↰

Long-short portfolio return and volatility

EXERCISE 01

Buy 20,000 of A and short 5,000 of B. Net capital is 15,000, so the weights are 4/3 and −1/3. The same data appear as YXC/LKJ.

AssetExpected returnVolatilityWeight
A7.000%16.000%1.33333
B10.000%20.000%-0.33333

Correlation = 0.22.

  1. A.Calculate portfolio expected return and volatility.
  2. B.Can these two risky assets form a zero-volatility portfolio?

A.

E[RP]=wAE[RA]+wBE[RB]E[R_P]=w_AE[R_A]+w_BE[R_B]
σP2=wA2σA2+wB2σB2+2wAwBρσAσB\sigma_P^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho\sigma_A\sigma_B
E[RP]=1.333333(0.07)+(−0.333333)(0.1)≈0.06E[R_P]=1.333333(0.07)+(-0.333333)(0.1)\approx 0.06
σP2=(1.333333)2(0.16)2+(−0.333333)2(0.2)2+2(1.333333)(−0.333333)(0.22)(0.16)(0.2)\sigma_P^2=(1.333333)^2(0.16)^2+(-0.333333)^2(0.2)^2+2(1.333333)(-0.333333)(0.22)(0.16)(0.2)
σP=0.04369778≈0.20904\sigma_P=\sqrt{0.04369778}\approx 0.20904

Expected return = 6.000%; volatility = 20.904%.

B.

No: their correlation is strictly between −1 and +1, so the covariance matrix is positive definite and no fully invested combination has zero variance.