Random portfolios: correlation clustering

We investigate whether two clustering techniques, k-means clustering and hierarchical clustering, can improve the risk-adjusted return of a random equity portfolio. We find that both techniques yield significantly higher Sharpe ratios compared to random portfolio with hierarchical clustering coming out on top.

Our debut blog post Towards a better benchmark: random portfolios resulted in a lot of feedback (thank you) and also triggered some thoughts about the diversification aspect of random portfolios. Inspired by the approach in our four-factor equity model we want to investigate whether it is possible to lower the variance of the portfolios by taking account of the correlation clustering present in the data. On the other hand, we do not want to stray too far from the random portfolios idea due to its simplicity.

We will investigate two well-known clustering techniques today, k-means clustering and hierarchical clustering (HCA), and see how they stack up against the results from random portfolios. We proceed by calculating the sample correlation matrix over five years (60 monthly observations) across the S&P 500 index’s constituents every month in the period 2000-2015, i.e. 192 monthly observations. We then generate 50 clusters using either k-means (100 restarts) or HCA on the correlation matrix and assign all equities to a cluster. From each of these 50 clusters we randomly pick one stock and include it in our portfolio for the following month. This implies that our portfolios will have 50 holdings every month. (None of the parameters used have been optimized.)

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The vertical dark-grey line represents the annualized return of a buy-and-hold strategy in the S&P 500 index.

The mean random portfolio returns 9.1% on an annualized basis compared to the benchmark S&P 500 index, which returns 4.1%. The k-means algorithm improves a tad on random portfolios with 9.2% while HCA delivers 9.5%. Not a single annualized return from the 1,000 portfolios is below the buy-and-hold return no matter whether we look at random portfolios, k-means or HCA.

We have added a long-only minimum variance portfolio (MinVar) to the mix. The 1,000 MinVar portfolios use the exact same draws as the random portfolios, but then proceed to assign weights so they minimize the portfolios’ variances using the 1,000 covariance matrices rather than simply use equal weights.

Unsurprisingly, MinVar delivers a mean return well below the others at 8.1%. The weights in the MinVar portfolios are only constrained to 0-100% implying that in any given MinVar portfolio a few equities may dominate in terms of weights while the rest have weights very close to 0%. As a result, the distribution of the annualized mean returns of the MinVar portfolios is much wider and 1.2% of the portfolio (12) are below the S&P 500 index’s annualized mean.

g2

The vertical dark-grey line represents the annualized Sharpe ratio of a buy-and-hold strategy in the S&P 500 index.

Turning to the annualized Sharpe ratio using for simplicity a risk-free rate of 0%, we find that a buy-and hold strategy yields a Sharpe ratio of 0.25. Random portfolios, meanwhile, yields 0.46 while k-means and HCA yield 0.49 and 0.58, respectively.

The k-means and HCA algorithms result in lower volatility compared to standard random portfoliios (see table below). While the random portfolios have a mean annualized standard deviation of 19.8%, k-means and HCA have mean standard deviations of 18.7% and 16.3% respectively (in addition to the slightly higher mean annualized returns). Put differently, it seems from the table below and charts above that k-means and in particular HCA add value relative to random portfolios, but do they do so significantly?

US_g3

Using a t-test on the 1,000 portfolios’ annualized means we find p-values of 0.7% and ~0% respectively, supporting the view that significant improvements can be made by using k-means and HCA. The p-values fluctuate a bit for the k-means algorithm with every run of the code, but the HCA’s outperformance (in terms of annualized mean return) is always highly significant. Testing the Sharpe ratios rather than mean returns yield highly significant p-values for both k-means and HCA.

We have looked at two ways to potentially improve random portfolios by diversifying across correlation clusters. The choice of k-means and HCA as clustering techniques was made to keep it as simple as possible, but this choice does come with some assumptions. Variance Explained – for example – details why using k-means clustering is not a free lunch. In addition, what distance function to use in the HCA was not considered in this post, we simply opted for the simple choice of: 1 – correlation matrix. We leave it to the reader to test other, more advanced clustering methods, and/or change the settings used in this blog post.

Portfolio size

So far we have only looked at a portfolio size of 50, but in the original post on random portfolios we also included portfolios of size 10 and 20. For completeness we provide the results below noting simply that the pattern is the same across portfolio sizes. In other words, HCA is the best followed by k-means and random portfolios – though the outperformance decreases with the sample size.

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Excess returns

The approach above uses the historical correlation matrix of the returns, but there is a case to be made for using excess returns instead. We have therefore replicated the analysis above using excess returns instead of returns, where excess returns are defined as the residuals from regressing the available constituent equity returns on the market returns (i.e. S&P 500 index returns) using the same lookback window of five years (60 observations).

Using the matrix of residuals we construct a correlation matrix and repeat the analysis above; we follow the same steps and use the same parameter settings. Disappointingly, the results do not improve. In fact, the results deteriorate to such a degree that neither k-means nor HCA outperform regardless of whether we look at the mean return or Sharpe ratio.

Postscript: while we use k-means clustering on the correlation matrix we have seen it used in a variety of ways in the finance blogosphere, from Intelligent Trading Tech and Robot Wealth, both of which look at candlestick patterns, over Turing Finance’s focus on countries’ GDP growth to MKTSTK’s trading volume prediction, to mention a few.

