# 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.

• Hi" I probably can't answer your question anyway but is the vector of $b_{i}$ what the person has to allocate at each period ? or is it the resulting allocation by the person ? Also, are each of the $n$ bets to be taken simultaneous or sequential ? Apologies for confusion but I've been looking at this topic lately so I'm curious if you're problem is similar to mine. Nov 7 '20 at 0:48
• (i) The task is to allocate the vector of $b_i$'s. (ii) Simultenous because changing one $b_i$ changes the others (due to $\sum_i b_i = B$). Also, the $b_i$'s are more like long-term invests where returns pay off with delay and $B$ is not influenced by the returns. Nov 10 '20 at 11:14
• Hi: Thanks for explanation. Still slightly confused. Shouldn't it be " the task is to allocate $B$" and not the vector $b_i$. What do the individual $b_i$ have to do with it ? Is each bet $i$ only allowed one $b_i$ ? Still, I'll send a link but don't know if it's relevant ? Nov 11 '20 at 12:01
• Hi: Not sure if this is relevant but it might be ? researchgate.net/publication/… Nov 11 '20 at 12:03
• I think the answer to my question is that each bet $i$ is allowed only one $b_i$. If that's the case, then what I sent above might be relevant. Let me know because there's also another link where a numerical algorithm is given for the problem. ( by a different person ). I don't want to send it unless it's relevant. Nov 11 '20 at 12:06

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".
• Yes, $R^* = \sum_i p_i \log f(b_i)$ is the objective function to be maximized (if this is what you mean). My returns are projected sales that depend on my invest $b_i$ but also on market numbers such as TAM, CAGR, market share which I called uncertain parameters. We can assume everything to be Gaussian, the $r_i$'s and the parameters, and $f(.)$ to be a (non-linear) differentiable function. We can use a first-order or unscented transform approach for error propagation from parameters to the $r_i$. Nov 10 '20 at 11:24
• A closed-form expression or algorithm to find the $b_i$'s that maximize $R^*$ -- sorry if this was not clear. If such a method came along with a way to trade off risk and return (risk possibly represented via the variance of R) that would be really great. Nov 10 '20 at 17:10