Open Midterm 2

Open Midterm 2#

FINM 36700 - 2025#

UChicago Financial Mathematics#

Scoring#

Problem

Points

1

30

2

20

2

20

2

30

Numbered problems are worth 5pts unless specified otherwise.

Submission#

You should submit a single Jupyter notebook (.ipynb) file containing all of your code and answers to Canvas.

Note: If any other files are required to run your notebook, please include them and only them in a single .zip file.

Data#

All data files are found in at the course web-book.

https://markhendricks.github.io/finm-portfolio/.

The exam uses the data found in commodity_factors.xlsx

  • sheet factors

  • sheet returns

Both tabs contain daily returns for a set of commodity futures from January 2010 to October 2025.

  • approximate 252 observations per year for purposes of annualization

Factors are

  • LVL: A level factor of commodity data

  • HMS: Hard Minus Soft commodities

  • IMO: Input Minus Output commodities

Returns are

  • various commodity futures across energy, metals, livestock, and agriculture

import pandas as pd

factors = pd.read_excel('./data/commodity_factors.xlsx', sheet_name='factors', index_col=0, parse_dates=True)
returns = pd.read_excel('./data/commodity_factors.xlsx', sheet_name='returns', index_col=0, parse_dates=True)

factors
LVL HMS IMO
Date
2010-01-05 0.001741 -0.003471 0.006597
2010-01-06 0.012889 0.016399 -0.020703
2010-01-07 -0.011484 0.011919 -0.001153
2010-01-08 0.000436 0.007519 0.000570
2010-01-11 -0.007377 0.006306 -0.000623
... ... ... ...
2025-10-27 -0.001392 -0.004590 0.007534
2025-10-28 -0.002979 -0.012895 0.002998
2025-10-29 0.008762 0.006220 0.006162
2025-10-30 0.019519 0.027967 -0.011074
2025-10-31 0.001617 -0.005249 -0.006371

3972 rows × 3 columns

returns
CL GC HO LE NG PL RB SB SI ZC ZL ZM ZS
Date
2010-01-05 0.003190 0.000358 0.001643 0.011127 -0.041978 0.008897 0.009789 0.000724 0.019553 0.000597 -0.004646 0.010759 0.002620
2010-01-06 0.017244 0.015920 0.004148 -0.004344 0.065993 0.013980 0.005459 0.027858 0.021484 0.007164 -0.000983 -0.004696 -0.001663
2010-01-07 -0.006251 -0.002465 -0.008896 -0.000291 -0.033783 0.000515 -0.000796 -0.014432 0.009360 -0.010077 -0.016720 -0.034287 -0.031176
2010-01-08 0.001089 0.004501 0.007648 -0.001164 -0.009817 0.007469 0.009555 -0.016786 0.006818 0.013174 -0.011503 -0.000652 -0.004667
2010-01-11 -0.002779 0.010982 -0.009181 -0.009030 -0.051313 0.015148 -0.005846 -0.028333 0.012190 -0.001182 -0.008601 -0.006845 -0.011106
... ... ... ... ... ... ... ... ... ... ... ... ... ...
2025-10-27 -0.003089 -0.028288 0.013732 -0.021070 0.041768 -0.009725 -0.001196 -0.034068 -0.037518 0.012995 0.009946 0.013941 0.024478
2025-10-28 -0.018920 -0.008921 -0.020073 -0.005790 -0.028181 -0.000887 0.002499 -0.006224 0.012091 0.007580 -0.010045 0.027834 0.010307
2025-10-29 0.005486 0.004412 0.015541 0.017143 0.009268 0.009068 0.025192 0.003479 0.012647 0.004630 -0.001990 0.007178 0.001855
2025-10-30 0.001488 0.004418 0.014726 0.016746 0.171801 0.010683 0.015048 -0.009709 0.014815 -0.008641 -0.010167 0.022352 0.010183
2025-10-31 0.006769 -0.004773 -0.011707 0.005632 0.042467 -0.023938 -0.005141 0.010504 -0.008962 0.002905 -0.019537 0.019011 0.007789

3972 rows × 13 columns

1. Commodity Returns and Factors#

1.1 Factor Summary Statistics#

For each of the three factors, report only the following summary statistics (rounded to at least 6 decimal places):

  • Annualized Mean Return

  • Annualized Volatility

  • Annualized Sharpe Ratio

1.2 Factor Correlations#

Calculate and report the correlation matrix between the three factors (rounded to at least 6 decimal places).

1.3 Interpretation#

Does the factor construction make sense given the correlations you observe?

1.4 Tangency Portfolio Weights#

Build a tangency portfolio using the three factors as assets.

Report the weights of each factor in the tangency portfolio, rounded to at least 6 decimal places. You may assume that the factors use excess return.

1.5 Interpretation#

What do the tangency portfolio weights suggest about the relative importance of each factor?

1.6.#

Estimate an autoregression of the factor LVL…

\[r_t = \gamma + \rho\, r_{t-1} + \epsilon_t\]

Only report \(\rho\) (rounded to at least 6 decimal places).

Does the LVL factor exhibit momentum?

2. Single Factor Model#

2.1 LVL Factor#

We want to test the hypothesis that:

\[ \mathbb{E}[r_{i}] = \beta_{i,LVL} \cdot \mathbb{E}[r_{LVL}] \]

Regress each commodity’s returns against the LVL factor, and report the mean absolute alpha, \(\bar{\alpha}\) and \(r^2\) across all commodities.

