# Portfolio allocation with a few twists

A similar question has been asked here, but this one is more complicated and has more constraints.

I'm trying to find an algorithm to solve the following (real-life) problem:

A customer has $M$ amount of money each month that he wants to invest according to her preferred asset allocation. Example

• 60% stocks
• 20% bonds
• 20% REITs

She has a list of "favourite" financial instruments and a preferred allocation in each of the asset classes, e.g.:

• Stocks: C (60%) and AAPL (40%)
• Bonds: AGG (70%) and LQD (30%)
• REITs: VNQI (100%)

Now, the algorithm would provide an investment advice (how many units and what financial instruments to buy) according to the following constraints:

1. Total sum of purchases with fees cannot exceed $M$ (amount of money).
2. Units are integer and indivisible, you can buy either one or two stocks but not 1.5. Since the customer ivests small(-ish) sums, this constraint becomes important. Example: $M$ is \$500 and AAPL is \$173 per unit!
3. Main goal is fee minimisation (again, since the sum invested is not large, fees become relatively large). Instruments have different fees that not flat-rate. E.g.: fees for buying C are "\$1 if transaction sum is <\$100, 0.30% otherwise", but for VNQI they're "\$10 if sum < \$2000, 0.40% otherwise".
4. Second goal is allocation that is approximately close to the "model allocation" above. Note that the model is concerned with asset classes, not individual instruments. That means the customer can buy only C or only AAPLto attain the desired 60% in Stocks. However, remember that there's also a preferred allocation within an asset class; if we decide not to buy C this month, then we'll need buy more of it next time to keep the stocks in balance.
5. Fee minimisation can be attained by limiting the number of transactions. That means we can decide to buy, e.g., C, AGG and VNQI only, and think about buying AAPL and LQD next month. We can go even further and decide not to buy a whole asset class at all if the first solution has fees that are too high (above some pre-determined limit). In that case we'll remember that decision and buy more of it next month.

Now, I've been thinking about this for some time; I had my CS lessons over fifteen years ago, so had to rack my brain :) My first thought was of Bin packing problem. Nah. Doesn't really fit.
Then I remembered about Multi-dimensional unbounded knapsack. Again, not what I was looking for.
Considered Quadratic programming and Dantzig's simplex algorithm for linear programming. Not exactly fitting...

Maybe there is a special algorithm / special case for this kind of problem? Or shall I invent a new one? :)

• Are you looking to have fun, or just get the project done. If the number of investments are very small, it may be reasonable to simply brute force the state-space. Feb 22 '18 at 0:34
• Thanks for the advice @CortAmmon; as for your question - both, actually :) If I wanted a quick-and-dirty solution I wouldn't post on CS, I'd ask on StackOverflow :) I'm looking for a theoretically right way to tackle this problem. Feb 22 '18 at 9:35
• @Alexander: Did you solve this problem and have an example on how you did it? I am also trying to do the same thing. :) (working on a simplistic rebalancing algo) Oct 1 '18 at 11:06