# lect11 - Real Portfolios Normal Random Variable The Normal...

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Real Portfolios Normal Random Variable The Normal Model Outline Real Portfolios Normal Random Variable Normal Probability Distribution Central Limit Theorem Probabilities as Areas The Normal Model Standardizing Using Normal Tables The Empirical Rule, Revisted 1 / 23 ISOM 2500 Lect 11: Real Portfolios; Normal Model I

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Real Portfolios Normal Random Variable The Normal Model Real Portfolios Recall the pink portfolio in Game II. Let’s now move beyond the Pink portfolio and consider portfolios of real stocks traded in the Hong Kong Stock Exchange. Two aspects need to be considered: Returns on the individual investments are typically not independent of each other. The probability distributions of the returns on the individual investments are unknown. Thus, We cannot assume independence. The unknown characteristics must be estimated from data. 2 / 23 ISOM 2500 Lect 11: Real Portfolios; Normal Model I
Real Portfolios Normal Random Variable The Normal Model 1_5returns.JMP contains the monthly (net) returns of Cheung Kong(0001.HK) and HSBC Holdings(0005.HK) from 08/1981-07/2011. Let’s try to form an equally-weighted portfolio 1 HSCK = 0 . 5 HSBC + 0 . 5 CheungKong . 1 As in the dice simulation, this portfolio rebalances after each month so that half of the value of the portfolio is kept in HSBC Holding and the other half in Cheung Kong. 3 / 23 ISOM 2500 Lect 11: Real Portfolios; Normal Model I

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Real Portfolios Normal Random Variable The Normal Model From these summaries we obtain the volatility-adjusted returns 2 as follows: Investment Mean Variance Mean – Var/2 HSBC Holdings 0.0106 0.0067 0.0073 Cheung Kong 0.0143 0.0122 0.0082 HSCK 0.0125 0.0079 0.0085 Once again, a portfolio offers an improvement in long-term gains over investing 100 % in either of the individual investments. For the equally-weighted portfolio of HSBC Holdings and Cheung Kong stocks, the mean return on the portfolio is just the average of the mean returns on the two stocks Mean(HSCK) = 0 . 0125 = 0 . 5 ( 0 . 0106 ) + 0 . 5 ( 0 . 0143 ) but for the variance we find: Var(HSCK) = 0 . 0079 6 = 0 . 5 2 ( 0 . 0067 ) + 0 . 5 2 ( 0 . 0122 ) which was the formula we used for Pink. What’s missing? 2 The volatility-adjusted return formula for gross returns from Lect 10 also holds for net returns. 4 / 23 ISOM 2500 Lect 11: Real Portfolios; Normal Model I
Real Portfolios Normal Random Variable The Normal Model Role of Covariance It turns out that returns on Cheung Kong and HSBC Holdings are dependent. The correlation between Cheung Kong and HSBC Holdings is 0.7094. The covariance between Cheung Kong and HSBC Holdings = ( 0 . 7094 )( 0 . 1106 )( 0 . 0818 ) = 0 . 00642 For the variance of a weighted sum, 0 . 0079 = 0 . 5 2 ( 0 . 0067 ) + 0 . 5 2 ( 0 . 0122 ) + 2 ( 0 . 5 )( 0 . 5 )( 0 . 00642 ) 5 / 23 ISOM 2500 Lect 11: Real Portfolios; Normal Model I

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Real Portfolios Normal Random Variable The Normal Model Beyond equally weighted portfolios of two stocks The pairwise portfolios that we considered put equal weight on the two stocks in the portfolio. There’s no need to divide the investment equally.
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