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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.

Prerequisites

  • 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.

Momentum

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.

Glitch
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.


Value

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.


Betting-Against-Beta

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.

Quality

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.

Size

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.

Seasonality

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.

Sentiment

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.

Multi-Factor

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.


Commodities

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

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

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!

Your Name :


Your Email: (required)


Why are you interested in quantitative finance, and what are your eventual goals? (required)


August 26, 2020

Free Sessions for September 2020 (Quantitative Finance)

Recent large losses in standard stock/bond portfolios emphasize the need for greater diversification and risk control. At the same time, students are quarantined at home without access to the educational resources they're accustomed to.

As a result, I'm offering free weekly quant finance sessions for the month of September. Please contact me using the form at the bottom of this page for more information.


Students who love math and have a deep-seated interest the stock market will want to check out the curriculum for my Quantitative Finance course, developed from thirteen years of experience reading finance research and four years working as a quant.

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  

Take a look at what others in the finance community are saying about my work:
"Just noticed that @ReformedTrader is bizarrely under-followed. He's curated, quoted, pasted, summarized, analyzed, organized and synthesized well over a hundred papers and articles on academic finance. Honestly, wtf are y'all reading if you aren't following his threads?" Adam ButlerCIO at ReSolve Asset Management (here as well)

"@ReformedTrader has put together an awesome series of Twitter “moments” that highlight research on risk premia, style premia, seasonality, and craftsmanship. Dig in." Corey HoffsteinCIO at Newfound Research and a member of Investopedia's Top 100 in finance (here as well)

"Gotta hand it to @ReformedTrader for his consistency in posting quality quant finance research links. Everyone who is interested in quant finance should follow him. Hidden gem." Pravit Chintawongvanich, Wells Fargo equity derivatives strategist

"Wow, good stuff. I didn't even know the Moments could be used like this. Really great reference and shows the power of info sharing and knowledge building on Twitter." Justin Carbonneaumanaging partner at Validea Capital Management

"Read this thread and become an expert on the quality factor. Fantastic work by @ReformedTrader." Chris Cain, Quantitative Researcher at Connors Research and author of The Alpha Formula (here as well) 
"I enjoy these long threads Darren does on some great financial books." Jim O'Shaughnessy, founder of O'Shaughnessy Asset Management and author of What Works on Wall Street

"Wow, Darren knows the paper way better than I do now." Cliff Asness, billionaire co-founder of AQR Capital Management, which makes hedge fund strategies available to investors at mutual-fund-level fees

The charts below, taken from a paper written by my friend Pim van Vliet, describe one of the strategies covered in the course.

Note the strong performance in each decade, including the dot-com bust (2000-2002) and 2008 crisis periods. The Conservative leg of the strategy owns low volatility, high momentum, and high buyback yield stocks, leading to more reliable returns than those of the market as a whole.

 

This chart is from The Conservative Formula: Quantitative Investing Made Easy, one of the papers covered in my Quant Finance course.

Please contact me about tutoring and mention the words "Finance Offer."

Your Name :


Your Email: (required)


Why are you interested in quantitative finance, and what are your eventual goals? (required)


August 15, 2020

SAT Practice Tests

Update: I've removed Practice Tests 2 and 4 from the list, since the College Board no longer recommends that students take those two tests.

Table of Contents

College Board SAT practice tests
Ivy Global SAT practice tests, including answer explanations
PSAT practice tests

Suggestions

The PSAT practice tests at the end of the list provide useful diagnostic scores until you hit 700 in either Verbal or Math. If you're worried that you'll run out of practice tests, start with the PSAT.

The College Board no longer recommends that students take Practice Tests 2 and 4, so I've removed those tests from the list below and replaced them with newly available Practice Tests 9 and 10.

If you find yourself making small mistakes or running out of time on one or more sections of the test, consider following an objective set of timing rules so that you won't feel rushed.

I strongly suggest printing practice tests out onto real paper. It's almost impossible to take notes, cross off answer choices, or double-check your bubbling unless you're working on paper!


College Board SAT Practice Test 1

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart (This document has the scoring instructions!)
Answer Explanations (These are the College Board's explanations. The SAT Black Book has better ones.)

I've written detailed answer explanations for some of the supporting evidence questions in SAT Practice Test 1.

