Monte Carlo Simulation for Alternative Asset Allocation
When things just aren't normal, you won't get far by assuming that they are.
What measure is most important in a computer-based simulation?
Not necessarily. Consider how a simulation is designed. Many variables are non-parametric in nature, negating the need for variance.
Unlikely; consider that it depends on the design of the simulation. Many variables are non-parametric in nature, negating the need for variance.
Right.
In a Monte Carlo simulation for alternative asset allocation, the researcher selects the variables of interest (or the risk factors if using those), and is free to design any distribution desired so that the model can run these assumptions forward and see what sort of asset class return scenarios occur.
What would be a logical final step?
That's not it. Your simulation won't tell you for sure which asset will provide the greatest returns. It just outputs what you give it.
Exactly.
Maybe you're working with an endowment fund and want a probability of meeting return objectives, or you need to estimate the funding level of a city's pension plan. These are the real values desired that the model will hopefully assist in estimating.
Unlikely. One main reason for Monte Carlo simulation is to be able to control distribution assumptions for non-normal returns, like those presented by alternative assets.
One popular method of modeling a non-normal distribution is to have two different normal distributions together: a "calm" normal distribution during regular times and a "storm" normal distribution during crises and periods of high volatility. What would be higher in the crisis times normal distribution?
No, the means come next. But a "storm" distribution would certainly have more dispersion, meaning a higher standard deviation.
Of course.
No, the area under any normal distribution curve is 1. But a "storm" distribution would certainly have more dispersion, meaning a higher standard deviation.
When you apply this idea to 1997–2017 quarterly returns of stocks and government bonds, some interesting things appear:
| | Equities | Government Bonds |
|-----|-----|-----|
| "Calm" Mean Return | 5.1% | 0.5% |
| "Storm" Mean Return | -3.1% | 2.4% |
| "Calm" Volatility | 5.5% | 1.9% |
| "Storm" Volatility | 13.7% | 3.8% |
Given the fact that stocks outperform government bonds overall, what would you assume about the correlation between these returns when a crisis hits?
Actually, it goes down.
That's right!
For bonds to outperform stocks in crises (no surprise there), they must show decent returns while stocks generally decline. This changes the correlation from 0.0 in "calm" times to -0.6 in "storm" times for this period.
With probabilities of the calm and the storm, you can then model each occurring through time, switching back and forth between each normal distribution. What would the overall distribution look like?
Not quite. Two normal distributions with very different volatility parameters won't sum to a normal distribution.
That's not it. Recall that a platykurtic distribution has a larger portion of observations within a couple standard deviations of the mean than does a normal distribution; that's not the case here.
Exactly.
The whole idea of putting the "calm" with the "storm" was to model the fat-tails distribution of returns overall, and that's what you'll have from your model by combining the two.
Now consider this sort of volatility modeling with forward-looking returns for various asset classes. Then a few portfolios can be modeled:
| Government Bond Portfolio | 50/50 Portfolio | Endowment Portfolio |
|:-----:|:-----:|:-----:|
| 100% Government Bonds | 50% Global Equities 50% Broad Fixed Income | 40% Global Equities 25% Hedge Funds 15% Private Equity 10% Private Real Estate 10% Commodities |
An investor considering all three funds would have what sort of investment horizon?
Not really. Considering some of the allocations, there's only one that makes sense.
Absolutely. A short-term investor wouldn't seriously consider an allocation to private equity. And here's what the simulation shows for results:
| | Government Bond Portfolio | 50/50 Portfolio | Endowment Portfolio |
|-----|:-----:|:-----:|:-----:|
| 10-year Geometric Mean Return | 2.3% | 5.6% | 7.0%
| Annual Total Return Volatility | 4.2% | 6.6% | 11.2% |
That's not it. Consider the suitability of some of the allocations, especially in the endowment portfolio.
Recall that the final step in the simulation is to get at the metric of interest, such as meeting a long-term return goal. Which portfolio does the model suggest will maximize that likelihood?
Not quite. Consider the risk and return of each in terms of what is most important for the objective. Maximizing the likelihood of meeting a substantial return objective will require substantial expected returns.
Yes.
It offers the highest mean returns, but of course the highest risk level as well, which translates into greater value at risk measures, maximum drawdown, etc. But any scenario analysis from here looking at a reasonable range of cumulative portfolio values will show that this allocation to alternative assets allows the portfolio to increase the probability of meeting the balance goals over time, and maximizing this probability is commonly the objective function of interest.
Maybe it's not the normal way to get there, but non-normality has its benefits!
Actually, this would *minimize* that likelihood; consider the risk and returns presented by the model.
To summarize:
[[summary]]
Variance
Standard deviation
Whatever the researcher chooses
Translating those scenarios into indicators
Choosing the highest returning asset
Applying mean–variance optimization to the result
Mean return
Standard deviation
Area under the curve
It goes up
It goes down
Normal
Platykurtic
Leptokurtic
Any
Long-term
Short-term
Government bond portfolio
50/50 portfolio
Endowment portfolio
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