Question

The idea of this exercise is to show that adding uncorrelated returns with the same variance lowers the total variance of a portfolio. In what follows, consider that there are many uncorrelated returns with variance equal to 0.10 each.

I) Assume that a portfolio is composed of two assets only. Asset 1 has a weight a∈[0,1] and asset 2 has a weight (1−a). Write the return's variance of this portfolio as a function of a.

Answer 1

The variance of a portfolio composed of two uncorrelated assets, each with a variance of 0.10 and weights a and (1−a), is calculated as 0.10 * (a^2 + (1 - a)^2).

Step-by-step explanation:

The question revolves around the concept of portfolio variance in finance, particularly when considering uncorrelated assets. If we have two uncorrelated assets each with a variance of 0.10, and we denote the weight of Asset 1 by a and the weight of Asset 2 by (1 - a), then the variance of the portfolio can be calculated using the formula of portfolio variance which is:

Variance(Portfolio) = a2 * Variance(Asset 1) + (1 - a)2 * Variance(Asset 2)

Given that both assets have a variance of 0.10 and are uncorrelated, the formula simplifies to:

Variance(Portfolio) = a2 * 0.10 + (1 - a)2 * 0.10

Simplifying this expression, we get:

Variance(Portfolio) = 0.10 * (a2 + (1 - a)2)