test

Characteristics of an Investment Portfolio The important parameters characterizing an investment portfolio are: * Expected return * Standard deviation Every basket of shares has two important parameters associated with it.

Deviation 7 Basket of shares (symbol: B) 7 Investment portfolio (symbol: P) 7 Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior

Portfolio, Expected Return, Standard Deviation 7 Basket of shares (symbol: B) 7 Investment portfolio (symbol: P) 7 Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a

Meaning of the Standard Deviation Representing Risk, Together with an Example In the framework of the example, we will consider three scenarios pertaining to a share whose expected return is 2%.

coefficient 28 Distinguishing between baskets with different correlation coefficients 30 Conclusion about reducing risk 34 Formula for finding the internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with

8 GRAPH Table 7 refers to the three baskets displayed on Curve C. In each basket, the internal composition of the shares and their parameters is listed: Table 7 Baskets Internal Composition of the Shares Expectation of the Basket - EB Standard Deviation of the Basket - ? B S1 S2 S1 100% 0 0.05 0.15 S2 0 100% 0.1 0.3 C 2/3 1/3 0.066 0 Calculation of the internal composition of Basket C is based on the formula for calculating the variance of a basket with two shares, which will be displayed

In other words, the fluctuation around the average of 2% expectation is relatively low.

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

Correlation coefficient 28 Distinguishing between baskets with different correlation coefficients 30 Conclusion about reducing risk 34 Formula for finding the internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45

Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28 Correlation coefficient 28 Distinguishing between baskets with different correlation coefficients 30 Conclusion

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28 Correlation coefficient 28 Distinguishing between baskets with different correlation coefficients 30 Conclusion

Shares, Investment Portfolio, Expected Return, Standard Deviation 7 Basket of shares (symbol: B) 7 Investment portfolio (symbol: P) 7 Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting

Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28 Correlation coefficient 28 Distinguishing between baskets with different correlation coefficients 30 Conclusion about reducing

Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend

The variance of the basket, Var(B), describes the overall fluctuation of Basket B. Exercise Given two shares, S5 and S6, from the example for calculating the covariance of two shares, Table 5 displays the data for the shares calculated in the example.

asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

The selected investment basket of every investor is located at the point at which the efficiency frontier is tangential to one of its indifference curves.

Formula for finding the internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios

Point a – a bank deposit with a 3% return Point b – a loan that guarantees the bank a 5% return Figure 3 Superior Baskets and Inferior Baskets Every basket in a plane, such as Basket A in Figure 4, is superior to all the baskets located in the rectangle of which Basket A is the upper left vertex, and of which the bottom side is the X axis.

risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23

The CAPM model uses the beta index (?) to determine the expected return of individual shares, because according to the model, only fluctuations in the return of individual share i, which are correlated with the fluctuations in the market portfolio, will add to the share’s expected return.

Investment Portfolio, Expected Return, Standard Deviation 7 Basket of shares (symbol: B) 7 Investment portfolio (symbol: P) 7 Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its

Standard Deviation 7 Basket of shares (symbol: B) 7 Investment portfolio (symbol: P) 7 Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior

Point a – a bank deposit with a 3% return Point b – a loan that guarantees the bank a 5% return Figure 3 Superior Baskets and Inferior Baskets Every basket in a plane, such as Basket A in Figure 4, is superior to all the baskets located in the rectangle of which Basket A is the upper left vertex, and of which the bottom side is the X axis.

In order to illustrate the meaning of the expected return and the standard deviation of a share, we use an example based on a sample of 360 monthly measurements of the share’s return (over a 30-year period from a calculation of 12 monthly measurements per year).

When A is superior to B, we say that it is more efficient than B. Figure 4 Division of the Plane into Quadrants (Figure 5) If we divide the plane into four quadrants denoted A, B, C, and D, we can state with certainty that every basket in Quadrant A is superior to every basket in Quadrant D. Financiers say: Every basket in Quadrant A is more efficient than every basket in Quadrant D. The use of the term more efficient is accepted in the profession.

A basket of shares will be denoted by the symbol B (portfolio).

Solution: A. The formula for the correlation coefficient (?) is: (1) ?1,2 = ?1,2/( ?1 * ?2) Multiplying both sides of the equation by ?1 * ?2, we obtain: ?1,2 * ?1 * ?2 = [?1,2/(

decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive

Point a – a bank deposit with a 3% return Point b – a loan that guarantees the bank a 5% return Figure 3 Superior Baskets and Inferior Baskets Every basket in a plane, such as Basket A in Figure 4, is superior to all the baskets located in the rectangle of which Basket A is the upper left vertex, and of which the bottom side is the X axis.

We assume that all the efficient portfolios in the field between rf and m are financed through equity.

percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

It is customary in every country, however, to equate it to a leading basket of shares.

Displaying and Characterizing Baskets The Expectation-Standard Deviation Plane A system of Cartesian coordinate axes in which the X axis represents standard deviation and the y axis represents expectation is called an expectation-standard deviation plane, or an expectation-risk plane.

symbol: B) 7 Investment portfolio (symbol: P) 7 Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier

We assume that the sample results have a normal probability distribution with an average of 2% (per month) and a standard error of 5%.

An Improved Investment Portfolio (abbreviated as an improved portfolio) (Denoted by P) – Definition We define an improved investment portfolio as an investment portfolio with two components: A basket of shares is located on the efficiency frontier (an efficient basket).

between percentages and percentage points 53 Practical use of the CML line 53 Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market

Average – a single number that reflects a centralized value of a group of numbers of similar character measured in the same measurement units.

Displaying and Characterizing Baskets The Expectation-Standard Deviation Plane A system of Cartesian coordinate axes in which the X axis represents standard deviation and the y axis represents expectation is called an expectation-standard deviation plane, or an expectation-risk plane.

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28 Correlation coefficient 28 Distinguishing between baskets with different correlation coefficients 30

Reminder: in order to go from a decimal fraction to percentages, multiply by 100.

efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28 Correlation coefficient 28 Distinguishing between baskets with different correlation

From the lenders’ perspective (lenders such as banks and financial institutions): state-guaranteed loans.

Symbols: Table 1 Expected Return Standard Deviation Basket of shares E(B) ?(B) Investment portfolio E(P) ?(P) Statistics of Finance The Expected Return and the Standard Deviation of an Individual Share – Example and Illustration A reminder: Expectation – the average obtained by sampling the entire population.

The weighting is according to their weight in the basket, as follows: E(B) = W1 * E(S1) + W2 * E(S2 ) If we substitute the figures from Table 2, we obtain for the expectation of Basket E: E(B) = (0.5 * 0.05 + 0.5 * 0.15) = 0.1 = 10% In simple language, the expectation of the basket is 10%.

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49 Distinguishing between Ordinary Investors and Risk-Seeking Investors 50 Slope

plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28 Correlation coefficient 28 Distinguishing

In order to illustrate the meaning of the expected return and the standard deviation of a share, we use an example based on a sample of 360 monthly measurements of the share’s return (over a 30-year period from a calculation of 12 monthly measurements per year).

