Algorithmic Alpha: Quantitative Back Testing of Downside Risk in a Concentrated 10-Stock Portfolio

Midterm: Investment Analytics and Technology Application

1 Introduction

1.1 Background and motivation

Diversification has long been the foundational principle of asset management. Guided by traditional asset pricing models, retail and institutional investors alike rely on broad market exposure to minimize idiosyncratic risk. Yet as passive index investing has come to dominate the market, excess returns (alpha) are increasingly associated with portfolio concentration. High-conviction investing holds that managers should commit capital only to their highest-probability ideas and deliberately deviate from benchmark weights. The trade-off is clear: fewer holdings may raise the potential for alpha, but they also expose the investor to deeper and more painful losses.

Two strands of evidence make the trade-off hard to settle. On one side, studies of active managers suggest that their most concentrated, highest-conviction positions are where skill shows up (Cohen et al., n.d.; Kacperczyk et al., 2005). On the other side, the distribution of individual stock returns is so skewed that a small portfolio of randomly chosen stocks is likely to miss the few firms that generate most of the market’s long-run gains (Bessembinder, 2018; Markov & Markov, 2022). Concentration therefore pays only if the rule used to pick the stocks carries real information, and it can hurt badly if it does not.

1.2 Problem statement

Institutional managers have used concentrated portfolios to exploit informational advantages, but there is little evidence on whether the same idea works as a strictly data-driven, rules-based framework that a retail investor can replicate. The problem addressed here is whether a portfolio of exactly ten stocks, chosen by a transparent rule, can keep downside risk, measured mainly by maximum drawdown, under control while remaining competitive with diversified benchmarks on a risk-adjusted basis.

1.3 Research questions

The three questions this project asks

• RQ1 (risk-adjusted return). Does a rules-based portfolio of ten stocks, chosen by price momentum, earn a higher Sharpe ratio and alpha than a capitalization-weighted benchmark (SPY) and an equal-weighted benchmark (RSP)?
• RQ2 (downside risk). Is its downside risk, measured by maximum drawdown and conditional value at risk (CVaR), materially worse than that of the benchmarks, especially in stress periods such as the COVID-19 crash of early 2020 and the 2022 bear market?
• RQ3 (selection value). Do the results survive out-of-sample testing and a comparison with randomly selected 10-stock portfolios, which would show whether the selection rule adds information beyond concentration itself?

1.4 Contribution and structure

The project makes three contributions. First, it centers the evaluation on downside outcomes (drawdown and tail loss) rather than on average return alone. Second, it builds the evaluation protocol the back-testing literature asks for: rules fixed in advance, an untouched out-of-sample period, a count of every variant tried, and a random-portfolio benchmark. Third, it delivers the whole workflow as reproducible R code that a retail investor could rerun. Section 2 reviews the literature on diversification, concentration, momentum, drawdown measurement, and back testing, and derives four testable hypotheses. Section 3 presents the data with defined sources, the research design, and a test dataset used for a simple preliminary analysis. References follow.

2 Literature Review

2.1 Diversification and the question of portfolio size

Modern portfolio theory begins with Markowitz (1952), who showed that holding imperfectly correlated assets lowers portfolio variance without a proportional sacrifice of expected return. Idiosyncratic risk can be removed by adding securities, so it earns no compensation, and a concentrated portfolio carries risk that an investor is not paid to bear. The practical question is how many securities are enough. Evans and Archer (1968) sampled random portfolios of increasing size and found that most of the reduction in dispersion had been achieved by around ten stocks. Statman (1987) countered that this ignores costs: weighing the marginal benefit of diversification against its marginal cost, he argued for roughly 30 to 40 stocks.

Later work pushed the number higher. Campbell et al. (2001) documented that, between 1962 and 1997, firm-level volatility rose relative to market volatility, so the number of stocks needed to reach a given level of diversification increased. Domian et al. (2007) used 1,000 large U.S. stocks and 20-year horizons and focused on shortfall risk, the chance that ending wealth falls below a target; they found that shortfall risk keeps falling even beyond 100 stocks. For this project the lesson is twofold. A ten-stock portfolio sits at the lower edge of the textbook range, and its outcomes are dominated by stock-specific results rather than by the market. Drawdowns and tail losses, not variance alone, are therefore the right quantities to examine.

