Cours marché financier - KSU



Dr. Ben Arab Mounira

CBA - DEPARTMENT OF FINANCE

CASE STUDIES

PORTFOLIO MANAGEMENT

.

RETURN & RISK

RETURN

The return on a stock or portfolio can be measured within certain or uncertain environment.

1. Return on certain environment

Return could be measured over one period or multi period.

1.1 One period 

Stock price equal to the net present value of all cash flows (dividends plus the ending stock price. Actuarial rate corresponds to the required stockholder yield,

P0 ( 1 + R ) = D1 + P1

Equally  :

R = D1/P0 + ( P1 – P0 )/ P0

With :

R : yield

D1 : Dividend

P0 : current price

P1 : sell price

In general the return is expressed as follow : Dividend plus capital gain or loss.

[pic]

Example

[pic]

[pic]

1.2 Multi-periods 

The return over a period is composed of a series of successive rates. the average return is measured through the arithmetic or geometric average or through the internal rate of return.

Arithmetic average rate :

Ra = ( R1 + R2 + (+ Rn ) / n

This measure is misleading overestimate the true value.

Internal rate of return :

It is a classical measure and encounter a potential mathematical problems. The major problem is when there is a conflict between NPV & IRR major. The conflict is the result of the implicit hypothesis toward reinvestment of intermediate cash flows. With NPV, the reinvestment is suppose to be the cost of capital however, the IRR, suppose cash flows to be reinvest at the same yield rate "R" .

Average geometric return :

( 1 + Rg )n = ( 1+ R1 ) ( 1 + R2 )… ( 1 + Rn )

2. Return in the uncertain environment:

2.1 Over one period

The return is measured by the expected return, that is, the mean of distribution :

E (R ) = ∑ Pi Ri

If we have historical data of the stock return, and if their distribution is stable, than :

E ( R ) = ( R1 + R2 + …+ Rn ) / n = ∑ Ri / n

2.2 Multi-period 

E(R) = [ ( 1 + E (R1) ) ( 1+ E (R2) ) … ( 1 + E (Rn) ) ]1/n - 1

2.3 Portfolio return

The return is equal to the weighted average of the individual stock..

Observations : Rp = ∑ wi Ri

Random variable : E ( Rp ) = ∑ wi E (Ri )

wi : weight of stock i

RISK

1. Risk of stock 

Definition : Financial risk is defined as the uncertainty related to the future value. it represents the dispersion of the return round the expected value.

Measure : it is measured by the variance.

σ2(R ) = ∑ pi [ Ri – E (R) ]2 = E [Ri – E ( R ) ]2

with:

[ Ri – E (R) ] = Δ from the expected value

pi : Probability of the variation

From historical data : The variance of the observed related to average would be the estimation of future variance .

σ 2 (R) = ∑ ( Ri – R )2 / ( n – 1)

But because it is more convenient to compare distances in the same dimension unit, the standard deviation, the square root of the variance, is most often used

The standard deviation of continuously compounded returns of an asset is often referred to as the volatility of the asset.

[pic]

38,3% of the observed return lie in the area [pic]

2/3 of the observed return lie in the area [pic]

95% of the observed return lie in the area [pic]

2. Risk of portfolio / Diversification

The inclusion of number of securities in portfolio reduces the risk of the later compare to individual risk..

Example, we consider 2 securities having respectively an expected return [pic] and [pic] and variance [pic] and [pic] (ort [pic] and [pic]). The square root is equal to standard deviation [pic] et [pic].

[pic] and [pic] are the weights (proportions) respectively invested in both securities.

The expected return of the portfolio is equal the weighted average of the individual expected return :

[pic]

[pic]

Risk of the portfolio

[pic]

[pic]

[pic]

[pic]

Or

[pic]

Standard deviation:

[pic]

Covariance.

[pic]

Note that for a simple of observations we divide by N-1 instead of N

Correlation

[pic]

[pic]

Very important : The correlation is the important piece of the diversification.

