Ice Cream

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Expected Return=E(R_(ASSET ) )= ∑_(i=1)^n▒〖(P_(i ) 〗 x R_(i ))=(P_(1 ) x R_1 )+(P_2 x R_2)

E(RICE CREAM) = (.50 x -.02) + (.50 x .20) = -.01 + .10 = .09 or, 9%

E(RFRISBEES) = (.50 x .06) + (.50 x .12) = .03 + .06 = .09 or, 9%

E(RUMBRELLAS) = (.50 x .15) + (.50 x .03) = .075 + .015 = .09 or, 9%

Variance=Var(R)=σ^(2 )= ∑_(i=1)^n▒〖{pi x [Ri-E(R) ]^(2 )}〗

Var(R_(ICE CREAM) )= .5 x (-.02- .09)^(2 )+ .5 x (.20- .09)^2= .0121

Var(R_FRISBEES )= .5 x (.06- .09)^(2 )+ .5 x (.12- .09)^2= .0009

Var(R_UMBRELLAS )= .5 x (.15- .09)^(2 )+ .5 x (.03- .09)^2= .0036 B.) The covariances and correlations for the returns on the two investment alternatives described are as follows:

〖〖Covariance〗_(ICE CREAM,FRISBEES)=σ〗_R1,2= ∑_(i=1)^n▒〖{P_(i ) [(R_(1,i )–E[R_1 〗])([R_(2,i )–E[R_2 ])}

〖Covariance〗_(ICE CREAM,FRISBEES)={.5[(-.02- .09)(.06-.09)+(.20-.09)(.12-.09)]}

= .5[(-.11)(-.03)]+ [(.11)(.03)]

= .0165 + .0165= .0033

〖Covariance〗_(ICE CREAM,UMBRELLAS)={.5[(-.02- .09)(.15-.09)+(.20-.09)(.03-.09)]}

= .5[(-.11)(.06)]+ [(.11)(-.06)]

= -.0033 + -.0033 = -.0066

Correlations= P_(R1,2 )= σR1,2/(σR1 σR2 )

〖Correlations〗_(ICE CREAM,FRISBEES)= .0033/((.11)(.03))= 1

〖Correlations〗_(ICE CREAM,UMBRELLAS)= (-.0066)/((.11)(.06))= -1

Given the correlation analysis, we would choose the Ice Cream and Umbrellas investment alternative due to its correlation of -1. This means that the returns on the two assets are perfectly negatively correlated. When the return on Ice Cream is positive, the return on Umbrellas will always be negative. When the return on Umbrellas is positive, the return on Ice Cream will always be negative. From an investor point of view, this asset combination is giving us the benefits of diversification.
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