The next in the series of star-performing investments is Warren Buffett's portfolio. In contrast with the previous portfolio of ETFs (mostly) aimed at replicating returns of Yale University's Endowment Fund, Warren Buffett's portfolio consists of stocks only. Mr. Buffett's 42-stock portfolio is structured asymmetrically with the biggest five holdings making up more than two-thirds of the entire asset allocation. The weighted-average dividend yield for the portfolio is approximately 2.2% which is 20+ basis points higher than S&P 500's. Warren Buffett is famous for delivering investment returns in excess of broad benchmarks': in simpler terms, he has consistently beaten the market in most years. This has largely been true, and the following illustration shows what the current asset allocation in his portfolio would have delivered had it remained unchanged for the last two decades:

The weighted portfolio has shown a positive alpha 70% of the time with an average annual return of almost 16%, compared with S&P 500's 9.0% (dividends excluded in both cases). What is more remarkable than that, is WB's statistical properties. During this time frame, Buffett's portfolio has demonstrated a standard deviation of annual returns slightly below 15%, while S&P 500 has experienced full 18.8%! In terms of risk/return, Buffett's current portfolio has shown a ratio below 1 (risk/return = 14.8%/15.7%), while S&P 500's ratio is above 2. WB's portfolio beta is 3% higher than the Index's. Whether Buffett applied Portfolio Theory while forming this portfolio or not we do not know, but his value/growth investment philosophy has definitely paid off.

## Analysis of Holdings

I will not go into details in regards to individual stocks and will only describe the top 5 holdings, since they employ the majority of the portfolio's capital (68%, to be almost precise). I am delighted to see that Warren Buffett holds at least three stocks that I have covered earlier in 2013: Exxon Mobil (NYSE:XOM), Conoco Phillips (NYSE:COP), and IBM (IBM).

The remaining one-third of capital has been allocated among 37 stocks. The following picture shows how skewed the distribution of weights is:

## Securities

*Wells Fargo and Co.* (NYSE:WFC)

Warren Buffett is significantly overweight on this particular stock. This diversified financial institution is the biggest in its peer group and shows attractive valuation metrics:

(Data courtesy of Google Finance)

WFC is third on the list in categories such as Return-on-Assets and Return-on-Equity. The above-inflation dividend yield is above the portfolio's average yield. In terms of other ratios, Wells Fargo is well-positioned. With this stock Buffett participates in the general recovery of financial intuitions, which, although have gained significantly since the market trough in 2008, have underperformed the broader market:

(Data courtesy of TD Waterhouse)

In fact, the graph shows that at the beginning of 2014 the sector has not recovered to the pre-crisis levels and remains roughly 20% underwater. Ostensibly, Warren Buffett believes there is room for expansion in the long term.

*Coca-Cola* (NYSE:KO)

There is lot of coverage on this company on Seeking Alpha and other popular web resources. What I wanted to point out here is that: (1) Coca-Cola has been on average less risky than Wells Fargo and (2) the two stocks have a very low covariance. The data in the model shows that Coca-Cola has a standard deviation of returns 3 percentage points lower than Well Fargo's. Of course, this came at a cost of ~6.5% difference in annual returns. On the other hand, the pop producer has a significantly larger dividend yield (2.84% vs. 2.59%) which can be more compelling to dividend-seekers.

The two stocks have an extremely low covariance of roughly 0.1%. The beauty of this figure is expressed in sample two-stock portfolio consisting of only Wells Fargo and Coca-Cola, equally weighted. While individually each one of the has shown risk exceeding 20% over the past 20 years, a portfolio of the two would have shown a standard deviation below 16% and a return of 12.5%. Both metrics are more attractive than S&P 500's standard deviation of 18.8% and an average return of 9.0%. Therefore, the two holdings that comprise over a third of the portfolio's capital already on their level tend to beat the market in both risk and return measures.

*International Business Machines*

IBM has taken the third place in Warren Buffett's portfolio. The stock is also ranked third among the top 5 holdings in terms of weighted-average returns. Although it has demonstrated a 3% alpha over Coca-Cola's average return, the score came at a cost of almost 600 basis points on the risk side of the equation. Coca-Cola and IBM have identically low betas (0.49) but Coca-Cola has a noticeably higher dividend yield than IBM. An equally-weighted portfolio consisting of only Coca-Cola and IBM would also show a smaller variance than the stocks by themselves, although the result is not as astonishing as in the case of Coca-Cola + Wells Fargo.

