September 1, 2020

Quantitative Finance Course

Update: I've recently started trading a combination of strategies from the course, including risk parity, long/short futures factors, long/short equity factors, and long/short volatility. Here's the live performance (March 2019 through August 2020):  

Return (CAGR): 18.5%/year  
CAPM Alpha: 23.2%/year  
Sharpe ratio: 1.09  
Sortino ratio: 3.45 = 2.44√2  
Gain to Pain ratio: 2.43 (raw returns); 2.09 (excess returns)  
CAPM Information Ratio: 4.38  

Here's the curriculum I've put together for Quantitative Finance students.

It starts at a fairly basic level (an introduction to diversification, trend following, and relative momentum) and culminates with advanced reading in academic papers.


  • A deep-seated interest in how financial markets work
  • An A grade in either precalculus or high school statistics (AP Calculus AB or AP Statistics preferred)
  • Ability to commit a minimum of six hours per week to finance homework

Introductory Books

The Ivy Portfolio (Meb Faber)

I've screened dozens of books and have chosen this one because of its readability and balanced introduction to a variety of topics.

Part I discusses how the well-known Yale and Harvard endowments invest and how individual investors can replicate those strategies using index funds.

Part II provides a brief introduction to private equity and hedge fund strategies.

Part III introduces a simple trend-following system that has historically increased risk-adjusted returns while cutting the size of large losses in half.

For example, adding a 10-month moving average filter to the S&P 500 increased its return from 9.3% to 10.6% and lowered its volatility from 15.6% to 11.9%, nearly doubling the risk-adjusted return. The maximum drawdown (largest peak-to-trough loss) was cut from 45% to 23%.

The book also discusses a 13F strategy that can be used to track the stocks that hedge funds buy, cloning their exposures without paying fees.

The Way of the Turtle (Curtis Faith)

Curtis Faith's book is the most accessible introduction to building trading strategies that I've seen.

Chapters 1 and 2 cover the trader's greatest enemy: himself and his own biases. We can reduce these effects by using objective trading strategies that have historically had consistent performance across asset classes, especially if the results survive small tweaks to the strategies' rules.

Chapters 3-8 introduces trend following trading, risk management, and basic statistical metrics that are used to evaluate strategies.

Chapters 9-10 discuss specific indicators that are used to build trend-following trading systems.

Chapters 11-14 go into some of the pitfalls of building systems, including overfitting, small sample sizes, and under-diversification.

The epilogue presents the full-fledged trend trading system that the author, one of the original Turtle traders, used when he was in his twenties working for Richard Dennis.

Strategy Research and Paper Trading

To practice screening for stocks, students initially sign up for StockRover's free plan.

When they're ready to start testing to start testing stock market strategies, they'll need to sign up for Portfolio123. Students that are serious about doing their own research will find the Ultimate plan to be most helpful, as it will allow them to use variable position sizing (as opposed to equally weighting all the stocks) when they run their tests.

Paper trading for stocks and options can be done for free through TD Ameritrade's ThinkorSwim platform. (It's the paperMoney feature.) Your broker probably also offers a paper-trading system, though it may not be as sophisticated as Thinkorswim's.

Students who are interested in futures trading should start with the free Web site FuturesBacktest.

Academic Research (organized by topic)

Within each topic, I've sorted the research so that the easiest material is at the top of the list.

Read These First

A friend asked me to summarize factor investing in a simple way. I didn't do a good job of it over lunch, so here's a (hopefully) better attempt.

Does Higher Risk Actually Lead to Higher Returns?
This is a thread for weird stuff that's hard to explain. In my opinion, there's a lot evidence against the random walk and Capital Asset Pricing Model interpretations of the Efficient Market Hypothesis. That suggests that finance has the same difficulties as other fields in finding a model that works both conceptually and empirically.

Passive Investing

The Rate of Return on Everything
Our new, comprehensive data set includes total returns for equity, housing, bonds, and bills in 16 advanced economies from 1870-2015, revealing new insights and puzzles.


AQR addresses criticisms of momentum by pointing to its strong premium, pervasive evidence (both through time and across markets), and negative correlation to value.

Adaptive Asset Allocation
Estimates of parameters for portfolio optimization based on long-term observed average values are inferior to estimates based on observations over much shorter time frames.
NOTE #1: You can play around with Portfolio Visualizer's mean-variance optimizer if you want to get a hands-on idea of what the paper is talking about.
NOTE #2: This is one of the papers featured in my $10,000 financial adviser offer.

