Markowitz model (old)
[ This page is slowly going away as part of a reorganizing of my investment pages. See here for its "missing" sections. ]
Simulation
Nevertheless, let's consider simulation via historical data. The basic idea is to simulate my life a large number of times, record the resulting utility, and then choose the financial strategy that maximizes the average (i.e. expected) utility.
The steps are basically
- Construct a probabilistic model of financial markets with some parameters.
- Construct a multivariable PDF for the model's parameters using historical data.
- Do the following a large number of times:
- Using the parameter PDF, randomly initialize a model.
- Run through a simulation of me following a particular financial strategy using the model to generate random returns.
- Record the resulting utility.
- Note the average utility.
- Repeat steps 3-4 with different financial strategy to find the best one.
Bootstrapping
The simplest model to use is just boostrapping: just assume real returns are independent each year and just sample the last ~100 years of returns with replacement.
The pros of this "model" is that it is extremely simple - heck it doesn't even have any parameters, which means we don't even have to do steps (1) or (2).
The major con is that it assumes independence between years. I attempted to verify this using data from my spreadsheet of historical returns. Using 125 years of data, I found no statistically significant correlation between real returns one year and the next for government bonds, gold, housing, or the S&P 500. Unfortunately, I did find a correlation for government bills (p<0.1%, r~0.2).
What's worse is that since just eye-balling graphs of returns, several financial markets look very different after 1954 than before. If we only consider years after 1954, both government bills and housing have significant year-to-year correlation (both p<0.1%, r~0.7 and r~0.6 respectively). [by the by, this fact strongly suggest the EMH doesn't apply to these markets].
Since these correlations are all positive, using bootstrapping to model returns will result in an underestimating of risk for these two asset classes, which will make the model suggesting buying more government bills and housing than is appropriate.
A second problem with bootstrapping is that inflation also has strong year-to-year correlations (p<0.1%, r~0.4). While this isn't super problematic for naive investing since its generally accounted for by just considering real returns, this can lead to significant problems if you want to allow for home ownership in your model, since a rise in inflation causes the real interest you pay for your house to fall while a fall in inflation can allow for refinancing your home.
In short, a good model can't reasonably assume independence between years - it needs to account for these cross-year correlations.
Bonds Are Losers
On the other hand, we do have one thing that makes the analysis easier: you shouldn't invest in bonds.
Not to beat a horse that's been beat to death by everyone with any interest in finance, but yields on government bonds are low. As of this writing they are literally negative in inflation-adjusted terms DFII10.
More generally, government bills and bonds have averaged 1.2% and 1.7% real returns over the last 125 years - compare this to housing's 5.5%.
Now, the conventional reply is that government bills and bonds are so much safer, that the lower rates are comparable - not so! The variance on annual housing returns during that time period was actually in between the variance of government bills and bonds.
Moreover, from the equation above, we can deduce that justify such a wide gap in returns can only be justified if $\epsilon \gt 200$, which is an absolutely absurd level of risk aversion.
Fine, you say, but government bills and bonds move opposite stocks, which means they can effectively have negative risk, right? Nope: the correlation between bill/bond returns and stock/housing returns are all positive.
Moreover, when you're saving up for retirement, you have significant human capital, which effectively means you're anti-leveraged, which means concerns about risk are severely underestimated by the equations above, which only concern themselves with financial capital.
In short, there really is really no rational reason to invest in treasury bonds. For this reason, we will restrict our analysis to housing, stocks, and gold.
Gold is a Losers
The average return for gold over the last 125 years has been 0.5% and its significantly more risky than bills, bonds, or housing - nearly as risky as stocks. However, unlike bills and bonds, gold's returns actually do negatively correlate with stock and housing returns, so it's worth at least considering.
Suppose we have only S&P 500 investments and we're considering adding gold. By taking the derivative of equation (2) we can figure out if non-zero gold holdings are optimal:
$$ E[G_i] - E[S_i] + \epsilon \cdot Var[S_i] - \frac{\epsilon}{2} Cov[S_i, G_i] $$where $G_i$ and $S_i$ represent gold and S&P 500 returns, respectively.
The difference between stock and gold returns has averaged 7.7%. The variance of stock returns averaged 3.5%. The covariance of stock and gold returns is -0.2%. In other words, non-zero gold holdings are only justified if $\epsilon \gt 2.1$ - much higher than credible estimates I've seen. Recall, further, that this is all ignoring human capital, which means this model is significantly more risk averse than we should be. Repeating this analysis with housing makes gold look even worse owing to housing's lower variance compared to both stocks and gold.
In short, gold is a no-go as well.
A Simple Model Inflation
So this leaves us with housing and stocks as asset classes (and maybe international stocks later if I can find data).
A brief analysis reveals that the previous two year of housing returns (but not S&P 500 returns) predict next year's housing returns. The parameters of the least squares linear model are
| Variable | Estimate | Standard Error | P-Value |
| housing[i-2] | -0.376 | 0.125 | 0.000 |
| housing[i-1] | +0.928 | 0.123 | 0.002 |
| y-intercept | +0.0237 | 0.006 | 0.000 |
So, more or less, our plan will be to bootstrap but make the adjustments given by the two (non constant) coefficients above.