*American Express* (NYSE:AXP)

Even though Wells Fargo and American Express can both be described as financial stocks, they operate in extremely different businesses. This is partially expressed in the relatively high negative covariance of returns between the two. Among the top five, AXP has delivered the highest returns, especially in 2013 (53.5%). On average, AXP has outperformed Wells Fargo by 31% over the last two decades (20.8% vs. 15.8%). On the other hand, it must be noted AXP carried 64% more price risk than Wells Fargo. AXP has a lower beta than WFC but also offers a significantly lower dividend yield, which is also a below portfolio's. American Express had only four "bad" years in the past twenty versus WFC's five. However, the average negative return was much lower for WFC than AXP. Overall, I believe that American Express deserves its spot among Buffett's top five.

*The Procter & Gamble Company* (NYSE:PG)

The last but not least on the list is Procter & Gamble. The diversified consumer goods producer and supplier has shown the least price risk among the five and has on average exceeded Coca-Cola's returns by 30 basis points. Also, the stock offers the highest yield in the list and has had the least "bad" years among the five (only 2 out of 20). Most importantly, in my opinion, is that Procter & Gamble is enormously diversified geographically and by type of business. I think it could be approximated to an ETF for investment purposes due to the large number of businesses inside the group. This leads to a conclusion that Warren Buffett actually owns several hundreds of businesses in its top 5 stocks.

The rest of the stocks show mixed results in comparison with the top five. It also seems that Warren Buffett allocated some capital towards speculative bets. For example, Lee Enterprises (NYSE:LEE), which posted a 195% gain in 2013, has shown a weighted-average return of 99% on a standard deviation of 281%! In total, there are five stocks that have shown a wild deviation of returns in excess of 100%. Readers are welcome to browse the Excel file to see these and other picks.

## Portfolio Analysis and Optimization

A lot has been said already about the portfolio from a statistical standpoint. In comparison with the broader market, Warren Buffett's investment vehicle is definitely a win in both returns and risk. However, readers are strongly encouraged to remember that these results are historical and, unless the tendencies remain the same on average, risks and returns may change unfavorably in the future. Unfortunately, I was not able to conduct a trend analysis on metrics such as covariance and correlation year-over-year or by selected time periods (there is simply too much data to work on). Overall, the 42x42 portfolio covariance matrix looks like this, zoomed out:

The snapshot is not meant to be readable because readers can examine it in the model itself. What I want to point readers' attention to is the amount of red color in the table. The red color indicates that two selected stocks have negative covariance, that is, on average tend to move in opposite directions. I do not know if Warren Buffett purposely selected stocks with these qualities but the result is quite remarkable: not only do the stocks have low correlations among each other but a lot of them also have negative correlations.

Once again, I have developed several asset allocation scenarios using samples from Warren Buffett's portfolio. The output is presented below:

Readers may choose scenarios according to their investment goals. They should remember that these options are based on previous data and may not be representative of future results. Apart from obvious risk-return figures I have included three metrics: Risk/Return Ratio (the lower the better), weighted-average dividend yield, and weighted-average beta. Readers should not forget about cash returns from dividends when they choose an investment option from the table.

Below is given a simple Efficient Frontier chart (axis interchanged) for a selection of Risk-Minimized scenarios. Also included are figures for the current Warren Buffett's portfolio and S&P 500:

(E(x) denotes expected return; SD(x) stands for standard deviation or market risk)

As one can see, Warren Buffett's portfolio together with the theoretically risk-minimized vehicles carries much less risk than S&P 500 for a given return. A portfolio with the highest standard deviation (the approximately equally-weighted option, furthest on the right of the Risk Minimized matrix) compensates abundantly for the extra ~1.5% of risk: expected return is more than doubled.

## Conclusion

Portfolio analysis goes beyond simple investigation of individual holdings. As readers can see, allocation techniques matter enormously and should not be overlooked. I encourage investors to go into some depth when creating a portfolio, especially their own. Setting up a specific goal, a mandate, is the cornerstone of personal investing. Statistics and financial analysis are vehicles that help drive investors to their goals but should not be prioritized over well-defined targets.

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