like to bet, which indisputably does not characterize a risk-averse investor.
  2. Investors have the same investment horizon, which is far from certain. For example, we can refer to investors who are ‘expecting’, say a 15% return from a share, while others would be satisfied having such a return after 2 or 3 years time.
  3. Investors’ target is to maximize the utility of wealth. This is not unrealistic as it is known from the economic theory – consumer behavior.
  4. All the investors use the same ‘tools’ in order firstly, to measure the volatility (risk), i.e. the standard deviation of return, and secondly, to measure the return. However, this is implausible because investors are likely to have different ‘opinion’ about the future rates of the ‘tools’ in question.
  5. The likelihood that all investors can lend and borrow any desired amount of funds at the risk-free interest rate. Certainly, this is not case. For example, a huge multinational company can, undoubtedly, much more easily, negotiate the terms of a loan, e.g. the interest rate, than a small domestic one.
  1. In practice, investors take into consideration the historically determined betas, which does not necessary reflect their future value. Therefore, it would be better to consider the required return, which is specified by the model, as a rough estimation of the actual one.

[1] The CAPM return is a good indicator regarding the company’s cost of equity capital (having first deducted the relevant taxes).


Risk Management – Part C

Manolis Anastopoulos @ University of Leicester - Risk Management

2.3 Risk (volatility) of a portfolio and the efficient frontier

The previous analysis concluded that the higher volatility increases the risk. We have also accepted in our analysis the regime of the risk–adverse investors, which mean the desire of minimizing the volatility in portfolios. Certainly, such a decision includes an effect in the opposite direction; if we limit the volatility (risk) we are at the same time limiting the prospective rate of return.

Consequently, in order to limit the volatility, it would be to our benefit to include in our portfolio securities with negative or low correlation, i.e. different risk’s securities in a way that prospective fluctuations of individual securities would cancel each other to a satisfactory extent. For instance, an increase in a specific raw material might be good for a producing company but not for a company that uses the specific raw material. Thus, it would be to our benefit to include both companies’ shares into our portfolio.

Therefore, it can be argued that through diversification, i.e. a combination of a satisfactory number of different assets or shares, we can attain a portfolio of a high average rate of return with less fluctuation. As stated in an recent article (JWH Journal 2005), ‘ As the number of stocks in the portfolio increases, there will be a reduction in the idiosyncratic risk or unique risk associated with the individual stocks, and the overall risk of the portfolio should more closely match the systematic risk of the market.’

Such a diversified portfolio that succeeds the highest average rate of return for a given level of risk, or, in other words, the portfolio that manages risk and return is called efficient portfolio.

Finally, we conclude that there are ‘efficient’ portfolios that ‘optimally’ balance risk and return. These optimal portfolios, according to the Modern Portfolio Theory (MPT), are lying across an upward-sloping curve called the efficient frontier of portfolios (fig. 2.3).


The efficient frontier -Manolis Anastopoulos


The above chart makes clear that for any given value of volatility (risk), investors can choose an ‘optimal’ portfolio that gives them the greatest possible rate of return.

Certainly, the portfolio, in order to give the maximum return for the amount of risk investors wish to bear, should lie on the efficient frontier, rather than lower in the interior of the curve. In addition, it is also worth mentioning, that as an investor moves higher up the curve, he undertakes proportionately more risk for a lower return, and, on the other hand, as he moves lower down the curve, he prefers a combination of low risk and low return.


2.3.1 Indifference curves and risk

Given that, as earlier stated, the majority of investors are risk–averse, it is anticipated that their indifference curves (x1, x2, z1, z2) will slope upwards from left to right (fig. 2.3). The curve’s slope indicates the fact that risk–averse investors are requesting proportionally more expected return than the risk level they are to undertake. On the contrary, the curve will slope downwards, when we consider risk–taking investors.

We conclude that - as Markowitz stated, investors ask for different portfolios of assets based on how risk-averse they are. For example, as illustrated in (fig. 2.3), an investor with an indifference curve of x1 (who is less risk-averse than the other with indifference curve of Z1) choose the point a that represents a portfolio with a less volatility (risk) and less expected return, as well.


2.3.2 Portfolios with equities and cash

When investing in risk-free assets, i.e. cash, we have a ‘pure’ return Rf, say 5%, with an σk (standard deviation) which equal to zero - point A in (fig.2.3.2). Certainly, we have the option, instead of investing in cash, to invest in ‘risky’ assets, i.e. securities or bonds, with a higher expected return and a higher σk, as well – point B in (fig. 2.3.2).

Certainly, investors should choose any point between A and B or right of B close to C, which means a combination of risk-free assets and a share mix.