*, [pic] securities are perfectly negatively correlated . Example, if the first increases by 1% the second decreases by the same percentage. In this special case:

[pic]

[pic]

standard deviation [pic]

* [pic] securities are perfectly positively correlated . Example, if the first increases by 1% the second increases as well by the same percentage. In this very special case :

[pic]

[pic]

Standard deviation [pic]

We can notice here that the standard deviation of the portfolio is equal to the weighted average of the individual standard deviations

* [pic] securities are independent. In that case :

[pic]

[pic]

[pic]

a- [pic] and correlation impact

|[pic] |[pic] |[pic][pic] |Observations |

|1 |[pic] |25% |No diversification |

|[pic] |[pic] |22,4% * |Risk of portfolio less than sigma |

| | | |od securities |

|0 |[pic] |17,68% |same |

|[pic] |[pic] |11,18%** |same |

|-1 |[pic] |0% |Risk is fully hedged |

* [pic]

** [pic]

b- [pic] and [pic] impact : Risk of portfolio depend on the risk of securities.

Example [pic] is equal to zero, [pic] =0,5. [pic].

as a consequence :[pic]

Taking different values of [pic] et à [pic], we have :

|[pic] |[pic] |[pic] |

|25% |25% |17,6% |

|15% |15% |10,6% |

|5% |5% |3,5% |

|25% |0% |12,5% |

|15% |0% |7,5% |

|5% |0% |2,5% |

c- [pic] number of securities

|N |[pic] |[pic] |[pic] |[pic] |

|2 |[pic] |[pic] |[pic] |17,6% |

|3 |[pic] |[pic] |[pic] |14,4% |

|4 |[pic] |[pic] |[pic] |12,5% |

| |[pic] |[pic] |[pic] |[pic] |

|10 |[pic] |[pic] |[pic] |7,9% |

|[pic] |[pic] |[pic] |[pic] |[pic] |

|100 |[pic] |[pic] |[pic] |2,5% |

|[pic] |[pic] |[pic] |[pic] |0% |

d- [pic] and the weight effect

Example [pic] =0, [pic].

Than : [pic]

Taking different values of [pic] et à [pic], we get:

|[pic] |[pic] |[pic] |

|0 |1 |25% |

|0,1 |0,9 |22,63% |

|0,2 |0,8 |20,61% |

|0,3 |0,7 |19,04% |

|0,4 |0,6 |18,03% |

|0,5 |0,5 |17,67% |

The optimal portfolio is when we have [pic] .

Conclusion

• Additive law is stable for the Expected return.

• Risk can be minimized through :.

- correlation

- number of securities (size) ;

- optimal weights ;

The diversification effect as a result of the combination of two securities can be measured by the reduction percentage of the variability attributed to the imperfect correlation of securities. The reduction is calculated according to maximum variability σpc (correlation = 1).

EDD en % = (σpc - σp ) / σpc

Maximum diversification

σp2 = ∑ xi2 σi2 + ∑ ∑ xi xj σi;j

How to calculate the risk with the variance-covariance Matrix

This method is recommended for large portfolio (more than 3 and 4 securities).

[pic]

A' is the horizontal weight vector in the above example : (0,5 0,5).

A is the vertical weight vector [pic]

C is the variance-covariance Matrix [pic].

on the diagonal we have variances and on both sides we have covariances. Notice that the covariance of 1with 2 is the same as 2 with 1., The matrix is therefore symmetric.

[pic]

Example

[pic]

[pic]

[pic]

= 0,03125*0,5 + 0,03125*0,5 = 0,3125

[pic]

MARKET MODEL

THE UNDERLYING IDEA IS TO EXPLAIN THE VARIABILITY OF THE SECURITY PRICE. THIS LATER SEEMS TO BE RELATED TO THE FLUCTUATION OF THE MARKET AS HOLE AND TO SPECIFIC EVENTS.

Examples of common market factors: the variability of the interest rate, inflation, Tax, presidential elections , event such as the fall of the Berlin Wall, peace or war in the Gulf ......

Examples of specific factors that may affect the prices of securities: the distribution of dividend, the adoption of an investment project, the announcement of a major PER, etc.

[pic]

[pic]

The fact that all the points are not exactly on the line highlights the importance of the unsystematic risk. the equation of the line that statistically fit the best points is obtained by applying the least squares method.

[pic] : Return of security i on t ;

[pic] : Return of the market on t ;

[pic] : the slope, specific parameter to each security i, indicates the relationship between the fluctuations of the stock i and the general fluctuations of the market index. It expresses the sensitivity to fluctuations in the value of the index. Often, it is called coefficient of the volatility of i and has the following expression [pic]

Betas are usually positive except for few stocks such as gold mines.

 [pic]    : Is the intersection of the regression straight line with the y-axis, it represents the profitability of the action when i market profitability is zero (= 0). It can be positive, negative or zero.

