Wednesday, March 16, 2011

Some efficient low volatility portfolios: the 1/N policy

From this post, I will present some of the low-volatility strategies that perform well in practice. I will start with the well-known 1/N policy: simply invest the same amount of money in each of the N assets considered by the investor.
All the ideas contained in this post are inspired by the following paper (my best wishes to the authors):
Optimal versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy? by DeMiguel, Garlappi and Uppal. Review of Financial Studies 22(5), 1915--1953 (2009).
There is no better way to start this post than reciting the beginning of the previous paper:

In about the fourth century, Rabbi Issac bar Aha proposed the following rule for asset allocation: “One should always divide his wealth into three parts: a third in land, a third in merchandise, and a third ready to hand.”

Although the 1/N policy, by construction, is not a low volatility strategy, it shares similar properties as the other low volatility strategies. That is, the 1/N portfolio performs better than market indexes in terms of risk and return.

For instance, over the long run the 1/N policy, when applied to the 500 stocks composing the S&P 500 index, attains a bit worse volatility than that of the index but a much better return.  That is, it obtains consistently better Sharpe ratios or risk-adjusted returns.
It is true we can obtain a bit better performance if we have the skill to forecast expected returns or if we estimate the associated covariance matrix with state-of-the-art techniques. This is especially true if we use daily or weekly data in our estimations. But the 1/N policy always outperforms the associated market index in terms of risk and return, and this is more than enough for many investors.
The best way to analyze the 1/N performance versus the market one is by looking at the MSCI Equal Weighted Indices. From this information, and over a 10-years period, the world 1/N policy obtained an annual return of 10.9%, versus the 5.7% return of the corresponding world index. That is, it almost doubles the market return. The volatility of the 1/N policy is worse, but only a bit: 14% versus 13.4% for the world index.
In other words, the Sharpe ratio (SR) of the 1/N policy was 0.78 and the SR of the world index was 0.42.
The same results are obtained for other asset compositions: Europe, USA, Japan, etc.
In order to confirm these findings, I have run my own experiments with the stocks composing the S&P 500.
But before, let me add a brief a comment regarding the rebalancing of the 1/N policy. In order to maintain the same weight in each asset, it is necessary to rebalance the portfolio from time to time. This is because the weights of the assets will change due to their price evolution.
But when do we need to rebalance the 1/N policy? Well, it depends on the investor. For instance, the mentioned MSCI equal-weighted indices are rebalanced quarterly. A passive investor may rebalance yearly. In contrast, a more active investor should consider rebalancing on a monthly basis to take advantage of price movements.
I have chosen a monthly frequency to rebalance the 1/N policy. This is the best trade-off I have found between return performance and transaction costs. Later, I will show the results corresponding to a yearly rebalancing frequency (thay are only a bit worse).
The back-test is as follows: at a given week from 2006 up to now, I consider the 1/N policy, and a week later I compute the corresponding portfolio return, net of transaction costs (40 bps) respect to the last portfolio composition. For the S&P 500 index, I consider the corresponding weekly return, but net of any transaction costs.
I repeat this procedure for every week in the last five years obtaining the following performance: portfolio-return mean and volatility, the Sharpe ratio, the Value-at-Risk, and the correlation of the 1/N policy with the market-index return.
As I said, I rebalance the 1/N policy every four weeks in the past. The performance is the following. Over the last five years, the 1/N strategy attained a volatility of 24% (versus 21% of the S&P 500), a 14% worse.
But this is more than compensated by the return performance. The annualized mean return of the 1/N policy (after proportional transaction costs of 40 bps were discounted) was 12% versus the 3.7% return of the S&P 500, more than three times better!
For the 1/N policy, this corresponds to an annualized SR of 0.49. On the other hand, the SR of the S&P 500 was 0.17, almost three times worse.
To have a visual insight of this performance, in the next figure you can see the cumulative returns, over the last 52 weeks, of the 1/N policy (always after transaction costs) and the S&P 500 (no costs).
Note the better performance of the 1/N policy in terms of cumulative returns and the similar levels of risk.It can be observed that always, the 1/N return is higher than that of the S&P 500.
Regarding the rebalance frequency, if we are a more passive investor and want to rebalance only every year, the corresponding results are roughly similar (although a bit worse). For instance, the annualized SR over the last five years of the 1/N strategy was 0.45 versus 0.13 for the S&P 500. That is, we lose 8% in risk-return performance if we rebalance every year instead of every month.
To see if the previous results are consistent over time, next figure shows the rolling excess-return of the 1/N policy respect to the S&P 500. The evolution is from 2006 up to now, and using rolling 52-weeks periods.
It is clear that the 1/N policy outperforms the market index over the last five years. But I prefer to take into account the risk when analyzing the return performance. Hence, next figure shows the evolution for the two portfolios of the annualized SR from 2006, over rolling 52-weeks periods.

