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Goals Based Investing Question - how do you get to a certain probability of success?

For goals-based investing first you might categorize goals/needs into Primary, Secondary and Aspirational and assign them required probabilities of success.  For example, a primary goal of having $5M in 10-years to retire may need to have a 95% probability of success.  Lets say they have $2.5M now, they’d need a 7.18% annual rate of return, but how can I design a portfolio that mathematically has a 95% chance of success in 10 years? 

Would I use the Value at Risk framework, ER(p)-1.65(Standard deviation)?  I’m not sure how to use that over a 10-year time frame.  I guess I’m asking, how do I determine what combination of expected return and standard deviation will give me a 95% probability of success over a given time period?  Is Monte Carlo the only way to do this?

Thank you for any help and feedback!

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I do a Monte Carlo. But important to remember that it does not account for fatter tails. Also the returns are randomized when returns are really sequential. For example if market drops 80 percent, the odds of it dropping further is smaller since prices are that much cheaper and cyclical weaknesses will more likely subside. 

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Need a lot more info brah. What assets are you using and what return distribution? How are you projecting the next 10y of returns? You will also probably end up with a large number of portfolios that satisfy your return objective. Which is the best one that you would choose? You might also get zero portfolios.Then what?

In the simplest kind of exercise: known normal-ish distribution, future returns and deterministic cross asset correlation, you don’t even need a simulation, as you can fully define the probability distribution using these variables and any combination of portfolio weights. 10y return is just subject to 10 years of annual drift and 10 years of annualized volatility. 

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Good point about Monte Carlo being random, rather than conditional probabilities.  Thanks for the feedback!

I was thinking of a 9 asset class diversified portfolio, assuming normal distribution.  I would probably use 10-year historical returns or projected returns based on capital market assumptions.  I think what I’m asking is how do I, “fully define the probability distribution using these variables and any combination of portfolio weights?”  Sounds like this might require a specialized software?  I thought Maybe I could choose from several portfolios I’ve already created, using the statistical characteristics I know about them such as standard deviation and expected return.  Thank you for your response, you clearly know what you’re talking about!