Optimal Asset Allocation

Question

Something I've recently gotten into is asset allocation optimization and using AI (Extended Kalman Filter) & statistics in this area. Anyway, I was ready to move 100% into stocks so I put 80% into VTSAX (Vanguard Total Market Admiral) & 20% into VGTSAX (Vanguard Total Intl Stock)...

Anyway, using an excel document I put together with different weights for different portfolios I'm getting results that are telling me that a portfolio of 100% VTSAX dominates the portfolio that I have now.

I was always under the assumption that as long as two securities are less than perfectly correlated (i.e. 1), that the standard deviation/risk would be less than if I had put 100% into either of the securities.

VTSAX & VGTSX have a correlation coefficient of .889 meaning that they move together, but not perfectly. Does it even makes sense for me to put into VGTSX or international stocks if I have a portfolio of 100% domestic stocks that dominates any combination involving international stocks.

Using the utility function: E[x] - .5*A*sig^2 results in the highest utility of 100% VTSAX.

Any thoughts or recommendations about moving 100% into VTSAX? I'm just curious if anyone else tracks their portfolios this way.

Answer 1

Generally a diversified portfolio will give you a better overall return --a couple of factors that may address what you are looking at -

1) Correlation - The correlation between your two funds is still very high -- it's partially a function of how global economies are related and many companies are now multi-national. It may help if you diversified into other types of products.

2) Diversification - Following up from before, you may want to also look into diversifying into some bonds, commodities, reits, etc. They will have a much smaller correlation with a total domestic stock fund.

3) Returns - I'm not sure if by dominate you mean that it has better overall returns, but the point of diversification is to to get you the highest returns. It's really the ability to limit the risk for the returns - this really translates to limiting the volatility. This may mean that overall your max returns could be lower-- ie: maybe VTSAX gives potential average returns between 3%-11%. A diversified portfolio may give you potential average returns of 5%-9%.

A similar article debating the merits of 'smart beta ETFs' if you are curious.

Hope that helps.

Answer 2

There are some good answers about the benefits of diversification, but I'm going to go into what is going on mathematically with what you are attempting.

I was always under the assumption that as long as two securities are less than perfectly correlated (i.e. 1), that the standard deviation/risk would be less than if I had put 100% into either of the securities.

While there does exist a minimum variance portfolio that is a combination of the two with lower vol than 100% of either individually, this portfolio is not necessarily the portfolio with highest utility under your metric. Your metric includes returns not just volatility/variance so the different returns bias the result away from the min-vol portfolio.

Using the utility function: E[x] - .5*A*sig^2 results in the highest utility of 100% VTSAX.

So here the Sharpe ratio (risk adjusted return) of the U.S. portfolio is so much higher than the international portfolio over the period tracked that the loss of returns from adding more international stocks outweigh the lower risk that you would get from both just adding the lower vol international stocks and the diversification effects from having a correlation less than one.

The key point in the above is "over the period tracked". When you do this type of analysis you implicitly assume that the returns/risk observed in the past will be similar to the returns/risk in the future. Certainly, if you had invested 100% in the U.S. recently you would have done better than investing in a mix of US/Intl. However, while the risk and correlations of assets can be (somewhat) stable over time relative returns can vary wildly!

This uncertainty of future returns is why most people use a diversified portfolio of assets. What is the exact right amount is a very hard question though.

Answer 3

There are a couple of reasons to diversify your assets. First, since we cannot predict which of our investments will perform best, we want to "cast our net" broadly enough to have something invested in what's going to be performing well.

Second, diversification isn't intended to provide the highest returns, but rather it is used to soften the effects of market volatility. By softening the downsides and lowering the overall volatility among our assets, returns are more consistent.

If a model does not address future downside risk it is only telling you part of the story. (Past performance does not guarantee... you get the picture)

Answer 4

When you have multiple assets available and a risk-free asset (cash or borrowing) you will always end up blending them if you have a reasonable objective function. However, you seem to have constrained yourself to 100% investment. Combine that with the fact that you are considering only two assets and you can easily have a solution where only one asset is desired in the portfolio. The fact that you describe the US fund as "dominating" the forign fund indicates that this may be the case for you. Ordinarily diversification benefits the overall portfolio even if one asset "dominates" another but it may not in your special case.

Notice that these funds are both already highly diversified, so all you are getting is cross-border diversification by getting more than one. That may be why you are getting the solution you are. I've seen a lot of suggested allocations that have weights similar to what you are using.

Finding an optimal portfolio given a vector of expected returns and a covariance matrix is very easy, with some reliable results. Fancy models get pretty much the same kinds of answers as simple ones. However, getting a good covariance matrix is hard and getting a good expected return vector is all but impossible. Unfortunately portfolio results are very sensitive to these inputs. For that reason, most of us use portfolio theory to guide our intuition, but seldom do the math for our own portfolio. In any model you use, your weak link is the expected return and covariance. More sophisticated models don't usually help produce a more reasonable result.

For that reason, your original strategy (80-20) sounds pretty good to me. Not sure why you are not diversifying outside of equities, but I suppose you have your reasons.