# Do riskier stocks earn higher returns?
### Analysing the risk-return tradeoff across S&P 500 sectors
*BEE2041 Empirical Project — University of Exeter*

*Austeja Siksnyte*
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import statsmodels.formula.api as smf
import os

PALETTE = {
"Technology": "#7F77DD",
"Healthcare": "#1D9E75",
"Finance": "#378ADD",
"Energy": "#EF9F27",
"Consumer": "#D85A30"
}

df = pd.read_csv("../data/processed/stock_metrics.csv")
log_returns = pd.read_csv("../data/processed/log_returns.csv",
index_col=0, parse_dates=True)

print(f"Dataset: {len(df)} stocks across {df['sector'].nunique()} sectors")
df[["annual_return","annual_volatility","sharpe_ratio"]].describe().round(3)
## Introduction

One of the oldest ideas in finance is deceptively simple: if you want higher returns, you have to accept higher risk. This intuition underpins the Capital Asset Pricing Model (CAPM), developed by Sharpe (1964) and Lintner (1965), which remains one of the most widely taught frameworks in financial economics. Capital Asset Pricing Model (CAPM) depicts the risk-return tradeoff by predicting that an asset's expected return should be a linear function of its exposure to market-wide risk. Stocks that move more dramatically with the market, or that are simply more volatile, should offer investors a higher expected return as compensation for bearing that uncertainty. But while this sounds straightforward, the real world is rarely so tidy. Markets are complicated, and sometimes taking more risk doesn’t actually lead to bigger rewards. How closely risk and return are linked can depend on the industry, time period, or even how you measure risk in the first place.

However, the empirical record is messier than the theory suggests. Fama and French (1992) famously found that beta is the CAPM measure of systematic risk. It had almost no power to explain cross-sectional variation in stock returns after controlling for size and book-to-market ratios. Research has documented what is now called the low-volatility anomaly: stocks with lower historical volatility have tended to outperform their higher-volatility counterparts on a risk-adjusted basis. This directly inverts the CAPM prediction.

This project examines the question using five years of daily price data for 25 large-cap S&P 500 stocks across five sectors. Rather than simply testing the overall relationship, we dig into whether the risk-return tradeoff differs across sectors and whether it holds up once sector composition is accounted for via regression with fixed effects. I used a mix of number crunching, regression analysis, and charts to gain a clearer picture of the risk-reward link. The goal is not to definitively resolve a debate that has occupied financial economists for decades, but to bring real data to bear on it in a transparent and replicable way.

## The data

To build my dataset, I gathered the daily closing prices for 25 different stocks, picking five from each of these S&P 500 sectors: Technology, Healthcare, Finance, Energy, and Consumer. I chose these stocks to reflect a mix of company sizes and business approaches within each sector. The price data came from Yahoo Finance, pulled using the yfinance Python package, and covers January 2019 through January 2024—giving me 1,258 trading days for every stock in the sample.

This five-year window is particularly rich in terms of differences in market conditions. It opens during a period of relative stability, passes through the COVID-19 volatility shock of February to March 2020, captures the bull market that followed through 2021, and then encompasses the Federal Reserve's aggressive interest rate-tightening cycle of 2022 to 2023. The fact that our sample spans such varied regimes adds both interest and complexity to the analysis: any patterns we find have survived conditions ranging from a global pandemic to the fastest rate-hiking cycle in decades.

From the raw price data I constructed three metrics for each stock. Daily log returns were computed as the natural logarithm of the ratio of consecutive closing prices. From these, we calculated annualised return by multiplying the mean daily log return by 252 trading days. Finally, annualised volatility is calculated by multiplying the daily log return standard deviation by the square root of 252. Finally, the Sharpe ratio divides annualised return by annualised volatility, giving a measure of return per unit of risk. A Sharpe ratio above zero means the stock earned a positive return relative to its risk, while values close to zero or negative indicate poor risk-adjusted performance. The table below summarises these metrics averaged across sectors.
summary = df.groupby("sector")[["annual_return","annual_volatility","sharpe_ratio"]].mean().round(3)
summary.columns = ["Avg Annual Return","Avg Volatility","Avg Sharpe Ratio"]
summary
## Do riskier stocks earn more?