[pic]: Is a residual random variable (specific parameter to action i) its standard deviation is a measure of specific risk. Indeed, as:

[pic]

then : [pic]

If the share price followed exactly the market, all the points should be perfectly aligned with the regression line. The dispersion around this line therefore is a measure of the variability.

Statisticians have another tool to determine to what extent the observations differ from the regression line: the coefficient of determination [pic], its value is between 0 and 1. This indicates how changes in equity are explained by changes in the market. A coefficient of determination of 1 would mean that all observations are on the regression line, which would mean that variations of the security return are fully and exclusively explained by changes in the market.

Thus the market model dislocate the total variability of the share in two:

- one component due to the effect of the market: the systematic risk (called non-diversifiable risk or market risk).

- The other is due to the specific characteristics of the share. This is called diversifiable or specific risk or none systematic (unsystematic) risk.

Market model and diversification

WE SAW IN PREVIOUS CHAPTER HOW THE VARIABILITY OF A PORTFOLIO CAN BE REDUCED THROUGH DIVERSIFICATION. THE RISK OF A SECURITY DEPENDS ON TWO COMPONENT, ITS MARKET RISK AND SPECIFIC RISK ([pic] ). IF WE BUILD A PORTFOLIO OF N SECURITIES EQUALLY WEIGHTED AND ASSUME THAT THEY ARE ALL INDEPENDENT, THEN THE, RISK OR THE PORTFOLIO VARIANCE WILL DEPEND ON TWO COMPONENTS:

[pic]

where [pic] and [pic]

It seems clear that when the number of securities included in the portfolio, N tends to infinity, the specific risk of the portfolio would be eliminated.

To show this result assume that [pic] the average value of the individual risk. As a result

.[pic]

Then [pic]

Portfolio risk is reduced to the systematic risk of each security with the market. It is indeed, thea risk that cannot be diversified (non-diversifiable risk).

Schematically it looks like this:

[pic]

Capital Asset Pricing Model

I. PORTFOLIO EFFICIENCY

CONSIDER TWO SECURITIES, THEIR EXPECTED RETURN AND RISK ARE RESPECTIVELY [pic] AND [pic] AND [pic] IS THEIR CORRELATION COEFFICIENT. THE EFFICIENT PORTFOLIO IS DEFINED BY THE OPTIMAL COMBINATION (A1, A2) OF SECURITY 1 AND 2 WHICH DOMINATES ALL OTHER COMBINATIONS. THAT IS TO SAY, THE COMBINATION THAT MAXIMIZES THE EXPECTED RETURN GIVEN A PARTICULAR LEVEL OF RISK. OR IN ANOTHER WAY THAT MINIMIZES THE RISK GIVEN A LEVEL OF EXPECTED RETURN.

[pic]

We assume :

• possible short sale,

• individuals prefer expected return and avoid risk

• no assympthotique behavior

• all individuals behave the same.

Then :

[pic]

[pic]

* If two securities are perfectly positively correlated, = 1,

[pic]

or : [pic]

The proportions to hold to eliminate completely the risk ([pic]) are

[pic] and [pic].

Exemple : if [pic] = 10% and [pic] = 25% : [pic] et [pic]

* If the two securities are perfectly negatively correlated, [pic] = -1

[pic]

[pic]

The proportions to hold to eliminate completely the risk ([pic]) are

[pic] and [pic].

Example if [pic] = 10% and [pic] = 25% : [pic]

Solution : a1 = 0.71 , a2 = 0.29 ; a1= -0.71 , a2 = 1.7

* If [pic] = 0

[pic]

[pic], [pic]

Example : [pic] et [pic] :

[pic]

[pic]

[pic] = 0,5

The expected return of the portfolio is the same as the individual security but the risk is divided by .

The choice of efficient portfolio depends on the correlation coefficient of the returns.

II. Efficient frontier

MARKOWITZ (1952) DEVELOPED A GENERAL METHOD OF SOLUTION OF THE PROBLEM OF PORTFOLIO STRUCTURE THAT INCORPORATES THE QUANTIFIED RISK TREATMENT AS IT HAS BEEN PRESENTED ABOVE. THIS METHOD OFFERS INVESTORS A SET OF "EFFICIENT" PORTFOLIOS THAT IS TO SAY, FOR A POSSIBLE OVERALL EXPECTED, HAVE THE LOWEST RISK AND VICE VERSA. THIS METHOD USES ONLY THE AVERAGE OF CONCEPTS FOR THE EXPECTED RETURN AND VARIANCE FOR THE UNCERTAINTY ASSOCIATED WITH THIS RETURN, HENCE THE NAME OF MEAN- VARIANCE CRITERION WAS ASSOCIATED WITH THE ANALYSIS OF MARKOWITZ.