###########################################################
# #
# 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 #
# #
###########################################################

library(TTR)
library(PerformanceAnalytics)
library(quadprog)
library(xtable)

# Function for finding the minimum-variance portfolio
min_var <- function(cov.mat, short = FALSE) {

if (short) {

aMat <- matrix(1, size.port, 1)
res <- solve.QP(cov.mat, rep(0, size.port), aMat, bvec = 1, meq = 1)

} else {

aMat <- cbind(matrix(1, size.port, 1), diag(size.port))
b0 <- as.matrix(c(1, rep(0, size.port)))
res <- solve.QP(cov.mat, rep(0, size.port), aMat, bvec = b0, meq = 1)

}

return(res)

}

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)

port.names <- c("Random", "K-means", "HCA", "MinVar")

# Array that stores performance statistics
perf <- array(NA, dim = c(3, draws, length(port.names)),
dimnames = list(c("Ret (ann)", "SD (ann)", "SR (ann)"), NULL, port.names))

# Storage array
ARR <- array(NA, dim = c(N, draws, size.port, length(port.names)),
dimnames = list(rownames(prices), NULL, NULL, port.names))

rows <- which(start.port == rownames(prices)):N

# Loop across time (months)
for (n in rows) {

cat(rownames(prices)[n], "\n")

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

# Forward and backward return for available equities
fwd.returns <- prices[n, cols]/prices[n - 1, cols] - 1
bwd.returns <- log(prices[(n - lookback):(n - 1), cols]/
prices[(n - lookback - 1):(n - 2), cols])

# Are these equities also available at n + 1?
cols <- which(is.na(fwd.returns) == FALSE & apply(is.na(bwd.returns) == FALSE, 2, all))

# Returns for available equities
fwd.returns <- fwd.returns[cols]
bwd.returns <- bwd.returns[, cols]

bwd.returns.index <- log(prices.index[(n - lookback):(n - 1), 1]/
prices.index[(n - lookback - 1):(n - 2), 1])

# Covariance and correlation matrices
cov.mat <- cov(bwd.returns)
cor.mat <- cor(bwd.returns)

# K-means on covariance matrix
km <- kmeans(x = scale(cor.mat), centers = size.port, iter.max = 100, nstart = 100)
hc <- hclust(d = as.dist(1 - cor.mat), method = "average")

for (d in 1:draws) {

samp <- sample(x = 1:length(cols), size = size.port, replace = FALSE)
opt <- min_var(cov.mat[samp, samp], short = FALSE)

ARR[n, d, , "Random"] <- fwd.returns[samp]
ARR[n, d, , "MinVar"] <- opt$solution * fwd.returns[samp] * size.port

}

hc.cut <- cutree(hc, k = size.port)

for (q in 1:size.port) {

ARR[n, , q, "K-means"] <- sample(x = fwd.returns[which(km$cluster == q)], size = draws, replace = TRUE)
ARR[n, , q, "HCA"] <- sample(x = fwd.returns[which(hc.cut == q)], size = draws, replace = TRUE)

}

}

ARR <- ARR[rows, , , ]

# Performance calculations
returns.mean <- apply(ARR, c(1, 2, 4), mean) - cost.port/100 ; N2 <- NROW(ARR)
returns.cum <- apply(returns.mean + 1, c(2, 3), cumprod)
returns.ann <- returns.cum[N2, , ]^(12/N2) - 1

std.ann <- exp(apply(log(1 + returns.mean), c(2, 3), StdDev.annualized, scale = 12)) - 1
sr.ann <- returns.ann / std.ann

returns.index <- c(0, prices.index[-1, 1]/prices.index[-N, 1] - 1)
returns.index <- returns.index[rows]
returns.cum.index <- c(1, cumprod(returns.index + 1))
returns.ann.index <- tail(returns.cum.index, 1)^(12/N2) - 1

std.ann.index <- exp(StdDev.annualized(log(1 + returns.index[-1]), scale = 12)) - 1
sr.ann.index <- returns.ann.index / std.ann.index

perf["Ret (ann)", , ] <- returns.ann * 100
perf["SD (ann)", , ] <- std.ann * 100
perf["SR (ann)", , ] <- sr.ann

results.mean <- as.data.frame(apply(perf, c(1, 3), mean))
results.mean$Benchmark <- c(returns.ann.index * 100, std.ann.index * 100, sr.ann.index)

print(round(results.mean, 2))

t.test(returns.ann[,2], returns.ann[, 1], alternative = "greater")
t.test(returns.ann[,3], returns.ann[, 1], alternative = "greater")
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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 &lt;- 1000
start_time &lt;- &quot;1999-12-31&quot;
freq_cal &lt;- &quot;MONTHLY&quot;

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

prices &lt;- as.matrix(prices)
prices_index &lt;- as.matrix(prices_index)
prices_ETF &lt;- as.matrix(prices_ETF)

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

#N &lt;- NROW(prices)

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

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

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

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

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

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

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

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

# Forward return for available equities
fwd_returns &lt;- fwd_returns[cols]

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

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

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

} # End of i loop

} # End of n loop

ARR[is.na(ARR)] &lt;- 0

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

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

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

perf[m, m2, &quot;Ret(ann)&quot;, ] &lt;- returns_ann * 100
perf[m, m2, &quot;SD(ann)&quot;, ] &lt;- std_ann * 100
perf[m, m2, &quot;SR(ann)&quot;, ] &lt;- sr_ann

} # End of m2 loop

} # End of m loop

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

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

# Print medians to screen
STAT_MED &lt;- apply(perf, c(1, 2, 3), median, na.rm = TRUE)
rownames(STAT_MED) &lt;- paste(&quot;Size &quot;, sizes, sep = &quot;&quot;)
colnames(STAT_MED) &lt;- paste(&quot;Cost &quot;, costs, sep = &quot;&quot;)
print(round(STAT_MED, 2))