The most direct way to examine the risk-return tradeoff is to plot volatility against return for each stock and fit a regression line. If CAPM holds, we would expect a clear positive relationship: stocks with higher annualised volatility should cluster towards the top of the chart. The scatter plot below does exactly this, with each stock coloured by sector and a dashed OLS regression line summarising the overall relationship.
m1 = smf.ols("annual_return ~ annual_volatility", data=df).fit()

fig, ax = plt.subplots(figsize=(9, 6))
for sector, grp in df.groupby("sector"):
ax.scatter(grp["annual_volatility"], grp["annual_return"],
color=PALETTE[sector], label=sector, s=100, alpha=0.85, zorder=3)
for _, row in grp.iterrows():
ax.annotate(row["ticker"],
(row["annual_volatility"], row["annual_return"]),
fontsize=8, xytext=(4,4), textcoords="offset points",
color=PALETTE[sector])

x_line = np.linspace(df["annual_volatility"].min(), df["annual_volatility"].max(), 100)
y_line = m1.params["Intercept"] + m1.params["annual_volatility"] * x_line
ax.plot(x_line, y_line, color="#2C2C2A", linewidth=1.5, linestyle="--",
label=f"OLS fit (b={m1.params['annual_volatility']:.2f}, p={m1.pvalues['annual_volatility']:.3f})")

ax.xaxis.set_major_formatter(plt.FuncFormatter(lambda x,_: f"{x:.0%}"))
ax.yaxis.set_major_formatter(plt.FuncFormatter(lambda y,_: f"{y:.0%}"))
ax.set_xlabel("Annualised Volatility", fontsize=12)
ax.set_ylabel("Annualised Return", fontsize=12)
ax.set_title("Volatility vs return: 25 S&P 500 stocks (2019-2024)", fontsize=13)
ax.legend(frameon=False, fontsize=9)
sns.despine()
plt.tight_layout()
plt.savefig("../figures/01_scatter_main.png", dpi=150)
plt.show()

print(f"Beta={m1.params['annual_volatility']:.3f}, p={m1.pvalues['annual_volatility']:.3f}, R2={m1.rsquared:.3f}")
There is a positive slope,this follows CAPM prediction. The regression coefficient suggests that for each additional
percentage point of annualised volatility, annual returns increase by roughly
the estimated beta. That said, the relationship is far from tight. The R-squared
is modest, meaning volatility alone explains only a limited share of the
variation in returns across stocks. Several stocks deviate substantially from
the regression line in both directions. Highlighting that firm-specific factors
play a large role in determining returns over a period.

It is also worth asking whether the positive slope reflects a genuine risk
premium or simply the fact that Technology stocks happen to sit in the
upper-right of the chart. Technology is both the most volatile and the highest
returning sector in our sample. If we removed Technology stocks entirely, the
slope might look quite different. This motivates the sector-level analysis
that follows.

## Which sectors show the strongest risk-return relationship?

Rather than treating the 25 stocks as a homogeneous group, we now run the
same simple regression separately within each sector. This tells us whether
the risk-return tradeoff is a consistent feature of each industry, or whether
it is driven by one or two sectors pulling the overall result. The bar chart
below summarises the estimated beta coefficient from each sector regression,
with the p-value and R-squared shown in the accompanying table.
sector_results = {}
for sector in df["sector"].unique():
sub = df[df["sector"] == sector]
if len(sub) >= 3:
m = smf.ols("annual_return ~ annual_volatility", data=sub).fit()
sector_results[sector] = {
"Beta": round(m.params["annual_volatility"], 3),
"P-value": round(m.pvalues["annual_volatility"], 3),
"R2": round(m.rsquared, 3),
"N": len(sub)
}

results_df = pd.DataFrame(sector_results).T.sort_values("Beta", ascending=False)
print(results_df.to_string())

fig, ax = plt.subplots(figsize=(8, 4))
colors = [PALETTE[s] for s in results_df.index]
bars = ax.barh(results_df.index, results_df["Beta"], color=colors, alpha=0.85)
ax.axvline(0, color="#888780", linewidth=0.8)
ax.set_xlabel("Volatility coefficient (beta)", fontsize=11)
ax.set_title("Risk-return relationship by sector", fontsize=13)
for bar, val in zip(bars, results_df["Beta"]):
offset = 0.02 if val >= 0 else -0.12
ax.text(bar.get_width() + offset, bar.get_y() + bar.get_height()/2,
f"{val:.3f}", va="center", fontsize=9)
sns.despine()
plt.tight_layout()
plt.savefig("../figures/02_sector_betas.png", dpi=150)
plt.show()
The results are significant. Technology shows the strongest positive beta, which suggests that in this sector, higher risk was linked to higher returns over the sample period. That fits the theory and matches the large risk-return gap between stocks like NVDA and INTC.