Volatility-Adjusted Momentum
Incorporating volatility estimates in constructing stock momentum leads to a Sharpe ratio increase (0.34 to 1.14) and strongly reduced crash risk. Similar improvements are found in corporate bonds.

Alpha Momentum and Alpha Reversal in Country and Industry Equity Indexes
Past short-term (long-term) alphas positively (negatively) predict future returns, subsuming their return-based counterparts.

Trend Following

We show that the returns of Managed Futures funds and CTAs can be explained by time series momentum strategies and we discuss the economic intuition behind these strategies. Time series momentum strategies produce large correlations and high-R-squares with Managed Futures indices and individual manager returns, including the largest and most successful managers. While the largest Managed Futures managers have realized significant alphas to traditional long-only benchmarks, controlling for time series momentum strategies drives the alphas of most managers to zero. We consider a number of implementation issues relevant to time series momentum strategies, including risk management, risk allocation across asset classes and trend horizons, portfolio re-balancing frequency, transaction costs, and fees.

Time Series Momentum
A diversified portfolio of time series momentum strategies across asset classes delivers abnormal returns with little exposure to standard asset pricing factors & performs best during extreme markets.

Trends Everywhere
AQR examines out-of-sample evidence for trend following in factors and alternative assets. They find that it works, provides substantial diversification, and isn't explained by vol targeting.

Enduring Effect of Time-Series Momentum on Stock Returns Over Nearly 100 Years
Trend following works in individual U.S. (1927-2014) and international (1975-2014) stocks without January losses or momentum crashes.

Two Centuries of Trend Following
Trend following on commodities/currencies/stock indices/bonds is profitable using spot data going back as far as 1800 (both before and after accounting for upward drift).

Dual Momentum – A Craftsman’s Perspective
Our objectives were twofold:
1.Verify the strength and robustness of the Dual Momentum concept and specifically the Global Equity Momentum strategy
2.Describe how to use ensemble methods to preserve expected performance while minimizing the probability of adverse outcomes

You Can't Always Trend When You Want
Trend following has continued to profit from market moves and diversification. However, the average size of global market moves has been more muted than usual.

The convexity profile for a short-term trend following system is most significant when measured over weekly or monthly horizons. Long-term systems, on the other hand, exhibit negative or insignificant convexity profiles over these horizons.

Cross-Sectional and Time-Series Tests of Return Predictability: What Is the Difference?
The difference between the performances of TS and CS strategies is largely due to a time-varying net-long investment in risky assets.

Do Stocks Outperform Treasury Bills?
The best-performing 4% of listed companies explain the net gain for the entire U.S. stock market since 1926; other stocks collectively matched Treasury bills.


Fact, Fiction, and Value Investing
With characteristic wit, AQR corrects misconceptions. Topics include concentration, profitability, momentum, value in other asset classes, composite metrics, and large caps.

Equal-Weight Benchmarking: Raising the Monkey Bars
In the period from 1969 to 2011, if you had picked stocks at random, there is a 99.9% chance you would have beaten the market. It is certainly remarkable that, at a time when the vast majority of hypothetical monkeys flinging darts at the financial pages outperformed, less than half of active managers managed to do so.
NOTE: This is the paper featured in my Introduction to Quantitative Finance page.

Combination Metric Backtest and Comparison of Value Deciles (1951 to 2013)
Over the long run, and with some regularity, cheap stocks tend to outperform more expensive stocks. The PB, PE and PCF ratios are all very useful metrics for sorting cheap stocks from expensive stocks, but we can’t know which will be the better bet at any given point in time. The combo spreads the risk of underperformance relative to any single metric, and, in doing so, generates reliable investment performance over the full period without lagging far behind the front-runner at any point.

Will Value Survive Its Long Winter?
Value and other premiums carry the risk of being negative for as long as a decade. Allocating to value, even when it is not providing a premium, reduces long-term portfolio risk.

Do 'Dogs of the World' Bark or Bite? Evidence From Single-Country ETFs
Mean reversion in financial markets is commonly accepted as a powerful force. This paper examines the performance of a simple mean-reversion-based strategy -- Dogs of the World -- designed to take advantage of return reversals in national equity markets. Both a simulated application of the strategy using indexes since 1971 and application using single-country ETFs since 1997 produces higher compounded average returns than those of a comparable market index. Although the Dogs strategy also produces higher volatility than the index, the information ratio for the strategy suggests that the return more than compensates. An advantage of this strategy is that its implementation using single-country ETFs is straightforward and inexpensive. [NOTE: This paper is no longer available at the SSRN link provided.]

