10.trace - University of California Los Angeles Department of Statistics Statistics C183/C283 Instructor Nicolas Christou Trace out the efficient

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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Trace out the efficient frontier Short sales allowed No riskless lending and borrowing Example On the next page we can see many combinations of three stocks (short sales are allowed). The characteristics of the stocks are: Stock 1 2 3 ¯ R 0.14 0.08 0.20 σ2 0.0036 0.0064 0.0400 The correlation coefficients are: ρ12 = 0.5, ρ13 = 0.2, ρ23 = 0.4. We assume the existence of two risk free rates to trace out the entire efficient frontier. Let Rf 1 = 0.05, and Rf 2 = 0.08. We will find the point of tangency for each one of the two risk free rates (points A and B ). • We begin with Rf 1 = 0.005 to find point A. We need to compute the zi ’s first: −1 0.0036 0.0024 0.0024 0.14 − 0.05 29.166667 = Σ−1 R1 = 0.0024 0.0064 0.0064 0.08 − 0.05 = −9.821429 . 0.0024 0.0064 0.0400 0.2 − 0.05 3.571429 ZA 3 i=1 zi The sum of the zi ’s is = 22.917. .166667 Therefore x1 = 2922.917 = 1.2727, x2 = −9.821429 = −0.4286, x3 = 22.917 Compute the mean and variance of the point of tangency A: 3.571429 22.917 = 0.1558. ¯ ¯ RA = xA R = 0.1751 2 σA = xA ΣxA = 0.005457. • Now we find the point of tangency (point B ) when Rf 2 = 0.08: We need to compute the zi ’s first: ZB 0.0036 = Σ−1 R2 = 0.0024 0.0024 The sum of the zi ’s is 3 i=1 zi 0.0024 0.0064 0.0064 −1 22.222222 0.14 − 0.08 0.0024 0.0064 0.08 − 0.08 = −11.904762 . 3.571429 0.0400 0.2 − 0.08 = 13.889. Therefore x1 = = 1.60, x2 = −11.904762 = −0.8571, x3 = 13.889 Compute the mean and variance of the point of tangency B : 22.222222 13.889 ¯ ¯ RB = xB R = 0.2069. 2 σB = xB ΣxB = 0.009134. • We also need the covariance between portfolios A and B : σAB = xA ΣxB = 0.006845. 1 3.571429 13.889 = 0.2571. • We treat now portfolios A and B as two “stocks”. Since we know their mean returns, variances, and covariance we can choose many combinations (allowing short sales) to trace the entire efficient frontier. This is shown on the last page. • Find the minimum risk portfolio (how much of each stock). Using the formulas that we discussed in class (look in your handout) we find x1 = 0.009134 − 0.006845 = 2.54, and x2 = −1.54. 0.005457 + 0.0009134 − 2(0.006845) Portfolio A consists of 1.2722 stock 1, -0.4286 stock 2, and 0.1558 stock 3. Portfolio B consists of 1.60 stock 1, -0.8571 stock 2, and 0.2571 stock 3. We conclude that the minimum risk portfolio consists of 0.7687 stock 1, 0.2314 stock 2, and -0.0002 stock 3. 0.32 The plot of many portfolios of the three stocks: q q q q 0.28 q q q q q q q q q q qq q qq q qq q q q qq q q q q q q q q q q q q q q q q qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q qq q q q q q q q qq q qq qq qq q q q q q q qq qq q q q q q qqq q q q q q q qq q q q q q qq q q q qq qqq q q q q q q q qq q q q q q q q qq q q q q qq qq q qq q q q qq q q q q q q qq qq q qq q q qq qq qq q qq q qq q q q q qq qq q q q qqq q q q q qq q qq qqq q q q q q qq q qqq q q q q q q q qq q q q q qq q q q q q q qqq q q q q qqq q q q qq qq qq q qq q q q q qq qq q q q q q qq q qq qq q q q q qqqqq qq q q q q qq qq q qq q qq qqq q q qq q qq q qq q q q q q q qq q q q q qq q qq q q q qq q q qq q qq q q q q qq q q qq q q qq q q qq q q q q qqqq q q q q qq qq q q q q q qq q q qq q q q q qq qqq q qq qqqq q qqq q q q