# Indifference between 2 risky asset

Consider the problem of an individual that must choose how much of his initial wealth w0 > 0 to allocate to a risky asset X. The risky asset X has n ≥ 2 possible return rates, namely r1, . . . , rn, and P rob(r = ri) = pi > 0 for all i ∈ {1, . . . , n}. The individual’s utility function over wealth u is increasing, strictly concave and differentiable everywhere. Suppose now that there exists a second independent risky asset Y that has the same possi1ble return rates happening with the same probabilities as X.. Is the existence of Y beneficial to the individual?

When I first read the question, I think if they both have exactly the same probability and return value, it shouldn't be an advantage for us. But at the same time, I am wondering if there is any other situation that I am missing and that could take advantage of this situation. I wonder your comments.

It can be shown that under the assumption of strict concavity of utility function the investor will choose to split her wealth equally in the two assets. This happens because the variance on returns decreases by investing in two assets (assuming that the returns on two assets are independent).

We can consider two scenarios: first when the individual invests only in $$X$$ and the second where she splits her investment between $$X$$ and $$Y$$. It can be shown that in second case the variance of return goes down.

Let $$r \equiv E(R_x) = E(R_y)$$ and $$\sigma^2 = Var(R_x) = Var(R_y)$$

Assume that in second case, she invests $$\alpha$$ fraction of wealth in $$X$$ and $$(1-\alpha)$$ fraction in asset $$Y$$. Clearly $$E(R) = r$$. However:

\begin{align} Var(R) &= Var(\alpha R_x + (1-\alpha)R_y)\\ &= (\alpha^2+(1-\alpha^2))\sigma^2 \tag{from independence} \\ &=(2\alpha^2+1-2\alpha) \sigma^2 \end{align}

The investor will choose $$\alpha$$ to minimize the term in braces which happens at $$\alpha=0.5$$ and so $$Var(R) = \sigma^2/2$$

Now it is a well know result that a portfolio with mean preserving spread is less preferred by risk-averse (concave utility) individuals. So the individual will choose to split her wealth equally among the two assets.

• I don't think the assumption of independence is necessary. The two can be decomposed into correlated and uncorrelated components (other than the degenerate case where the uncorrelated component is zero, in which one is indifferent and one essentially doesn't have two different assets) the correlated components can be ignored as they don't affect the ordering of expected utility, and then the logic used in the independent case can be applied to the uncorrelated component. The proof is simpler in the independent case, however. – Acccumulation Dec 16 '20 at 19:42