Industry Long-Term Return Reversal
Long-term reversals (3-10 year formations): all U.S. stocks are grouped into 48 industries, and the industries are ranked and traded L/S against each other.


We present a model with leverage and margin constraints that vary across investors and time. We find evidence consistent with each of the model’s five central predictions: (1) Since constrained investors bid up high-beta assets, high beta is associated with low alpha, as we find empirically for U.S. equities, 20 international equity markets, Treasury bonds, corporate bonds, and futures; (2) A betting-against-beta (BAB) factor, which is long leveraged low beta assets and short high-beta assets, produces significant positive risk-adjusted returns; (3) When funding constraints tighten, the return of the BAB factor is low; (4) Increased funding liquidity risk compresses betas toward one; (5) More constrained investors hold riskier assets.

Leverage Aversion and Risk Parity
Consuming the high risk-adjusted returns of safer assets requires leverage. Risk parity portfolios exploit this opportunity by equalizing the risk allocation across asset classes.

Fact and Fiction about Low-Risk Investing
This article presents five facts and dispels five fictions about low-risk investing within equities and across other asset classes.

Betting Against Correlation
What drives BAB - leverage constraints or behavioral effects? AQR digs in by dividing BAB into B.A.Correlation and B.A.Volatility factors.


Accounting for both quality and value yields dramatic performance improvements over traditional value strategies. Gross profitability is particularly powerful, especially for large cap stocks and long-only investors.

Higher-quality stocks have higher prices (P/B) on average, but not by a very large margin. A quality-minus-junk (QMJ) factor earns significant risk-adjusted returns in the U.S. and globally.


We examine many claims about the size effect and aim to clarify some of the misunderstanding surrounding it by performing simple tests using publicly available data.

Short-Term Reversal

Only the component of short-term reversal which isolates reaction to recent “nonfundamental” price changes is significant and positive; it generates a risk-adjusted return 3x as large.


Halloween and turn-of-the month (TOM) are the strongest calendar effects, fully diminishing the other three (the January effect, weekend effect and holiday effect) to zero.

* Stocks, L/S factors, commodities, and country indices have seasonality
* Factor seasonalities are comparable in size to factors themselves
* Stock seasonality may originate from factor seasonality

Using a sample of 97 stock return anomalies, we find that returns are 6 times higher on earnings announcement days. Results are consistent with biased expectations being at least partially corrected.


The Short of It: Investor Sentiment and Anomalies
Following high levels of market-wide sentiment:
* Anomalies in individual stocks tend to be stronger
* The short leg becomes more profitable
* The long leg's returns are not affected

Why Do Short Interest Levels Predict Stock Returns?
Short sellers are sophisticated investors who have value relevant information about firms and position themselves in stocks with deteriorating fundamentals.

Want Smart Beta? Follow the Smart Money: Market and Factor Timing Using Relative Sentiment
We present a real-time, cross-asset, positions-based relative sentiment indicator derived from the COT report to predict the U.S. equity market.


A simple formula based on low vol, high net payout yield, and 12-1 momentum provides "full and efficient exposure to the most important factor premiums" worldwide.

Value and Momentum Everywhere
Value and momentum are more positively correlated across asset classes than the asset classes are themselves. However, value and momentum are negatively correlated both within and across asset classes.

Superstar Investors
Identify structural edges and commit to seeing them through inevitable periods of underperformance. As each of our superstars shows, “merely good” edges over time compound to great long-term performance.

Buffett's Alpha
AQR finds that Berkshire Hathaway's large 13% CAPM alpha and 0.79 Sharpe ratio become insignificant after controlling for BAB and QMJ. Warren Buffett may have been the first multifactor investor.
NOTE #1: Here's a thread about Berkshire Hathaway's factor loadings. You can also check them yourself at Portfolio Visualizer.
NOTE #2: This is one of the papers featured in my $10,000 financial adviser offer.

Death of Diversification Has Been Greatly Exaggerated
Factor diversification is the best answer for the many investors whose risk is dominated by stock market directionality and who will take the time to understand the approach.

