Corporate Finance

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Corporate Finance

Situation 2

ρ 12 is –1
σp^2 = 116.64 + 116.64 – 233.28 = 0

Yes! The portfolio variance is 0. But it is unlikely that you’ll find perfectly negatively correlated stocks
in real life.

Situation 3

ρ 12 is less than 1 (say 0.5)
σp^2 = 349.92
σp= 18.7 percent

Negative or less than perfect correlation (covariance) reduces the risk of the portfolio, positive covariance
does not.
We have arbitrarily chosen 60 percent and 40 percent as weights. Suppose we alter it to 50 percent and
50 percent. Let ρ 12 be 0.3:

σp^2 = (0.5 × 18)^2 + (0.5 × 27)^2 + 2 × 0.5 × 0.5 × 0.3 × 18 × 27

= 336.15

σp= 18.3 percent

E(Rp) = (0.5 × 21.48) + (0.5 × 16.56)

= 19.02 percent

We can construct a large number of portfolios with different weighting schemes. But is there one best
portfolio that minimizes portfolio variance?

σp^2 = X 12 × σ 12 + X 22 × σ 22 + 2X 1 × X 2 × Cov 12

= X 12 × σ 12 + (1 – X 1 )^2 × σ 22 + 2X 1 × (1 – X 1 ) × ρ 12 × σ 1 × σ 2

Differentiating the above equation with respect to X 1 and equating to zero:

δσp^2 /δX 1 = 2X 1 × σ 12 + (2 × 1 – 2)σ 22 + 2ρ 12 × σ 1 × σ 2 – 4 × ρ 12 × σ 1 × σ 2 = 0.

Solving for X 1 :

X 1 = [σ 22 – ρ 12 × σ 1 × σ 2 ]/[σ 12 + σ 22 – 2ρ 12 × σ 1 × σ 2 ]

where

σ 1 = 18 percent,

σ 2 = 27 percent, and

ρ 12 = 0.3.

On substitution we get X 1 = 76 percent and X 2 (= 1 – X 1 ) = 24 percent.