Undoubtedly, as we move to the right, we increase the expected return of our portfolio and the standard deviation as well. As a result, an investor might think of borrowing an (X) amount at the risk-free rate of say 4% in order to buy a share mix of the same amount (X) that theoretically will offer him a higher expected return, say 12%. In such a case he would benefit from the difference of 8% (12% - 4%).

Finally, as illustrated in (fig.2.3.2), the straight line ABC, - which is called the Capital market line, can be regarded as the ‘new’ efficient frontier. As stated in his book (King 1999), ‘The capital market line shows the highest possible expected return for each possible degree of risk, once lending or borrowing at the risk-free rate is permitted.’

Portfolios with equities and cash - Manolis Anastopoulos


The point B, where the line that starts from the Rf touches the ‘original’ efficient line, represents the market portfolio, which is the ‘target’ of all investors, or, in other words, – as stated in his book (King 1999:84), ‘All investors want an identical portfolio of risky assets which, on its own, would take them to B.’

However, such a market portfolio - which is a risky portfolio – that succeeded through diversification to eliminate all the risk factor that was possible to eliminate, should (among others) contain all companies’ shares quoted on the stock exchange, which is undoubtedly unattainable. The aforementioned unattainable concept of including all shares can be overcome by selecting a maximum of 20 shares, this number of which has been proved to reduce significantly the risk that was possible to eliminate.

As stated in their book (Lumby & Jones 2002), ‘ Several studies have shown that constructing a randomly selected portfolio of shares consisting of only between fifteen and twenty different securities results in the elimination of around 90% of the maximum amount of risk which it would be possible to eliminate through diversification.’

This theory that actually expanded the Markowitz work, since it ‘offered’ the option to investors of including risk-free assets in their portfolios, was introduced by James Tobin in 1958. The theory called separation theorem.



Risk Management – Part B

Manolis Anastopoulos @ University of Leicester - Risk Management

2.1 Return and risk

2.1.1 Return

Undoubtedly, the value of a claim is determined from its return (cash flow) over the holding period. Certainly, the level of the risk that is associated with the expected return is negatively related to the value of the claim. For example, a cash flow prepared for a vessel of 25 years of age has incorporated a great risk or uncertainty (expected earnings are very sensitive to a possible downturn of freights because of the vessel’s age), which results in lowering the value in question.

Finally, what investors actually consider is the present value (PV), which according to the basic valuation model, states that the value of any asset is ‘the present value of all future returns (cash flows) it is expected to provide over the relevant time period.’

More specifically, we can calculate the PV of a dated fixed-interest security (PDFIS), using, as stated in his book (King 1999), the following equation:

C: Annual coupon payment

n: A maturity value

M: Years to maturity

r1, r2 and r n: The annual rates of return required on loans of one, two and N years to the security issuer.

In many cases, investors anticipate not only to earn income but to make capital gain, as well. In such a case they consider the holding – period return, i.e. when possible price rise is incorporated in the expected return, as the following formula:


                        RH = [D + (P1 – P0)] / P0                                                                                                  (2.1.b)


D: Dividend or coupon payment over the period

P1: The price at the end of the period

P0: The current price.

Although theoretically the above equation is of good use, in practice there are two disadvantages. First, we are not quite sure about the accuracy of the P1, and secondly, D can not always be regarded a ‘guaranteed’ and ‘safe’ income payment. As a result, what we are actually doing is to forecast the P1 (the price at the end of the period), and the RH, as well. Consequently, we undertake risk whenever we are trying to forecast the expected return.


2.1.2 Risk

As stated earlier, risk is the possibility of having a final (actual) return that differs from the return that was expected (our initial estimation). In other words, it is the uncertainty that is related to the variability of returns.

There are risk free investments, e.g. a government bond that guarantees its holder the X amount after Z days. On the contrary, due to the variability of their returns, there are very risky investments, e.g. a holder of a company’s stocks.


2.2 Risk Measurement

In order to decide under uncertain conditions, e.g. investing in the stock market, it is helpful to measure the risk that is embodied in our decision. In addition to its range[1] and to probability[2], we have to apply statistical analysis for quantitative measurements of the risk, or, in other words, of the likely variability of its returns. More specifically, we can utilize the standard deviation[3]an indicator (equation 2.2.a) that measures the dispersion around the expected value[4] (equation 2.2.b), and the coefficient of variationan indicator that measures the relative dispersion or relative risk, useful in comparing the risk with different (2.2.c) expected values.

kj : return for the  jth outcome

Prj : Probability of occurrence of the jth outcome

n: number of outcomes considered


Finally, it will be to the benefit of the prospective investor to choose the lowest risk asset, comparing the coefficient of variation rather than the standard deviation. This is due to the fact that the coefficient of variation also considers the standard deviation (σk) and the expected value (k), as well. For example, the asset A, with CV = 0.75, σk = 4.5% and k = 6% although has a lower σk has a higher CV compared to asset B, with CV = 0.5, σk = 5% and k = 10%, which, due to its lower CV (0.5 vs 0.75), should be preferred.




