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Scenario probabilities, return moments and beta

EXERCISE 01
StateProbabilityZ returnMarket return
110.000%30.000%12.000%
225.000%20.000%10.000%
350.000%15.000%8.000%
415.000%-20.000%6.000%
  1. A.Compute means, variances, standard deviations, covariance and correlation.
  2. B.Calculate Z’s beta.

A.

Use A for Z and B for Market. Returns are expressed as decimals in the calculations.

E[RA]=0.1(0.3)+0.25(0.2)+0.5(0.15)+0.15(−0.2)=0.125E[R_A]=0.1(0.3)+0.25(0.2)+0.5(0.15)+0.15(-0.2)=0.125
E[RB]=0.1(0.12)+0.25(0.1)+0.5(0.08)+0.15(0.06)=0.086E[R_B]=0.1(0.12)+0.25(0.1)+0.5(0.08)+0.15(0.06)=0.086
Statep × (A − mean A)²p × (B − mean B)²p × (A − mean A)(B − mean B)
10.003062500.000115600.00059500
20.001406250.000049000.00026250
30.000312500.00001800-0.00007500
40.015843750.000101400.00126750
Var(A)=0.0030625+0.00140625+0.0003125+0.01584375=0.020625Var(A)=0.0030625+0.00140625+0.0003125+0.01584375=0.020625
Var(B)=0.0001156+0.000049+0.000018+0.0001014=0.000284Var(B)=0.0001156+0.000049+0.000018+0.0001014=0.000284
σA=0.020625,σB=0.000284\sigma_A=\sqrt{0.020625},\qquad\sigma_B=\sqrt{0.000284}
E[R]=∑spsRs,Var(R)=∑sps(Rs−E[R])2E[R]=\sum_s p_sR_s,\quad Var(R)=\sum_s p_s(R_s-E[R])^2
Cov(A,B)=∑sps(As−E[A])(Bs−E[B])Cov(A,B)=\sum_s p_s(A_s-E[A])(B_s-E[B])
MeasureZMarket
Mean12.500%8.600%
Variance0.0206250.000284
Standard deviation14.361%1.685%
Cov(A,B)=∑sps(As−E[A])(Bs−E[B])=0.00205Cov(A,B)=\sum_s p_s(A_s-E[A])(B_s-E[B])=0.00205
ρAB=0.002050.020625×0.000284≈0.847028\rho_{AB}=\frac{0.00205}{\sqrt{0.020625\times 0.000284}}\approx 0.847028

Covariance = 0.002050; correlation = 0.847028.

B.

βZ=Cov(Z,M)/Var(M)\beta_Z=Cov(Z,M)/Var(M)
βZ=0.002050.000284≈7.21831\beta_Z=\frac{0.00205}{0.000284}\approx 7.21831

Beta = 7.21831.