Sorry for the delay in posting. I've been away since the subsequent conversation ensued.Originally Posted by Billy
Billy, I wrestle with this concept often. Is holding one security at 100% in my portfolio, such as the IWM or the SPY, "diversified", or does proper diversification require multiple holdings? I think this is what you are asking. I don't know the answer as I bang this out but allow me to think through it as I write...
When we talk about diversification, indirectly I think we're talking about a reduction in risk. Ideally, we're also looking for maintaining/increasing potential returns while reducing risk, but at the end of the day, I think the goal is to minimize risk while maintaining some constant (predictable) level of potential gains. I add this latter part because the "risk free return" (RFR) reference used in many calculations is a 3 month T-bill, which is assumed to have 0 volatility at some nominal return.
"Diversification Benefit" is something that the financial types talk about when going from a portfolio containing n=1 security to a portfolio containing n=m (m=many ~>15, 20, 50, etc.). The "benefit" portion comes from a reduction in risk as we add stocks. Offsetting this benefit are transactional costs, e.g., it's far cheaper to manage 1 security than to mange 50. So, returns are enhanced as n ---> 1, but risk increases as n ---> 1.
IWM is the market, for all intensive purposes. In support of this statement here is the correlation of IWM with other markets:
The correlation with the Dow is 0.87, with the S&P500 0.92, and the NASDAQ 0.95. Being the "market", we can assume that it is diversified, e.g., addition of any further stocks to IWM will not result in a significant change in the correlation with other broad markets. Put another way, we can assume that the addition of another security to IWM will not change the volatility of IWM in a significant manner.
Correspondingly, holding a single position that is 100% in IWM certainly represents a portfolio that contains the market. For the portfolio to be diversified, we need to successfully show that additional positions in the portfolio do not reduce volatility and return. Put another way, for a given level of estimated return, can we find a security that has greater or equivalent estimated return but lower volatility?
First of all, it's REALLY hard to find a company, or companies, that fit this bill. With my processing power, I can process about 55 stocks at a time, and I went through over 600 stocks in 10 batches and didn't find one that had higher return at the same or less volatility than IWM. Here's a representative chart:
I've circled IWM -- you can see that there are no stocks with higher return and lower volatility -- so it's the combination of all of these stocks, with their relative correlations, that give IWM higher return but lower volatility.
Take a look at this next figure:
Look at the left -- you can see IWM, and you can see a purple diamond just below IWM. The blue line represents all the possible combinations of these 54 stocks in terms of weighting, and the one "closest" to the IWM is chosen. The purple diamond shows that this portfolio of 54 stocks can be made to look like IWM with specific weightings, which is what managers of big funds attempt to accomplish.
Hence, I think we can conclude that:
1) Holding 100% IWM is to be considered fully diversified within your portfolio, e.g., you're not going to improve your Diversification Benefit from n = 1 (because transaction costs are minimized, which maximizes takehome gain), and
2) out of the universe of 1800+ small-cap stocks, there are NO stocks that give an expected return higher than IWM at an equivalent or lower volatility (although I only checked ~600 of them).
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IWM was the hard nut, GDX is easier.
Answering the question of whether a portfolio holding 100% GDX is diversified portfolio follows along the same lines as above. If we can show that the addition of a security improves/maintains the portfolio return while reducing volatility, then we can say GDX is not a diversified holding.
Take a look at the following chart, which is the expected returns/volatility of the components of GDX, sans LIHR:
I've circled GDX to make it stand out on the chart. As you can see, there are INDEXES (^RUT, ^IXIC, and ^GSPC) that have lower volatility at equivalent or higher gain. Furthermore, because ^RUT has the highest expected return but is lower in volatility, then the IWM proxy should be considered as the additional security so that we can maximize return yet realize a lower volatility than GDX alone.
From the chart above we can conclude that owning a portfolio that was comprised of 100% GDX is not diversified -- there are additional securities that can be added which reduce risk (volatility) while improving gain.
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Extra Credit ( I'm on a roll now and am learning a few things ... )
The next question in my mind is "what allocation of IWM and GDX minimizes volatility?". This is the same as asking what combination minimizes my pain to drawdown?
Take a look at this chart:
The unlabeled yellow diamond in the middle of the picture is the result of 100 shares being allocated to GDX, and 100 shares being allocated to IWM. The blue line should extend down there but for some reason it does not. I'll look into that later.
It's clear from the blue line that there is yet another combination that has higher gain but minimum volatility. That allocation ends up being:
73% IWM
27% GDX
This is telling me that all things being equal, using the price data that has a ~ 100d moving average applied (see RiskMetrics document provided earlier), that a minimum-risk portfolio exists with IWM and GDX in the allocations above.
Of course, maximizing expected return means placing all your eggs in IWM, since it has the highest return yet the lower volatility of the two, or IWM = 100% and GDX = 0%.
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Extra Extra Credit ...
Since Pascal is working on XLE, let's see what the allocations for a IWM-GDX-XLE triplet should be in order to minimize risk in the portfolio, again using data through 6/20/11:
So, we see that XLE lies halfway between IWM and GDX in terms of volatility as well as expected return. The unlabled yellow diamond is a 100-share contribution in each security. The end of the blue line just above the unlabled diamond is the place where risk (volatility) is minimized -- and this corresponds to an allocation of
IWM: 69%
GDX: 23%
XLE: 8%
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So, in summary,
- trading 100% in IWM maximizes your expected return and can be considered a fully diversified holding
- trading 100% in GDX reduces your potential gain as well as introduces additional volatility to your portfolio
- trading 73% in IWM and 27% in GDX minimizes your portfolio volatility, at the cost of some expected return
I hope this has been helpful -- I've learned a few things, especially that intuition isn't always correct (my assumptions about GDX standing alone).
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