I've also written a detailed answer explanation for #29 in section 4 (calculator-based math).

College Board SAT Practice Test 3

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart (This document has the scoring instructions!)
Answer Explanations (These are the College Board's explanations. The SAT Black Book has better ones.)

College Board SAT Practice Test 5

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations

College Board SAT Practice Test 6

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations

College Board SAT Practice Test 7

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations

College Board SAT Practice Test 8

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations

College Board SAT Practice Test 9

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations

College Board SAT Practice Test 10

Test (Print this out onto real paper!)
Answer Sheet
Essay
Answers and Scoring Chart
Answer Explanations


IvyGlobal online SAT Practice Test 1

Test, Answer Sheet, and Scoring Chart (Print this out onto real paper!)
Online Scoring

Answer Explanations
Note: This practice test has an answer key error. The answer to Critical Reading question #18 is A, not D.

IvyGlobal's SAT questions are accurate, and their answer explanations are very well-written. Their practice tests are an excellent way to learn by doing, especially if you don't have a tutor.

IvyGlobal online SAT Practice Test 2

Test, Answer Sheet, and Scoring Chart (Print this out onto real paper!)
Online Scoring

Answer Explanations


College Board PSAT/NMSQT Practice Test 1

Test (Print this out onto real paper!)
Answer Sheet
Answers and Scoring Chart
Answer Explanations

College Board PSAT/NMSQT Practice Test 2

Test (Print this out onto real paper!)
Answer Sheet
Answers and Scoring Chart
Answer Explanations

IvyGlobal online PSAT Practice Test

Test, Answer Sheet, and Scoring Chart (Print this out onto real paper!)

Additional Practice

IvyGlobal's practice tests are almost as good as College Board tests, and the IvyGlobal answer explanations are detailed and complete. You'll find links to IvyGlobal's materials (11 practice tests in print and 2 online) in my list of recommended SAT books.

Use Khan Academy to practice for SAT Critical Reading. The answer explanations aren't great, and the practice questions haven't been tested as extensively as real SAT questions have, but it's still College Board material. If you really want to challenge yourself, you can prep for SAT Literature Subject Test and AP English Language, which make SAT Reading questions seem like child's play.

ACT English and SAT Grammar/Writing are nearly identical. The SAT gives you more time per question, but it also gives you a slightly higher proportion of passage interpretation and adding/deleting sentence questions. Khan Academy is also good here.

You can find several SAT Math practice resources in my list of recommended SAT booksKhan Academy also provides decent practice, but its answer explanations may leave you a bit frustrated.



July 17, 2020

SAT Math Level 2 Subject Test: The Best Prep Books

Update: I've updated the errata section for my review of 320 SAT Math Subject Test Problems - Level 2 (Steve Warner).

The Official SAT Subject Study Guide: Math Level 2

This book has four official College Board Math Level 2 practice tests.

Tests 3 and 4 are the same as the tests in the old edition of the College Board's Math Level 1 & 2 book. Test 3 is also the same as the test included in the Official Guide for All SAT Subject Tests.

The great news is that Tests 1 and 2 are completely new. They haven't been published before in any form, and they're even not the same as the two official but unpublished College Board Math Level 2 tests that are floating around.

Pros
Official material is a true confidence builder. Every question you get wrong contains skills you need to practice.

Most prep books have poorly written questions, answer key errors, and questions that are unrealistically easy, difficult, or off-topic. If you get questions wrong or run out of time on unofficial tests, you'll have trouble figuring out whether the fault lies with you or with the book you're using.

Most students who take practice tests for Math Level 1 and Math Level 2 find that they do better on Level 2 because of its generous curve. Based on the raw-to-scaled score conversion tables in the book, a raw score of 43/50, 40/50, 44/50 and 43/50 will get you perfect 800's on the first, second, third, and fourth Math Level 2 practice tests, respectively. To get a perfect score on Math Level 1, you usually need a raw score of 49/50 or 50/50.

Cons
The book only offers practice tests, warm-up questions, and answer explanations for the practice tests. It doesn't address strategy in any kind of detail.


SAT Math 2 Prep Black Book (Mike Barrett)

Barrett's method focuses on shortcuts and calculator tricks to get you through problems you're not sure how to do. He includes very detailed answer explanations for the problems in two College Board practice tests.

