43. Practice problems.

43. Practice problems. - University of California Los...

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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Practice problems Problem 1 You are given the folowing data on 7 stocks: Stock i 1 2 3 4 5 6 7 ¯ Ri 0.15 0.20 0.18 0.12 0.10 0.14 0.16 σi 0.10 0.15 0.20 0.10 0.05 0.10 0.20 ¯ R −R Using these data we ranked the stocks based on the excess return to standard deviation ratio ( i σi f ) to compute the entries in the next table. Assume Rf = 5% and that the average correlation coefficient is ρ = 0.50. Stock i 1 2 5 6 4 3 7 ¯ Ri −Rf σi 1.00 1.00 1.00 0.90 0.70 0.65 0.55 ¯ Rj −Rf i σj j =1 ρ 1−ρ+iρ 0.5000 0.3333 0.2500 0.2000 0.1667 0.1429 0.1250 1.00 2.00 3.00 3.90 4.60 5.25 5.80 Ci 0.5000 0.6667 0.7500 ??? 0.7668 0.7502 0.7250 a. Find the missing value C4 . b. What is the composition of the optimum portfolio assuming no short sales? c. What is the expected return and standard deviation of the combination of the optimum portfolio with the risk free asset (80% and 20%)? Show this combination on the graph of expected return against standard deviation. Problem 2 You are given the following data: Stock i 1 2 3 ¯ Ri 0.29 0.19 0.08 σi 0.03 0.02 0.15 a. Assume short sales are allowed, Rf = 0.05, and ρ = 0.5. Rank the stocks based on the excess return to standard ¯ deviation ratio, find the cut-off point C ∗ , and find the optimum portfolio. b. The above solution could have been found using the techniques that discussed earlier in class through the following: Z = Σ−1 R = 0.00090 0.00030 0.00225 0.00030 0.00040 0.00150 0.00225 0.00150 0.02250 −1 0.29 − 0.05 0.19 − 0.05 0.08 − 0.05 = 280.00 320.00 −48.00 Explain what you see here and verify that the solution is the same as with part (a). . Problem 3 Suppose the single index model holds. Also, short sales are allowed and there is a risk free rate Rf = 0.002. For 3 stocks the following were obtained based on monthly returns for a period of 5 years: Stock i 1 2 3 α 0.01 0.04 0.08 β 1.08 0.80 1.22 σ2 0.003 0.006 0.001 2 ¯ The expected return and variance of the market are Rm = 0.10 and σm = 0.002 for the same period. a. Suppose that the optimum portfolio consists of 30% of stock 1, 50% of stock 2, and 20% of stock 3. What is the β of this portfolio. b. Suppose that you are a portfolio manager and you have $500000 to invest in this optimum portfolio on behalf of a client. In addition this client wants to invest another $300000 by borrowing this amount at the risk free rate Rf = 0.002. What is the expected return and standard deviation of this portfolio. Show it on the expected return standard deviation space. c. What is the covariance between the portfolio of part (a) and the market? d. If the client wants to allocate 60% of his initial funds in the optimum portfolio and the remaining 40% in the risk free asset, what would be the expected return and standard deviation of this position? e. What is the covariance between stock 1 and the market? Problem 4 2 Assume that σm = 10, Rf = 0.05. You are also given β1 = 1, β2 = 1.5, β3 = 1, β4 = 2, β5 = 1, β6 = 1.5, β7 = 2, β8 = 0.8, β9 = 1, β10 = 0.6. The table below shoes the procedure for finding the cut-off point C ∗ . Stock i 1 2 3 4 5 6 7 8 9 10 ¯ Ri −Rf βi ¯ (Ri −Rf )βi σ 2i 10.0 8.0 7.0 6.0 6.0 4.0 3.0 2.5 2.0 1.0 ¯ (Rj −Rf )βj i j =1 σ2 0.20 0.45 0.35 2.40 0.15 0.30 0.30 0.10 0.10 0.06 j (a) 0.65 1.00 3.40 3.55 3.85 4.15 4.25 4.35 4.41 2 βj i j =1 σ 2 j 2 βi σ 2i 0.02000 0.05625 0.05000 