Despite the restrictions, criterion (Mean, Var) has enjoyed considerable success and constitute the springboard of all modern financial theory.

[pic] As report on the graph (Figure 1a), each individual security is represented by its characterized risk and expected return. Combining these securities on portfolios, for any given return we can reduce the risk. Combining these in different proportions we obtained a set of portfolios, (Figure 1b). The optimal combination is typically called efficient frontier, represented by the curve AB (Figure 2). These optimal or efficient portfolios are such that for a given level of risk they maximize expected return or inversely for a level of expected return, they minimize risk.

[pic]

This method makes it possible to determine a set of efficient portfolios,( which for each level of risk, it maximizes expected return), but it does not indicate the best portfolio for every investor. The one who desires higher profitability has to incur higher risk. So we have to interject the attitude of investors to risk. Knowing his preference we can find for him the optimal portfolio at each point of tangency of utility curves and the efficient frontier.

III. Capital market line

IN THE PREVIOUS ANALYSIS, WE CONSIDERED ONLY RISKY INVESTMENT. HOWEVER YOU CAN INVEST AS WELL IN RISK-FREE ASSET FOR A YIELD R.

Let's choose a portfolio Y : A combination of risk-free asset and a stock portfolio with an expected return E([pic]) and risk σ ([pic]) with proportions p and (1 - p) respectively. Expected return and risk of the portfolio will be then :

[pic]

substituting equation (2) in (1), we get :

[pic]

risk premium value

free + * of

rate risk risk

As we can see the relationship between expected return and risk is linear. Thus any combination of a risky portfolio and a risk-free asset can be represented by a straight line in the space return, risk. As in the following figure, the line rotates to stop at the point M (portfolio market). The slope of the line is[E(Rm) - r] /.σm. [pic]

[pic]

capital market line cml

[pic]

All securities are included in the portfolio M (market portfolio) in a proportion equal to that each represents in the total market capitalization.

At equilibrium, investors hold the market portfolio in different proportions depending on their degree of risk aversion.

IV. CAPM

THE CAPITAL ASSET PRICING MODEL (CAPM) IS DUE TO THE WORK OF MARKOWITZ [1952/1959] ON PORTFOLIO THEORY AND OWES ITS DEVELOPMENT TO SHARPE [1964] LINTNER [1965] AND MOSSIN [ 1966]. DESPITE THAT DEVELOPMENT WORK CONTINUES, THIS MODEL REPRESENTS TO DATE THE MOST WIDELY USED APPROACH FOR THE ASSESSMENT OF RISKY ASSETS

Hypothesis

Several hypothesis have been undertaken : It is assumed that the decision makers have homogeneous expectations, they have a quadratic utility function, in other words or stochastic returns of any risky asset are completely described by the mean and variance of the distribution. It is also assumed that the risk-free borrowing and lending rates are available at the same rate Rf, and that the capital markets are perfect in the sense that there are no transaction costs or taxes.

1. No transaction costs,

2. Infinitely divisible and marketable assets

3. No tax

4. Individuals behave according to the model of Markowitz (they agree on normality returns "Mean, Variance, Covariance"

5. lending and borrowing rate are equal

6. Homogeneous Anticipation

7. Perfect competition (an individual cannot influence the market)

8. The amount of outstanding securities is fixed (no additional issuing during the period).

Therefore, it can be shown that in a world at a time when all investors have homogeneous expectations, each asset would be evaluated as:

E(Ri) = Rf + βi [E(Rm) - Rf] (1)

where [pic]

Rm is a random variable describing the rate of return of holding the market portfolio M, R is the random variable describing the rate of return from holding the security i, σm is the standard deviation of Rm, σmi is the covariance of the random variables Ri and Rm.

[pic] is the contribution to the risk of an additional security in an already diversified portfolio.

As we have already seen, the risk of a security consists of two parts:

* The specific risk

* Systematic risk

The financial market does not reward specific risk since individuals can get rid of (hence the name diversifiable risk). It rewards the systematic risk represented by the covariance with the market (also referred to as market risk).