[1] The range is calculated by subtracting the pessimistic outcome from the optimistic outcome.

[2] The probability is the change or likelihood that a given outcome will happen.

[3] Rule of thump: A higher standard deviation means a higher risk and a higher possible return.

[4] The expected value is the most likely return on a given asset.


Risk Management – Part A

Manolis Anastopoulos @ University of Leicester - Risk Management


It is a historically known fact that people instead of keeping their cash with a zero return, decided to lend their surplus funds from the moment that the borrowers were willing to offer them a return.

On the other hand, as we know from economics, in order to acquire something we have to pay the price of having less of something else. As stated in their book, (Parking & King 1992), ‘More corn can be grown only by paying the price of having less cloth’.

In other words, it is a prerequisite for the surplus units to loose certainty, i.e. to accept risk, in order to gain return. This is the meaning of ‘risk’, the probability of ‘losing money’.

However, it can be generally argued that the majority of investors like to avoid risk whenever possible; they are risk–averse. As stated in their book, (Gitman & Madura 2001), ‘Given two choices with similar expected returns, they (the investors) prefer the less risky one.’  Consequently, investors are willing to accept increased risk only if they will be compensated with an increased rate of return. In other words, there is always a positive link between any claim’s uncertainty (risk) attached to its future performance and the required rate of return.


Portfolio Theory - Modern Portfolio Theory (MPT)

Investors, before the 1950s - based on the aforementioned ‘rules’ - used to focus on assessing the risk of individual securities in order to construct their portfolios[1]. For example, if an investor considers that the bank’s XX stocks were offering a satisfactory risk – reward characteristics, then he might construct his portfolio entirely of the stocks in question.

Later, in 1952, Harry Markowitz with his paper ‘Portfolio Selection’[2] introduced what is known as Modern Portfolio Theory.  Based on the concept of ‘diversification’ of the risk and ‘correlation’ of securities, he proposed that investors should select portfolios taking into consideration the portfolio’s overall risk – reward characteristics. More specifically, he argued that instead of tracing the risk’s level of individual securities, investors should measure their portfolios’ overall risk.


[1] Portfolios are usually constructed by different assets, such as: stocks, mutual funds, bonds and cash.

[2] Markowitz’s paper ‘Portfolio Selection’ published in 1952 by ‘The Journal of Finance.’

Market failure ~ Negative externalities

According to Vilfredo Pareto (1848-1923) the economy allocate resources efficiently (Pareto's efficiency) when resources are used in such way that it would not be possible to use them in a different way to make someone better without making someone else worse off.

However it is well known that market mechanism (equilibrium D+S) fail to take into account all the external costs and benefits in providing and /or consuming the goods. As a result markets failed to supply the socially optimal amount.

The competitive market forces (D+S) will not produce the quantities of goods where the price (P) reflects the marginal benefit (utility) of consumption. This in turn leads to over/under consumption of the goods, e.g. allocate inefficiency.

The existence of externalities - positive and negative, (ie a cost or benefit arising from any activity which does not accrue to the person or organisation carrying on the activity cost or benefit arising from any activity which does not accrue to the person or organisation carrying on the activity) is one of the types of market failure.

Here below we will shortly analyze the negative externalities.

In the diagram 1, we assume that the CO2 is the product in question. Both demand curve (Dprivate) and supply curve (Sprivate) ignoring the social cost are taking into consideration only the private benefits and costs resulting to the provision of the Qpriv quantity (overprovision, that cause the global warming problem) at the price of Ppriv (point a).

On the contrary, if the external costs to the society (additional costs to rectify the global warming – pollution, if it is possible) could be esteemed and added to the companies’ cost, a new supply curve (S private+externality) has to be drawn to the left of the initial one. This is due to the incensement of the cost, which shift the supply curve to the left. Thus, the equilibrium point will move from point a to point b, that in turn will reflect a new increased price (Psoc) and a reduced quantity (Qsoc) that will reduce or eliminate the problem of global warming.



Negative externalitiies


  1. Firms do not always include in their cost the “external cost” when focusing maximizing its profits.
  2. Output will be Qpriv instead of the social optimum Qsoc where MSC=MCB.
  3. This over-production will cause external cost to the society, which will not be reflected in firm’s cost or the price paid by consumers.
  4. The negative externality is shown as the (vertical distance) between Sprivate + externality and Sprivate.
  5. Price and output is set where private cost equals private benefits.
  6. BUT, third parties suffer from this → NEGATIVE externalities from consumption.

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