Towards a better equity benchmark: random portfolios

Random portfolios deliver alpha relative to a buy-and-hold position in the S&P 500 index – even after allowing for trading costs. Random portfolios will serve as our benchmark for our future quantitative equity models.

The evaluation of quantitative equity portfolios typically involves a comparison with a relevant benchmark, routinely a broad index such as the S&P 500 index. This is an easy and straightforward approach, but also – we believe – sets the bar too low resulting in too many favorable research outcomes. However, we want to hold ourselves to a higher standard and as a bare minimum that must include being able to beat a random portfolio – the classic dart-throwing monkey portfolio.

A market capitalization-weighted index, such as the S&P 500 index, is inherently a size-based portfolio where those equities which have done well in the long run are given the highest weight. It is, in other words, dominated by equities with long-run momentum. Given that an index is nothing more than a size-based trading strategy, we should ask ourselves whether there exist other simple strategies that perform better. The answer to this is in the affirmative and one such strategy is to generate a portfolio purely from chance. Such a strategy will choose, say, 20 equities among the constituents of an index each month and invest an equal amount in each position. After a month the process is repeated, 20 new equities are randomly picked, and so forth.

The fact that random portfolios outperform their corresponding indices is nothing new. David Winton Harding, the founder of the hedge fund Winton Capital, even went on CNBC last year to explain the concept, but he is far from only in highlighting the out-performance of random portfolios (e.g. here and here).

To demonstrate the idea we have carried out the research ourselves. We use the S&P 500 index – without survival bias and accounting for corporate actions such as dividends – and focus on the period from January 2000 to November 2015. We limit ourselves in this post to this time span as it mostly covers the digitization period, but results from January 1990 show similar results. Starting on 31 December 1999 we randomly select X equities from the list of S&P 500 constituents that particular month, assign equal weights, and hold this portfolio in January 2000. We then repeat the process on the final day of January and hold in February and every subsequent month until November 2015.

g1

In the chart above the annualized returns for 1000 portfolios containing 10, 20, and 50 equities are depicted (the orange line represents the S&P 500 index). A few things are readily apparent:

  • The mean of the annualized returns is practically unchanged across portfolio sizes
  • The standard deviation of the annualized returns narrows as the portfolio size increases
  • All three portfolio sizes beat the S&P 500 index
  • 6.5% of the 1000 random portfolios of size 10 have an annualized return which is lower than that of the S&P 500 index (4.2%). Of the portfolios with sizes 20 and 50 the percentages are 0.9% and 0%, respectively. In other words, not a single of the 1000 random portfolios of size 50 delivers a annualized return below 4.2%.

So far we have talked one or many random portfolios without being too specific, but for random portfolios to work we need a large number of samples (i.e. portfolios) so that performance statistics, such as the annualized return, tend toward stable values. The chart below shows the cumulative mean over the number of random portfolios (size = 10), which stabilizes as the number of random portfolios increases (the orange line represents the mean across all 1000 portfolios).

g2

All three portfolios beat the S&P 500 index in terms of annualized return, but we must keep in mind that these portfolios’ turnovers are high and hence we need to allow for trading costs. The analysis is thus repeated below  for 1000 random portfolios with 50 positions with trade costs of 0%, 0.1% and 0.2% (round trip).

g3

Unsurprisingly the mean annualized return declines as trading costs increase. Whereas not a single of the 1000 random portfolios of size 50 delivered an annualized return below 4.2% without trade costs , 2 and 40 portfolios have lower returns when trade costs of 0.1% and 0.2%, respectively, are added. Put differently, even with trade costs of 0.2% (round trip) every month a portfolio of 50 random stocks outperformed the S&P 500 index in terms of annualized return in 960 of 1000 instances.

g4

Weighing by size is simple, and we like simple. But when it comes to equity portfolios we demand more. Put differently, if our upcoming quantitative equity portfolios cannot beat a randomly-generated portfolio what is the point? Therefore, going forward, we will refrain from using the S&P 500 index or any other appropriate index and instead compare our equity models to the results presented above. We want to beat not only the index, we want to beat random. We want to beat the dart-throwing monkey!

