Question

) In a market with three assets, the portfolios P₁ = (0.6, 0.3, 0.1) and P₂ = (- 0.2, 0.5, 0.7) lie on the Minimum Variance Set. The portfolios have returns 12% and 4% respectively.

a) Find the portfolio on the MVS with return 14%.

b) Does the portfolio P = (0.1, 0.4, 0.5) lie on the MVS ? Explain.

Answer 1

Part (a) let the portfolio on MVS with return 14% is made up of "p" portion of P1 and (1 - p) portion of P2.

Hence, 14% = p x 12% + (1 - p) x 4% = 0.08p + 0.04

Hence, p = (0.14 - 0.04) / 0.08 = 1.25

Hence the desired portfolio = 1.25 x P1 + (1 - 1.25) x P2 = 1.25 x (0.6, 0.3, 0.1) - 0.25 x (-0.2, 0.5, 0.7) = (0.8, 0.25, -0.05)

Part (b)

If portfolio P lies on MVS then it must be expressed as a combination of P1 & P2

Hence, let P be made up of "p" portion of P1 and (1 - p) portion of P2.

Hence, P = p x P1 + (1 - p) x P2

or, (0.1, 0.4, 0.5) = p x (0.6, 0.3, 0.1) + (1 - p) x (-0.2, 0.5, 0.7) = (0.6p - 0.2 + 0.2p, 0.3p + 0.5 - 0.5p, 0.1p + 0.7 - 0.7p) =

(0.8p - 0.2, 0.5 - 0.2p, 0.7 - 0.6p)

Equating term by term, 0.1 = 0.8p - 0.2

hence, p = (0.1 + 0.2) / 0.8 = 3/8

Then, 0.4 = 0.5 - 0.2p; Hence, p = (0.5 - 0.4) / 0.2 = 1/2

And finally, 0.5 = 0.7 - 0.6p; Hence, p = (0.7 - 0.5) / 0.6 = 1/3

Hence, we are getting different values of p for each of the three assets combination. Hence there is no unique mix of P1 and P2 that results into P. Hence, the portfolio P = (0.1, 0.4, 0.5) does not lie on the MVS