0.40000 0.02500 0.07500 0.10000 0.04000 0.05000 0.06000 0.02000 0.07625 0.12625 0.52625 0.55125 0.62625 (b) 0.76625 0.81625 0.87625 Ci 1.67 3.69 4.42 5.43 (c) 5.30 5.02 4.91 4.75 4.52 a. Find the three missing values, (a), (b), (c) in the table above. b. If short sales are not allowed find the cut-off point C ∗ and the value of z1 . c. If short sales are allowed find the cut-off point C ∗ and the value of z1 . d. Find the correlation coefficient between stock 1 and the market. Problem 5 For three stocks you are given the following data based on the single index model: ¯ Stock R β σ2 A 0.0051 0.94 0.0033 B 0.0120 0.61 0.0038 C 0.0160 1.12 0.0046 Below you are given the solution to the problem (the point of tangency) when short sales are allowed and Rf = 0.005. Z=Σ −1 R= 0.00489048 0.00103212 0.00189504 0.00103212 0.00446978 0.00122976 0.00189504 0.00122976 0.00685792 −1 0.0051 − 0.005 0.0120 − 0.005 0.0160 − 0.005 = −0.883563202 1.327096101 1.610164293 . 3 The sum of the zi ’s is z = 2.053697192 and therefore the xi ’s are: i=1 i x1 = −0.4302, x2 = 0.6462, x3 = 0.7840. The above is one way to solve the problem. We can also solve the problem by ranking the stocks based on the excess return to beta ratio. 2 a. Verify the entries in the table below that will allow you to find the C ∗ . You will also need σm = 0.0018. Stock i 2 3 1 αi 0.0059 0.0048 -0.0043 ¯ Ri 0.61 1.12 0.94 0.0120 0.0160 0.0051 0.0038 0.0046 70.0033 Ri −Rf ˆ βi ˆ σ2 i ˆ βi 0.0114754098 0.0098214286 0.0001063830 ˆ ¯ (Ri −Rf )βi ˆ σ2 i 1.12368421 2.67826087 0.02848485 i ˆ ¯ (Rj −Rf )βj ˆ σ2 j 1.123684 3.801945 3.830430 j =1 ˆ β2 i ˆ σ2 i 97.92105 272.69565 267.75758 ˆ β2 j 2 j =1 σˆ j 97.92105 370.61670 638.37428 i Ci 0.001719548 0.004105009 0.003208254 b. Assume short sales are allowed. Find C ∗ and use it to find the composition of the optimum portfolio (point of tangency). Your answer should be exactly the same as above. c. Assume short sales are not allowed. Find C ∗ and use it to find the composition of the optimum portfolio. d. Compute the mean return and standard deviation of the portfolios in (b) and (c) and place them (approximately) on the graphs below. Your answer should be the point of tangency in both cases. Note: The first graph allows short sales, while the second graph does not. e. Write down the expression in matrix form that computes the covariance between the portfolio of part (b) and 1 1 the equally allocated portfolio ( 3 A, 1 B, 3 C ). No calculations, just the expression! 3 f. Consider the portfolio of part (b). Suppose that you want to place 60% of your funds in portfolio (b) and invest the other 40% in the risk free asset. Find the mean return and standard deviation of this new portfolio and show it on the first graph. g. You have $2000 to invest in portfolio (b). In addition you borrow another $1000 to invest in portfolio (b). Show the position of this portfolio on the first graph (approximately). 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qq q q −0.025 −0.01 Expecrted return Short sales allowed (question (d)): 0 0.04 0.1 0.16 0.22 0.28 0.34 q 0.4 Risk (standard deviation) 0.017 Short sales not allowed (question (d)): 0.011 0.008 0.005 0 0.002 Expecrted return 0.014 q q q qq q qq q qq qqq qq q q qq q qq q q qqq qq q q qq q q qq qq qq qqq qq qqqq q qqq qq q q qq q qq q q qq q qq q qqq q qq q qq q qqqqq qq q q q q qq q q q qq q q q qqq qq q q q q qqq qq q q q q qqq q q q q qqq q q q q q q q q qqq qq q q qq q q q qq q q q q q qq q q qq q q q qqq q qq