Healthcare and Consumer look different. Their betas are small and not statistically significant, so we cannot say that riskier stocks in these sectors reliably outperformed safer ones. This may be because volatility here often reflects short-term news or market sentiment rather than true long-term uncertainty.

Finance is the most interesting case. Its beta is clearly negative, meaning that within our sample, more volatile financial firms actually earned lower returns. This may be because financial volatility often comes from credit risk, regulation, or interest rates, which the market does not reward in the same way as technology risk.

## Does volatility still predict returns after controlling for sector?

The sector-level analysis already hints that much of the overall positive
relationship between volatility and returns is driven by between-sector
differences rather than within-sector risk pricing. To test this directly,
we estimate a second OLS model that adds sector fixed effects as control
variables. The fixed effects absorb all variation associated with sector
membership, so the coefficient on volatility in model 2 tells us whether
more volatile stocks outperform within their own sector, holding sector
constant. This is a cleaner and more demanding test of the risk-return
hypothesis than the simple cross-sectional model.
m2 = smf.ols("annual_return ~ annual_volatility + C(sector)", data=df).fit()

comparison = pd.DataFrame({
"No controls": {
"Beta (volatility)": round(m1.params["annual_volatility"], 3),
"p-value": round(m1.pvalues["annual_volatility"], 3),
"R2": round(m1.rsquared, 3),
"N": int(m1.nobs)
},
"Sector controls": {
"Beta (volatility)": round(m2.params["annual_volatility"], 3),
"p-value": round(m2.pvalues["annual_volatility"], 3),
"R2": round(m2.rsquared, 3),
"N": int(m2.nobs)
}
}).T
print(comparison.to_string())
comparison
Adding sector fixed effects increases the R-squared noticeably, confirming that knowing which sector a
stock belongs to helps substantially in predicting its return. More importantly,
the coefficient on volatility changes between models. If this effect becomes much weaker in model 2, it suggests that the simple positive relationship in model 1 was mostly driven by sector mix. In other words, high-volatility sectors may also be high-return sectors, rather than volatility itself causing higher returns within each sector. This is an important result because it means investors should not assume that more volatility automatically means more return inside every industry.

## Risk-adjusted performance: the Sharpe ratio by sector

Raw returns are an incomplete picture. A stock that returned 40% in a year
sounds impressive until you learn it had 80% annualised volatility. This means
the investor endured enormous swings to get there. The Sharpe ratio corrects
for this by expressing returns relative to the risk taken to achieve them.
Sectors with high Sharpe ratios are genuinely efficient at converting risk
into return, while sectors with low Sharpe ratios are exposing investors to
volatility without adequate compensation. Comparing the two models shows that adding sector effects explains more of the variation in returns. The change in the volatility coefficient shows how much of the relationship was due to sector mix rather than risk being priced within sectors.
order = df.groupby("sector")["sharpe_ratio"].median().sort_values(ascending=False).index

fig, ax = plt.subplots(figsize=(8, 5))
sns.boxplot(data=df, x="sector", y="sharpe_ratio", order=order,
hue="sector", palette=PALETTE, ax=ax, linewidth=0.8, legend=False)
ax.axhline(0, color="#888780", linewidth=0.8, linestyle="--")
ax.set_xlabel("")
ax.set_ylabel("Sharpe ratio", fontsize=11)
ax.set_title("Risk-adjusted returns by sector (2019-2024)", fontsize=13)
sns.despine()
plt.tight_layout()
plt.savefig("../figures/03_sharpe_by_sector.png", dpi=150)
plt.show()
The Sharpe ratio distribution across sectors tells a more nuanced story than
raw returns alone. Technology not only delivered the highest average returns
but also maintained competitive Sharpe ratios, suggesting investors were
meaningfully compensated for the volatility they accepted. This contrasts with
a naive view that high-volatility sectors must offer poor risk-adjusted returns.

Healthcare, despite more modest raw returns, shows relatively tight
and positive Sharpe ratios. This reflects the defensive characteristics of
pharmaceutical and insurance firms: low volatility means even modest returns
translate into acceptable risk-adjusted performance. For a risk-averse
investor, Healthcare's steady efficiency may be more appealing than
Technology's higher but choppier rewards.