2.2 Evidence that concentration can add value

A second literature asks whether managers who concentrate their holdings perform better. Kacperczyk et al. (2005) found that mutual funds concentrated in a few industries outperformed more diversified funds, which they linked to managers exploiting informational advantages. Brands et al. (2006) reached a similar conclusion for Australian equity funds: performance was positively related to concentration, and the relationship was stronger for stocks on which managers held overweight positions. Huij and Derwall (2011) studied global equity funds and suggested that the benefit of concentration comes mostly from the breadth of the underlying strategies, not simply from taking large bets.

The most direct evidence comes from Cohen et al. (n.d.), who identify a manager’s “best ideas” as the positions with the largest overweight relative to the market. They report that these positions outperform the market and the manager’s remaining holdings, and they argue that managers have incentives to keep adding positions that contribute little alpha. Together these studies support the premise behind a ten-stock portfolio, that conviction can be valuable. All of them, however, rely on human managers whose selection skill is unobserved and cannot be replicated by a simple rule. Whether a mechanical rule can supply a comparable informational edge is the open question here.

2.3 Skewness and the cost of missing the winners

Concentration has a hidden cost when returns are skewed. Bessembinder (2018) examined the lifetime returns of U.S. common stocks in the CRSP database since 1926 and reported that about 58 percent of them failed to beat one-month Treasury bills over their lifetimes, while the best-performing four percent or so of companies accounted for the net wealth the market created above Treasury bills. He concluded that this helps explain why poorly diversified active strategies often underperform market averages. Markov and Markov (2022) reach a related conclusion in a log-normal framework: long-horizon stock returns are strongly right-skewed, the typical return of a small random portfolio is lower than that of the equal-weighted index, and portfolios of more than 20 stocks are often needed before the typical outcome matches the index.

The implication is direct. A small portfolio is likely to miss the few big winners unless its selection rule carries information, and a randomly chosen ten-stock portfolio should be expected to lag. This is why the study compares the rule-based portfolio with randomly selected portfolios of the same size (hypothesis H4) rather than only with an index.

2.4 Momentum as a rules-based selection signal

Price momentum is a natural, transparent candidate for such a rule. Jegadeesh and Titman (1993) showed that buying past winners and selling past losers over horizons of three to twelve months produces significant abnormal returns, and Carhart (1997) found that a momentum factor accounts for much of the apparent persistence in mutual fund performance. Most of this evidence, however, concerns long-short portfolios of many stocks, not a long-only portfolio of ten.

Momentum also carries serious downside risk. Daniel and Moskowitz (2016) document momentum crashes, in which the strategy suffers severe losses when markets rebound sharply after prolonged declines. Barroso and Santa-Clara (2015) find that the risk of momentum varies widely over time but is predictable, and that scaling exposure by recent realized volatility nearly doubles the Sharpe ratio and virtually eliminates crashes. The strategy studied here is long-only, equal-weighted, and not volatility-scaled, so it is exposed to momentum risk in a raw form. That makes it a demanding but informative test of drawdown.

Trading costs are a further concern. Lesmond et al. (2004) argued that, after realistic trading costs, the apparent profits of relative-strength strategies largely disappear. Novy-Marx and Velikov (2016) show that anomalies with monthly turnover below about 50 percent can keep significant net returns, and that a buy/hold spread, which sets a stricter bar for entering a position than for keeping one, is the most effective cost-mitigation technique. A monthly top-ten momentum portfolio will turn over substantially, so results are reported net of costs at several cost levels, and a buy/hold-spread variant is a natural extension.

2.5 Measuring downside risk

Maximum drawdown, the largest peak-to-trough fall in portfolio value, is the primary risk measure because it reflects what an investor holding the portfolio would actually have experienced. Magdon-Ismail and Atiya (2004) study its statistical behavior and relate it to a portfolio’s return, volatility, and horizon. Because it is determined by a single episode, maximum drawdown is a noisy statistic. Chekhlov et al. (2005) therefore propose conditional drawdown, the average of the worst fraction of drawdowns, which includes maximum drawdown and average drawdown as limiting cases. This project reports maximum drawdown with bootstrapped confidence intervals and complements it with conditional drawdown and CVaR.