It measures the dispersal of the samples around the average, where the deviation is the distance of the individual measurement from the average.

composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49

Example of an investment portfolio: a portfolio worth $10,000: $5,000 invested in a bank deposit and $5,000 invested in General Motors shares.

Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares

Indifference curves do not intersect.

List of Symbols E(Si) - the expectation of share Si Wi - the weight of share Si in the basket (W is short for weight) B - the basket of shares E(B) - the expectation of the basket ?(B) - the standard deviation of the basket Calculation of the Standard Deviation of a Basket of Shares Preliminary Background In order to calculate the standard deviation of a basket of shares, we must first become familiar with two statistical concepts: 1.

representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the

example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of

Calculating the Expectation and Standard Deviation of a Basket of Shares The explanation is accompanied by an example of a basket containing two shares, denoted S1 and S2.

Only when the correlation between the shares is complete is it impossible to disperse the risk, i.e. to create an investment basket with less risk.

Symbols: Table 1 Expected Return Standard Deviation Basket of shares E(B) ?(B) Investment portfolio E(P) ?(P) Statistics of Finance The Expected Return and the Standard Deviation of an Individual Share – Example and Illustration A reminder: Expectation – the average obtained by sampling the entire population.

Scenario A: ? = 5% Scenario B: ? = 10% Scenario C: ? = 20% In each scenario, the probability distribution is normal (see Figure 1) Figure 1 – The Distribution of the Share under the Three Scenarios The center of the curve is exactly 2% - the expectation - because this number is the expected return, or the average return, and the normal probability distribution is symmetric.

internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49

The parameters of the shares and their relative weight in the basket are listed in Table 2.

As we go upward from point S1 along the dotted segment, the weight of share S2 in each basket increases at the expense of Share S1.

Example: If ?1,2, the covariance between S1 and S2, is 0.3, and ?3,4, the covariance between S3 and S4, is 0.4, it cannot be concluded that the connection between S1 and S2 is stronger than the connection between S3 and S4, just as it cannot be stated that a profit of $10 on Share S1 is preferable to a profit of $8 on Share S2.

Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket

From the lenders’ perspective (lenders such as banks and financial institutions): state-guaranteed loans.

The idea is that fluctuations in the return on share i that are specific to the individual share, and not connected to fluctuations in the market as a whole, will not add to the return of a shareholder in i.

A risk-averse investor will not like such fluctuation.

Expected Return, Standard Deviation 7 Basket of shares (symbol: B) 7 Investment portfolio (symbol: P) 7 Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14

Recession 2.

The baskets on the efficiency frontier are theoretical baskets; no one knows what they include.

Question: Which of the two shares has a greater chance of generating a negative return of 20% (meaning that the share price drops)?

6 (a reminder) Shares E ? Share S1 0.05 0.15 Share S2 0.10 0.30 Assume Table 8 from the same example: Table 8 (a reminder) Baskets Internal Composition of the Shares Expectation of the Basket - E Standard Deviation of the Basket - ? S1 S2 S1 100% 0 0.05 0.15 S2 0 100% 0.1 0.3 C1 80% 20% 0.06 0.134 Let C1 be a basket of two shares with an internal composition such that the variance and the risk are a minimum (minimum variance basket) A. How did we derive the internal composition of the

Formula 1 will appear as: ?2p = W12 * ?21 + Wrf 2* ?2rf + 2 * W1 * W2 * ?1,rf The use of rf instead of a bond simplifies Formula 1.

inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28 Correlation coefficient 28 Distinguishing between baskets with

Bonds are popular because they are negotiable, and usually make it possible to borrow money at a lower price than the regular interest on bank loans.

Shlomo Simanovski Finance for Advanced Students Meitav Self Learning Book about Financing for Beginners Strolling to the College Level Meitav Self Learning Publishing Ltd.

We need the term “improved portfolio” solely to simplify the explanation of the methodology.

Table 3 Scenario 2 Probability of the Scenario – 80% The Return Scenario 1 Probability of the Scenario – 20% The Return ? (calculated) E (calculated) The Shares ? ? ? ? ? 0.14 0.08 0.024 0.128 0.18 0.10 0.032 0.164 Explanation of the table: Column 1 – The names of the shares.

Figure 22 GRAPH Examples of the Three Scenarios An Example of Scenario 1: Every year, the Chalice Company distributes a fixed $100 dividend.

41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49 Distinguishing between Ordinary Investors and Risk-Seeking Investors 50 Slope of the CML line, called the risk premium

fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive

The area under each curve reflects the chance of obtaining the return under the area.

Table of Contents Introduction – Basket of Shares, Investment Portfolio, Expected Return, Standard Deviation 7 Basket of shares (symbol: B) 7 Investment portfolio (symbol: P) 7 Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing

Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28 Correlation coefficient 28 Distinguishing between baskets with different correlation coefficients 30 Conclusion about reducing risk 34 Formula for finding the internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment

The upper frontier curve connects all the baskets with the highest expectation for each standard deviation.

individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

Point a – a bank deposit with a 3% return Point b – a loan that guarantees the bank a 5% return Figure 3 Superior Baskets and Inferior Baskets Every basket in a plane, such as Basket A in Figure 4, is superior to all the baskets located in the rectangle of which Basket A is the upper left vertex, and of which the bottom side is the X axis.

in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend Payments 63 Appendix 68 Calculating the Expectation of a Share – The Accepted Practice in Exercises 68 Glossary

50 Slope of the CML line, called the risk premium 53 Distinguishing between percentages and percentage points 53 Practical use of the CML line 53 Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets

We assume that the sample results have a normal probability distribution with an average of 2% (per month) and a standard error of 5%.

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

Symbols: Table 1 Expected Return Standard Deviation Basket of shares E(B) ?(B) Investment portfolio E(P) ?(P) Statistics of Finance The Expected Return and the Standard Deviation of an Individual Share – Example and Illustration A reminder: Expectation – the average obtained by sampling the entire population.

13 Scenarios Probability of the Scenario Return under the Scenario Scenario’s Contribution to the Expectation 1 2 3 4 Recession 20% (= 0.2) -0.02 [0.2 * (-0.02)] = -0.004 = -0.4% Stability 50% (= 0.5) 0.06 [0.5 * (0.06)] = 0.03 = 3% Prosperity 30% (= 0.3) 0.10 [0.3 * (0.1)] = 0.03 = 3% Total 100% 0.056 = 5.6% - The expectation of the share Glossary of Concepts Bond – a promissory note that confers ownership of the sum of money listed on it, which the issuer of the note will pay in the

Practical use of the CML line 53 Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value

A decrease in rf causes an increase in the slope of the CML line and a decrease in the risk and expectation of the market basket (basket m1).

Figure 9 GRAPH In Basket C1, ?B is minimal, but it is not zero.

Symbols: Table 1 Expected Return Standard Deviation Basket of shares E(B) ?(B) Investment portfolio E(P) ?(P) Statistics of Finance The Expected Return and the Standard Deviation of an Individual Share – Example and Illustration A reminder: Expectation – the average obtained by sampling the entire population.

Column 5 applies to Share S5 with the same format as Column 4.

