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.

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

Thank you very much.
I was very unclear with the question 2, and you have clarified it with
good research pages.

Thanks.