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Jaaziah Kish Dy Galing BSA 3
- Given below is a bivariate distribution for the random variables x and y. (a) Compute the expected value and the variance for x and y, The marginal distribution of X is given as, P(X=30)=0. P(X=40)=0. P(X=50)=0. The marginal distribution of Y is given as, P(Y=50)=0. P(Y=60)=0. P(Y=80)=0. The expected value of X will be given as, E(X)=∑xP(X=x)=(30×0.5)+(40×0.3)+(50×0.2) x = So the expected value of X is 37. Now, E(X2)=∑ x/x 2P(X=x) =(302×0.5)+(402×0.3)+(502×0.2) = So the variance of X will be given as, V(X)=1430−(37) = The variance of X is 61.
The expected value of Y will be given as, E(Y)=∑y/y P(Y=y) =(50×0.5)+(60×0.3)+(80×0.2)= So the expected value of Y is 59. Now, E(Y2)=∑y/y 2P(Y=y) =(502×0.5)+(602×0.3)+(802×0.2) = So the variance of Y will be given as, V(Y)=3610−(59) = The variance of Y is 129. (b) Develop a probability distribution for x+y. The probability distribution for Z=X+Y will be given as, Z= X+Y P(X+Y) 80 0. 100 0. 130 0. (c) Using the result of part (b), compute E(x+y) and Var(x+y). The expected value of X+Y will be given as, E(Z=X+Y)=∑z/z P(Z=z)
So the correlation coefficient is 0.98075. Since the coefficient is positive, X and Y are positively related to each other. (e) Is the variance of the sum of x and y bigger than, smaller than, or the same as the sum of the individual variances? Why? The sum of individual variances of X and Y is given as, V(X)+V(Y) =61+ = The variance of X+Y is 364. So the variance of the sum of X and Y is bigger than the sum of individual variances. This occurs because the variables are not independent, i.e., they are correlated, as expressed in the following variance property: V(aX+bY) =a^2 V(X)+b^2 V(Y)+2abCov(X,Y) The random variable X takes the values 30, 40, 50 and the random variable Y tales the values 50, 60, and 80. The joint distribution is given as, x Y=50 Y=60 Y= 30 0.5 0 0 40 0 0.3 0 50 0 0 0.
- A person is interested in constructing a portfolio. Two stocks are being considered. Let x 5 percent return for an investment in stock 1, and y 5 percent return for an investment in stock 2. The expected return and variance for stock 1 are E(x) 5 8.45% and Var(x) 5 25. The expected return and variance for stock 2 are E(y) 5 3.20% and Var(y) 5 1. The covariance between the returns is s xy 5 23.
a. What is the standard deviation for an investment in stock 1 and for an investment in stock 2? Using the standard deviation as a measure of risk, which of these stocks is the riskier investment? Stock 1 Stock 2 SDx =√v(x) SDY = √v(y) =√ 25 =√ 1 =5 = Stock 1 is the riskier investment. b. What is the expected return and standard deviation, in dollars, for a person who invests $500 in stock 1? E(x) = 500 (8.45%) SD = 500 (5%) = 42.25 % = 25% c. What is the expected percent return and standard deviation for a person who constructs a portfolio by investing 50% in each stock? Expected Percent Return = 0.5(8.45) + 0.5(3.20) = 4.225 + 1. = 5. SD = √((0.5(5))^2 +(0.5(1)^2 +2(0.5(0.5)(-3)) = √6.25+0.25+(-1.5) = √ 5 = 2. d. What is the expected percent return and standard deviation for a person who constructs a portfolio by investing 70% in stock 1 and 30% in stock 2? Expected Percent Return = 0.7 (8.45)+0.3(3.20) = 5.915 + 0. = 6. SD = √((0.7(5))2+(0.3(1))2+2(0.7)(0.3)(-3)) = √ 12.25+0.09+(-1.26) = √11. = 3.
Cov (X,Y) = -0.32 √0.19452 (0.02132) = -0. P(Y,Z) = Cov(Y,Z) / √Var(Y) Var(Z) -0.04 = Cov(Y,Z) / √0.02132 (0.23172) Cov(Y,Z) = -0.04 √0.02132 (0.23172) = -0. b. Construct a portfolio that is 50% invested in an S&P 500 index fund and 50% in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio? Solution: E[0.5X + 0.5Y] = 0.5(0.0504) + 0.5(0.0578) = 0. Var(0.5X+0.5Y) = 0.52(0.19452)+0.52(0.02132)+2(0.5)(0.5)(-0.0013257) = 0. sd(0.5X + 0.5Y) = √0.00890814 = 0. c. Construct a portfolio that is 20% invested in an S&P 500 index fund and 80% invested in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio? Solution: E[0.2X+0.8Y] = 0.2 (0.0504)+0.8(0.0578) = 0. Var(0.2X+0.8Y) = 0.22(0.19452)+0.82(0.02132)+2(0.2)(0.8)(-0.0013257) = 0. sd(0.2X+0.8Y) = √0.00137935 = 0. d. Construct a portfolio that is 80% invested in an S&P 500 index fund and 20% invested in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio? Solution: E[0.8Y+0.2Z] = 0.8(0.0504)+0.2(0.0578) = 0. Var(0.8Y+0.2Z) = 0.8^2 (0.1945^2 )+0.2^2 (0.0213^2 )+2(0.8)(0.2)(-0.0013257) = 0. sd(0.8Y+0.2Z) = √0.02380528= 0.
e. Which of the portfolios in parts (b), (c), and (d) has the largest expected return? Which has the smallest standard deviation? Which of these portfolios is the best investment alternative? Solution: Part (d) has the largest expected return. Part (c) has the smallest standard deviation. f. Discuss the advantages and disadvantages of investing in the three portfolios in parts (b), (c), and (d). Would you prefer investing all your money in the S&P 500 index, the core bonds fund, or one of the three portfolios? Why? Those who wish to be more conservative with their money should invest in portfolios, whilst those who want a larger return should be more ready to lose their money because those portfolios have higher standard deviations. Part (c) is a fairly prudent investment, but the payout is very low. Part (b) would be a moderately risky investment since the standard deviation is higher than in part (c) but lower than in part (d) (d). Part (d) is regarded as a more aggressive investment due to the higher predicted returns.
