Risk Metrics#
1. Risk Metrics of a Stock#
This problem uses data from data/spx_returns_weekly.xlsx.
Choose any stock to evaluate below.
1.1 Return Moments#
Report the moments of the returns. Annualize the mean and volatility.
mean
volatility
skewness
(excess) kurtosis
Note that the pandas function for kurtosis already reports excess kurtosis.
1.2 Maximum Drawdown#
Report the maximum drawdown for each return series.
If we resampled this data to monthly and recalculated the maximum drawdown, do you think it would be larger or smaller (in magnitude)?
1.3 Quantiles#
Report the quantiles of the series. Use .describe() for a useful summary.
Report the 5th quantile scaled by standard deviation.
How much sampling error is in the mean return? Is this likely to cause much sampling error in the quantile estimation?
1.4 Comparison#
Try repeating 1.1-1.3 for another asset.
2. Time Aggregation#
Use the data in ../data/risk_etf_data.xlsx.
Use the price series to calculate monthly returns. (You may find df.resample('ME').last() helpful.)
2.1 Bitcoin and Measuring Covariation Risk#
People often say that bitcoin has disappointed by being highly correlated to SPY.
Report the correlation between BTC and SPY for both daily and monthly returns.
What do you conclude?
2.2 Betas#
For each series, calculate its beta to SPY.
Estimate the regression with an intercept (alpha) but no need to report it.
How do these betas compare to the daily return betas seen in the note?
2.3 Time Scaling to Higher Moments#
Report the skewness and kurtosis for the monthly returns.
How do these compare to the daily skewness and kurtosis measures?
What do you conclude?
Solutions#
1. Risk Metrics of a Stock#
Use AAPL, with META, NVDA, and TSLA for the comparison in 1.4.
The common sample uses weekly total returns from spx_returns_weekly.xlsx.
Annualize the mean by 52 and volatility by \(\sqrt{52}\).
573 weekly observations, 2015-07-10 to 2026-06-26
1.1 Return Moments#
| Mean | Volatility | Skewness | Excess kurtosis | |
|---|---|---|---|---|
| Stock | ||||
| AAPL | 24.9% | 27.8% | -0.18 | 1.80 |
| META | 23.2% | 35.7% | 0.04 | 3.49 |
| NVDA | 64.5% | 45.4% | 0.35 | 1.49 |
| TSLA | 43.7% | 58.0% | 0.56 | 1.63 |
1.2 Maximum Drawdown#
| Weekly sampling | Monthly sampling | |
|---|---|---|
| Stock | ||
| AAPL | -34.6% | -31.1% |
| META | -76.0% | -73.4% |
| NVDA | -65.9% | -61.4% |
| TSLA | -72.2% | -66.8% |
Monthly sampling gives a smaller or equal maximum drawdown in magnitude. It can miss a peak or trough observed in the weekly path. The comparison retains the same initial wealth and terminal observation. June 2026 ends at the final weekly observation, June 26; the monthly grid includes this partial month.
1.3 Quantiles#
| Stock | AAPL | META | NVDA | TSLA |
|---|---|---|---|---|
| Weekly statistic | ||||
| Mean | 0.48% | 0.45% | 1.24% | 0.84% |
| Volatility | 3.85% | 4.96% | 6.30% | 8.04% |
| Minimum | -17.53% | -23.70% | -20.05% | -25.86% |
| 5% | -5.49% | -7.23% | -8.58% | -11.70% |
| 25% | -1.78% | -2.21% | -2.60% | -4.24% |
| 50% | 0.55% | 0.47% | 1.23% | 0.35% |
| 75% | 2.72% | 3.01% | 4.84% | 5.34% |
| 95% | 6.47% | 8.21% | 11.19% | 14.24% |
| Maximum | 14.73% | 24.48% | 30.20% | 33.34% |
| Mean | SE of mean | 5% quantile | Quantile / volatility | |
|---|---|---|---|---|
| Stock | ||||
| AAPL | 0.48% | 0.16% | -5.49% | -1.43 |
| META | 0.45% | 0.21% | -7.23% | -1.46 |
| NVDA | 1.24% | 0.26% | -8.58% | -1.36 |
| TSLA | 0.84% | 0.34% | -11.70% | -1.46 |
Under independent, identically distributed returns, \(\operatorname{SE}(\bar r)=s/\sqrt{N}\).