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49 Distinguishing between Ordinary Investors and Risk-Seeking Investors 50 Slope of the CML line, called the risk premium 53 Distinguishing between percentages and percentage points 53 Practical use of the CML line 53 Simple investors and risk-seeking investors – further explanation

The expected return, called the expectation for short, is denoted E. The standard deviation of the return, which constitutes a risk indicator, and is also called risk, denoted ?.

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

Examples: basket of shares, basket of bonds, basket of options, basket of commodities, and so forth.We will deal mainly with baskets relating to shares and refer to them as “baskets” for short.

On the other hand, assume that investor B has a share portfolio composed of a share of a company that grows oranges and a share of a company that makes heaters.

Example The Tel Aviv 100 basket of shares includes the 100 shares on the Tel Aviv Stock Exchange with the highest market value.

The tangent point is the market basket.

Figure 22 GRAPH Examples of the Three Scenarios An Example of Scenario 1: Every year, the Chalice Company distributes a fixed $100 dividend.

an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend Payments 63 Appendix 68 Calculating the Expectation of a Share – The Accepted Practice in Exercises 68 Glossary of Concepts

Note: The required return, Ke, can be found, among other ways, with the help of the SML formula in the preceding section (i.e. the required return for the risk).

If the correlation coefficient of Shares S1 and S2 is 0.8 and the correlation coefficient of Shares S3 and S4 is 0.4, then the connection (i.e. to what degree one variable 1 enables us to predict variable 2) between Shares S1 and S2 is stronger than the connection between Shares S3 and S4.

which reflects a still larger standard deviation, is lower in the center than Curve B, and the area underneath it is spread out more to the sides than in Curve B. This means that the chances of obtaining a return distant from the 2% average are even greater than for Curve B. The normal probability distribution is symmetric, meaning that both the chances of obtaining a return higher than the average and the chances of obtaining a return lower than the average increase as the standard

Meaning of the Standard Deviation Representing Risk, Together with an Example In the framework of the example, we will consider three scenarios pertaining to a share whose expected return is 2%.

selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49 Distinguishing between Ordinary Investors and Risk-Seeking Investors 50 Slope of the CML line

We assume that the sample results have a normal probability distribution with an average of 2% (per month) and a standard error of 5%.

A Risk-Free Asset and its Placement in a Plane A risk-free asset is an investment with some expectation whose standard deviation is 0 (zero), meaning that we are certain to obtain the expectation.

In order to go from percentages to a decimal fraction, divide by 100.

Table 3 Scenario 2 Probability of the Scenario – 80% The Return Scenario 1 Probability of the Scenario – 20% The Return ? (calculated) E (calculated) The Shares ? ? ? ? ? 0.14 0.08 0.024 0.128 0.18 0.10 0.032 0.164 Explanation of the table: Column 1 – The names of the shares.

Point A – represents an efficient portfolio that contains rf and m at some internal ratio between them.

the CML line 53 Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based

Prosperity We will continue the explanation through examples.

Since a large sample is involved, we assume that the sample distribution represents the probability distribution of the population (the population is all the monthly observations measuring the share’s return from the time it was first issued up to infinity).

Example of an investment portfolio: a portfolio worth $10,000: $5,000 invested in a bank deposit and $5,000 invested in General Motors shares.

The weighting is according to their weight in the basket, as follows: E(B) = W1 * E(S1) + W2 * E(S2 ) If we substitute the figures from Table 2, we obtain for the expectation of Basket E: E(B) = (0.5 * 0.05 + 0.5 * 0.15) = 0.1 = 10% In simple language, the expectation of the basket is 10%.

Average – a single number that reflects a centralized value of a group of numbers of similar character measured in the same measurement units.

Shlomo Simanovski Finance for Advanced Students Meitav Self Learning Book about Financing for Beginners Strolling to the College Level Meitav Self Learning Publishing Ltd.

(CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49 Distinguishing between Ordinary Investors and Risk-Seeking Investors 50 Slope of the CML line, called the risk premium 53 Distinguishing between percentages and percentage points 53 Practical use of the CML line 53 Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms

Distinguishing between baskets with different correlation coefficients 30 Conclusion about reducing risk 34 Formula for finding the internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free

Covariance = 0 When there is no connection between the directions of the responses of each of the shares to any scenario, we say that the covariance is 0.

The covariance tests how two shares respond to the same scenario.

Equation 1 (calculation of the expectation): EP = Wm * Em + (1 – Wm)rf Equation 2 (calculation of the standard deviation): ?p = Wm * ?m Isolating Wm in the second formula, we get: Wm = ?p/?m

Example 1 Share S1 belongs to a construction company named AA whose expectation is 10%.

The forces here are opposite, and superiority therefore depends on the specific investor – whether he puts greater emphasis on a high return or on lower risk.

Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22

We have chosen to use the term “basket of shares,” not “share portfolio,” in order to distinguish it from the term “investment portfolio,” which we will immediately define.

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In order to decide which profit is better, we must obtain additional information about the prices of the shares.

A capital market line (CML) A CML is a curve starting at the rf point and tangential to the efficiency frontier.

Since a large sample is involved, we assume that the sample distribution represents the probability distribution of the population (the population is all the monthly observations measuring the share’s return from the time it was first issued up to infinity).

Figure 2 - Expectation-Risk Plane Displaying a Basket of Shares in a Plane Every point in Figure 2 represents a basket, whose location in the plane is determined by its two parameters: Point a represents a basket whose parameters are E = 5% and ? = 3%.

According to the growth model, the theoretical price of the share equals the present value of the dividend receipts that it generates from now on, while the price of capital for the firm, denoted Ke, is used for discounting.

Shape of the Frontier Curve It is customary to sketch the frontier curve as a concave curve.

Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22

Remember: The “efficient portfolios” contain the market basket (a well dispersed basket that contains all the assets in the market that involve some risk), together with a certain proportion of a risk-free asset.

Shlomo Simanovski Finance for Advanced Students Meitav Self Learning Book about Financing for Beginners Strolling to the College Level Meitav Self Learning Publishing Ltd.

1 Return under Scenario 1 Probability of Scenario 2 Return under Scenario 2 E(S6) = [0.2 * 0.10] + [0.8 * 0.18] = 0.164 Calculation of the standard deviation of each share: Probability of Scenario 1 Return under Scenario 1 Expected Return Probability of Scenario 2 Return under Scenario 2 Expected Return Var (S5) = 0.2 * (0.08 - 0.128)2 + 0.8 * (0.14 - 0.128)2 = 0.000576 ?(S5) = √(0.000576) = 0.024 = 2.4% [0.000576 = 5.76 * 10-4] The standard deviation (?) is the square root

Solution W5 (the weight of Share S5 in the basket) is 20%.

Example 2 Share S3 belongs to a luxury tourism company whose expectation is 10%.

The measuring units of the expectation and the risk are percentages.

The best-known bonds are mortgages, in which an individual borrows money from a bank or government agency, and pays his debt in installments, plus interest.

The point of intersection is rf.

The expected return, called the expectation for short, is denoted E. The standard deviation of the return, which constitutes a risk indicator, and is also called risk, denoted ?.

Shape of the Frontier Curve It is customary to sketch the frontier curve as a concave curve.

The area under each curve reflects the chance of obtaining the return under the area.

