One way to make money with money is to sell insurance to people who want to protect their existing investments. On average, you win, but there is a chance you may lose.
Consider the three financial instruments above. Each instrument protects the same asset, so if you buy, e.g., $100 of Instrument 2, then there are only two possible outcomes after one week: You either walk away with $105 with 50% probability or you are left with only $96 with the same probability. Of course, if we invest different amounts of money in all three instruments, then there are 8 potential outcomes. The three different bets are independent from each other; there is no correlation.
Our initial budget is $10,000. We offer these instruments every week for 2 years, so for 104 weeks in total. We need to decide what percentage of our current budget we invest in each instrument. This percentage will remain constant for the entire 2 years.
Our objective is twofold:
- Maximize our expected budget at the end of two years. It should be at least $15,000, but it would be great if we can triple our money.
- Maximize the average budget in the 10% worst scenarios. The CVaR-10% should be no less than $8,000 and ideally be close to $12,000.
Can you advise how best to invest our money?
Hints: Ask yourself these three questions:
- Is there a single best instrument?
- Is it better to invest in just one instrument or should you invest in multiple?
- Is it advisable to use your entire remaining budget every week?
Seeker suggests investing
- 2.326% in Instrument 1,
- 59.83% in Instrument 2, and
- 2.078% in Instrument 3.
35.76% remain as cash on hand. With this strategy, we expect, on average, an end budget of $18,399 with a CVaR-10% of $8,892.
With this setting, we expect Instrument 1 to contribute ~$3,330, Instrument 2 ~$4,280, and Instrument 3 ~$770 to our expected profit. It would be wrong to conclude that there was one dominating investment, though. Even though they are independent, the diversified investment makes it less likely that we lose simultaneously on all of them. And this allows us to keep less cash on hand and invest more in each individual instrument than would be advised in isolation.
What this puzzle teaches us again is that, with uncertain futures, outcomes always vary. A solver that does not make this distribution of outcomes explicit falls short for real-world applications. Not only does the user lack insight in the risk that they are taking on. The solver is also not able to shape the distribution of outcomes to fit the objectives of the user.
That is why using point estimates and simulation outside optimization are the wrong approaches. The assessment of outcome distributions must, necessarily, happen inside the optimization.
And that is why we built InsideOpt Seeker for you. It is the only commercial optimization solver that does exactly this for you.