

Subject:
Method to calculate Beta coefficient
Category: Business and Money > Finance Asked by: staedtlerga 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: liviofloresga 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=1Wa; then: E(RP) = (Wa)*[E(Ra)] + (1Wa)*[E(Rb)] And Variance = (Wa)^2*(STDa)^2 + (1Wa)^2*(STDb)^2 + + 2*(Wa)*(1Wa)*(STDa)*(STDb)*[Correlation(a,b)] STDP = sqrt(Variance)  100% in option a: E(RP) = (1)*[9%] + (11)*[8%] = 9% Variance = (1)^2*(7.56)^2 + (11)^2*(3.75)^2 + + 2*(1)*(11)*(7.56)*(3.75)*(0.5) = = 57.1536 STDP = sqrt(57.1536) = 7.56  67% in option a: E(RP) = 0.67*9% + (10.67)*8% = 8.67% Variance = (0.67)^2*(7.56)^2 + (10.67)^2*(3.75)^2 + + 2*(0.67)*(10.67)*(7.56)*(3.75)*(0.5) = = 20.91947229 STDP = sqrt(20.91947229) = 4.574  33% in option a: E(RP) = 0.33*9% + (10.33)*8% = 8.33% Variance = (0.33)^2*(7.56)^2 + (10.33)^2*(3.75)^2 + + 2*(0.33)*(10.33)*(7.56)*(3.75)*(0.5) = = 6.26849829 STDP = sqrt(6.26849829) = 2.503  0% in option a: E(RP) = 0*9% + (10)*8% = 8% Variance = (0)^2*(7.56)^2 + (10)^2*(3.75)^2 + + 2*(0)*(10)*(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 = (97)/(107) = 2/3 Beta of b: Beta_b = (87)/(107) = 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 = (95)/(105) = 0.8 Beta of b: Beta_b = (85)/(105) = 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/lclibrary.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, liviofloresga 
staedtlerga
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. 

There are no comments at this time. 
If you feel that you have found inappropriate content, please let us know by emailing us at answerssupport@google.com with the question ID listed above. Thank you. 
Search Google Answers for 
Google Home  Answers FAQ  Terms of Service  Privacy Policy 