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Session 8

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Session 8

  1. 1. 7/17/2012 Occurrence of these breakdown may be assumed to constitute a Poisson process. NORMAL APPROXIMATION TO Average no. of breakdowns per month (based on last 5 years data) POISSON is 150/60 = 2.5. Similarly, a Poisson distribution with large µ can be Assuming that there is no change in this random process (due to change in the policy), we would compute the probability of approximated by an appropriate NORMAL distribution having 2 breakdowns given that there are six breakdowns in the last couple of months. If this probability is low, we should come to the conclusion that the change of policy is likely to have caused more breakdowns in the past couple of weeks; and hence Q. What should be the mean and variance of this NORMAL caused the production to fall during Raj’s vacation. distribution? Normal approximation of Risk and Return from Investment Binomial n=2 n=5 n=10 600 450 350 400 500 300 350 250 400 300Frequency Frequency Frequency 250 200 300 200 150 200 150 100 100 100 50 50 0 0 0 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Bin Bin Bin n=20 n=25 250 200 180 200 160 140 Frequency Frequency 150 120 100 100 80 60 50 40 20 0 0 0 0.08 0.16 0.24 0.32 0.4 0.48 0.56 0 1 2 3 4 5 0. 0. 0. 0. 0. Bin Bin 25 1
  2. 2. 7/17/2012 Mean & Standard deviation of RETURN Covariance and Correlation between (Stock X) return from Stocks X and Y State of the X = Return Y= Return (x- mu_X) economy p = Prob. stock X stock Y *(y-mu_Y) x*yState of the p= X = Return from 1 0.05 15 -5 231 -75 economy Probability stock X (in %) (x -mu_X)^2 x^2 2 0.05 -5 15 31 -75 1 0.05 15 121 225 3 0.1 5 25 -189 125 4 0.5 35 5 -99 175 2 0.05 -5 961 25 5 0.3 25 35 -19 875 3 0.1 5 441 25 4 0.5 35 81 1225 mean 26 16 -61 355 5 0.3 25 1 625 variance 139 199 std dev 11.79 14.11 check mean 26 139 815 covariance -61 355-26*16= -61 correlation -0.367 variance 139 815-26^2= 139 std dev 11.79 Expected Value / Standard Covariance and Correlation between deviation of return from Stock returns from Stock A and Stock B R B : retu rn fro m S to ck B is an o th er ran d o m variab le RA : return from Stock A is a random variable E x p ected valu e o f retu rn fro m S to ck B : Expected value of return from Stock A: E (RB ) = ∑b i × p i = µ B , w h ere i E ( RA ) = ∑ ai × pi = µ A , where b i = retu rn fro m S to ck B w h en sta te o f th e eco n o m y is i i V arian ce o f retu rn fro m S to c k A : pi = P(state of the economy is i) σ 2 = E (RB − µ B )2 = B ∑ (b i − µ B ) 2 × pi = ∑ (b ) i 2 × pi − µ 2 B i i ai = return from Stock A when state of the economy is i C o varian ce b etw een retu rn s fro m S to ck A an d S to ck B : Variance of return from Stock A: σ A B = E [ ( R A − µ A )( R B − µ B ) ] = ∑ (a i − µ A ) × ( bi − µ B ) × p i i σ 2 = E ( RA − µ A ) 2 = ∑ (ai − µ A ) 2 × pi = E [R A RB ] − µ Aµ B = ∑a i × bi × p i − µ A × µ B A i i C o rrela tio n b etw een retu rn s fro m S to ck A an d S to ck B : = E ( RA ) 2 − µ 2 = ∑ (ai ) 2 × pi − µ 2 A A ρ AB = σ AB i σA × σB Mean and Standard deviation of Average Return and Std. Dev of return RETURN from stock Y weight for from a Portfolio stock 0.8 0.2 P= portfolio (P-mu_P)^2State of the p= Y= Return from State of the X= Return Y= Return return economy p = Prob. stock X stock Y economy Probability stock Y (in %) (y -mu_Y)^2 y^2 1 0.05 15 -5 11 169 1 0.05 -5 441 25 -1 625 2 0.05 -5 15 2 0.05 15 1 225 3 0.1 5 25 9 225 3 0.1 25 81 625 4 0.5 35 5 29 25 4 0.5 5 121 25 5 0.3 25 35 27 9 5 0.3 35 361 1225 mean 26 16 24 77.4 mean 16 199 455 variance 139 199 77.4 std dev 11.79 14.11 8.80 variance 199 455-16^2= 199 covariance -61 0.8*26+0.2*16 std dev 14.11 correlation -0.367 0.8^2 * 139 + 0.2^2 * 199+ 2*0.8*0.2 *(-61) 2
  3. 3. 7/17/2012 Risk and Return from Portfolio: Diversification of Portfolio Portfolio: Out of 1 unit, put w unit in A and (1- w) in B Return from portfolio: P = w × RA + (1- w) × RB Expected value of return from portfolio: E ( P) = w × µ A + (1- w) × µ B Variance of return from portfolio σ 2 = w2 × σ A + (1 − w) 2 × σB + 2w(1 − w)σ AB P 2 2 If the two returns are negatively correlated (negative covariance) The portfolio may have reduced risk (standard deviation) Portfolio Diversification and Asset allocation• The benefit of ‘diversification’• The above formulae can be elegantly formulated using matrix• Computation using Excel – Portfolio Diversification – Data file• Optimization technique 3

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