Problem set 7 helpful hint

Here is a helpful hint for parts 1D and 2C of problem set 7. As we showed last week in class, the weights for the minimum variance two-asset portfolio can be found by applying the following ratio in order to determine w1; upon calculating w1, then w2 = 1 – w1. (source: p. 7 of http://fin4335.garven.com/fall2019/lecture12.pdf):

{w_1} = \displaystyle\frac{{\sigma _2^2 - {\sigma _{12}}}}{{\sigma _1^2 + \sigma _2^2 - 2{\sigma _{12}}}}.

Furthermore, in cases where correlation is equal to either -1 or 1, the weighting scheme shown above guarantees that the minimum variance combination is riskless. Therefore, if the expected return on a riskless two-asset portfolio is either greater or less than the expected return on a riskless asset, there’d be a riskless arbitrage opportunity. In such cases, one would want to buy the higher yielding investment and short the lower yielding investment. Although no risk is involved, we assume (here as well as throughout the course) that marginal utility of wealth is positive for all decision-makers.

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