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The variance of the basket, Var(B), describes the overall fluctuation of Basket B. Exercise Given two shares, S5 and S6, from the example for calculating the covariance of two shares, Table 5 displays the data for the shares calculated in the example.

For each scenario, we calculate the difference between the return and the expectation for each of the shares, and multiply the differences for the two shares.

Do not duplicate, copy, photocopy, translate, store in a database, broadcast, or record in any manner whatsoever, or through any electronic, optical, or other mechanical media, any part whatsoever of the material in this book.

line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend Payments 63 Appendix 68 Calculating the Expectation of a Share – The Accepted Practice in Exercises 68 Glossary of Concepts 69 Formulas Page 73 List of Symbols 75 Introduction – Basket of Shares, Investment Portfolio, Expected Return, Standard Deviation Basket of Shares (symbol: B) Any collection of investments based on a single investment instrument is called a basket, with the name of the instrument added

Shlomo Simanovski Finance for Advanced Students Meitav Self Learning Book about Financing for Beginners Strolling to the College Level Meitav Self Learning Publishing Ltd.

Meaning of the Standard Deviation Representing Risk, Together with an Example In the framework of the example, we will consider three scenarios pertaining to a share whose expected return is 2%.

The result is called the scenario’s contribution to the expectation.

reducing risk 34 Formula for finding the internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48

The strength of the connection between different pairs of shares can be compared by using a statistical tool called the correlation coefficient, which will be explained immediately.

Risk-Seeking Investors 50 Slope of the CML line, called the risk premium 53 Distinguishing between percentages and percentage points 53 Practical use of the CML line 53 Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity

Calculating the Variance and Standard Deviation of the Improved Portfolio Introduction In general, the formula for calculating the variance of an investment portfolio (that is not an improved portfolio) with two assets (for example: an efficient basket + bonds) corresponds to Formula 1 Formula 1: ?2p = W12 * ?21 + W22 * ?22 + 2 * W1 * W2 * ?1,2 List of Symbols P is the improved investment portfolio (not included in Formula 1).

Calculate NPV (net present value) of the dividend flow in the first period.

of Scenario 1 Return under Scenario 1 Probability of Scenario 2 Return under Scenario 2 E(S6) = [0.2 * 0.10] + [0.8 * 0.18] = 0.164 Calculation of the standard deviation of each share: Probability of Scenario 1 Return under Scenario 1 Expected Return Probability of Scenario 2 Return under Scenario 2 Expected Return Var (S5) = 0.2 * (0.08 - 0.128)2 + 0.8 * (0.14 - 0.128)2 = 0.000576 ?(S5) = √(0.000576) = 0.024 = 2.4% [0.000576 = 5.76 * 10-4] The standard deviation (?) is the square

Examples: basket of shares, basket of bonds, basket of options, basket of commodities, and so forth.We will deal mainly with baskets relating to shares and refer to them as “baskets” for short.

The slope is [(Em – rf)/?m] (in Figure 19, the slope of the CML line is highlighted).

From the lenders’ perspective (lenders such as banks and financial institutions): state-guaranteed loans.

Table of Contents Introduction – Basket of Shares, Investment Portfolio, Expected Return, Standard Deviation 7 Basket of shares (symbol: B) 7 Investment portfolio (symbol: P) 7 Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing

In the framework of this section, we will examine how the parameters of the basket change following changes in the internal weight of its shares, given three scenarios for the correlation coefficient between the pair of shares in the basket: Scenario 1: ?1,2 = -1 Scenario 2: ?1,2 = 0 Scenario 3: ?1,2 = +1 Shares S1 and S2 In each scenario, Shares S1 and S2 are different shares belonging to different sectors.

Figure 11 GRAPH Formula for Finding the Internal Composition of a Basket with a Minimum Standard Deviation Mathematicians have developed a formula for a basket containing two shares, S1 and S2 (or any two securities of any type), that enables us to calculate the internal composition between them with the lowest ?.

Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28 Correlation coefficient 28 Distinguishing between baskets with different correlation coefficients 30 Conclusion about reducing risk 34 Formula for finding the internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41

Do not duplicate, copy, photocopy, translate, store in a database, broadcast, or record in any manner whatsoever, or through any electronic, optical, or other mechanical media, any part whatsoever of the material in this book.

As a measure of the systematic risk of the individual share with the market portfolio, the beta will be on the X axis.

The formula is based on the beta index (?).

13 Scenarios Probability of the Scenario Return under the Scenario Scenario’s Contribution to the Expectation 1 2 3 4 Recession 20% (= 0.2) -0.02 [0.2 * (-0.02)] = -0.004 = -0.4% Stability 50% (= 0.5) 0.06 [0.5 * (0.06)] = 0.03 = 3% Prosperity 30% (= 0.3) 0.10 [0.3 * (0.1)] = 0.03 = 3% Total 100% 0.056 = 5.6% - The expectation of the share Glossary of Concepts Bond – a promissory note that confers ownership of the sum of money listed on it, which the issuer of the note will pay in the

The product of the differences under each scenario is multiplied by the probability that the scenario will occur.

risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend Payments 63 Appendix

Calculating the Expectation and Standard Deviation of a Basket of Shares The explanation is accompanied by an example of a basket containing two shares, denoted S1 and S2.

The weighting is according to their weight in the basket, as follows: E(B) = W1 * E(S1) + W2 * E(S2 ) If we substitute the figures from Table 2, we obtain for the expectation of Basket E: E(B) = (0.5 * 0.05 + 0.5 * 0.15) = 0.1 = 10% In simple language, the expectation of the basket is 10%.

Question: Which of the two shares has a greater chance of generating a negative return of 20% (meaning that the share price drops)?

Calculating the Covariance of a Basket with Two Shares ?2(B) = Var(B) The variance of a basket with two shares depends on the variance (or the standard deviation, which equals the square root of the variance) of each share separately and the covariance of the two shares.

Distinguishing between Ordinary and Risk-Seeking Investors For ordinary investors, the investment in an efficient portfolio is financed solely from equity (Figure 17).

Correlation coefficient Covariance – COV [Symbol: ?1,2 (for short: or COV1,2)] Covariance can apply to any pair of investment assets (commodities, shares, bonds, etc.)

Examples: basket of shares, basket of bonds, basket of options, basket of commodities, and so forth.We will deal mainly with baskets relating to shares and refer to them as “baskets” for short.

Commercial use of the material in this book is absolutely prohibited.

This is an acceptable way for a country, company, or agency to borrow money from the public.

58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend Payments 63 Appendix 68 Calculating the Expectation of a Share – The Accepted Practice in Exercises 68 Glossary of Concepts 69 Formulas Page 73 List of Symbols 75 Introduction – Basket of Shares, Investment Portfolio, Expected Return, Standard Deviation Basket of Shares (symbol: B) Any collection of investments based on a single investment instrument

In order to illustrate the meaning of the expected return and the standard deviation of a share, we use an example based on a sample of 360 monthly measurements of the share’s return (over a 30-year period from a calculation of 12 monthly measurements per year).

standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend Payments 63 Appendix 68 Calculating the Expectation of a Share – The Accepted Practice in Exercises

An Improved Investment Portfolio (abbreviated as an improved portfolio) (Denoted by P) – Definition We define an improved investment portfolio as an investment portfolio with two components: A basket of shares is located on the efficiency frontier (an efficient basket).