Investing with Style
Value, Momentum, Carry and Defensive deliver positive returns with low correlation in out-of-sample tests across a multitude of asset classes and time periods using very liquid securities.

Strategic Allocation to Commodity Factor Premiums
Portfolios of commodity factor premiums exhibit significantly better risk-adj performance than the commodity market portfolio and add value to a stock/bond portfolio.

Two Centuries of Commodity Futures Premia: Momentum, Value and Basis
* Commodity factors work in hand-collected pre-sample data (back to 1877)
* Long commodities + factor tilts earn higher Sharpes and are still able to hedge inflation

Global Factor Premiums
New sample evidence reveals that the large majority of global factors are strongly present under conservative p-hacking perspectives, with limited out-of-sample decay.

Factor Performance 2010-2019: A Lost Decade?
There appears to be a clear dichotomy in recent factor performance: while generally accepted factors struggled, various factors that are considered to be inferior or redundant remained effective.

Exploiting Commodity Momentum Along the Futures Curves
Momentum strategies that invest in contracts on the futures curve with the largest expected roll-yield or the strongest momentum earn higher risk-adjusted returns.

Ignored Risks of Factor Investing
The risks of factor investing are usually understated (perhaps, severely so), and the diversification benefits tend to be overstated.

Factor Momentum and the Momentum Factor
Factors are positively autocorrelated [and time-series momentum strategies work on them]. Momentum is not a distinct risk factor; it aggregates the autocorrelations found in other factors.

Factor-Based Commodity Investing
A multi-factor portfolio combining the momentum, basis, basis-momentum, hedging pressure and value commodity factor portfolios outperforms widely used commodity benchmarks.

Craftsmanship Alpha

Leveraged Trading (Robert Carver)
* Systematic > discretionary
* Be careful with leverage
* Execution costs and financing spreads matter
* Adjust signals for volatility
* Measure position sizes in dollar × standard deviation units
* Diversification works wonders: add markets and create ensembles

Seemingly inconsequential design decisions can actually matter a lot. The skillful targeting and capturing of style premia may constitute a form of alpha on its own.

Long-Only Style Investing: Don't Just Mix, Integrate
AQR compares "mixing" styles via stand-alone portfolios and "integrating" by combining signals during the stock selection process. Integrating is better b/c it minimizes the impact of constraints.

Role of Shorting, Firm Size, and Time on Market Anomalies
AQR dives into the sources of alpha for L/S size, value, and momentum and whether the alphas have weakened over time.

When Equity Factors Drop Their Shorts
* The alphas of equity factor short legs are subsumed by those of the long legs.
* Only the short (not long) legs of HML, low vol, and low beta are subsumed by FF5.

Shorting Costs and Profitability of Long–Short Strategies
Shorting costs amount to almost 40 percent of gross long–short returns. If other trade-related transaction costs were considered, long–short profits would be reduced further.

Equity Factors: To Short Or Not To Short, That is the Question
A long-short implementation leads to better risk-adjusted returns than its hedged long-only counterpart, at least when AUM are not too large.


Returns of commodity futures indexes have, on average, been positive over the long run. Commodities are a potentially attractive asset class in portfolios of stocks and bonds.

Conquering Misperceptions about Commodity Futures Investing
Three misperceptions:
(1) Commodities are a play on commodity prices
(2) Commodity prices provide an inflation hedge
(3) Commodity markets can absorb abundant capital

Commodity Futures Risk Premium: 1871–2018
* New database (newspaper data)
* Non-surviving contracts have a minor impact
* Unlike stocks, median commodity has a positive return
* Long-only, carry, and trend strategies work but have prolonged drawdowns


Carry predicts returns cross-sectionally and in time series for a host of different asset classes, including global equities, global bonds, commodities, US Treasuries, credit, and options.

Anatomy of Commodity Futures Risk Premia
A single factor, high-minus-low from basis sorts, explains the cross-section of spot premia. Two additional basis factors are needed to explain the term premia.

Tax Alpha

Tax-managed factor tilts (β=1) generated average tax alphas of 1.6%-1.9%/year. Alpha for tax-managed indexing was 2.3%/year. This can be attributed to loss harvesting and the tax rate differential.

Tax Benefits of Separating Alpha from Beta
The turnover of a strategy that separates α from β is concentrated on the long-short component and enables the deferral of capital gains on the passive market component.