ADDENDUM: A couple of comments noted that we must use an an equal weight index to be consistent with the random portfolio approach. This can, for example, be achieved by investing in the Guggenheim S&P 500 Equal Weight ETF, which has yielded 9.6% annualized since inception in April, 2003 (with an expense ratio of 0.4%). The ETF has delivered a higher annualized return than that of the random portfolios (mean) when trading costs are added.

In other words, if you want to invest invest equally in the S&P 500 constituents, there is an easy way to do it. We will continue to use random portfolios as a benchmark as it is a simple approach, which our models must beat and we can choose the starting date as we please.

###########################################################
# #
# INPUT: #
# prices: (months x equities) matrix of close prices #
# prices_index: (months x 1) matrix of index close prices #
# tickers: (months x equities) binary matrix indicating #
# whether a stock was present in the index in that month #
# #
###########################################################

draws <- 1000
start_time <- "1999-12-31"
freq_cal <- "MONTHLY"

N <- NROW(prices) # Number of months
J <- NCOL(prices) # Number of constituents in the S&P 500 index

prices <- as.matrix(prices)
prices_index <- as.matrix(prices_index)
prices_ETF <- as.matrix(prices_ETF)

# Narrow the window
#prices <- prices[-1:-40, , drop = FALSE]
#prices_index <- prices_index[-1:-40, , drop = FALSE]
#prices_ETF <- prices_ETF[-1:-40, , drop = FALSE]

#N <- NROW(prices)

# Combinations
sizes <- c(10, 20, 50) ; nsizes <- length(sizes) # Portfolio sizes
costs <- c(0, 0.1, 0.2) ; ncosts <- length(costs) # Trading costs (round trip)

# Array that stores performance statistics
perf <- array(NA, dim = c(nsizes, ncosts, 3, draws),
dimnames = list(paste("Size", sizes, sep = " "), paste("Cost", costs, sep = " "),
c("Ret(ann)", "SD(ann)", "SR(ann)"), NULL))

# Loop across portfolio sizes
for(m in 1:nsizes) {

# Storage array
ARR <- array(NA, dim = c(N, sizes[m], draws))

# Loop across time (months)
for(n in 1:(N - 1)) {

# Which equities are available?
cols <- which(tickers[n, ] == 1)

# Forward return for available equities
fwd_returns <- prices[n + 1, cols]/prices[n, cols] - 1

# Are these equities also available at n + 1?
cols <- which(is.na(fwd_returns) == FALSE)

# Forward return for available equities
fwd_returns <- fwd_returns[cols]

# Loop across portfolios
for(i in 1:draws) {

# Sample a portfolio of size 'sizes[m]'
samp <- sample(x = cols, size = sizes[m], replace = F)

# Store a vector of forward returns in ARR
ARR[n, , i] <- fwd_returns[samp]

} # End of i loop

} # End of n loop

ARR[is.na(ARR)] <- 0

# Loop across trading costs
for(m2 in 1:ncosts) {

# Performance calculations
returns_mean <- apply(ARR, c(1, 3), mean) - costs[m2]/100
returns_cum <- apply(returns_mean + 1, 2, cumprod)
returns_ann <- tail(returns_cum, 1)^(percent_exponent/N) - 1

std_ann <- exp(apply(log(1 + returns_mean), 2, StdDev.annualized, scale = percent_exponent)) - 1
sr_ann <- returns_ann / std_ann

perf[m, m2, "Ret(ann)", ] <- returns_ann * 100
perf[m, m2, "SD(ann)", ] <- std_ann * 100
perf[m, m2, "SR(ann)", ] <- sr_ann