Risk-adjusted performance is measured with the Sharpe ratio (Sharpe, 1994). Because monthly returns are short, non-normal, and serially dependent, differences in Sharpe ratios are tested with the studentized block bootstrap of Ledoit and Wolf (2008), confidence intervals use the stationary bootstrap of Politis and Romano (1994), and alphas are estimated against the Fama and French (1993) factors and the Carhart (1997) momentum factor with Newey and West (1987) standard errors.

2.6 Statistical hazards of back testing

Moving from discretionary management to algorithmic execution also introduces statistical hazards. Harvey and Liu (2015) warn that back-tested Sharpe ratios are inflated when many rules or parameters are tried on the same data, and they propose discounting reported Sharpe ratios for the number of trials. Bailey and López de Prado (2014) develop the deflated Sharpe ratio to correct for selection bias, back-test overfitting, and non-normal returns. Bailey et al. (2014) add that the number of configurations tried is rarely reported, so investors cannot judge the degree of overfitting, and they suspect that many published back tests are misleading. Harvey et al. (2016) argue that, after hundreds of tested factors, a newly proposed factor should clear a t-statistic hurdle above 3.0 rather than the conventional 2.0.

Evidence on what happens after discovery points the same way. McLean and Pontiff (2016) studied 97 variables that had been shown to predict stock returns and found that portfolio returns were 26 percent lower out of sample and 58 percent lower after publication. A signal as widely known as momentum may therefore be weaker in the recent sample than in the original studies. Finally, sample construction matters. Brown et al. (1992) show that samples truncated by survivorship can create the appearance of return predictability. The analogous problem here arises from building the universe out of today’s large companies, and it is addressed in the limitations and robustness sections. Taken together, these papers imply that a credible back test must fix its rules in advance, report how many variants were examined, and be evaluated out of sample.

2.7 Synthesis, research gap, and hypotheses

Table 1 summarizes the studies that shape the design. Taken together, the literature shows that concentrated portfolios can generate alpha when managers have information (Brands et al., 2006; Cohen et al., n.d.; Huij & Derwall, 2011; Kacperczyk et al., 2005), that concentration without information is likely to lag a broad equal-weighted index (Bessembinder, 2018; Markov & Markov, 2022), and that momentum offers a rules-based source of information that is exposed to crash risk and trading costs (Daniel & Moskowitz, 2016; Jegadeesh & Titman, 1993; Lesmond et al., 2004). Most of this work, however, studies human discretionary managers or abstract return distributions. This project addresses the gap by testing the implementation viability of a fully rules-based 10-stock portfolio, focusing on drawdown and tail-risk measures, with out-of-sample validation and a random-portfolio benchmark to guard against the overfitting that Harvey and Liu (2015) describe.

Table 1. Key studies and their role in this project

Theme	Study	Evidence	Role in this project
Portfolio size	Evans and Archer (1968)	Most dispersion reduction achieved by about ten stocks	Lower bound for the portfolio size
Portfolio size	Statman (1987)	About 30 to 40 stocks once costs are considered	Argues against ten stocks
Portfolio size	Campbell et al. (2001)	Firm-level volatility rose from 1962 to 1997; more stocks needed	Idiosyncratic risk is larger today
Portfolio size	Domian et al. (2007)	Shortfall risk keeps falling beyond 100 stocks	Motivates tail-risk focus
Concentration	Kacperczyk et al. (2005); Brands et al. (2006); Huij and Derwall (2011)	Concentrated funds tend to outperform	Conviction can add value
Concentration	Cohen et al. (n.d.)	Managers’ best ideas outperform their other holdings	Premise of a top-ten portfolio
Skewness	Bessembinder (2018); Markov and Markov (2022)	A few stocks create most wealth; random small portfolios lag	Random-portfolio test (H4)
Momentum	Jegadeesh and Titman (1993); Carhart (1997)	Winners keep winning over 3 to 12 months	Selection rule
Momentum risk	Daniel and Moskowitz (2016); Barroso and Santa-Clara (2015)	Crashes after market rebounds; risk is predictable	Stress windows; drawdown focus
Costs	Lesmond et al. (2004); Novy-Marx and Velikov (2016)	Costs erode momentum profits; low turnover helps	Cost sensitivity (0, 10, 25 bp)
Back testing	Harvey and Liu (2015); Bailey and López de Prado (2014); Bailey et al. (2014); Harvey et al. (2016); McLean and Pontiff (2016)	Overfitting, multiple testing, post-publication decay	Out-of-sample split; deflated Sharpe ratio

The literature motivates four hypotheses, which are tested in the final project against the decision rules in Table 5.