q q qqq qqq q q q q qq q q q qq qqqq q qq q q qq q q q q q qq q q qqq qqqqqq q qq q q q q qq q q qq q q qqqqq qq qq qqqqq q qq qq qqqq q q q q qq q qq qq qq q qq q q q q qqq q q q q qq qqq qq q qqq q q q qq qqq q q q q q qq qq q q qq q q qq qqq qq qq q qq qqq q q q qqqq q qq q qqq qqq qq q q qqq qq q qq qq qqqqq qq q qq qq q q qqqq q qq q qq q qq q qq qq qq qqqq qqqqq qqqq q q q qqqq qqq q qq q qqqq qqq q qqqq qqqq qqq q qqqq q qqq qq qqqq q qqqq qq qqq qqqq qqqqqq qq q qqqqqq 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qqqq qq q qqq qq q qq q q q q qq qqqq qq qq qq q qq q qq q qqqq qqq q qqq q qqq qqq qq qq qq q q q qq qqq q qqq q qq q qq qq qq q q q 0 0.01 0.02 0.03 0.04 0.05 0.06 Risk (standard deviation) 0.07 0.08 0.09 Problem 6 Using the constant correlation model we completed the table below on 12 stocks. Assume Rf = 0.05 and average correlation ρ = 0.45. Stock i 1 2 3 4 5 6 7 8 9 10 11 12 ¯ Ri 0.27 0.31 0.16 0.15 0.33 0.27 0.19 0.13 0.16 0.12 0.08 0.06 σi 0.031 0.042 0.023 0.021 0.059 0.061 0.039 0.029 0.051 0.038 0.022 0.028 ¯ Ri −Rf σi ρ 1−ρ+iρ 7.097 6.190 4.783 4.762 4.746 3.607 3.590 2.759 2.157 1.842 1.364 0.357 0.450 0.310 0.237 0.191 a =? 0.138 0.122 0.108 0.098 0.089 70.082 0.076 ¯ Rj −Rf i σj j =1 7.097 13.287 18.070 22.832 b =? 31.184 34.774 37.532 39.689 41.531 42.895 43.252 Ci 3.194 4.124 4.280 4.372 c =? 4.318 4.229 4.070 3.883 3.701 3.510 3.271 a. Find the three missing numbers a, b, c in the table above. b. Find the cut-off point C ∗ if short sales are not allowed. c. Find the cut-off point C ∗ if short sales are allowed. d. Write down the expression in matrix form that computes the variance of the portfolio when short sales are allowed. No calculations. ¯ e. You are given a new stock with R = 0.055, σ = 0.025. What will change when short sales are allowed and when short sales are not allowed in terms of the portfolio allocation. Briefly explain your answer without doing all the calculations. Problem 7 Answer the following questions: a. Assume that the variance of security A is 0.16 and the variance of security B is 0.25. The variance of a portfolio consisting of 50%A and 50%B is 0.0525. What is the covariance between securities A and B ? ¯ ¯ b. Assume Rf = 0.05 and two stocks A, B with RA = 0.14, RB = 0.10. Suppose the point of tangency to the efficient frontier (the one constructed using the two stocks), consists of 60%A and 40%B . Let’s say that you want to build a portfolio by combining the risk free asset and portfolio G to obtain expected return 11%. Determine the percentages of your investment in the risk free asset and in portfolio G. c. Consider the data from part (b). Suppose you want to build a portfolio with expected return 0.10 by combining the risk free asset and portfolio G. What is the composition of this portfolio in the risk free asset, in A, and in B? ¯ ¯ d. Suppose two stocks have the following: RA = 0.14, σA = 0.06, RB = 0.08, σB = 0.03. What value of ρAB will force you to invest everything in the least risky asset? e. Consider the data from part (d). If short sales are allowed, what composition of A and B will minimize the risk of the resulting portfolio if ρAB = 0.80? Problem 8 Explain how you would trace out the efficient frontier using the data of problem 3 when short sales are allowed but no riskless lending and borrowing. You do not need to perform any calculations, but you must show a graph and the inputs of the method you are using. ...
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