Portfolio Utility First-Order Conditions for Risky Asset Weights

# Utility Theory and portfolio optimization - Proof of a lemma


# Utility Theory and portfolio optimization - Proof of a lemma












I have a question on the following problem from chapter 9 of D. Luenberger, Investment Science, International Edition:

> (Portfolio Optimization) Suppose an investor has utility function $U$. There are $n$ risky assets with rates of return $r_i$, $i=1,2,...,n$, and one risk-free asset with rate of return $r_f$. The investor has initial wealth $W_o$. Suppose that the optimal portfolio for this investor has (random) payoff $x^*$. Show that $$E[U^{'}(x^*)(r_i-r_f)]=0$$ for $i=1,2,...,n.$

I'm finding this a very hard problem, as I don't see how I can prove this analytically using the formulas for $E[XY]$ or $Cov[X,Y]$ in some straightforward way.

It is not mentioned whether or not the risky assets are correlated or not which also raises my level of uncertainty about this problem.

One thought is that the optimal portfolio should be such that $U(x^*)$ is maximized, but this does not mean that $U^{'}(x^*)=0$ even then since $U$ is monotone increasing.

I've tried to look at some simple examples in Excel to see if this identity holds true in practice, but I'm finding it very difficult to show this even if I use some simple utility function like a linear utility function.

In fact, if I assume that $U(x)=x$ as a simple starting step, then $U^{'}(x)=1$ for all $x$, and then the equation would read $E[r_i-r_f]=0$, which isn't necessarilly true.

I've tried to think about it from a few different angles but I just can't process it. Is there anyone who would know how to show this lemma true or false?

Thanks in advance.

## Answer by user32416 (score 6, accepted)

https://quant.stackexchange.com/a/20710

This is the well known Euler's equation for optimality. The trick here is to setup the budget constraint correctly. Your initial wealth $W_0$ is irrelevant. The terminal (risky) wealth is, $$W = W_0( 1 + \pi_1 (R_1 - r_f) + \ldots + \pi_n (R_n - r_f) )$$ (Check that this can be written this way), where $\pi_i \in \mathbf{R}$ is the weight allocated to asset $i$.

Your optimization problem is then simply maximize expected utility, $$ \sup_{ \pi_1, \ldots, \pi_n } E[ U(W) ]$$.

Take first order conditions above, assuming that you can interchange the order of integration and differentiation (there are technical conditions to ensure that this will hold, and if you want to see the details, https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign. But in finance, these conditions practically always hold). Once you've done that, you'll see your desired result.

Note: Your example of risk neutrality (i.e. $U(x) = x$) is precisely the case when the optimization problem is mute. In this case, when the investor does not care about risk, he will simply invest an infinite amount of wealth into the asset with the highest excess return $E[R_i - r_f]$ and short an infinite amount into everything else. That is why in that special case, your equation will not hold. You need that $U$ is a utility function that represents a risk averse agent.

## Answer by James (score 2)

https://quant.stackexchange.com/a/20713

Edit: The original follow fill in the details answer to my answer by James is still wrong (despite many hints). I'll just go in to fix it to avoid detracting future readers of this post.

Just to fill in the details from the answer that has been accepted already:

It is required to maximize $$\sup_{ \pi_1, \ldots, \pi_n } E[ U(W) ] = \sup_{ \pi_1, \ldots, \pi_n }\;\int ... \int U(W)dF(r_1,\ldots, r_n)$$

So for example in the two asset case the first order condition for $\pi_1$ is

$$0 = {\partial \over \partial \pi_1} \int_{\mathbf{R}} \int_{\mathbf{R}} U(W^*)dF(r_1,r_1) \Big\vert_{ \pi_1 = \pi_1^*, \pi_2 = \pi_2^*}$$, where $W^*$ is the optimal terminal portfolio, evaluated at the optimal portfolios $\pi_i^*$; in the original problem, this notation would be $x^*$. To be even more explicit, $$ W^* = W_0(1 + \pi_1^* (r_1 - r_f) + \pi_2^* (r_2 - r_f) )$$ Note to James: It would be very helpful for your sake if you could understand what is the difference between a random return $R_i$ and its realization $r_i$.

If one can interchange the order of differentiation and integration, then we can rewrite the FOC as, $$ 0 = \int_{\mathbf{R}} \int_{\mathbf{R}}U'(W^*)(r_1-r_f) dF(r_1,r_2)\Big\vert_{ \pi_1 = \pi^*_1, \pi_2 = \pi_2^*}$$

Similarly for $\pi_1$ this is done.