Hypotheses

• H1 (risk-adjusted return). The 10-stock strategy has a higher Sharpe ratio than SPY and RSP. Momentum evidence supports it; the skewness evidence says concentration alone will not.
• H2 (downside risk). The strategy’s maximum drawdown and 5% CVaR are not larger than those of SPY. The evidence on idiosyncratic risk and momentum crashes suggests this is the hypothesis most likely to be rejected.
• H3 (persistence). The strategy’s alpha remains positive out of sample, after controlling for market, size, value, and momentum factors. Post-publication decay (McLean and Pontiff, 2016) is the main threat.
• H4 (selection value). The strategy’s Sharpe ratio is above the 75th percentile of randomly selected 10-stock portfolios drawn from the same universe, which would show that the rule adds information beyond concentration.

3 Data

3.1 Data sources

All data come from free, publicly available sources and are downloaded programmatically in R so that the analysis can be reproduced. Table 2 lists each source, what it provides, and how it is accessed.

Table 2. Data sources

Data	Content	Source	Access
Stock prices	Daily dividend- and split-adjusted closing prices of the candidate stocks	Yahoo Finance	quantmod::getSymbols(src = "yahoo")
Benchmarks	SPY (cap-weighted S&P 500) and RSP (equal-weighted S&P 500) adjusted prices	Yahoo Finance	quantmod::getSymbols(src = "yahoo")
Factor data	Market excess return, SMB, HML, momentum, risk-free rate (monthly)	Kenneth R. French Data Library	Download of the CSV files from the library website
Stock universe	Candidate universe: S&P 100 constituents (full list in an appendix of the final project)	S&P Dow Jones Indices	Fixed list compiled by the author

The full study uses the whole universe; the test dataset in Section 3.8 is a small subset used to check the pipeline.

3.2 Sample period and universe

The study period runs from January 2016 to December 2025, which gives 120 monthly observations. Prices from mid-2015 are also collected so that the first six-month momentum signal is available in January 2016. To limit overfitting, the sample is split into an in-sample period (January 2016 to December 2020) and an out-of-sample period (January 2021 to December 2025). The in-sample period is used only to examine robustness (Section 3.7); the strategy parameters are fixed beforehand from the literature and applied unchanged to the out-of-sample period.

The candidate universe is the S&P 100, a fixed list of large U.S. companies. A stock is eligible at a formation date only if it has at least seven months of price history, enough to compute the momentum signal. Compiling the list from today’s index constituents introduces survivorship and look-ahead bias, because firms that fell out of the index are missing. The bias is discussed in Section 3.9. If point-in-time constituent lists can be obtained through the university’s data subscriptions, they will be used as a robustness check.

3.3 Variables and cleaning

Table 3. Variable definitions

Variable	Definition	Source
Monthly return \(r_{i,t}\)	Simple return from month-end to month-end of the adjusted closing price of stock \(i\)	Yahoo Finance
Momentum score \(M_{i,t}\)	Cumulative return over months \(t-6\) to \(t-1\), skipping month \(t\)	Computed
Risk-free rate \(r_{f,t}\)	One-month Treasury bill rate	French Data Library
Factors	Market excess return, SMB, HML, momentum	French Data Library
Portfolio return \(r_{p,t}\)	Return of the 10-stock portfolio after trading costs	Computed

Daily prices are converted to month-end values, and all series are aligned to month-end dates. Adjusted prices already account for dividends and splits. Observations with missing prices are not filled in; a stock with an incomplete history at a formation date is simply excluded from selection at that date.

3.4 Strategy design

The strategy follows five rules, all fixed before any results are examined.

• At each month-end \(t\), compute each eligible stock’s cumulative return over the previous six months, skipping the most recent month to avoid short-term reversal (Jegadeesh & Titman, 1993): \(M_{i,t} = P_{i,t-1} / P_{i,t-7} - 1\), where \(P\) is the month-end adjusted price.
• Rank all eligible stocks by this momentum score and select the top ten.
• Assign equal weights of 10% to the selected stocks and hold them for the following month. Weights are never optimized.
• Rebalance monthly and deduct a trading cost of 10 basis points per unit of value traded (each purchase or sale): \(r_{p,t} = r^{gross}_{p,t} - c \sum_i |w_{i,t} - \tilde{w}_{i,t}|\), where \(\tilde{w}_{i,t}\) is the weight the holding has drifted to just before rebalancing and \(c = 0.0010\).
• Record monthly portfolio returns, holdings, and turnover in a transaction log for later analysis.