Energy shows the widest dispersion of any sector, with some stocks delivering
excellent Sharpe ratios and others performing poorly. This is consistent with
the commodity-driven nature of energy returns: oil price swings affect all
energy firms simultaneously. Highly dependent on strategy, asset mix and geographic exposure.
Results show that choosing the right stock matters significantly compared to sector membership being a reliability signal.

## Stock rankings: who delivered the best returns?

The chart below ranks all 25 stocks by annualised return, making it easy to
see which individual stocks drove performance and whether top performers
tended to be high or low volatility names.
# Import the missing Patch class
from matplotlib.patches import Patch

ranked = df.sort_values("annual_return", ascending=True)

fig, ax = plt.subplots(figsize=(8, 9))
colors = [PALETTE[s] for s in ranked["sector"]]
ax.barh(ranked["ticker"], ranked["annual_return"], color=colors, alpha=0.85)
ax.axvline(0, color="#888780", linewidth=0.8)
ax.xaxis.set_major_formatter(plt.FuncFormatter(lambda x,_: f"{x:.0%}"))
ax.set_xlabel("Annualised Return", fontsize=11)
ax.set_title("All 25 stocks ranked by annualised return (2019-2024)", fontsize=13)
# Now Patch is properly imported and can be used to create legend elements
legend_elements = [Patch(facecolor=PALETTE[s], label=s) for s in PALETTE]
ax.legend(handles=legend_elements, frameon=False, fontsize=9)
sns.despine()
plt.tight_layout()
plt.savefig("../figures/05_ranked_returns.png", dpi=150)
plt.show()
## How does volatility evolve over time?

Volatility is not a static property of a stock. It spikes during market stress
and compresses during calm conditions. The 30-day rolling volatility chart
below illustrates this for three stocks representing different risk profiles:
NVDA (high-risk Technology), AMZN (mid-risk Consumer), and JNJ (low-risk
Healthcare).
available = [t for t in ["NVDA", "AMZN", "JNJ"] if t in log_returns.columns]
rolling_vol = log_returns[available].rolling(30).std() * np.sqrt(252)

fig, ax = plt.subplots(figsize=(10, 4))
colors_rv = ["#7F77DD", "#D85A30", "#1D9E75"]
for ticker, color in zip(available, colors_rv):
ax.plot(rolling_vol.index, rolling_vol[ticker],
label=ticker, color=color, linewidth=1.2)
ax.yaxis.set_major_formatter(plt.FuncFormatter(lambda y,_: f"{y:.0%}"))
ax.set_ylabel("Annualised volatility (30-day rolling)", fontsize=11)
ax.set_title("Volatility over time: high vs low risk stocks (2019-2024)", fontsize=13)
ax.legend(frameon=False)
sns.despine()
plt.tight_layout()
plt.savefig("../figures/04_rolling_volatility.png", dpi=150)
plt.show()
The time series paints a vivid picture of how differently these three stocks
have experienced market stress over the sample period. The most dramatic
feature is the volatility spike of February and March 2020, when COVID-19
began to spread globally and equity markets collapsed. NVDA's 30-day rolling
volatility briefly surged past 100% on an annualised basis, a level that
implies daily price moves of around 6% becoming routine. Even JNJ, one of
the most defensive large-cap stocks in the US market, saw its rolling
volatility roughly triple during this episode, illustrating how systemic
shocks override individual stock characteristics.

A second, smaller but sustained elevation in volatility is visible from
late 2021 onwards across all three stocks. This corresponds to the shift in
Federal Reserve policy from quantitative easing to aggressive rate hikes.
Higher interest rates are particularly damaging to valuations of
long-duration growth assets:such as NVDAs value being concentrated on expected future earnings, compressed by higher discount rates.
This explains why NVDA shows a noticeably wider volatility
gap relative to AMZN and JNJ in the 2022 to 2023 period compared to the
preceding years.

What the chart also shows is that the relative ordering of the three stocks by volatility is highly
persistent. NVDA consistently sits above AMZN, which consistently sits above
JNJ, regardless of the market regime. This means that whilst the absolute
level of volatility varies dramatically over time, the ranking of stocks by
risk is stable.

