1. Lecture 1- Part 2 Risk Management and Derivative by Stulz, Ch:2 Expected Return and Volatility

2. A key measure of investors’ success is the rate at which their funds have grown Holding-period return (HPR) of shares is composed of capital gain and dividend HPR = (P1-Po + Cash Dividend)/Po This definition assumes end of period returns and ignores re-investment of income Knowing risk and return of securities and portfolios

3. If the return on a stock is fixed, there will be 100% probability (Certaininty)that the return will be realized, like in bonds and T-bills In stocks, return is not fixed so the probability of all likely outcomes should be assessed Probability is the chance that the specified outcome will occur Return Distributions

4. Probability distribution is the specification of likely outcomes and the probability associated with each outcome Suppose we expect that PPL can give either 10%, 20% or -5% return. So we have three possible outcomes, if we associate chances of occurrence with each return, then it becomes probability distribution

5. Expected return is the single most likely outcome from a PD It is calculated by taking a weighted average of all possible return outcomes E(R) = ΣRiPi Expected Return

6. An Example Oil PricesReturnProbabilityRiPi Flat0.100.300.03 Rise0.20.50.1 Fall00.200 Sum.13

7. Variance of a random variable is a statistical tools that measures how the realization of the random variable are distribute around their expected values In other words it measures risk Variance = Σ[Ri-E(R)] 2 Pi Variance of Returns (Risk)

8. Standard Deviation: taking square root of.00600, we get value of 0.077 or 7.7% ReturnProbab ility RiPi[ Ri-E(R)] Σ[Ri-E(R)] 2 Σ[Ri-E(R)] 2 Pi 0.100.300.03-.03.0009.00027 0.20.50.1.07.0049.00245 00.200-.13.0169.00338 Sum.13.00600 Variance of Returns (Risk)

9. The cumulative distribution function of a random variable y specifies, for any number y, the probability that the realization of the random variable will be no greater than y For POL, a reasonable estimate of the stock return volatility is 9.2% with expected return of 13%, the following table show cumulative distributions functions for different levels of returns Cumulative Distribution Function

11. CDF can easily be calculated with MS Excel Put the equal sign in a cell = Open parenthesis and give x value (x means the level of return for which you want cumulative probability) Then give the mean return value, Then the standard deviation value And finally write TRUE and close parenthesis How to Calculate CDF

12. Taking values from the table in the previous slide, CDF is.59 with the 20% return level It means that there is 59% probability that return on POL will be less than 20% An investor has Rs.100,000 investment in POL and he wants that he does not lose more than Rs.30000 of his investment, what is the probability of this occurrence Interpretation

13. To calculate an average rate on a combination of stocks, we simply take the weighted average return of all stocks E(Rp) = Σwi E(Ri) Wi = Weight of the security in the poftfolio E(Rp) = The expected return on the portfolio Return of a Portfolio

14. Calculating portfolio return Calculating weights: PPL = 20000/60000 =.33 [FFC = 30000/6000 =.5] [Lucky = 10000/60000 =.16 StockValueReturnWeightΣwi E(Ri PPL2000015%.334.95 FFC3000012.56 Lucky1000010.161.6 Sum12.55

15. Calculating Portfolio Risk Risk of the porftolio is not the weighted average risk of the individual securities Rather it is determined by three factors – 1.the SD of each security – 2. the covariance between the securities – The weights of securities in the portfolio

16. Diversification By combining negatively correlated stocks, we can remove the individual risks of the stocks Example: Pol face the risk of falling oil prices PIA face the risk of rising oil prices By combining these two stocks, reduction in return in one stock due to change in oil price is compensated by increase in return of the other stock However, all of market risk cannot be eliminated through diversification

17. Investors should select portfolios on the basis of expected return and risk A portfolio is efficient if: – 1 it has the smallest level of risk for a given return or – 2. largest return for a given level of risk To select efficient portfolios, investors should find out all portfolios opportunities set i.e find out risk and return set for all portfolios Efficient Frontier

18. Example given in the Excel File Steps: 1. Calculate securities return 2. calculate portfolio returns 3. Find portfolio risk 4. Make different portfolios by changing weights of the securities 5. Find risk and return of each portfolio developed in step 4 6. Plot the risk and return of these portfolios 7. Find the minimum variance portfolio 8. Portfolios above the minimum variance portfolios are efficient Efficient Frontier