A Summary Exercise Figure 7 displays an efficiency frontier.

Do not duplicate, copy, photocopy, translate, store in a database, broadcast, or record in any manner whatsoever, or through any electronic, optical, or other mechanical media, any part whatsoever of the material in this book.

Figure 2 - Expectation-Risk Plane Displaying a Basket of Shares in a Plane Every point in Figure 2 represents a basket, whose location in the plane is determined by its two parameters: Point a represents a basket whose parameters are E = 5% and ? = 3%.

Despite the simplicity of the formula for calculating the correlation coefficient, the statistical knowledge required to understand it is beyond that required for holders of MBA degrees.

When A is superior to B, we say that it is more efficient than B. Figure 4 Division of the Plane into Quadrants (Figure 5) If we divide the plane into four quadrants denoted A, B, C, and D, we can state with certainty that every basket in Quadrant A is superior to every basket in Quadrant D. Financiers say: Every basket in Quadrant A is more efficient than every basket in Quadrant D. The use of the term more efficient is accepted in the profession.

In professional jargon, we say that Danny is more risk averse than Chaim.

Another definition is negative growth in the economy (i.e. a decrease in output) in two consecutive quarters.

The entity that issued the bond (a government or company) originally received the money from the bond purchaser, and undertook to repay it in the future.

The entity that issued the bond (a government or company) originally received the money from the bond purchaser, and undertook to repay it in the future.

A larger slope means that the investor is more conservative.

The return is usually unstable and not constant, and it is therefore customary to report an annual return.

The following are examples of risk-free assets: From the savers’ perspective: (a bank deposit, government bond, etc.).

Profit – when a company has a surplus of revenue over expenses, the difference between the two reflects profit (in contrast to loss, which reflects a surplus of expenses over revenue).

Since a large sample is involved, we assume that the sample distribution represents the probability distribution of the population (the population is all the monthly observations measuring the share’s return from the time it was first issued up to infinity).

13 Scenarios Probability of the Scenario Return under the Scenario Scenario’s Contribution to the Expectation 1 2 3 4 Recession 20% (= 0.2) -0.02 [0.2 * (-0.02)] = -0.004 = -0.4% Stability 50% (= 0.5) 0.06 [0.5 * (0.06)] = 0.03 = 3% Prosperity 30% (= 0.3) 0.10 [0.3 * (0.1)] = 0.03 = 3% Total 100% 0.056 = 5.6% - The expectation of the share Glossary of Concepts Bond – a promissory note that confers ownership of the sum of money listed on it, which the issuer of the note will pay in the

Basket F. Figure 6 On the other hand, we are unable to determine the superiority or inferiority of the efficient baskets to each other (were we able to determine that one of two baskets was inferior, that basket would not be an efficient basket), i.e. we do not know whether C is superior to A. Basket C has more risk, but also a higher return.

The sum of the results obtained under the two scenarios is the covariance.

Symbols: Table 1 Expected Return Standard Deviation Basket of shares E(B) ?(B) Investment portfolio E(P) ?(P) Statistics of Finance The Expected Return and the Standard Deviation of an Individual Share – Example and Illustration A reminder: Expectation – the average obtained by sampling the entire population.

share at Time 0, i.e. today, is the sum NPV1 + NPV3: P = NPV1 + NPV3 = 175.7 +585.2 = $760.90 Appendix Calculating the Expectation of a Share – The Accepted Practice for Exercises In most exercises, students are asked to calculate the expectation of some share on the basis of the following information: Referring to Various Scenarios in Some Time Period and the Probability that They Will Occur The scenarios usually involve 2 to 3 economic situations, such as: a recession, economic stability

Therefore: P (the share price) = D0/Ke = 100/0.1 = 1,000 An Example of Scenario 2: Messy Ltd. yesterday distributed a $1,000 dividend.

The correlation between these shares is negative, meaning that if the year is hot, the investor will make a larger profit on the shares of the company that grows oranges, but will earn a smaller profit on the shares of the company that makes heaters, because the share of this company will drop in value during a hot year.

The parameters of the shares and their relative weight in the basket are listed in Table 2.

The product of the differences under each scenario is multiplied by the probability that the scenario will occur.

We assume that the sample results have a normal probability distribution with an average of 2% (per month) and a standard error of 5%.

Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the

Introduction – Basket of Shares, Investment Portfolio, Expected Return, Standard Deviation 7 Basket of shares (symbol: B) 7 Investment portfolio (symbol: P) 7 Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A

Distinction between Percentages and Percentage Points When the return rises from 5% to 6%, then the increase in percentages is: (6-5)/5 * 100 = 20% The increase in percentage points is: 1 percentage point (equal to the difference between the percentages at the two times (6% - 5%).

List of Symbols S1 is the first share S2 is the second share ?1,2 is the covariance of Shares S1 and S2 ?1 is the standard deviation of S1 ?2 is the standard deviation of S2 The correlation coefficient can receive only values between –1 and 1.

+ ?2(S1) – 2 * ?1,2 * ?1 * ?2] We know from the data that: ?1,2 = 0 (this is the given scenario) ?(S1) = 0.15 (Table 6) ?(S2) = 0.3 (Table 6) Therefore: W1 = [0.302 – 0* 0.15 * 0.30]/[0.302 + 0.152 – 2 * 0 * 0.15 * 0.30] = 0.302/[0.302 + 0.152] = 0.8 = 80% Meaning that the portfolio with the minimum variance will consist of 80% Share S1 (W1 = 80%) and 20% Share S2 (W2 = 100% - 80% = 20%).

The investor is indifferent to a choice between any of the baskets on the indifference curve.

deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49 Distinguishing between Ordinary

This means that for each increase in risk, the investor demands an additional return.

points 53 Practical use of the CML line 53 Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating

Its return is constant under any scenario.

We assume that the sample results have a normal probability distribution with an average of 2% (per month) and a standard error of 5%.

and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend Payments

As we go upward from point S1 along the dotted segment, the weight of share S2 in each basket increases at the expense of Share S1.

Example: If ?1,2, the covariance between S1 and S2, is 0.3, and ?3,4, the covariance between S3 and S4, is 0.4, it cannot be concluded that the connection between S1 and S2 is stronger than the connection between S3 and S4, just as it cannot be stated that a profit of $10 on Share S1 is preferable to a profit of $8 on Share S2.

The two companies, AA and BB, are in the same industry, and it is reasonable for them to be affected in the same direction in a recession and at prosperous times.

If the correlation coefficient of Shares S1 and S2 is 0.8 and the correlation coefficient of Shares S3 and S4 is 0.4, then the connection (i.e. to what degree one variable 1 enables us to predict variable 2) between Shares S1 and S2 is stronger than the connection between Shares S3 and S4.

Answer: Basket A is superior to Basket D, and we therefore say that Basket A is more efficient than Basket D. Question: Which basket is superior, A or E?

The calculation is based on the formulas adjusted to simple investors appearing in the first row of Table 11.

The two parameters work in opposite directions: a higher expectation is an advantage, while a higher standard deviation is a disadvantage.