We can see that most of the time, the SR of the 1/N strategy is larger than that of the S&P 500.

Finally, in the next graph, we show the evolution of the annualized tracking error of the 1/N policy (respect to the S&P 500) from 2006 up to now.

It can be observed that the mean tracking-error is around 6%, not too low. This allows for better expected returns.
As a conclusion, it is well-established that the 1/N policy dominates the corresponding market indexes, showing it attains consistently better risk-adjusted returns.
To finish this post, I would like to throw up the following question: why do you think the 1/N policy outperforms market indexes in practice?
I do not have a clear answer. But one of the reasons may be the 1/N policy gives more weight to small stocks (compared to cap-weighted benchmarks). But I would like to know your opinions, they will be very welcome.

Edited (April 4th, 2011): Some conclusions from this blog and other forums:

·         Equal-weighted indexes put more weight on small and value stocks than cap-weighted index. Hence, as long as these stocks tend to outperform large or growth stocks, equal-weighted indexes will tend to outperform the corresponding market ones.

·         Rebalancing frequency may be another reason for the better performance. That is, rebalancing frequency may have a larger impact on equal weights than on cap weights.


  1. None of the stocks in the S&P500 are considered small cap stocks, so I don't think that the 1/N strategy is tapping in to the small cap effect. Given your rebalance frequencies, which are relatively high, I'd say 1/N amounts to a short-horizon Value strategy: you're buying more of stocks that have been beat down in price in excess of their fundamentals (over sold). Once the market realizes its overraction, it corrects and you see an outsized return.

  2. John, it makes sense and seems reasonable. Thanks for the comment.

  3. Actually I view the LV strategy as 'smaller' stocks market cap average around 20B vs. 80B for the S&P 500. So no it is not small cap like a R2000 but it is smaller then the S&P 500.

    Also, if you run back tests to limit turn to say quarterly or even annually the returns are still great. Especially risk adjusted (Sharpe)!

  4. David, I agree some “size” anomaly is helping the equal-weighted strategy.

    Actually, I have run the 1/N policy with a yearly rebalancing and the results are only a bit worse (always in terms of SR), but as you said they are still great.

    But we need to take care about this type of anomaly: past good performance does not assure the same future performance…

  5. Curious your thoughts on equal-value (e.g. above) versus equal-weight (e.g. RSP)?

  6. Some time ago I read an article wich analazed this matter. I do not remember the analized index and the period of time, but their conclusion: 86% of low volatility portfolios are explaned by the small and value effect. I do not remember if the analysis used the 1/N concept.

    What I want to say is that it would be of interest to know wich extent the performance of a 1/N volatility portfolio it is explaned by the small and value effect.

  7. Wonder if the french data library can
    be of any help?

    Can you apply LV to his data series?

    My guess it would be poosible to answer
    the small and value issue. Also shed some
    light on overall robustness over time.


  8. Prazor,

    Thanks for the suggestion. Indeed, we have tried the datasets in the K. French library in our academic papers. The insights are practically the same. I prefer to use stocks in this blog because the assets in French library are not tradable, but I agree with you they can give a light about the small and value effects.

    Best regards.