Expected Log Return for Budgeted Allocation Across Independent Investments

# Optimal investment strategy problem with competing bet-sizing options and limited budget


# Optimal investment strategy problem with competing bet-sizing options and limited budget












Apologies for a potentially naive question and unusual wording. I am from another field and would be very grateful for help!

I am looking for the optimal investment strategy that maximizes an overall expected return with multiple uncertain invest events and stochastic returns. Let strategy/portfolio $\mathcal{S} = \{b_1,b_2,...,b_n\}$ be a tuple of $n$ invests and $\sum_i b_i = B$ the investor's budget. The uncertainty that an invest occurs is characterized by probability $p_i$ where all invest events are mutually independent. If invest $b_i$ occurs, its return is $r_i = f_i(b_i)$ with $f_i(.)$ being a known mapping with uncertain parameters $\theta_i$. Returns are mutually uncorrelated.

My question is:

How to solve for the optimal strategy/portfolio $\mathcal{S^*}$ that maximizes the overall expected return $R^*$ under budget $B$ in a sequence of investment decisions?

As far as I understand, the Kelly criterion is similar in that it gives an optimal bet sizing strategy to invest wager $b$ with payoff odds given as a fraction of the invest (i.e. $f(b)$ is a linear function with zero intercept), and that there is only a payoff with probability $p$. It differs, however, in that it assumes a single invest, whereas here we have $n$ competing investment options under a limited budget. The extension by Smoczynski and Tomkins (2010) is unsuitable as it considers mutually exclusive investments, such as in horse races, where our problem has mutually independent investments. Also, in our problem, and unlike some portfolio choice problems, the investor has no option to divest e.g. into a risk-free asset.

I am also unsure about the treatment of time. Similar to the Kelly criterion and intertemporal portfolio choice problems, where bets are placed over a sequence of runs, in this problem, the investor has to continuously take decisions as parameters $\theta$, returns $r$ and the budget $B$ change (asynchronously) over time.

My application is a research portfolio for which I seek a soundly derivable strategy to distribute head count on research topics.

## Answer by Igor Rivin (score 1)

https://quant.stackexchange.com/a/59195

If I understand correctly, you are looking for the optimal allocation $b_1, \dotsc, b_n$. You can compute the expected log return for each asset - it is $p_i \log f(b_i),$ so you optimize the sum $\sum_i p_i \log f(b_i).$ I am not sure what you mean by "uncertain parameters".