3.5 Performance and risk metrics

Let \(V_t\) denote portfolio value at month \(t\). The drawdown is the percentage fall from the running peak, and the maximum drawdown is its largest value over the sample:

\[DD_t = 1 - \frac{V_t}{\max_{s \le t} V_s}, \qquad MDD = \max_t DD_t .\]

The Sharpe ratio is computed from monthly excess returns and annualized:

\[SR = \sqrt{12}\,\frac{\overline{r_p - r_f}}{\sigma(r_p - r_f)} .\]

The 5% conditional value at risk is the average of the worst 5% of monthly returns, \(CVaR_{5\%} = -E[\,r_p \mid r_p \le q_{5\%}\,]\), where \(q_{5\%}\) is the 5th percentile of monthly returns. Table 4 lists all metrics.

Table 4. Performance and risk metrics

Metric	Definition	Purpose
Annualized return (CAGR)	Geometric average return per year	Absolute performance
Annualized volatility	Standard deviation of monthly returns times \(\sqrt{12}\)	Total risk
Sharpe ratio	Annualized mean excess return divided by annualized volatility (Sharpe, 1994)	Risk-adjusted return
Sortino ratio	Mean excess return divided by downside deviation	Penalizes only downside
Maximum drawdown	Largest peak-to-trough decline in portfolio value (Magdon-Ismail & Atiya, 2004)	Primary downside risk
Conditional drawdown (5%)	Average of the worst 5% of drawdown observations (Chekhlov et al., 2005)	Drawdown beyond one episode
Calmar ratio	CAGR divided by maximum drawdown	Return per unit of drawdown
CVaR (5%)	Average of the worst 5% of monthly returns	Tail risk
Drawdown duration	Longest time from a peak to its recovery	Recovery risk
Beta and turnover	Regression beta vs. SPY; average monthly turnover	Market exposure; trading cost

3.6 Statistical inference

Because monthly returns are short, non-normal, and serially correlated, inference relies on methods designed for those features. Table 5 links each hypothesis to its evidence, test, and decision rule.

• Sharpe ratio difference. The studentized circular block bootstrap test of Ledoit and Wolf (2008) with 5,000 resamples, comparing the strategy with each benchmark (H1).
• Drawdown uncertainty. A stationary bootstrap (Politis & Romano, 1994) builds confidence intervals for maximum drawdown and CVaR (H2).
• Alpha estimation. Time-series regressions of monthly excess returns on the market factor (CAPM), the Fama and French (1993) three factors, and a four-factor model that adds momentum (Carhart, 1997). Standard errors are Newey and West (1987) heteroskedasticity- and autocorrelation-consistent (H3).
• Random-portfolio test. 1,000 portfolios of ten stocks drawn at random from the same universe each month, with identical weights, rebalancing, and costs. The strategy’s Sharpe ratio and maximum drawdown are located within this distribution to obtain an empirical percentile (H4; Markov & Markov, 2022).
• Multiple-testing adjustment. The deflated Sharpe ratio (Bailey & López de Prado, 2014) uses the total number of parameter variants examined (Harvey & Liu, 2015).

Table 5. From hypotheses to evidence

	Evidence (output)	Test	Supported if
H1	Sharpe ratios and growth-of-$1 chart, full and out-of-sample	Ledoit-Wolf block bootstrap vs. SPY and RSP	Strategy Sharpe is higher and p < 0.10
H2	Maximum drawdown, CVaR, underwater chart, stress windows	Stationary bootstrap confidence intervals	Strategy interval does not lie above SPY’s
H3	Alpha table (CAPM, FF3, four-factor)	Newey-West t-test on out-of-sample alpha	Alpha > 0 and p < 0.10
H4	Histogram of random-portfolio Sharpe ratios	Empirical percentile of the strategy	Strategy is above the 75th percentile

Thresholds are set before the analysis and applied to every result, whether favorable or not. Given the warning of Harvey et al. (2016) about multiple testing, a result with a t-statistic between 2 and 3 will be described as suggestive rather than established.

