# Proving that constant absolute risk aversion and relative risk aversion implies independence of initial wealth

I was able to prove that for a portfolio with one risk-free asset and one risky asset, if the Arrow-Pratt measure of absolute risk aversion is constant (i.e., constant absolute risk aversion, CARA), then the dollar amount invested in the risky asset does not depend on the agent's wealth. Similarly, with constant relative risk aversion (CRRA), the proportion of wealth an agent invests in the risky asset also does not depend on the agent's wealth.

Now, how can I prove the more general statement that with CARA and $N$ risky assets, the amounts of wealth invested in each of the $N$ risky assets is independent of the agent's initial wealth? As well as the analogous statement for CRRA?

• We generally ask not to crosslist questions on different stacks. You have crosslisted this on the quant SE: quant.stackexchange.com/questions/35926/… Otherwise this question is on topic. – Kitsune Cavalry Sep 6 '17 at 19:20
• Yes, I was referred to Econ SE as you can see in the comments, hence why I posted it here. Do I need to delete my Quant SE post? – user40333 Sep 6 '17 at 20:01
• It would be prudent to remove your question from quant.SE then. I'll see if I can take a look at this question in a bit – Kitsune Cavalry Sep 6 '17 at 20:02

The foundation for my exposition comes from Mas-Colell's examples in Ch. 6.

The maximization problem for your specific question can be generalized pretty easily. Consider the case with one risky asset and one riskless asset. Let $\beta$ be the wealth invested in the safe asset, normalized to 1 dollar per dollar invested. Let $\alpha$ be the wealth invested in the risky asset, which has some random payout $z$ such that:

$$\int z \ \text{d}F(z) > 1$$

So that the mean return is greater than the riskless asset. So we express our maximization problem as:

$$\max \ \int u(\alpha z + \beta) \ \text{d}F(z) \\ \text{s.t.} \quad\alpha + \beta = w$$

You can take advantage of the fact that $w - \alpha = \beta$ $\implies \alpha z + \beta = w + \alpha(z - 1)$ and find first order conditions. If $u$ is concave (risk averse) then Kuhn-Tucker FOCs which combine to make:

$$\int u'(w + \alpha(z - 1))\cdot(z - 1) \ \text{d}F(z) = 0 \quad \text{iff} \quad \alpha \in (0, w)$$

So for the generic case, you can do the same setup with $N$ risky assets and one riskless asset that is better than whatever other riskless assets are out there. Let's normalize it again to payout of 1.

So the maximization is now:

$$\max \int u(\alpha_1 z_1 + \cdots + \alpha_N z_N + \beta) \ \text{d}F(z_1, \cdots z_N) \\ \text{s.t.} \quad \alpha_1 + \dots + \alpha_N + \beta = w$$

Notes:

If you have already done the easy case, this general case should not be so bad, just some more work. For other readers, I'll state the definitions of constant absolute and relative risk aversion below, respectively.

$$r_A(x) = -\frac{u''(x)}{u'(x)} = n \quad \forall x$$ $$r_R(x) = -\frac{x \cdot u''(x)}{u'(x)} = n \quad \forall x$$