Basket F. Figure 6 On the other hand, we are unable to determine the superiority or inferiority of the efficient baskets to each other (were we able to determine that one of two baskets was inferior, that basket would not be an efficient basket), i.e. we do not know whether C is superior to A. Basket C has more risk, but also a higher return.

line, called the risk premium 53 Distinguishing between percentages and percentage points 53 Practical use of the CML line 53 Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases

The forces here are opposite, and superiority therefore depends on the specific investor – whether he puts greater emphasis on a high return or on lower risk.

Shlomo Simanovski Finance for Advanced Students Meitav Self Learning Book about Financing for Beginners Strolling to the College Level Meitav Self Learning Publishing Ltd.

On the other hand, assume that investor B has a share portfolio composed of a share of a company that grows oranges and a share of a company that makes heaters.

interpretation 28 Correlation coefficient 28 Distinguishing between baskets with different correlation coefficients 30 Conclusion about reducing risk 34 Formula for finding the internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved

If the correlation coefficient of Shares S1 and S2 is 0.8 and the correlation coefficient of Shares S3 and S4 is 0.4, then the connection (i.e. to what degree one variable 1 enables us to predict variable 2) between Shares S1 and S2 is stronger than the connection between Shares S3 and S4.

It is therefore impossible to devise a portfolio with these two shares that will reduce the risk.

The table lists the initial equity of each investor, the composition of his efficient portfolio, and for the risk-seeking investors, the amount of the loan that they received.

Conclusion about reducing risk 34 Formula for finding the internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier

The entity that issued the bond (a government or company) originally received the money from the bond purchaser, and undertook to repay it in the future.

In most cases, the distinction is unimportant, because what we mean is clear in the context.

between Ordinary Investors and Risk-Seeking Investors 50 Slope of the CML line, called the risk premium 53 Distinguishing between percentages and percentage points 53 Practical use of the CML line 53 Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise

Symbols: Table 1 Expected Return Standard Deviation Basket of shares E(B) ?(B) Investment portfolio E(P) ?(P) Statistics of Finance The Expected Return and the Standard Deviation of an Individual Share – Example and Illustration A reminder: Expectation – the average obtained by sampling the entire population.

i.e. today, is the sum NPV1 + NPV3: P = NPV1 + NPV3 = 175.7 +585.2 = $760.90 Appendix Calculating the Expectation of a Share – The Accepted Practice for Exercises In most exercises, students are asked to calculate the expectation of some share on the basis of the following information: Referring to Various Scenarios in Some Time Period and the Probability that They Will Occur The scenarios usually involve 2 to 3 economic situations, such as: a recession, economic stability, prosperity, drought

Example of an investment portfolio: a portfolio worth $10,000: $5,000 invested in a bank deposit and $5,000 invested in General Motors shares.

The probability of the scenario appears in the heading.

As a result, the last element in the formula becomes 0, and is erased. ?2rf = 0 (the standard deviation of rf = 0).

44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49 Distinguishing between Ordinary Investors and Risk-Seeking Investors 50 Slope of the CML line, called the risk premium 53 Distinguishing between percentages and percentage points 53 Practical use of the CML line 53 Simple

We assume here that the risk-free interest rate is the same for borrowers and lenders, and is equal to rf.

We assume the total investment, beyond 100% of the market basket, is funded through a loan whose return and standard deviation are the same as those of rf.

Standard error – the standard deviation obtained from a sample.

When the correlation coefficient is -1, a risk-free basket can be assembled (Basket C).

different correlation coefficients 30 Conclusion about reducing risk 34 Formula for finding the internal composition of a basket with the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment

Positive Covariance A positive covariance is a situation in which two shares respond in the same direction to same scenarios, relative to their expectations.

Each measurement is called an observation.

The indifference curves reflect the investor’s risk aversion and benefit, which also appear in the standard deviation-expected return plane.

The two parameters work in opposite directions: a higher expectation is an advantage, while a higher standard deviation is a disadvantage.

We will focus on shares: Share S1 and Share S2.

?(S5) = 0.024 ?(S6) = 0.032 ?5,6 = 7.68 * 10-4 Therefore: ?5,6 = ?5,6/[ ?(S5) * ?(S6)] = 7.68 * 10-4/(0.024 * 0.032) = 1 Distinguishing between Baskets with Different Correlation Coefficients Introduction Any change in the internal weight of the shares in the basket in effect creates a new basket with different E(B) and ?(B) parameters.

Table 10 The Investors Category of the Investors Equity in Dollars Loan Components of the Efficient Portfolio Explanations The Market Basket rf $$ % of Equity $$ % of Equity 1 2 3 4 5 6 7 8 9 A Ordinary 1,000 (100%) 0 200 20% 800 80% B Ordinary 1,000 (100%) 0 400 40% 600 60% C Risk-seeking 1,000 (100%) 200 (20%) 1,200 120% 0 -20% The market basket constitutes 120% of the equity.

The amount of the debt that the borrower undertakes to repay to the bond owner, plus interest, on the redemption date is listed on the bond.

The amount of the debt that the borrower undertakes to repay to the bond owner, plus interest, on the redemption date is listed on the bond.

Table 11 demonstrates the way to calculate the parameters in the investment portfolio of C and D when the parameters m and rf are as follows: E ? m 0.12 0.08 rf 0.1 0 Table 11 – Calculation of the Parameters of Investors C and D Calculation of the Expectation (E) Calculation of the Standard Deviation (?)

Chaim’s selected basket is C. Danny’s selected basket is A. Figure 15 Danny’s indifference curve is steeper, which indicates that he is more conservative than Chaim.

The two companies, AA and BB, are in the same industry, and it is reasonable for them to be affected in the same direction in a recession and at prosperous times.

We will explain this very soon.

is the sum NPV1 + NPV3: P = NPV1 + NPV3 = 175.7 +585.2 = $760.90 Appendix Calculating the Expectation of a Share – The Accepted Practice for Exercises In most exercises, students are asked to calculate the expectation of some share on the basis of the following information: Referring to Various Scenarios in Some Time Period and the Probability that They Will Occur The scenarios usually involve 2 to 3 economic situations, such as: a recession, economic stability, prosperity, drought, plentiful

The following are examples of risk-free assets: From the savers’ perspective: (a bank deposit, government bond, etc.).

Meaning of the Standard Deviation Representing Risk, Together with an Example In the framework of the example, we will consider three scenarios pertaining to a share whose expected return is 2%.

the minimum standard deviation 36 Indifference Curves 39 Shape of the curves 40 The selected basket of investments 41 Improved Investment Portfolio (abbreviated as improved portfolio) 43 Calculating the expectation 44 Calculating the variance and standard deviation of the improved portfolio 45 Replacing bonds with a risk-free (rf) asset (in the investment portfolio) 46 Improved Efficiency Frontier (CAPM model) 48 Efficient portfolios and their financing 49 Risk-seeking investors 49

the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend Payments 63 Appendix 68 Calculating the Expectation of a Share – The Accepted

Correlation Coefficient Symbol: ?(S1,S2), or ?1,2 in abbreviated form (? is a Greek letter pronounced “roe”) The correlation coefficient is a statistical tool that can be used to measure the strength of the covariance between two shares.

