Portfolio Analysis

1. Portfolio Analysis `putting all eggs in a basket is not desirable'

2. Meaning • collection or combination of financial assets (or securities)such as shares, debentures and government securities • Held for `investment' purposes and not for `consumption' purposes.

3. Risk –Return Characteristics of individual Stock in portfolio • Historical return of individual stock is measured using either A) Holding Period Return B) Arithmetic Average c) Geometric Average

4. Holding Period Return • Most simplest • Holding Period Return = Income + (End of Period Value – Initial Value) / Initial Value • Annualized HPR =  {[(Income + (End of Period Value – Initial Value)] / Initial Value+ 1}1/t – 1 • Annualized HPR is used to compare return of assets held for different time

5. Holding Period Return • Stock A that was held for three years, during which it appreciated from 100 Rs to 150 Rs and provided Rs 5 in dividend each year, or Stock B that went from Rs 200 to Rs 320 and generated Rs 10 in dividendeach four years. Which performed better? • HPR of A= (3*5+(150-100))/100= 65 % • HPR of B= (10 *4+ (320-200))/200= 80% • Annualized HPR of A=(0.65+1)^(1/3)-1= 18.1% • Annualized HPR of B= (1+0.8)^(1/4)-1= 15.8 %

6. Holding Period Return • Returns for regular time periods such as quarters or years can be converted to a holding period return through the following formula: • (1 + HPR) = (1 + r1) x (1 + r2) x (1 + r3) x (1 + r4) where r1, r2, r3and r4are periodic returns. • Thus, HPR = [(1 + r1) x (1 + r2) x (1 + r3) x (1 + r4)] – 1

7. Holding Period Return • Your stock portfolio had the following returns in the four quarters of a given year: +8%, -5%, +6%, +4% • HPR for your stock portfolio = [(1 + 0.08) x (1 – 0.05) x (1 + 0.06) x (1 + 0.04)] – 1 = 13.1%

8. Arithmetic Average • Sum of returns of period and divided by`n‘ • Fails to consider time value of money • differential treatment of positive and negative return. • For instance if a stock price increases from Rs. 10 to Rs. 20 in period 1 and declines back to Rs. 10 in period 2, the Arithmetic average return is still positive value of 25%

9. Geometric Return • The geometric average return is based on the compound value and is also called time-weighted average return Where R1, R2 , R3 are HPR of year 1 ,2 and 3

10. Example • Five years back, you have applied and was allotted 100 shares of a company at the rate of Rs. 50 per share (Face Value Rs. 10). The price at the end of each year along with annual dividend per share received from the stock are as follows:

11. Example

12. Risk Profile • Assets with same expected return might have different risk profile Both A and B has same expected rate of return but Risk of A is greater as it has wider variability

13. Risk Return Profile of Portfolio • Return: Return is still the expected return, but for a portfolio the return will be the average return from all the assets held in the portfolio • Risk: Risk is still the variance (or standard deviation) of the expected returns from the portfolio. However, the risk of a combination of assets is very different from a simple average of the risk of individual assets. • Its possible the variance of a portfolio of two assets maybe less than the variance of either of the assets themselves

14. Expected Return of Portfolio • R(p) = the expected return of the portfolio; • Xi = the proportion of the portfolio's initial fund invested in asset ‘i’ • Ri = the expected return of asset `i'; • n = the number of assets in the portfolio.

15. Risk of Portfolio • where ‘σij’ denotes the covariance of returns between asset i and asset j. • Closely related to covariance is the statistical measure known as correlation. The relationship is given by • Where ρij is correlation coefficient

16. Risk of Portfolio • Suppose there are three stocks X, Y, Z with weight Wx, Wy and Wz. Then double summation is obtained by NOTE: More the number of stocks more complicated it becomes

17. Risk of Portfolio • Adding More Securities reduces the risk • For example if Initially there are three equally weighted stocks • The sum of the diagonal cells is equal to sum of the variance of three securities multiplied by (1/3)^2 • If we add one more security. Then sum will be (1/4)^2 • Thus (1/4)^2 < (1/3)^2

18. Risk of Portfolio • As we keep on adding the portfolio the multiplier keeps on decreasing and sum of diagonal reaches towards zero but not zero (asymptote) • Diversification means the spreading of investments over more than one asset with a view to reduce the portfolio's risk (i.e., the variability of expected portfolio returns).

19. What kind of security to be included in Portfolio • More than the number of securities, what matters in reducing the risk of the portfolio is the kind of securities included in the portfolio • risk of the portfolio can be reduced to zero if the correlation between the assets included in the portfolio is equal to minus 1

20. Correlation and Portfolio Risk

22. When number of Security increases • For every securities in the form a two security portfolio and find the portfolio return and risk for various combinations • Above plot shows six securities • on the dashed line. There are four portfolios offering same risk but different returns • Return of Portfolio Y is greater and return of Portfolio X is least

23. Efficient Frontier • If we keep on removing the inefficient portfolio then we can obtain one smooth efficient curve (AB) • Based on Risk appetite of the investor he may be at point A or B

24. Efficient Frontier • Obtaining Efficient frontier is a class of Constraint Satisfaction Problem (CSP) • Can be obtained using Quadratic programming

25. Efficient Frontier As number of securities increase it becomes more and more complicated

26. Markowitz Portfolio Selection Model • the model was presented by Harry Markowitz briefly in 1952 and later in a complete book entitled Portfolio Selection-Efficient Diversification of Investments (1959).