Typically, investors in the security and derivative markets must choose between adjusting prospective gains for increasing risks. The recent worldwide credit and financial crises demonstrate the need of risk measurement in portfolio management and modern finance. Risk measurement is essential in the derivatives sector since insufficient risk analysis can lead to derivative mispricing, which can lead to underestimate of typical risks such as credit risks and market risks.

This study investigates the importance of risk measurement in portfolio management and modern finance. Of importance in this research is the Markowitz seminal work of 1952 (Markowitz 1952, p.77) and other contributions to the risk measurement context such as the Efficient Market Hypothesis, APT and CAPM.

Markowitz (1952) Portfolio Theory (MPT)

According to Markowitz (Markowitz 1952, p.80), The Markowitz Theory came to light in 1952 when an article titled Portfolio Selection was published in the Journal of Finance in 1952. The portfolio selection focused on the interrelationship between the expected rate of return and the risk of individual securities as measured by correlation. The Markowitz Theory highlights on how risk-averse investors can construct portfolios to optimize or maximize expected return based on a given level of market risk (Markowitz 1952, p.76).

Based on the theory, it is possible to construct an efficient frontier of the optimal portfolios that is capable of offering the maximum possible expected return for a given level of risk. The MPT is viewed as a single period approach to mean that at the beginning of each period, an investor must make a decision in which particular security to invest and hold securities until the end of the period hence the theory is equivalent to selecting an optimal portfolio from a set of possible portfolios.

Using the MPT, the investor is capable of constructing a portfolio of multiple assets that are able to maximize returns for a given level of risk. The MPT makes the assumption that the investors are risk averse which implies that they will always select the less risky investments to the risky investments (Markowitz 1952, pp.76-79). The investors can always reduce their exposure to the individual asset risks through diversification. In the MPT, investors use indifference curves to select the most desirable portfolios.

The indifference curves represent the investors preferred risks hence is drawn putting expected returns on the vertical axis and the risks on the horizontal axis (Markowitz 1952, p.34). The indifference curves have two main features. First, all portfolios that lie on a given indifference curve are equally desirable to an investor and two, an investor has an infinite number of indifference curves. The MPT, therefore, operates with the assumptions that the investors prefer high levels of returns to low levels of returns (assumption of non-satiation) and investors are risk averse( the assumption of risk aversion).

Efficient Set Theorem

MPT tries to solve the dilemma of an investor sorting out the infinite number of portfolios. The theorem states that an investor chooses an optimal portfolio from the set that maximizes the expected returns for varying level of risk and offers minim risk for varying levels of expected returns (Tobin 1958, pp.69-74). A portfolio that accurately reflects the capital market line is considered a Markowitz efficient portfolio.

The diagram below can be used to explain the efficient set theorem.

Optimal Portfolio

Efficient Frontier

Rf

Risk Efficient set

The feasible set is the opportunity set from which the efficient set of portfolios can be identified. The Capital Market line (CML) is the tangent line that the efficient frontier makes with the risk free investment and hence this is the point with the optimal portfolio. The capital market line depicts the level of additional returns above the risk free rate for the changes in each level of risks. The CPL model is however faced with challenges of inaccuracies in the depiction of portfolio information. The investor should therefore choose a portfolio inside the feasibility or efficient set.

The MPT risk is measured by;

µp = X N i=1 wiµi ,

then the portfolio’s variance σ 2 p is

σ 2 p = X N i=1 X N j=1 σijwiwj ,

where • N is the number of assets in a portfolio;

• i,j are the asset indices and i, j ∈ {1, ..., N} ;

• wi is the asset weight, subject to the constraints: 0 ≤ wi ≤ 1, X N i=1 wi = 1;

• σij is the covariance of asset i with asset j;

• µi is the expected return for asset i.

Contributions of Sharpe, Treynor, Lintner and Mossin and how their Individual Works Improve Markowitz’s Initial Theorem

Sharpe’s measure.

This measure of performance evaluation compares the risk premium of a portfolio to the portfolios standard deviation of returns (Sharpe 1964, p.430).

Sharpe’s measure = total portfolio return – risk free rate RP-RF/δP

Standard deviation of portfolio

Sharpe’s measure applies CML and the high the measure the better the performance.

Treynor’s measure

Treynor’s measure makes use of portfolio beta and focuses only on the non- diversifiable risks (Treynor 1961, p.123). It is the risk premium per unit of non-diversifiable risk.

Treynor’s measure = total portfolio return- risk free rate

Portfolio beta

It applies SML and the higher the Treynor’s measure the better the performance.

Lintner’s and Mossin’s measures

The works by Lintner and Mossin (Lintner 1965, pp.15-19; Mossin 1966, pp.771-777) focused on polishing the Capital asset pricing model. the capital asset model is used in the determination of the theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. Lintner and Mossin designed the model in such a way that it was capable of taking into account the asset's sensitivity to non-diversifiable risk as well as the expected return of the market and the expected return of a theoretical risk-free asset.

Empirical testing of CAPM

According to Fama and MacbethJ (Fama and Macbeth 1973, pp.607-630), the following table can be used to determine the empirical tests of CAPM.

