**By
John Dowdee
:**
One of the central tenets of asset allocation is to select a
diversified portfolio. The idea is intuitive; you do not keep all
your eggs in one basket. Everyone talks about being diversified but
there is little discussion on how you measure the degree of
diversification. Surprisingly, when I began to research this topic,
I found that there is no universally accepted formula for assessing
diversification. In fact, the whole topic of diversification is
typically clouded with hand waving and qualitative judgments.

In finance, diversification means reducing portfolio risk by
investing in a variety of assets that do not move in lock-step with
one another. There are many types of portfolio risk (interest rate
risk, event risk, etc.) but for this analysis, I will define the
portfolio risk to be synonymous with volatility. Diversification
can then be characterized as the degree that you reduce the
volatility of a portfolio by selecting assets.

To be "diversified," you want to choose assets such that when
some assets are down, others are up. In mathematical terms, you
want to select assets that are uncorrelated (or at least not highly
correlated) with each other. In my definition of diversification,
correlations will always be zero or positive. If the correlations
are negative, I consider this to be "hedging" rather than
diversifying.

I will discuss two methods of measuring diversification in a
quantitative manner. Neither method is ideal but I have found that
both are helpful in assessing if a portfolio is diversified.

It is well known that if you combine assets into a portfolio,
the overall volatility (i.e. the standard deviation) is less than
the average of volatilities associated with each of the components.
So one measure of diversification is the percentage decrease
achieved in the portfolio volatility as compared to the weighted
average of the component volatilities. This is similar to the
definition that has been offered in other Seeking Alpha articles
such as
this one
.

As an example, consider an equally weighted portfolio (which I
will call Portfolio 1) that consists of the following
ETFs
:

- [[SPY]]: SPDR S&P 500 Index
- [[VO]]: Vanguard Mid-Cap
- [[VB]]: Vanguard Small Cap
- [[VGK]]: Vanguard MSCI Europe
- [[VPL]]: Vanguard MSCI Pacific
- [[VWO]]: Vanguard FTSE Emerging Markets

On the surface this would appear to be a well-diversified
portfolio with Large Caps, Mid-Caps, and Small Caps along with
international stocks from Europe, the Pacific, and Emerging
Markets.

To quantify the diversification, I plotted the annualized rate
of return in excess of the risk free rate of return (called Excess
Mu on the charts) against the historical volatility for each of the
ETFs. The Smartfolio 3 program (smartfolio.com) was used to
generate this plot based on three years of historical data. This
data is shown in Figure 1.

*(click to enlarge)*

**Figure 1: Risk Reward for Portfolio 1**

The volatility for each component is summarized in Figure 2 and
the average of these volatilities is 22.8%. When these ETFs are
combined into a portfolio, the resulting overall portfolio
volatility is 21.7%, a reduction of 5%. This is not a large
reduction in volatility, which suggests that this portfolio is not
very diversified. This conclusion is validated by observing that
the correlations among components are very high (see Figure 3).

**Figure 2: Portfolio 1 Volatility**

**
***(click to enlarge)*

**Figure 3: Portfolio 1 Correlation Matrix**

An alternative way to measure diversification is to compare your
portfolio with the "maximum diversified" portfolio. I define a
maximum diversification portfolio as one where all components are
perfectly uncorrelated. Such a portfolio is not realizable in the
real world but we can gain insight by comparing our portfolio with
this idealized portfolio.

For a "maximum diversification" portfolio, all the correlation
coefficients are zero so the portfolio variance is just the
weighted sum of the individual variances. If the constituents are
equally weighted, the portfolio standard deviation (i.e. the
volatility) is calculated as: (square root of the sum of variance)
divided by number of components.

For Portfolio 1, the volatility associated with maximum
diversification is computed as 9.4% (see Figure 2). The overall
volatility of Portfolio 1 was previously computed as 21.7%. This is
long way from the ideal maximum diversification volatility. This is
another indication that this is not a well-diversified
portfolio.

In the current environment, finding a truly diversified
portfolio is difficult since most assets have tended to move
together during the bull market that began in 2009. This is not
necessarily an issue and will likely not create any problems as
long as the bulls stay strong. I understand that correlations
change over time and an uncorrelated portfolio today may not be
uncorrelated tomorrow. As with all things financial, you have to
keep your eyes wide open and make adjustments to avoid risk.
However, I am fearful that assets that rise together may also fall
together, so I have continued to search for ETFs and CEFs that are
not highly correlated.

As an example, let's consider Portfolio 2, another equally
weighted portfolio consisting of the following ETFs and Closed End
Funds (CEFs):

- SPY: SPDR S&P 500
- [[ACG]]: Alliance Bernstein Income Fund
- [[DBS]]: PowerShares DB Silver
- [[FXA]]: CurrencyShares Australian Dollar Trust
- [[IBB]]: iShares Nasdaq Biotechnology
- [[PCY]]: PowerShares Emerging Market Sovereign Debt

This portfolio goes further afield than the previous portfolio
by including both commodity and currency assets. So the question is
whether this portfolio is more diversified than Portfolio 1.

Figure 4 shows the Risk Reward plot for Portfolio 2, based again
on 3 years of history. This portfolio delivers a rate of return
similar to the S&P 500 but has substantially less volatility.
The diversification statistics are summarized in Figure 5. The
average volatility of the components is 16.8%. Combining these
assets into a portfolio reduces the overall portfolio volatility to
11.8%, a reduction of almost 30%, much larger than for Portfolio 1.
The volatility of the maximum diversification portfolio is 8%,
which is not that far below the overall portfolio volatility. These
measures are indicative of good diversification, which is verified
by the correlation matrix shown in Figure 6. The correlations among
SPY, IBB, and FXA are relatively high but the correlations among
ACG, DBS, and PCY are relatively low. This illustrates that a few
low correlations can be very beneficial in lowering the overall
volatility of a portfolio.

*(click to enlarge)*

**Figure 4: Risk Reward for Portfolio 2**

**
**

**Figure 5: Portfolio 2 Volatility**

**
***(click to enlarge)*

**Figure 6: Portfolio 2 Correlation Matrix**

To summarize, using quantitative measurements, Portfolio 2 is
more diversified than Portfolio 1. Please note that I am not saying
that Portfolio 1 is a "bad" portfolio. There are many ways to judge
the suitability of a portfolio and diversification is only one of
many factors to consider.

I also fully realize that my analysis of diversification is
based on a set of assumptions that are only approximations to the
real world. The data I presented is from the last three years,
which was a tremendous bull market that is unlikely to repeat over
the next three years. I welcome discussion on how diversification
may or may not be a useful indicator for assessing portfolio risk
going forward.

**Disclosure:**
I am long [[DBS]], [[FXA]], [[PCY]]. I wrote this article myself,
and it expresses my own opinions. I am not receiving compensation
for it. I have no business relationship with any company whose
stock is mentioned in this article.

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