q q qq q q q q q q q qqq q q q q q q q qq q qq q q qq q q q q q qq q q qq q q q q q qqq qqqqqqq q q qq qqqqqqq qq q qq q q q q q q q qqq qq q qqq q q q q q qq q q qq q qq q q q q qq q q q q qq qq qq qqqqq q qqqqq q qqq qqq qq q q q q q q qq q qq qqq q q q qq q q q q q qq q q q qq qq qq q qq qq qqqq qqqqqqqq q qq q q qq qqq q q qq qq q q q q qqqqq q q qq qqq q qq qqqq qq qq q q qq q q q qq qq q q q q qq q q q qq q q q q q qqqqqq qq q qq qqqqq q qq qq qq qqqqqq qq q q qqqq qq q q qq q q qq qq q qq q q q qq qq q q q q q qq q q q q qq q qq q q q q q q q qq q qq q qq q qqq q qqq q q q q q q qq q q q q q qqq q q qqq q q qq q qqq q qq q q q q q qq q q q q qq qq q q qq q q q q q q q q q q qq q q qq qq qq q q qq q q q qq qq q qq q q qq q q q qq q q qq q q q q q q q q qq q q qqqq qqq q q qq q qq qqqq q qq q q q q q qq q q qq qqq q qq q q q q qq q qq q q q qq q q q qq qq q q q qqqq q q qq qqq qqqq qq qqqqq q q q q qq q q q qq q q q qq q qq q qq q qq q q qq qqq q q q q q qq q q q q q qq q q q q q qq qq q q q qqqq q q q q q q q qq q q q q q qq q q q q q qq qqq q q qqqq qq qq qq q qqqq q q q q q qq q qq q q q q qqq q qq q q q q qq q q q qq qq q qq q q q qq q q qq q q q q qq q qq q qq qqq q q q q q q qq q q q q qqqq q qqq q q q qq q q q qqq q q qqqqq q q qq qqq q q qq qqqq q q q qq q q qq qq q q q qqqqqq qqqq q q q q q qq qqq q q q qq qq q qq q q qq q q q q qq q q q qq qq qq qq qq q q q q qqq qq q q qq q qqq qqq q qqq q q q q q q q q qq q q q q q q qq q q qqqq q q q q q q q q q qq q qqq q qqq q q q q q q qq q q q q q qq qq q q q q q q q qq q q qq qq q q q q q qq q q q qq q q qq q qq q q qq qq q q q q q qq q q qq q q q q q q q q q q qq q qq q qq q q q qq q qq q q qq qq q q q q qq q q qq q q q q q qq q q q qq q qqq q qq q q q qq q q q qq q q q q q qq q q q q q qq q q q qq q q qq q q q qq q q q q qq q q q q q q q q qq q q q q qqq q q q q q q q qq q q qq q q qq q q q q q q q qq q q qq qq q q q q q q q qq q qq q q q qq q q qq q q qqq qq q q q q q q q q q q q q q qqq q q q q q q qqq q qq q q q q q q q qq q q q q q qq q q q qq q q q qq qq q qq qq qq q q q q q q qq q qq q q q qq q q q q qq q q qqq qq qq q qq q q q q q q q q qq qq qq q qq q q q q qq q q qqqq q q qq q qq q q q q qq q q q qq q q q q qq q qq qq q q q q qq qq q q q q q 0.04 0.08 0.12 0.16 0.20 q 0.00 Expected return 0.24 qq q q q 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Portfolio standard deviation 2 q q 0.16 q qq q 0.18 q qq q q q qq 0.20 0.32 Trace out the efficient frontier: The plot below was constructed using many combinations of portfolios A and B allowing short sales. 0.20 B 0.04 0.08 0.12 0.16 A 0.00 Expected return 0.24 0.28 q q q qq qq q qq qq qq qq qq qq qq q qq q qq qq q q q q q qq qq qq qq qq qq qq qq q qq qq q q qq qq qq qq qq qq qq qq qq qq q q q q qq qq qq qq qq qq q qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq q q qq q qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq q q q qq qq qq qq qq qq qq qq qq qq q qq qq qq qq qq qq qq qq qq q qq qq qq qq qq qq qq qq qq qq qq qq qq q q q q q q q qq q qq qq qq q qq qq qq q q qq qq qq q qq q q q q q q q q q q q qq q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Portfolio standard deviation 3 0.16 0.18 0.20 ...
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This note was uploaded on 06/02/2011 for the course STATS 183 taught by Professor Nicolas during the Spring '11 term at UCLA.

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