Investment type

Nominal return

Real return

Risk premium (over T-bills)

Standard deviation

Short-term T-bills

3.7%

0.5%

0.0%

3.4%

Intermediate T-bills

5.1

1.9

1.4

5.6

Long-term T-bills

5.4

2.2

1.7

8.6

Corporate bonds

5.4

2.2

1.7

8.5

Large-cap stocks

10.4

7.1

6.6

20.8

Small-cap stocks

12.1

8.7

8.2

35.3

The CAPM was developed by William F. Sharpe in 1964 (Sharpe 1964, p.433) as a development of the MPT developed earlier in 1952. CAPM specifies the relationship between the risk and the required rate of return on assets when the assets are held in a diversified manner. According to CAPM, the Expected return = rate of investment in a free security + a risk premium.

After Sharpe’s initial works, the first major study of the security risks was attributed by King (King 1966, p.157). King’s study triggered substantial research into the tendency of stocks to cavalry, for reasons other than their sensitivity to the overall market. King, in his model, tried to show how the share prices tend to fluctuate in line with the market.

CAMP E(r) = Rf + (Rm–Rf)Bp.

The difference between the return actually earned on a portfolio and the returns expected from a portfolio. Lintner’s measure = Actual portfolio return – Expected return on CAMP. Rp-E(Rp) (Lintner 1965, p. 29). CAPM takes into account the systematic risk and the as systematic risk. CAPM, therefore, operates under several assumptions such as: investors are risk-averse, the capital market is efficient and perfect, taxes and transaction costs are irrelevant, there is a risk-free rate at which an investor can transact and all investors are rational.

Black (Black 1972, p.446) shows how the CAPM model needs to be adapted in the event of the unavailability of the riskless borrowing. Treynor and Black (Treynor 1961, p.45; Black 1972, p. 450) show how best to construct such portfolios by linking the CAPM with Sharpe’s (Sharpe 1964, p.431) index model. The earliest rigorous test of the CAPM was performed by Black (Black 1972, p.452) and they found out that the cross sectional plots of the mean excess returns on the portfolios against the estimated betas indicate that the relation between mean excess return and beta was linear.

The study by Fama and MacBeth (Fama nd Macbeth 1973, p.625) focused on the direct testing of the validity of the zero-beta CAPM. The CAPM equation represents a straight line having an intercept of risk free rate and the slope of beta using the Security Market Line (SML). Unlike the MPT, the CAPM uses beta as the main measure of portfolio risk.

The following table gives an interpretation of the co-efficient beta (Gibbons 1982, p.15).

Beta

Direction

Interpretation

2.0

Same

The security risk of securities is twice higher than market risk

1.0

Same

The security risk of securities equals the market risk

0.50

Same

The security risk of securities is twice lower than market risk

0

Zero

The security risk is not influenced by the market risk

-0.50

Opposite

The security risk of securities is twice lower than market risk

-1.0

Opposite

The security risk of securities equals the market risk

-2.0

Opposite

The security risk of securities is twice higher than market risk

Criticisms of CAPM

The criticisms of CAPM mainly originate from the assumptions. These criticisms are: One, the assumption of the zero transaction cost is impossible since many investments involve transaction costs. Two, investments are subject to capital gain taxes hence the assumption of zero taxes is false. Three, investors tread in securities and derivatives with different expectations which go against the assumption of homogenous investor expectation and four, beta cannot be used as the complete measure of risk since there are other types of risks such as the inflation risk and the liquidly rest. Five, investors do not borrow at risk-free rates since in reality, risky investments come along with even higher costs and finally, there are no risk –free assets in both securities and the derivative markets.

The primary turning point in the empirical testing of the CAPM was the Roll (Roll 1977, p.137) critique. Roll (Roll 1977, p. 141) demonstrated that the market, as defined in the CAPM, is not a single equity market but an index of all the wealth. Merton (Merton 1973, p.868), on the other hand, developed an Interpol CAPM, based on the assumptions that time flows continuously and this was contradictory to the previous assumptions set on the CAPM. According to Merton (Merton 1973, p.873), static CAPM hardly holds in a dynamic setting. However, as Roll’s critique started sinking, Ross (Ross 1976, p.345) developed the arbitrage pricing theory (APT) as an alternative model that could potentially overcome the CAPMOs problems while still retaining the underlying message of the latter. Based on the APT, only a small number of systemic influences affect the average long-term returns of securities. Roll and Ross (Roll and Ross 1980, p.1097), however, reconciled Merton’s ICAPM with the classical CAPM by highlighting the dichotomy between wealth and consumption. He showed that the agent’s preference has to be defined over his or her consumption. Breeden's model which became known as the Consumption CAPM (CCAPM) allows assets to be priced with a single beta as in the traditional CAPM.

Conclusion

To sum up, the paper has elaborately discussed the concepts of CAPM, MPT, and Efficient Market Hypothesis among other concepts that explain the importance of the risk measurement to an investor who trades in securities and the derivative markets. Besides, the assumptions of the portfolio theories and the criticisms have been explained which makes the report important for potential investors.

References

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