} # End of m2 loop

} # End of m loop

# Index and ETF returns
returns_index <- prices_index[-1, 1]/prices_index[-N, 1] - 1
returns_ava_index <- sum(!is.na(returns_index))
returns_index[is.na(returns_index)] <- 0
returns_cum_index <- c(1, cumprod(1 + returns_index))
returns_ann_index <- tail(returns_cum_index, 1)^(percent_exponent/returns_ava_index) - 1

returns_ETF <- prices_ETF[-1, 1]/prices_ETF[-N, 1] - 1
returns_ava_ETF <- sum(!is.na(returns_ETF))
returns_ETF[is.na(returns_ETF)] <- 0
returns_cum_ETF <- c(1, cumprod(1 + returns_ETF))
returns_ann_ETF <- tail(returns_cum_ETF, 1)^(percent_exponent/returns_ava_ETF) - 1

# Print medians to screen
STAT_MED <- apply(perf, c(1, 2, 3), median, na.rm = TRUE)
rownames(STAT_MED) <- paste("Size ", sizes, sep = "")
colnames(STAT_MED) <- paste("Cost ", costs, sep = "")
print(round(STAT_MED, 2))
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20 thoughts on “Towards a better equity benchmark: random portfolios

  1. Interesting idea! I can see this type of benchmark being useful in disproving the value of a strategy that selects stocks from a universe if you then generated the random portfolios from the same universe (e.g., strategy is worse or no better than a random selection from the universe). I realize you weren’t concerned with why the random portfolios outperformed the index, but do you think any of it could be explained by equal vs. market cap weighting? Did you try any market cap weighted random portfolios or comparing the portfolios to the equally weighted index?

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    1. Hi Adam,

      The equal vs. market cap weighting is key as the addendum to the blog post explains. This is to say, the Guggenheim S&P 500 Equal Weight ETF has delivered a higher return since inception in 2003 – also risk-adjusted. Thank you for your comment.

      Like

  2. You need to use equal weighted version or S&P 500 as benchmark, to be consistent with random portfolios. I.e. each random portfolio equally weights its constituents.

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    1. Hi Investor Curious,

      You are correct. An equal weight S&P index beats not only the standard S&P 500 index, but also random portfolios. We have updated to blog post to reflect this. Thank you for your comment.

      Like

  3. Thank you both for your comments, Adam and Investor Curious. The comments about (an) equal weights (index) are definitely valid and we will have a look at it in an upcoming blog post.

    Happy New Year!

    EDIT: Instead of a new blog post, we have added a bit to this one.

    Like

    1. Hi Shane,

      Thank you for your comment. We haven’t looked at daily rebalancing, but using weekly instead of monthly rebalancing results in annualized returns of 8.8%, 3.3% and -2% with trading costs of 0%, 0.1% and 0.2% respectively (portfolio size of 50).

      The Sharpe ratios (with risk free rate at 0%) are 0.37, 0.14 and -0.08 respectively. The Sharpe ratios (SR) for the monthly model are higher due to both lower volatility and lower costs. Even with no trading costs the SR is 0.43 for the monthly model and 0.37 for the weekly model (though that is not a statistically significant difference).

      Like

  4. Thanks for sharing your research, Mr Beta. I am also of the opinion that random portfolios can be a useful benchmark in evaluating strategy performance. If you or your readers are interested, I explored this idea by generating random trades that mimic the trade frequency and holding period of the strategy being evaluated and applied it to a foreign exchange strategy. The concept could equally be applied to an equities portfolio: http://robotwealth.com/benchmarking-backtest-results-against-random-strategies/

    Like

    1. Hi Kris,

      It is a great paper, Andrew W. Lo, Harry Mamaysky, and Jiang Wang (2000), you are referring to in your blog post. These statistical tests are absolutely essentiel in order not to be fooled by randomness. Everytime you backtest a strategy you only have one sample based on the historical dataset which creates a lot of statistical issues.