Mean uncertainty: the table reports one standard error in weekly return units. Multiply by 52 for the standard error of the annualized mean.
Quantile estimation: the empirical quantile is computed directly from returns; it does not use the estimated mean. A location shift moves the quantile one-for-one. The mean’s standard error is small relative to the magnitude of the 5% quantile here.
Tail uncertainty: this does not imply a precisely estimated quantile. Its sampling error also depends on the number of observations near the tail. Serial dependence can invalidate the independent-return standard-error calculation.
1.4 Comparison#
Among these four stocks, TSLA has the highest volatility and META the deepest weekly drawdown. Skewness and excess kurtosis measure different features of the return distribution.
2. Time Aggregation#
Use the supplied daily total returns in risk_etf_data.xlsx and calculate monthly
returns from its adjusted month-end prices. These prices include distributions.
Daily returns use SPY trading dates; monthly prices retain each asset’s calendar.
Daily: 2,384 observations, 2017-01-04 to 2026-06-30
Monthly: 113 observations, 2017-02-28 to 2026-06-30
2.1 Bitcoin and Measuring Covariation Risk#
| Daily | Monthly | |
|---|---|---|
| Pair | ||
| BTC with SPY | 0.26 | 0.33 |
BTC’s correlation with SPY is 0.26 daily and 0.33 monthly. The correlation depends on the return horizon and sample; neither estimate alone establishes reliable protection in an equity selloff.
2.2 Betas#
| Daily | Monthly | |
|---|---|---|
| Asset | ||
| SPY | 1.00 | 1.00 |
| VEA | 0.82 | 0.86 |
| UPRO | 2.99 | 3.09 |
| GLD | 0.09 | 0.13 |
| USO | 0.54 | 0.65 |
| FXE | 0.06 | 0.14 |
| BTC | 0.97 | 1.61 |
| HYG | 0.35 | 0.39 |
| IEF | -0.04 | 0.08 |
| TIP | 0.02 | 0.18 |
| SHV | -0.00 | 0.00 |
BTC’s beta rises from 0.97 daily to 1.61 monthly, the largest change in this sample.
The slope with an intercept equals covariance with SPY divided by SPY’s variance. Beta has no mechanical annualization factor. Daily and monthly estimates can differ because aggregation changes the measured covariance and variance, and the monthly sample has fewer observations. BTC also trades on weekends; its calendar-month endpoints can differ from SPY’s final trading date.
2.3 Time Scaling to Higher Moments#
| Daily | Monthly | |||
|---|---|---|---|---|
| Skewness | Excess kurtosis | Skewness | Excess kurtosis | |
| Asset | ||||
| SPY | -0.31 | 14.51 | -0.44 | 0.49 |
| VEA | -0.79 | 14.90 | -0.42 | 1.23 |
| UPRO | -0.39 | 15.28 | -0.46 | 1.03 |
| GLD | -0.63 | 7.67 | 0.13 | 0.52 |
| USO | -1.02 | 14.35 | -0.39 | 7.43 |
| FXE | 0.16 | 1.38 | 0.31 | -0.26 |
| BTC | 0.04 | 6.49 | 0.66 | 0.33 |
| HYG | 0.13 | 28.43 | -0.99 | 5.69 |
| IEF | 0.19 | 3.30 | -0.03 | 0.08 |
| TIP | 0.36 | 15.52 | -0.84 | 3.79 |
| SHV | 0.63 | 3.27 | 0.16 | -1.28 |
Monthly excess kurtosis is lower for every series. Skewness does not uniformly move toward zero; several estimates change sign.
For a sum of \(m\) independent, identically distributed returns, skewness scales by \(1/\sqrt{m}\) and excess kurtosis by \(1/m\). Compounded returns and dependent daily returns need not follow these rules. The monthly sample also has much less information about extreme observations. Estimate these moments at the horizon of interest.