# 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?

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.

• So just to be clear, you're saying that U'(x*) can be treated here as differentiation with respect to one of the r_i and not x itself? I'm thinking hard and almost see the answer from what you've posted. I just don't see exactly how the details work out to give the final result. Thanks though this helps a lot. – James Sep 11 '15 at 22:50
• Okay I just wrote it out now with pen and paper and I can see how it works. Of course x itself is a vector of the (r_i)s in the first place so U'(x) is always a partial differentiation. I have the result now! Thanks again! – James Sep 11 '15 at 23:03
• Yes I see now like you said the tough part is just to make sure that all the integration stuff works out precisely in terms of calculus theory. – James Sep 11 '15 at 23:06
• The integration and differentiation stuff will in general always work out in finance (especially for these static models), since you usually use differentiable utility and non-weird distributions and moment assumptions. But in dynamic portfolio theory, one has to be far more careful about these issues, but that's beyond the scope of this post. – user32416 Sep 11 '15 at 23:19
• And I guess it's worth noting that I guess it doesn't matter whether or not the assets are uncorrelated or not. Thanks for the speedy answers too. Just one more thing. In the final equation, is U'(x*) with respect to the r_i value or with respect to the payoff value? I'm rethinking it now and I'm not sure. Shouldn't it be with respect to r_i since that was the partial differentiation made to get to the answer? – James Sep 11 '15 at 23:31

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.

• Unfortunately this is incorrect, and it starts from the first equality that you'd started. In the case of $(R_1, \ldots, R_n)$ with $n$ assets, the expectation is simply not that. It is rather, for any $(\pi_1, \ldots, \pi_n) \in \mathbf{R}^n$, $E U(W) = \int \cdots \int U( W_0 (1 + \pi_1(r_1 - r_f) + \ldots + \pi_n(r_n - r_f) ) dF(r_1, \ldots, r_n)$, where $F$ here is the multivariate CDF of $R_1, \ldots, R_n$. – user32416 Sep 12 '15 at 19:26
• but dF(x) = f(x)dx isn't it? Doesn't that hold in the multivariate case? Or is there something wrong with doing it this way for the integration? – James Sep 12 '15 at 20:31
• There are two issues at hand: (1) Whether $dF(x_1, \ldots, x_n) = f(x_1, \ldots, x_n) dx_1 \cdots dx_n$ depends whether a Radon-Nikodym derivative exists (i.e. usually in probability contexts, we call $f$ the PDF). There are many cases where $f$ does not exist. (2) Read my comment more carefully --- in your follow up answer, you are integrating with respect to $\pi_i$'s, which make no sense, since $(\pi_1, \ldots, \pi_n) \mapsto E[U(W)]$. – user32416 Sep 12 '15 at 20:38
• I wasn't aware of the details here. Thanks for pointing them out. I assume everything is okay though once you replace what I have with your $dF(r_i,...,r_n)$? – James Sep 12 '15 at 21:17
• And just to save us some trouble, this is the full first order conditions for the $k$th portfolio, $0 = \frac{\partial}{\partial \pi_k} EU(W) \vert_{ \pi_1 = \pi_1^*, \ldots, \pi_n = \pi_n^*} = \frac{\partial}{\partial \pi_k} \int \cdots \int U(W_0(1+\pi_1(r_1−r_f)+ \ldots +π_n(r_n−r_f) )dF(r_1,…,r_n) \vert_{ \pi_1 = \pi_1^*, \ldots, \pi_n = \pi_n^*}$, where $\pi_i^*$ are the optimal portfolios. – user32416 Sep 12 '15 at 23:37