The return is usually unstable and not constant, and it is therefore customary to report an annual return.

Another definition is negative growth in the economy (i.e. a decrease in output) in two consecutive quarters.

There is no economic justification for taking a loan in order to invest in rf, because the cost of the loan equals the expected profit from it (we have assumed that the parameters of the loan are equal to those of rf).

The calculation is based on the formulas adjusted to simple investors appearing in the first row of Table 11.

Doing this in formula 6, we obtain the following equation: (7) Var(B) = W12 * ?12 + W22 * ?22 + 2 * W1 W2 * ?1,2 * ?1 * ?2 We now substitute the given weights, the data from Table 6, and ?1,2 = 0, which is given: Var(B) = ?B2 = 0.82 * 0.152 + 0.22 * 0.302 + 2 * 0.8 * 0.2 * 0 * 0.15 * 0.30 = 0.018 ?B = √0.018 = 0.134 Indifference Curves The curve connecting all the baskets generating the same level of benefit for the investor is called an indifference curve.

Scenario A: ? = 5% Scenario B: ? = 10% Scenario C: ? = 20% In each scenario, the probability distribution is normal (see Figure 1) Figure 1 – The Distribution of the Share under the Three Scenarios The center of the curve is exactly 2% - the expectation - because this number is the expected return, or the average return, and the normal probability distribution is symmetric.

Note: The required return, Ke, can be found, among other ways, with the help of the SML formula in the preceding section (i.e. the required return for the risk).

between percentages and percentage points 53 Practical use of the CML line 53 Simple investors and risk-seeking investors – further explanation 54 Calculating the expectation and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security

We have chosen to use the term “basket of shares,” not “share portfolio,” in order to distinguish it from the term “investment portfolio,” which we will immediately define.

In the framework of this section, we will examine how the parameters of the basket change following changes in the internal weight of its shares, given three scenarios for the correlation coefficient between the pair of shares in the basket: Scenario 1: ?1,2 = -1 Scenario 2: ?1,2 = 0 Scenario 3: ?1,2 = +1 Shares S1 and S2 In each scenario, Shares S1 and S2 are different shares belonging to different sectors.

Do not duplicate, copy, photocopy, translate, store in a database, broadcast, or record in any manner whatsoever, or through any electronic, optical, or other mechanical media, any part whatsoever of the material in this book.

In order to illustrate the meaning of the expected return and the standard deviation of a share, we use an example based on a sample of 360 monthly measurements of the share’s return (over a 30-year period from a calculation of 12 monthly measurements per year).

illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21

= 10%), on the other hand, the area under the curve is spread out more to the sides, compared with Curve A. Curve B is lower than curve A. As stated above, the area under the curve reflects the probability of obtaining the return under that area.

The slope of CML is denominated in the percentage of the return added to the portfolio following an addition of 1% to its risk, i.e. the added return represents the premium required for each additional 1% of risk.

The difference in the value of an investment is derived from a number of sources, such as interest, net profit from a business, capital gain, and so forth.

Covariance – a measure of the connection between two random variables.

charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard Deviation of a Basket of Shares 20 Calculating the expectation of a basket of shares 20 Calculating the standard deviation of a basket of shares 21 Covariance 21 Positive covariance 22 Negative covariance 22 Covariance = 0 22 Calculating the covariance 23 Calculating the covariance of a basket with two shares 26 Covariance – interpretation 28 Correlation

According to the growth model, the theoretical price of the share equals the present value of the dividend receipts that it generates from now on, while the price of capital for the firm, denoted Ke, is used for discounting.

The two parameters work in opposite directions: a higher expectation is an advantage, while a higher standard deviation is a disadvantage.

Examples: basket of shares, basket of bonds, basket of options, basket of commodities, and so forth.We will deal mainly with baskets relating to shares and refer to them as “baskets” for short.

Profit – when a company has a surplus of revenue over expenses, the difference between the two reflects profit (in contrast to loss, which reflects a surplus of expenses over revenue).

This is an acceptable way for a country, company, or agency to borrow money from the public.

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= ?2(B) = W12 * ?2(S1) + W22 * ?2(S2) + 2 * W1 * W2 * ?(S1,S2) List of Symbols Var(B) is the variance of Basket B ?2(B) is the square of the standard deviation of Basket B W1 is the weight of Share S1 in the basket W2 is the weight of Share S2 in the basket ?2(S1) is the square of the standard deviation of Share S1 ?2(S2) is the square of the standard deviation of Share S2 ?(S1,S2) is the covariance of Shares S1 and S2 Important: The covariance ?(S1,S2) describes the degree of common variation

Regardless, in practice, only the composition of [the market basket + rf] (called an “efficient portfolio”) is used.

The SML formula, which gives the expected return of an individual share for each ? level of the share, is: (8) Eri = rf + ? * (Erm – rf) If the profit and dividend do not vary over the years, the formula for calculating the value of a share is: (9) D0/Ke If profits and dividends grow each year at some constant rate g, the formula for calculating the value of the share is: (10) D0 * [1 + g]/(Ke – g) List of Symbols rf is a risk-free asset.

S1 and S2: Var(B) = ?2(B) = W12 * ?2(S1) + W22 * ?2(S2) + 2 * W1 * W2 * ?(S1,S2) List of Symbols Var(B) is the variance of Basket B ?2(B) is the square of the standard deviation of Basket B W1 is the weight of Share S1 in the basket W2 is the weight of Share S2 in the basket ?2(S1) is the square of the standard deviation of Share S1 ?2(S2) is the square of the standard deviation of Share S2 ?(S1,S2) is the covariance of Shares S1 and S2 Important: The covariance ?(S1,S2) describes the degree

Research has been conducted in various countries to define and characterize these baskets, but no successes whatsoever in this endeavor are known.

Figure 18 GRAPH References to Percentages The percentages accompanying loans and the market basket in Figure 2 are based on the definition of total equity as 100%.

two shares, S1 and S2: Var(B) = ?2(B) = W12 * ?2(S1) + W22 * ?2(S2) + 2 * W1 * W2 * ?(S1,S2) List of Symbols Var(B) is the variance of Basket B ?2(B) is the square of the standard deviation of Basket B W1 is the weight of Share S1 in the basket W2 is the weight of Share S2 in the basket ?2(S1) is the square of the standard deviation of Share S1 ?2(S2) is the square of the standard deviation of Share S2 ?(S1,S2) is the covariance of Shares S1 and S2 Important: The covariance ?(S1,S2) describes

The uniform symbols make it possible to see the general picture easily.

We assume here that the risk-free interest rate is the same for borrowers and lenders, and is equal to rf.

Characteristics of an investment portfolio 7 Statistics of Finance 9 Expected return and standard deviation of an individual share – example and illustration 9 Use of decimal fractions instead of percentages 10 Meaning of the standard deviation representing risk, together with an example 10 Displaying and Characterizing Baskets 13 A risk-free asset and charting its position in a plane 14 Superior and inferior portfolios 15 The efficiency frontier 18 Calculating the Expectation and Standard

Curve C in Figure 8 (part of which is dotted) represents the entire range of efficient baskets that can be obtained under various compositions of Shares S1 and S2.

