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Q: Method to calculate Beta coefficient ( Answered ,   0 Comments ) Question
 Subject: Method to calculate Beta coefficient Category: Business and Money > Finance Asked by: staedtler-ga List Price: \$20.00 Posted: 29 Sep 2006 23:22 PDT Expires: 29 Oct 2006 22:22 PST Question ID: 769639
 ```Question: There are two options to invest \$60,000 with correlation coefficient with -0.50. Option a: expected return=9%, standard deviation=7.56% Option b: expected return=8%, standard deviation=3.75% Part I: create table with portfolio expected return and standard deviation for 100%, 67%, 33%, and 0% in option a. Part II: If kRF=7.0% and kM=10.0%, what are the beta coefficients for Option a and b? Part III: If leverage with risk free rate of 5.0%, what are the portfolio beta with 40% of \$60,000 allocated to option a, b=35%, and RF=25%? Thank you in advance.``` Subject: Re: Method to calculate Beta coefficient Answered By: livioflores-ga on 01 Oct 2006 23:15 PDT Rated: ```Hi!! Part I: create table with portfolio expected return and standard deviation for 100%, 67%, 33%, and 0% in option a. If we call: E(RP) = expected return on the portfolio E(Ra) = expected return on option a E(Rb) = expected return on option b STDa = standard deviation of a STDb = standard deviation of b STDP = standard deviation of the portfolio Wa = weight or proportion of option a in the portfolio Wb = weight or proportion of option b in the portfolio ; then: E(RP) = (Wa)*[E(Ra)] + (Wb)*[E(Rb)] Variance = (Wa)^2*(STDa)^2 + (Wb)^2*(STDb)^2 + + 2*(Wa)*(Wb)*(STDa)*(STDb)*[Correlation(a,b)] = Since Wa+Wb=1 ==> Wb=1-Wa; then: E(RP) = (Wa)*[E(Ra)] + (1-Wa)*[E(Rb)] And Variance = (Wa)^2*(STDa)^2 + (1-Wa)^2*(STDb)^2 + + 2*(Wa)*(1-Wa)*(STDa)*(STDb)*[Correlation(a,b)] STDP = sqrt(Variance) - 100% in option a: E(RP) = (1)*[9%] + (1-1)*[8%] = 9% Variance = (1)^2*(7.56)^2 + (1-1)^2*(3.75)^2 + + 2*(1)*(1-1)*(7.56)*(3.75)*(-0.5) = = 57.1536 STDP = sqrt(57.1536) = 7.56 - 67% in option a: E(RP) = 0.67*9% + (1-0.67)*8% = 8.67% Variance = (0.67)^2*(7.56)^2 + (1-0.67)^2*(3.75)^2 + + 2*(0.67)*(1-0.67)*(7.56)*(3.75)*(-0.5) = = 20.91947229 STDP = sqrt(20.91947229) = 4.574 - 33% in option a: E(RP) = 0.33*9% + (1-0.33)*8% = 8.33% Variance = (0.33)^2*(7.56)^2 + (1-0.33)^2*(3.75)^2 + + 2*(0.33)*(1-0.33)*(7.56)*(3.75)*(-0.5) = = 6.26849829 STDP = sqrt(6.26849829) = 2.503 - 0% in option a: E(RP) = 0*9% + (1-0)*8% = 8% Variance = (0)^2*(7.56)^2 + (1-0)^2*(3.75)^2 + + 2*(0)*(1-0)*(7.56)*(3.75)*(-0.5) = = 14.0625 STDP = sqrt(14.0625) = 3.75 Proportion Proportion of Portfolio Portfolio of of Expected Standard option a option b Return Deviation 100% 0% 9.00% 7.56 67% 33% 8.67% 4.574 33% 67% 8.33% 2.503 0% 100% 8.00% 3.75 ---------------------------- Part II: If kRF=7.0% and kM=10.0%, what are the beta coefficients for Option a and b? According to CAPM: E(Rs) = kRF + Beta_s*(kM - kRF) ==> Beta_s = (E(Rs)-kRF) / (kM - kRF) Beta of a: Beta_a = (9-7)/(10-7) = 2/3 Beta of b: Beta_b = (8-7)/(10-7) = 1/3 --------------------- Part III: If leverage with risk free rate of 5.0%, what are the portfolio beta with 40% of \$60,000 allocated to option a, b=35%, and RF=25%? Wa = 0.40 Wb = 0.35 Wrf = 0.25 If risk free rate change from 7% to 5% the betas of a and b will change also: Beta of a: Beta_a = (9-5)/(10-5) = 0.8 Beta of b: Beta_b = (8-5)/(10-5) = 0.6 Beta of the risk free asset is zero. The beta of a portfolio is the weighted average of the individual asset betas where the weights are the portfolio weights: Beta_P = 0.40*0.8 + 0.35*0.6 + 0.25*0.0 = 0.53 --------------------- For further reading see the following articles: "CAPM - The Capital Asset Pricing Model": http://www.teachmefinance.com/capm.html "Modern portfolio theory - Wikipedia, the free encyclopedia": http://en.wikipedia.org/wiki/Modern_portfolio_theory "Understanding Beta and Market Risk - INVESTOR SOLUTIONS - THE STRATEGIC ADVANTAGE": http://www.investorsolutions.com/lc-library.cfm?show=detail&articleID=424&artcategory=1 "Portfolios of Stocks": http://www.econ.ucsb.edu/~sparendo/134a/lecture11.pdf "Portfolios of Two Assets" by William F. Sharpe: http://www.stanford.edu/~wfsharpe/mia/rr/mia_rr5.htm Search strategy: "Expected Return on a Portfolio" capm "beta of a Portfolio" "beta of a Portfolio" "risk free asset" I hope this helps you. Please do not hesitate to request for a clarification if you find something unclear (like a typo or a miscalculation) and/or incomplete before rate this answer. Best regards, livioflores-ga```
 staedtler-ga rated this answer: and gave an additional tip of: \$2.00 ```Thank you very much. I was very unclear with the question 2, and you have clarified it with good research pages. Thanks.```  