How can I simulate portfolio risk (diversification) with a 'Wheel of Fortune' like investment options/returns?

Question

Say I have 6 possible investment options with the following probability of success and the corresponding returns:

| Investment | Probability of Success | Return |
|    I1      |           0.5          |   2x   |
|    I2      |           0.3          |   6x   |
|    I3      |           0.1          |  10x   |
|    I4      |          0.06          |  20x   |
|    I5      |          0.02          |  40x   |
|    I6      |          0.02          |  40x   |

What I wish to know what is the best 'investment portfolio' (i.e., the one with the least risk - i.e., there is not cut-off for least anything would do for now). I wish to 'simulate' such a portfolio and its expected return, for the investment. I don't know if it's possible but I'm trying to see if it answers some questions:

1. If the initial investment capability is $100,000 how should I go about investing in these options i.e., how much to invest in which option so that I do end up with 'some' profit with minimal risk of loss?
2. If it's not possible to do the above using a simulation then is it possible to 'see' what would be an optimal portfolio? i.e., where all to invest?
3. If neither of the above are possible then what is the best way of simulating such a scenario to gauge the 'risk of investment'?

I basically want a graphical view of how things would look like i.e., the risk of such an investment. I'd like to simulate this in Excel using Monte-Carlo, but I just can't seem to understand 'what' to do so as to observe the associated risk with the portfolio or what would form a good portfolio with minimal risk.

UPDATE: At any given point in time ONLY ONE investment will be a success since only 1 of the options can happen at time 't'. So if I invest \$1 in each initiative and it's I1 that comes out the winner, I'll land up with \$2 but a net loss of \$4. Implying a bad investment. But at the same time I could land up with \$40 too but with a very low probability. I basically wish to simulate something like that to see if it's even 'worth it' - basically a visual cue/proof of the same.

Answer 1

The general optimization problem for portfolio management is the following:

$$ \min x Q x $$

where $x$ is the allocation vector of your problem, and $Q$ is the covariance matrix of all your possible investments.

In your example, you can compute the expectation $E[x]$, the variance $Var[x]=E[x^2]-E[x]^2$ and the covariane $cov[x,y]=E[xy]-E[x]E[y]$ pretty easily for each bet $x$, the latter giving you the $Q$ needed for the problem.

As @cardinal said in his comment, if you do not add any constraint, the optimal portfolio is in that case the one with the lowest variance and hence, the one with no position at all.

First of all, for the problem to make sense, you need to add:

$$x_i \geq 0 \quad \forall i$$

Moreover, you would want to add is a constraint which is the following:

$$\sum_i x_i=n$$

which means that you must invest the total amount $n$ in the game.

But this will still not give you an interesting example.

In fact you want to put another constraint on the expected return of your portfolio of bets, which is given as follows:

$$ \mu x \geq \bar{\mu}$$

Where $\mu$ is the vector of the expectations of all the bets and $\bar{\mu}$ is the minimum expected return you want from the portfolio.

Finally, in order to have an interesting result, you would like to choose $\bar{\mu}$ to be equal to on of the expectation of the bets. You should be able to find that, for the same expected return, you could have a lower volatility by investing in different bets.

You can run this in MATLAB using fmincon for example, or you could simply create an excel spreadsheet and use the solver.

Monte-Carlo is not really useful here, it's just modern portfolio theory. For a graphical representation, just draw an efficient frontier.