      We will publish a blog post soon showcasing an equity factor model based on four known equity factors. In our backtest, we benchmark the strategy against 200 bootstrapped portfolios (resampling of selection universe).

      In a short follow-up post we will discuss and highlight why bootstrapping is absolutely essential. If someone came to us with a strategy and it was not benchmarked against bootstrapped strategies then we would reject the strategy in a split second.

      Like

  5. Random portfolios (or rules) can be used to validate backtested strategies but I think you have to consider how the benchmark is being used. Portfolio construction and performance attribution have different requirements than statistical validation of a backtest.

    In the majority of situations, a market-weight benchmark is the only logical choice because they can be simultaneously held by every investor and require no rebalancing (in theory).

    I’m not even sure I’m comfortable calling bootstrap samples a “benchmark” since that isn’t really the intention.

    Like

  6. Hi Ryan,

    Thanks for your comment. I interpret your comment as partly a comment to my answer and partly the overall post.

    Let me start with your last statement, which I think refers to my comment. I can see now, that I may have been imprecise in my comment. What we do, and will be explained in the upcoming post, is to bootstrap our four-factor equity model 200 times creating a good approximation of the underlying distribution af portfolio returns. This distribution is then compared against 200 random portfolios. If our strategy’s distribution is significantly different from the random distribution then we are closer to a strategy ready for production.

    Your comment about benchmarks are valid to some degree. The industry adopted definition of a benchmark is…

    – Specified in advance: The benchmark is specified prior to the start of the evaluation period.
    – Appropriate: The benchmark is consistent with the manager’s investment style or area of expertise.
    – Measurable: The benchmark’s return is readily calculable on a reasonably frequent basis.
    – Unambiguous: The identities and weights of securities are clearly defined.
    – Reflective of current investment opinions: The manager has current knowledge of the securities in
    the benchmark.
    – Accountable: The manager is aware and accepts accountability for the constituents and
    performance of the benchmark.
    – Investable: It is possible to simply hold the benchmark

    Managing investment portfolios: A dynamic process (CFA institute investment Series), Third edition, John L. Maginn, Donald L. Tuttle, Jerald E. Pinto, Dennis W. McLeavey

    In that regard you are right that S&P 500 is a better benchmark than random portfolios.

    However, I would add, that this is only correct for a long-only equity manager with a large set of securities overlapping with the benchmark or a portfolio with an investment policy statement containing many constraint about tracking error, active weights etc.

    If you are a small fund with almost no restrictions and have a concentrated approach (maybe only 20 stocks in the portfolio) then I would argue random portfolio is a better benchmark, because it is closer in nature to the underlying strategy.

    Lastly, we know that random beats S&P 500 so if your portfolio significantly beats random you are definitely home safe against S&P 500.

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  7. Hello
    Interesting article. My english and statistical knowledge is unfortunatly not very profound. I hope you understand my question.
    Did you regress your results to the fama french factors? Could the better returns of random and equal weighting strategies be contributed to the small cap , value, momentum or investment factors?
    There are many smart people arround who do not favour equal weighting and as mentioned the aggregate of all investors can not equal weighting.
    How would a random cap weighted strategy perform?
    Thank you for your answers.
    Regards from Switzerland
    Jakob

    Like

  8. You said in one of the comments that you looked at weekly rebalancing but what about only annual rebalancing. Your strategy would be very tax inefficient in taxable accounts and an annual rebalance could solve that. Could you show annual results so investors could see if this was a case of a few great years or consistent out performance.

    Like

    1. Hi David,

      Thank you for your comment.

      Using end-of-year rebalancing the portfolios have delivered annualized returns of 7.5%, 7.6%, and 7.8% for portfolio sizes 10, 20, and 50, assuming round-trip trading costs of 0.2% for the period 2000-2015. This compares with 3.8% for the index in the same period.

      Annualized volatilities are higher than when using monthly rebalancing at 24.8%, 23.6%, and 22.9% respectively.

      Like

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