3.7 Robustness and stress analysis

Robustness is examined along four dimensions. First, the momentum lookback is varied (3, 6, and 12 months). Second, the portfolio size is varied (5, 10, 20, and 30 stocks) to show how risk and return change as concentration falls. Third, trading costs are set to 0, 10, and 25 basis points. Fourth, performance is reported separately for stress windows, namely the COVID-19 crash (February to March 2020) and the 2022 bear market, to show how the strategy behaves when drawdown matters most. All variants are counted for the deflated Sharpe ratio.

3.8 Test dataset and preliminary analysis

For the preliminary analysis, a small test dataset is used. It contains ten large-cap U.S. stocks that all have a full price history over the sample period (AAPL, MSFT, AMZN, GOOGL, META, NVDA, AVGO, LLY, JPM, XOM) and the two benchmarks, SPY and RSP, with monthly returns from January 2016 to December 2025. The ten stocks are fixed in advance and are not selected by the momentum rule. The test dataset exists only to verify data quality and to describe returns and risk.

The preliminary analysis has four steps: data quality checks, a summary table for an equal-weighted portfolio of the ten test stocks against SPY and RSP (risk-free rate set to zero at this stage), a correlation matrix of the ten stocks, and growth-of-$1 and drawdown charts.

library(quantmod)
library(xts)
library(PerformanceAnalytics)
library(ggplot2)
library(knitr)

tickers <- c("AAPL","MSFT","AMZN","GOOGL","META","NVDA","AVGO","LLY","JPM","XOM")
bench   <- c("SPY","RSP")
all_sym <- c(tickers, bench)

# Download one symbol at a time; return NULL if Yahoo Finance rejects the request
get_px <- function(s) {
  Sys.sleep(0.5)
  x <- tryCatch(
    getSymbols(s, src = "yahoo", from = "2015-06-01", to = "2025-12-31",
               auto.assign = FALSE),
    error = function(e) NULL)
  if (is.null(x)) NULL else Ad(x)
}
px_list <- setNames(lapply(all_sym, get_px), all_sym)

failed <- names(px_list)[sapply(px_list, is.null)]
if (length(failed) > 0) {
  stop("Download failed for: ", paste(failed, collapse = ", "),
       ". Render again; Yahoo Finance sometimes rejects requests.")
}

adj <- do.call(merge, px_list)
colnames(adj) <- all_sym
monthly <- adj[endpoints(adj, on = "months")]
ret <- na.omit(Return.calculate(monthly, method = "discrete"))["2016/2025"]

3.8.1 Data quality checks

quality <- data.frame(
  Check = c("Monthly observations per series", "First month", "Last month", "Missing values"),
  Value = c(nrow(ret), format(index(ret)[1]), format(index(ret)[nrow(ret)]), sum(is.na(ret)))
)
kable(quality, caption = "Data quality checks")

Data quality checks

Check	Value
Monthly observations per series	120
First month	2016-01-29
Last month	2025-12-30
Missing values	0

3.8.2 Summary statistics

port <- Return.portfolio(ret[, tickers], weights = rep(0.1, 10),
                         rebalance_on = "months")
colnames(port) <- "Test10"
comp <- merge(port, ret[, bench])

summ <- data.frame(
  ret = apply(comp, 2, function(x) Return.annualized(x, scale = 12)),
  vol = apply(comp, 2, function(x) StdDev.annualized(x, scale = 12)),
  shp = apply(comp, 2, function(x) SharpeRatio.annualized(x, scale = 12)),
  mdd = apply(comp, 2, function(x) maxDrawdown(x)),
  cvr = apply(comp, 2, function(x) -ES(x, p = 0.95, method = "historical"))
)
tab <- data.frame(
  Series            = c("10-stock test portfolio", "SPY", "RSP"),
  `Ann. return`     = sprintf("%.1f%%", 100 * summ$ret),
  `Ann. volatility` = sprintf("%.1f%%", 100 * summ$vol),
  `Sharpe ratio`    = sprintf("%.2f", summ$shp),
  `Max drawdown`    = sprintf("%.1f%%", -100 * summ$mdd),
  `CVaR (5%)`       = sprintf("%.1f%%", -100 * summ$cvr),
  check.names = FALSE
)
kable(tab, align = c("l", rep("r", 5)),
      caption = "Preliminary summary statistics, test dataset, 2016-2025")