The investor is indifferent to a choice between any of the baskets on the indifference curve.

The probability of the scenario appears in the heading.

Correlation coefficient Covariance – COV [Symbol: ?1,2 (for short: or COV1,2)] Covariance can apply to any pair of investment assets (commodities, shares, bonds, etc.)

In professional jargon, we say that Danny is more risk averse than Chaim.

Conclusion about Reduction of the Risk When ?1,2, the correlation coefficient between the shares is completely opposite, it is possible to arrive at a basket with zero risk (Basket C).

Figure 5 The baskets in the upper envelope are the most efficient (Figure 6) A large number of baskets are placed in a plane in Figure 6.

The best-known bonds are mortgages, in which an individual borrows money from a bank or government agency, and pays his debt in installments, plus interest.

For learning purposes, the distinction achieves order and simplicity.

Its return is identical in any natural situation (scenario).

This is an acceptable way for a country, company, or agency to borrow money from the public.

Example 1 Share S1 belongs to a construction company named AA whose expectation is 10%.

Do not duplicate, copy, photocopy, translate, store in a database, broadcast, or record in any manner whatsoever, or through any electronic, optical, or other mechanical media, any part whatsoever of the material in this book.

The Growth Model – Estimating the Value of a Share on the Basis of Dividend Payments The growth model shows a simple way to estimate the price of a share according to the growth rates of its dividend.

Figure 10 GRAPH Conclusion about Reducing the Risk When ?1,2 = 1, i.e. when perfect correlation exists between the two shares, the risk rises as the basket includes more of the share with the higher risk.

List of Symbols E(Si) - the expectation of share Si Wi - the weight of share Si in the basket (W is short for weight) B - the basket of shares E(B) - the expectation of the basket ?(B) - the standard deviation of the basket Calculation of the Standard Deviation of a Basket of Shares Preliminary Background In order to calculate the standard deviation of a basket of shares, we must first become familiar with two statistical concepts: 1.

Three points on the CML curve are notable (Figure 17): Point rf – represents an efficient portfolio containing 100% rf and 0% m.

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Calculating the Covariance The calculation is made in three stages, as seen in the example whose information is listed in Table 3.

Every basket on an indifference curve lower than U120 is inferior to it, and a basket on an indifference curve higher than U120 is not available (there are no available baskets above the efficiency frontier).

The table lists the initial equity of each investor, the composition of his efficient portfolio, and for the risk-seeking investors, the amount of the loan that they received.

Calculate NPV of the second period for the current time (Time 0).

Characteristics of an Investment Portfolio The important parameters characterizing an investment portfolio are: * Expected return * Standard deviation Every basket of shares has two important parameters associated with it.

The variance of the basket, Var(B), describes the overall fluctuation of Basket B. Exercise Given two shares, S5 and S6, from the example for calculating the covariance of two shares, Table 5 displays the data for the shares calculated in the example.

The difference between a bond and a different loan is that the bond can be transferred and sold.

Only when the correlation between the shares is complete is it impossible to disperse the risk, i.e. to create an investment basket with less risk.

Characteristics of an Investment Portfolio The important parameters characterizing an investment portfolio are: * Expected return * Standard deviation Every basket of shares has two important parameters associated with it.

There is no economic justification for taking a loan in order to invest in rf, because the cost of the loan equals the expected profit from it (we have assumed that the parameters of the loan are equal to those of rf).

The improved investment portfolio contains a 50% weight of the efficient basket, while the remaining 50% is invested in a risk-free asset.

and standard deviation in an efficient portfolio of risk-seeking investors 55 The Use of the Terms “Improved Portfolio” and “Efficient Portfolio” 57 The use of the term “baskets” 57 A compromise in wording for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend Payments 63 Appendix 68 Calculating the Expectation of a Share – The Accepted Practice

for the sake of simplicity 57 Changes in rf assets 58 Several Emphases on the CML Line 59 The SML (security market line) 59 The Growth Model – Estimating the Value of a Share Based on the Dividend Payments 63 Appendix 68 Calculating the Expectation of a Share – The Accepted Practice in Exercises 68 Glossary of Concepts 69 Formulas Page 73 List of Symbols 75 Introduction – Basket of Shares, Investment Portfolio, Expected Return, Standard Deviation Basket of Shares (symbol: B) Any collection

It is usually reflected by a drop in prices, reduced volumes of production and commerce, and lower employment.

Example: If ?1,2, the covariance between S1 and S2, is 0.3, and ?3,4, the covariance between S3 and S4, is 0.4, it cannot be concluded that the connection between S1 and S2 is stronger than the connection between S3 and S4, just as it cannot be stated that a profit of $10 on Share S1 is preferable to a profit of $8 on Share S2.

Scenario 2 Expected Return Var (S6) = 0.2 * (0.10 - 0.164)2 + 0.8 * (0.18 - 0.164)2 = 0.0010224 ?(S6) = √(0.0010224) = 0.032 = 3.2% [0.0010224 = 1.0224 * 10-3] Calculation of the covariance ? 5,6: The general formula for calculating the variance of a basket with two shares is: Var(B) = ?(B)2 = W12 * ?2(S1) + W22 * ?2(S2) + 2 * W1 * W2 * ? 1,2 We have already found the variance of each share, and we will now calculate the covariance ? 1,2 (in our example ? 5,6) A list of the three operations

Since a large sample is involved, we assume that the sample distribution represents the probability distribution of the population (the population is all the monthly observations measuring the share’s return from the time it was first issued up to infinity).

List of Symbols S1 is the first share S2 is the second share ?1,2 is the covariance of Shares S1 and S2 ?1 is the standard deviation of S1 ?2 is the standard deviation of S2 The correlation coefficient can receive only values between –1 and 1.

In other words, the fluctuation around the average of 2% expectation is relatively low.

The investment portfolios located on the CML becomes an improved alternative to the efficient frontier, except for the market basket itself (m), which is common to both of them.

Figure 13 GRAPH The Selected Investment Basket Efficiency frontier – when there is a point on the graph that is definitely superior to another point (for example, two points with the same variance, while one of them has a greater expectation than the other).

Nevertheless, it would be more accurate to mention the parameters of the efficient portfolio in the following way: “Every point on the CML line represents an efficient portfolio whose two parameters (expectation and standard deviation) correspond to the location of the point in a plane.”

Return – the difference in the value of an investment at intervals of time.

For example, when Em = 0.12, ?m = 0.08, and rf = 0.10, the equation of the line is: EP = 0.10 + [(0.12 – 0.10)/0.08] * ?p If we calculate the slope, we get: EP = 0.10 + 0.25 * ?p In this equation, the expectation (EP) is a function of the risk (?p).

We will consider two common scenarios: 1.

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Figure 2 - Expectation-Risk Plane Displaying a Basket of Shares in a Plane Every point in Figure 2 represents a basket, whose location in the plane is determined by its two parameters: Point a represents a basket whose parameters are E = 5% and ? = 3%.

Shape of the Frontier Curve It is customary to sketch the frontier curve as a concave curve.

The covariance tests how two shares respond to the same scenario.

The amount of the debt that the borrower undertakes to repay to the bond owner, plus interest, on the redemption date is listed on the bond.