# Compute Utility From Portfolio Holdings Over Time

I have a dataset comprising daily stock holdings for individual investors over a one year-period. I only know about the individuals' investment in stocks. I have no information on any other wealth of the individuals. However, investors might rebalance their portfolios and buy or sell stocks increasing or decreasing their overall portfolio value.

Now, I want to compute the utility perceived from the portfolio wealth an investor holds over the time period of one year. How would you approach this task?

My approach (so far) shown with an example for one investor:

On day $$t$$, an investor holds $$N_t$$ assets in his portfolio. Each asset has a dollar value of $$x_i$$ with $$i=1,\dots,N_t$$. Let $$w_i = x_i/\left( \sum_{i=1}^{N_t}x_i\right)$$ denote the fraction asset $$i$$ has in the portfolio on day $$t$$. I would compute the weighted utility on day $$t$$ as:

$$U_t = \sum_{i=1}^{N_t} w_i \cdot u(x_i),$$

where $$u(\cdot)$$ may be any common utility function.

Since I want to evaluate over $$T$$ time periods, I would then compute an overall utility as:

$$U = \frac{1}{T}\sum_{t=1}^T U_t$$

Is this approach of intraday weighting of utility and averaging over time correct?

I haven't worked with utility computation so far and would be thankful for suggestions how to improve this approach or what pitfalls need to be considered.

Thanks!

• it's not clear why you need to calculate 'utility' at all. a somewhat obvious stand-on for utility here would simply be portfolio value, potentially excluding non-equity in the absence of data for them. please provide additional context. – Chris May 9 at 22:38
• In Brandt's Parametric Portfolio Policies (2006) they maximize $\frac{1}{T}\sum_{t=1}^T u(r_{p,t+1})$ which is basicaly the same as what you have, except they have the return as argument to u(), not the dollar value of the portfolio. – Alex C May 10 at 18:43