Icfai P.a.application Of Portfolio Theories

Application of Portfolio Theories in Investment Risk Appraisal Introduction: When a firm holds many projects in its portfolio, often, the cash flows of one of the projects may vary substantially in tandem with the cash flows of one or more other projects of the firm. In a typical case of forward integration, the revenues from both the projects will increase or decrease depending on the demand for final product. The cash flows from the two projects, therefore, have a high degree of correlation to each other. Similarly, there are also projects that influence the cash flows of each other negatively. The correlation between cash flows of different projects also comes to play when new projects are proposed. If a firm takes on projects that have a high degree of correlation with the existing projects, its total risk will increase as the cash flows from all the projects will increase or decrease at the same time.

By the same logic a new project with negative correlation will reduce the total risk of the firm as the fluctuations of the cash flows from one project will be offset by those from another. The covariation (varying together) of the cash flows sometimes leads to a project that is unviable in isolation becoming viable. A project that gives a negative NPV may become attractive as part of the portfolio of assets. The formula for the expected return given the possible returns and their probabilities is: n E(Rp) = ∑Xj E(Rj) ---------------- Eqn. 1 j =1 Where E(Rp) : Expected return on the portfolio Xj : Proportion of the asset E(Rj) : Expected return from asset j

While the calculation of the expected return on the portfolio is fairly simple, calculation of the standard deviation, which is the most commonly used measure of risk, is difficult. We cannot fine the standard deviation of the change in returns on a portfolio by weighing the standard deviations of the individual assets of the portfolio with their proportions in the portfolio. This problem has been addressed by the Modern Portfolio theory propounded by Harry Markowitz. Portfolio Theory: According to the portfolio theory, the standard deviation of a portfolio can be found using the formula. N N σp = √{ ∑ ∑ Xi Xj ρij σi σj} --------------- Eqn. 2 i=1 j=1 Where, σp = Expected portfolio standard deviation Xi = Proportion of asset i in the portfolio Xj = Proportion of asset j in the portfolio

ρij = Correlation coefficient between i and j σi = Standard deviation of security i σj = Standard deviation of security j N = Total no. of securities in the portfolio The above formula indicates that all (i) assets of the portfolio should be paired, (ii) the proportions, and covariances of each pair should be multiplied and (iii) the products thus arrived at should be added. Expanding the formula for two assets 1 and 2, we get σp = √{ X1² σ 1² + X2² σ 2² + 2 X1 X2 ρ 12 σ 1 σ 2} -----------Eqn. 3 For three assets, the expansion of the formula will be: σp = √{X1² σ 1² + X2² σ 2² + X3² σ 3² + 2 X1 X2 ρ 12 σ 1 σ 2 + 2X1X3 ρ 13 σ 1 σ 3 + 2X2X3 ρ 23 σ 2 σ 3 }--------------Eqn. 4 Using the above formula the variance of a portfolio of any number of assets can be calculated.

A Simplified version of the formula: N N

σp = √{ ∑ ∑

Xi Xj Covij }

Illustration: A firm is evaluating the return and risk on a combination of two projects. The characteristics of the two projects are as follows: P1 P2 E(Rj) 30% 35% σj 7% 1% Xj 0.50 0.50 The correlation between the two projects is –0.30. Find: (i) Expected return from the combination (ii) Expected portfolio standard deviation

Solution: (i) The return from the combination is: n E(Rp) = ∑Xj E(Rj) = 0.50 x 0.30 + 0.50 x 0.35 j =1 = 0.325 or 32.5% (ii) The expected portfolio standard deviation is: Using equation 3, N

σp

= √{ ∑

N-1 N Xj² σ j ²

+ 2 ∑ ∑ Xi Xj ρ ij σ i σ j}

j =1

j=1 j=1

= √{ 0.50² x 0.07² + 0.50² x 0.01² + 2x 0.50 x 0.50 x –0.30 x 0.07 x 0.01} = √{ 0.00114 } = 0.034 0r 3.4%

Exercise for practice: A firm is considering four projects, 1, 2, 3, and 4. The proportion in which it wants to invest its total funds is 0.20, 0.30, 0.35 and 0.15. The risk-return characteristics of the projects are: 1 2 3 4 E(Ri )% 10 15 20 22 σi % 2 4 5 4 The correlation coefficients of the projects with each other are: 1 2 3 4 1. 1.00 0.12 0.65 0.75 2. 0.12 1.00 0.70 0.35 3. 0.65 0.70 1.00 0.80 4. 0.75 0.35 0.80 1.00 Calculate: (Use Equation 1 and 4) a. The return from the combination of assets (Ans. 16.80%) b. The expected portfolio standard deviation (Ans. 3.4% )