The authors construct basic value and momentum strategies. The long-short portfolio has a tax BENEFIT of 0.3%, which can be increased to 6.1% through tax-aware stock selection and timing.

Relaxing the long-only constraint results in a large increase in their tax benefits in particular due to an increase in the character benefit.

Understanding the Tax Efficiency of Relaxed-Constraint Equity Strategies
* Tax benefits of relaxed-constraint (130/30) equity strategies (character and deferral components)
* Potential impact of the Tax Cuts & Jobs Act of 2017

Volatility Targeting

The authors look at a strategy that uses volatility to adjust the leverage of individual factors (SMB, HML, Mom, RMW, CMA, Carry) as well as country equity indices.

This paper looks at volatility targeting over equities, futures and bonds. Volatility is generally autocorrelated, BUT targeting works better for equity and credit than Treasuries and commodities.


Options as a Strategic Investment (Lawrence McMillan)

Selling a call against stock does NOT provide a reliable cushion against declines. The cushion comes from the call's negative delta and its exposure to the volatility risk premium, not from the premium that you initially collect.

Delta-hedged covered calls outperform unhedged covered calls. This is the case in equity indices throughout the world. The VRP appears to add more diversification than the equity component of the strategy.

Equity index covered calls have historically provided attractive risk-adjusted returns largely because they collect equity and volatility risk premia. However, they also embed exposure to an uncompensated risk, a naïve equity market reversal strategy.

(1) Long-short trend-following and (2) covered calls each have higher Sharpes than buy-and-hold but negative correlations to each other. [This paper is no longer available at the original SSRN download page.]

Collaring the Cube: Protection Options for a QQQ ETF Portfolio
Diagonal collars on QQQ, 3/1999-3/2008, generated higher returns, lower volatility, lower drawdowns, and lower kurtosis than QQQ itself. Purchases were made at ask and sales at bid.

Embedded Leverage
Many financial instruments are designed with embedded leverage such as options and leveraged exchange traded funds (ETFs). Embedded leverage alleviates investors' leverage constraints and, therefore, we hypothesize that embedded leverage lowers required returns. Consistent with this hypothesis, we find that asset classes with embedded leverage offer low risk-adjusted returns and, in the cross-section, higher embedded leverage is associated with lower returns. A portfolio which is long low-embedded-leverage securities and short high-embedded-leverage securities earns large abnormal returns, with t-statistics of 8.6 for equity options, 6.3 for index options, and 2.5 for ETFs. We provide extensive robustness tests and discuss the broader implications of embedded leverage for financial economics.

Portfolio Strategies for Volatility Investing
The degree of VIX futures contango has been shown to hold predictive power over volatility returns. This study proposes a conditional strategy which allocates to market and volatility risk.

Risk Premia and the VIX Term Structure
A single principal component, Slope, predicts the excess returns of S&P 500 variance swaps, VIX futures, and S&P 500 straddles for all maturities to the exclusion of the rest of the term structure.

Risk Parity is Not Short Volatility (Not That There's Anything Wrong with Short Volatility)
The two strategies’ similarities are overstated, and we find no empirical evidence to support the claimed hidden exposure.

Volatility Trading (Euan Sinclair)

Real Estate

Modeling private real estate is not straightforward. We consider theoretical arguments, historical average returns, and forward-looking yields.

Financial Advisors

Misguided Beliefs of Financial Advisors
Advisors trade frequently, chase returns, prefer expensive, actively managed funds, and underdiversify. Their net returns [alphas] of −3%/year are similar to their clients'.

More Strategies

I stay abreast of current research via the FinTwit community and maintain lists of the more interesting developments below (organized by topic):
  1. Benchmarks
  2. Value, Long-Term Reversal, Low Volatility, and Quality
  3. Trend-Following, Cross-Sectional Momentum, and Carry
  4. Volatility Risk Premium
  5. Calendar and Diagonal Spreads
  6. Real Estate, Commodities, and Bonds
  7. Seasonality, Sentiment, Macro, and Short-Term Mean Reversion
  8. Craftsmanship Alpha
  9. Tax Alpha
  10. Sarlacc Pits (things to avoid buying)
  11. Market Psychology and Bubbles
You'll also find a very useful collection of research summaries at CXO Advisory, though you'll have to pay for a membership if you want to access any material written after July 2017.

If you're interested in learning more, please use the form below to contact me. I look forward to meeting you!

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