## Correlation between risk and return metrics

The heatmap below shows the pairwise correlations between our three key
metrics across all 25 stocks, giving a clean statistical summary of how
strongly the variables relate to one another.
corr = df[["annual_return","annual_volatility","sharpe_ratio"]].corr()
corr.columns = ["Return","Volatility","Sharpe"]
corr.index = ["Return","Volatility","Sharpe"]

fig, ax = plt.subplots(figsize=(6, 5))
sns.heatmap(corr, annot=True, fmt=".2f", cmap="RdYlGn",
center=0, ax=ax, linewidths=0.5, annot_kws={"size": 12})
ax.set_title("Correlation matrix: return, volatility and Sharpe ratio", fontsize=13)
plt.tight_layout()
plt.savefig("../figures/06_correlation_heatmap.png", dpi=150)
plt.show()
All three stocks experienced a dramatic volatility spike in early 2020
coinciding with the COVID-19 pandemic. NVDA's volatility briefly exceeded
100% on an annualised basis while JNJ remained comparatively contained.
The post-2022 period shows elevated volatility across all stocks, reflecting
uncertainty from rising interest rates, which disproportionately affected
growth stocks like NVDA whose valuations depend heavily on discounted future
cash flows.

The ranking of stocks by volatility is relatively stable over time, with NVDA
consistently above AMZN, which sits above JNJ. This persistence justifies
using time-averaged volatility as a stock characteristic, even if it masks
short-run variation.
## Conclusion

This analysis examined whether the textbook risk-return tradeoff holds across 25 large-cap US stocks between January 2019 and January 2024. With the use of OLS regression, sector fixed effects, and rolling volatility analysis, I found results that are consistent with theory in some respects but reveal important heterogeneity across sectors and time periods. Across all stocks, higher volatility is positively associated with higher returns, offering partial support for the CAPM prediction. However, this relationship is substantially driven by sector composition rather than a universal risk premium. When sector fixed effects are added, the R-squared rises considerably, confirming that sector membership is a major determinant of returns and that the simple cross-sectional relationship overstates the strength of the risk-return tradeoff.

The sector-level analysis reveals striking differences. Technology is the one sector where the risk-return tradeoff appears relatively robust, with economically meaningful and positive coefficients. Finance shows a negative beta, suggesting within-sector volatility is not rewarded in financial stocks, likely because bank volatility is driven by credit and regulatory risk rather than priced market risk. Energy and Consumer sectors show near-zero relationships, consistent with the idea that idiosyncratic firm-level volatility in these industries are largely diversifiable and therefore not compensated by the market.

There are several important limitations to acknowledge. First, the sample of 25 stocks is small by the standards of empirical asset pricing research, which limits statistical power and means individual outliers can heavily influence results. Second, the sample suffers from survivorship bias: all stocks are large-cap S&P 500 constituents that survived the full five-year period. Firms that were delisted, went bankrupt, or were acquired during this time are excluded entirely. These firms would disproportionately have had high volatility and poor returns, meaning our estimated risk-return relationship is likely biased upward. Third, the sample period is dominated by two unusual episodes: the COVID-19 shock of 2020 and the aggressive Federal Reserve tightening of 2022 to 2023. Both events had asymmetric effects across sectors, potentially distorting the estimated sector-level betas.

Future research could address these limitations in several ways. Extending the sample to a broader universe of stocks, including smaller firms and international markets, would improve generalisability. Using a longer time horizon spanning multiple business cycles would reduce the influence of any single episode. Incorporating additional risk factors such as the Fama-French size and value factors would allow a cleaner test of whether volatility commands a premium after accounting for other well-known return predictors.

Despite these limitations, the findings offer a nuanced and empirically grounded view of the risk-return relationship. The tradeoff is real but not uniform. Sector membership matters as much as volatility level in determining whether risk is rewarded. Investors seeking to exploit a volatility premium would do better to focus within high-growth sectors like Technology rather than assuming the premium applies uniformly across the entire market.

Data: Yahoo Finance via yfinance API. Period: January 2019 to January 2024. Analysis: OLS regression with and without sector fixed effects, 30-day rolling volatility. All code available at the GitHub repository linked above.
print("Analysis complete.")
print(f"Stocks analysed: {len(df)}")
print(f"Sectors: {sorted(df['sector'].unique().tolist())}")
print(f"Figures saved: {sorted(os.listdir('../figures/'))}")