Preliminary summary statistics, test dataset, 2016-2025

Series	Ann. return	Ann. volatility	Sharpe ratio	Max drawdown	CVaR (5%)
10-stock test portfolio	32.0%	18.8%	1.70	-26.7%	-10.2%
SPY	14.8%	15.1%	0.98	-23.9%	-9.2%
RSP	11.6%	16.6%	0.70	-26.7%	-10.4%

3.8.3 Correlations among the ten test stocks

cdf <- as.data.frame(as.table(cor(ret[, tickers])))
names(cdf) <- c("A", "B", "r")
ggplot(cdf, aes(A, B, fill = r)) +
  geom_tile(colour = "white") +
  geom_text(aes(label = sprintf("%.2f", r), colour = r > 0.6), size = 3) +
  scale_colour_manual(values = c("TRUE" = "white", "FALSE" = "#1f3a5f"), guide = "none") +
  scale_fill_gradient(low = "#eef2f9", high = "#1f3a5f", limits = c(0, 1), name = "Correlation") +
  coord_fixed() + labs(x = NULL, y = NULL) +
  theme_minimal(base_size = 12) +
  theme(panel.grid = element_blank())

Correlation of monthly returns, test stocks

3.8.4 Growth of $1 and drawdowns

pal  <- c("Test10" = "#d95f02", "SPY" = "#1f3a5f", "RSP" = "#1b9e77")
labs_ <- c("Test10" = "10-stock test portfolio", "SPY" = "SPY", "RSP" = "RSP")
to_long <- function(x, value) {
  d <- data.frame(Date = index(x), coredata(x))
  d <- reshape(d, direction = "long", varying = colnames(x), v.names = value,
               timevar = "Series", times = colnames(x), idvar = "Date")
  d$Series <- factor(d$Series, levels = names(pal))
  d
}
nice <- theme_minimal(base_size = 12) +
  theme(legend.position = "bottom", legend.title = element_blank(),
        panel.grid.minor = element_blank())

ggplot(to_long(cumprod(1 + comp), "Value"), aes(Date, Value, colour = Series)) +
  geom_line(linewidth = 0.9) +
  scale_colour_manual(values = pal, labels = labs_) +
  scale_y_continuous(labels = function(x) paste0("$", x)) +
  labs(x = NULL, y = "Value of $1 invested") + nice

Growth of $1, 2016-2025 (test dataset)

ggplot(to_long(Drawdowns(comp), "Drawdown"), aes(Date, Drawdown, colour = Series)) +
  geom_hline(yintercept = 0, colour = "grey70") +
  geom_line(linewidth = 0.9) +
  scale_colour_manual(values = pal, labels = labs_) +
  scale_y_continuous(labels = scales::percent) +
  labs(x = NULL, y = "Drawdown") + nice

Drawdowns from the previous peak, 2016-2025 (test dataset)

3.9 Limitations

Several limitations should be acknowledged. Using a fixed list of current large-cap stocks introduces survivorship and look-ahead bias (Brown et al., 1992), which may flatter the strategy; the direction of this bias will be stated alongside every result, and point-in-time constituents will be used if they can be obtained. The 2016 to 2025 sample is dominated by a long bull market led by large technology stocks, so the findings may not generalize to other regimes, and the early-2020 and 2022 stress windows are only two episodes. Monthly data hides intra-month drawdowns, and the transaction-cost assumption ignores taxes and market impact. Because momentum is widely known, its return may have decayed in recent data (McLean & Pontiff, 2016). Finally, a single back test provides one historical path and cannot establish that the results will persist.

References

Bailey, D. H., Borwein, J. M., López de Prado, M., & Zhu, Q. J. (2014). Pseudo-mathematics and financial charlatanism: The effects of backtest overfitting on out-of-sample performance. Notices of the American Mathematical Society, 61(5), 458–471. https://doi.org/10.1090/noti1105

Bailey, D. H., & López de Prado, M. (2014). The deflated Sharpe ratio: Correcting for selection bias, backtest overfitting, and non-normality. The Journal of Portfolio Management, 40(5), 94–107.

Barroso, P., & Santa-Clara, P. (2015). Momentum has its moments. Journal of Financial Economics, 116(1), 111–120. https://doi.org/10.1016/j.jfineco.2014.11.010

Bessembinder, H. (2018). Do stocks outperform Treasury bills? Journal of Financial Economics, 129(3), 440–457. https://doi.org/10.1016/j.jfineco.2018.06.004

Brands, S., Brown, S. J., & Gallagher, D. R. (2006). Portfolio concentration and investment manager performance. International Review of Finance, 5(3–4), 149–174.