Annualize the alpha.

Output exactly the following two numbers rounded to 6 decimal places:

  • Mean Absolute Annualized Alpha across all commodities

  • Mean R-squared across all commodities

2.2. Interpretation#

If our hypothesis were true, what would you expect the values of \(\bar{\alpha}\) and \(\bar{r^2}\) to be?

2.3 Cross-Sectional Test#

Let’s test the one-factor LVL model directly. From 2.1, we already have what we need:

  • The dependent variable, (y): mean excess returns from each of the commodities.

  • The regressor, (x): the market beta for each commodity from the time-series regressions.

Then we can estimate the following equation:

\[ \underbrace{\mathbb{E}\left[\tilde{r}^{i}\right]}_{n\times 1\text{ data}} = \textcolor{ForestGreen}{\underbrace{\eta}_{\text{regression intercept}}} + \underbrace{{\beta}^{i,\text{LVL}};}_{n\times 1\text{ data}}~ \textcolor{ForestGreen}{\underbrace{\lambda_{\text{LVL}}}_{\text{regression estimate}}} + \textcolor{ForestGreen}{\underbrace{\upsilon}_{n\times 1\text{ residuals}}} \]

Report exactly the following 3 numbers (rounded to at least 6 decimal places):

  • The R-squared of this regression

  • The intercept estimate, \(\hat{\eta}\)

  • The regression coefficient \(\lambda_{LVL}\)

2.4.#

Does your time-series or cross-sectional estimate give a higher premium to the LVL factor?

3. Trading the Model#

3.1 Beta Estimation#

For each commodity, report their LVL beta rounded to at least 6 decimal places.

Display a table of these betas, sorted from lowest to highest.

Remember#

We estimated the betas in the time-series regression in 2.1.

Hint#

Use df.sort_values(by=<YOUR_DF>, ascending=True) to sort your results.

3.2 Portfolio Formation#

Regardless of your answer to 3.1, allocate your portfolio as follows:

  • Go long GC, LE, and ZM

  • Go short CL, HO, and RB

Go long 1 and short 0.25.

That is, your portfolio should be: $\( r_{port} = 1 \cdot \left( \frac{1}{3} \cdot r_{GC} + \frac{1}{3} \cdot r_{LE} + \frac{1}{3} \cdot r_{ZM} \right) - 0.25 \cdot \left(\frac{1}{3} \cdot r_{CL} + \frac{1}{3} \cdot r_{HO} + \frac{1}{3} \cdot r_{RB}\right) \)$ Report the last 5 daily returns of your betting against beta portfolio, rounded to at least 6 decimal places.

3.3 Performance Evaluation#

For your portfolio, report the following performance statistics (rounded to at least 6 decimal places):

  • Annualized Return

  • Annualized Volatility

  • Annualized Sharpe Ratio

3.4#

For your portfolio, test the hypothesis that its premium can be explained by the LVL factor alone:

\[ \mathbb{E}[r_{port}] = \beta_{port,LVL} \cdot \mathbb{E}[r_{LVL}] \]

Report exactly the following 3 numbers (rounded to at least 6 decimal places):

  • Annualized Alpha

  • LVL Beta

  • R-squared for the regression

4. Multi-Factor Model#

We now want to test a multi-factor model using LVL and HMS and IMO as factors:

\[ \mathbb{E}[r_{i}] = \beta_{i,WTI} \cdot \mathbb{E}[r_{LVL}] + \beta_{i,HMS} \cdot \mathbb{E}[r_{HMS}] + \beta_{i,IMO} \cdot \mathbb{E}[r_{IMO}] \]

4.1 Time Series Test#

Estimate the time series test of this pricing model. Regress each commodity’s returns against the three factors, and report the following for each commodity (rounded to at least 6 decimal places):

  • Annualized Alpha

  • LVL, HMS, and IMO Betas

  • R-squared

4.2#

Report the annualized Sharpe ratio (rounded to at least 6 decimal places) of the tangency portfolio formed from

  • the individual commodities.

  • the three factors

What should be true of the Sharpe ratios if the factor pricing model is accurate?

4.3 Cross-Sectional Test#

Run the cross-sectional test of this multi-factor model:

\[ \mathbb{E}\left[\tilde{r}^{i}\right] = \lambda_{0} + \lambda_{LVL} \cdot \beta_{i,LVL} + \lambda_{HMS} \cdot \beta_{i,HMS} + \lambda_{IMO} \cdot \beta_{i,IMO} + \nu_{i} \]

Report exactly the following numbers (rounded to at least 6 decimal places):

  • \(\lambda_{0}\)

  • \(\lambda_{LVL}\)

  • \(\lambda_{HMS}\)

  • \(\lambda_{IMO}\)

  • \(r^2\) of the cross-sectional regression

  • MAE of the cross-sectional regression

Annualize the MAE.

4.4 Interpretation#

Do the results of the cross-sectional test support the multi-factor model?

4.5 Risk Premia#

Compare the risk premia (\(\lambda\)’s) from the cross-sectional test to the average returns of each factor. Report exactly the following table (rounded to at least 6 decimal places):

  • Average return of each factor

  • Risk premia from the cross-sectional test

Annualize the estimates.

4.6 Interpretation#

What do you observe from the comparison of average returns and risk premia?

Theoretically, what could cause the risk premium of a factor to deviate from its average return?

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