[pic]

The beta of a security reflects the sensitivity to general movement of the market portfolio (economic movements, such as the change in interest rates in the banking sector, the change in the price of oil ...). Hence the beta reflects the systematic risk.

therefore, security with beta higher (below) than 1 would be appointed as more (less) volatile than the market.

CONCLUSION

* The risk / return relationship is linear;

* Only a fraction of the risk is rewarded: systematic risk. The specific risk can be eliminated through portfolio diversification;

* It is an ex ante model since it is based on expectations.

* It is a mono-periodical model.

V. SCOPE OF THE CAPM

FIRST AREA: WHEN A COMPANY ISSUES CAPITAL, THE YIELD REQUIRED BY SHAREHOLDINGS (MINIMUM RETURN) IS DETERMINED BY THE CAPM WHICH TRANSLATE THE EQUILIBRIUM CAPITAL MARKET RETURN. TO MAINTAIN ITS SHAREHOLDING, A FIRM MUST BE ABLE TO PROVIDE THIS MINIMUM. THEREFORE THE CAPM IS A VERY USEFUL TOOL FOR THE EVALUATION OF THE COST OF EQUITY.

Second area: investment choices (like the IRR ...) if a company have to choose between different alternative project having a specific internal rate of return. If this project is fully funded by equities which have a cost ke :

ke = E(Ri) = Rf + βi [E(Rm) - Rf]

Then project that should be selected, that have net present value, are those who have IRR> Ke.

Third area: portfolio management. To over-perform the market, portfolio manager who believes that market is not efficient, try to select under or overvalued securities.

If the anticipated return on one security according to financial analysts > than the expected return (according the SML), the security is undervalued. Portfolio manager will buy this security to take advantage of this undervaluation due to the error of the market.

In the opposite scenario, if the anticipated return on one security according to financial analysts < than the expected return (according the SML), the security is overvalued. Portfolio manager will sell short this security to take advantage of this overvaluation due to the error of the market. When the market adjusts (that is, it incorporates private information possessed by the analysts) and price decreases, the manager bought the security and the deliver it. He would thus gain the difference between the sales price (early period) ant the purchase price (ending period).

VI. empirical validation of capm

THE GREATEST DIFFICULTY WITH THE EMPIRICAL VALIDATION OF THE CAPM COMES FROM ITS FORMULATION IN TERMS OF EXPECTATIONS AND NOT IMPLEMENTED. THUS, AN EXPECTED RETURN IS NOT ALWAYS ACHIEVED.

From a statistical point of view, this introduces an error that should be zero on average. However, if it is very difficult to observe and measure market expectations, it is very easy to observe past rate of returns.

If the equation : [pic]

applies to expected returns, we can expect that the rate of return on each of the securities actually realized (ex post) to be:

[pic]

[pic] is the residual variable that reflects the specific or diversifiable risk of the security and that would be eliminated if it was integrated in a well-diversified portfolio.

So if the model is valid and if the rates of return are observed for securities and during different periods of time, ,[pic] should be equal on average to zero.

Therefore, the first step we need to test the CAPM into its stochastic form. This empirical version of the CAPM is expressed in terms of ex post return.

Note that there is a very significant difference between the theoretical model and the empirical model (ex ante/ ex post). The former can have only a positive slope, while the second can have a negative slope.

Note also that the stochastic version of the CAPM is very close to the regression line or market model postulated empirically (first chapter):

[pic]

Therefore CAPM implies that the expected value of [pic] is equal to [pic].

It is also convenient to talk about excess rate of return (above risk-free interest rates). The market relationship becomes:

[pic]

in this case [pic] should be equal zero. In other words, no securities or portfolio should have systematically a rate of return (in excess of the interest rate) higher or lower beta times the market rate of return (in excess of the interest rate).

-----------------------

VAN

VAN 2

VAN 1

R 2

R1

R

Diversifiable risk

Systemac risk

return of the security

market return

least square line

Individual securities

Combined securities

Efficient frontier

The line rotates to stop at the point M

Slope of the line

This is the risk premium

The composition of the risky portfolio is independent of individual preferences

The line rotates to stop at the point M

Individuals use risk free rate to maximize their risk averse preferences

Markowitz efficient frontier

[pic]

Aggressive stocks: amplification of moving upward or downward

Perfect positive correlation Perfect negative correlation Null correlation

Positive correlation Negative correlation

Systematic risk Non systematic risk

Non diversifiable risk Diversifiable risk

Market risk Pur risk

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