Brown, S. J., Goetzmann, W., Ibbotson, R. G., & Ross, S. A. (1992). Survivorship bias in performance studies. The Review of Financial Studies, 5(4), 553–580.

Campbell, J. Y., Lettau, M., Malkiel, B. G., & Xu, Y. (2001). Have individual stocks become more volatile? An empirical exploration of idiosyncratic risk. The Journal of Finance, 56(1), 1–43. https://doi.org/10.1111/0022-1082.00318

Carhart, M. M. (1997). On persistence in mutual fund performance. The Journal of Finance, 52(1), 57–82.

Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). Drawdown measure in portfolio optimization. International Journal of Theoretical and Applied Finance, 8(1), 13–58. https://doi.org/10.1142/S0219024905002767

Cohen, R. B., Polk, C., & Silli, B. (n.d.). Best ideas [Working paper]. Harvard Business School and London School of Economics. https://personal.lse.ac.uk/polk/research/BestIdeas.pdf

Daniel, K., & Moskowitz, T. J. (2016). Momentum crashes. Journal of Financial Economics, 122(2), 221–247.

Domian, D. L., Louton, D. A., & Racine, M. D. (2007). Diversification in portfolios of individual stocks: 100 stocks are not enough. The Financial Review, 42(4), 557–570. https://doi.org/10.1111/j.1540-6288.2007.00183.x

Evans, J. L., & Archer, S. H. (1968). Diversification and the reduction of dispersion: An empirical analysis. The Journal of Finance, 23(5), 761–767.

Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3–56.

French, K. R. (n.d.). Kenneth R. French Data Library. Tuck School of Business, Dartmouth College. https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

Harvey, C. R., & Liu, Y. (2015). Backtesting. The Journal of Portfolio Management, 42(1), 13–28. https://doi.org/10.3905/jpm.2015.42.1.013

Harvey, C. R., Liu, Y., & Zhu, H. (2016). … and the cross-section of expected returns. The Review of Financial Studies, 29(1), 5–68. https://doi.org/10.1093/rfs/hhv059

Huij, J., & Derwall, J. (2011). Global equity fund performance, portfolio concentration, and the fundamental law of active management. Journal of Banking & Finance, 35(1), 155–165. https://doi.org/10.1016/j.jbankfin.2010.07.032

Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. The Journal of Finance, 48(1), 65–91.

Kacperczyk, M., Sialm, C., & Zheng, L. (2005). On the industry concentration of actively managed equity mutual funds. The Journal of Finance, 60(4), 1983–2011. https://doi.org/10.1111/j.1540-6261.2005.00785.x

Ledoit, O., & Wolf, M. (2008). Robust performance hypothesis testing with the Sharpe ratio. Journal of Empirical Finance, 15(5), 850–859.

Lesmond, D. A., Schill, M. J., & Zhou, C. (2004). The illusory nature of momentum profits. Journal of Financial Economics, 71(2), 349–380. https://doi.org/10.1016/S0304-405X(03)00206-X

Magdon-Ismail, M., & Atiya, A. F. (2004). Maximum drawdown. Risk, 17(10), 99–102.

Markov, M., & Markov, V. (2022). The impact of big winners on passive and active equity investment strategies (arXiv:2210.09302). arXiv. https://arxiv.org/abs/2210.09302

Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91. https://doi.org/10.2307/2975974

McLean, R. D., & Pontiff, J. (2016). Does academic research destroy stock return predictability? The Journal of Finance, 71(1), 5–32. https://doi.org/10.1111/jofi.12365

Newey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703–708.

Novy-Marx, R., & Velikov, M. (2016). A taxonomy of anomalies and their trading costs. The Review of Financial Studies, 29(1), 104–147. https://doi.org/10.1093/rfs/hhv063

Politis, D. N., & Romano, J. P. (1994). The stationary bootstrap. Journal of the American Statistical Association, 89(428), 1303–1313.

Sharpe, W. F. (1994). The Sharpe ratio. The Journal of Portfolio Management, 21(1), 49–58.

Statman, M. (1987). How many stocks make a diversified portfolio? Journal of Financial and Quantitative Analysis, 22(3), 353–363.