Since Barrett de-emphasizes content review, you may have trouble breaking above 750. It's great, however, if you use it in conjunction with another book that teaches you how to solve problems more traditionally.

Pros
This book's greatest strength is its focus on strategic guessing. It's faster to cross off three or four answer choices and choose from what's left than it is to solve problems traditionally. Unless you're very good, you'll need to use strategic guessing for the first forty questions in order to garner enough time for questions 41-50, which tend to be much harder.

Most of Barrett's answer explanations are one to two pages per question, so his book can really help if you don't have time to find a tutor.


Cons
Barrett's book doesn't contain any practice problems. You have to get the College Board's Math Level 2 study guide: all of his answer explanations are for the problems from that book.

Keep in mind that Barrett's answer explanations are tied to the old edition of the aforementioned study guide. That means you'll either have to use that edition or match his explanations up with tests 3 and 4 of the new edition yourself.


Cracking the SAT Math 2 Subject Test

This is a good all-around study guide. It contains content review, useful strategies, and decent practice tests.

Pros
The practice tests have no answer key errors. They're not quite the same as official practice tests (some of the problems lack elegant solutions and will take you longer than 30-60 seconds to solve), but the differences aren't large enough to keep you from getting an 800 on the real thing.

The book's helpful content review chapters can keep you from feeling lost when you're trying to re-learn your entire high school math curriculum.

Cons
The Princeton Review is all about giving you what you need and not one iota more. It's a good idea to supplement this book with the extra practice material in Steve Warner's book.


320 SAT Math Subject Test Problems - Level 2 (Steve Warner)

I use this as a textbook in my own tutoring for students who score 700 or higher on practice tests. It's a huge set of practice problems with detailed answer explanations.

Pros
The practice material is very similar to real Math Level 2 tests.

The problems in this book are arranged by topic and difficulty level, so students who don't need any content review can jump straight to the chapters that contain what they want to work on.

The answer explanations provide more than one way to do each problem, and the fastest method is marked with a star.

Cons
The book doesn't provide any content review. Dr. Warner does define terms like range and domain in his answer explanations, but his book doesn't have an index. You'll need to label important pages with Post-It notes.

In order to finish on time, you should first eliminate as many answer choices as you can and then decide whether to solve each problem in thirty seconds, guess from the remaining answer choices, or skip the problem and come back later. Dr. Warner's answer explanations don't talk about eliminating answer choices, so you may find that you run out of time if you solve problems the way he does.

If you're scoring below 700 on practice tests, start with the Princeton Review's book and come back to Dr. Warner's book later.

Errata
#108 on page 101 has two correct answers: (D) and (E).

The answer to #65 on page 164 is 55, which is not one of the answer choices.

The answer to #98 on page 173 is (A), not (B). Since the first term is k0, k4 is actually the fifth term in the sequence.

The answer to  #99 on page 173 is 13/4, which doesn't match any of the answer choices.

The answer to #156 on page 140 is 9.54, not 8.43. Dr. Warner arrived at the wrong answer because he plugged in 115° for ∠SOT instead of 145°.

#16 on page 149 doesn't give you enough information to solve the problem. Assume that the heights of the two cones are equal, and you'll get the correct answer.

The answer to #112 on page 176 is (E), not (A). The complex number z + 2 can be in either Quadrant I or Quadrant II, depending on how big its real component is, so i(z+2) can be in either Quadrant II or Quadrant III. In fact, if z = -2+i,  the answer ends up lying on y-axis, which isn't in any of the four quadrants.

The answer to #135 on page 184 should be "none of the above." Since a square root can't be negative, f(g(x)) will never be equal to -1, and the quantity a + b is an imaginary number.

#156 on page 192 should read 0 < θ < π/2, not 0 < x < π/2.

#160 on page 193 is written in an unclear way, as it's not evident whether the order of the positive integers matters when you're adding them together.


Barron's SAT Subject Test: Math Level 2

Barron's practice tests are harder than real College Board tests, and I'd only recommend them if you really want to challenge yourself.

That said, I've changed my view on this book over the past couple of years. Most of my students want a perfect 800 on the test and constantly seek out difficult practice questions.

If you really like math and think the hardest questions are the most fun, even when the answer explanations aren't perfect, this could be the right book for you.

If you decide to try the practice tests, add 50 points to your score in order to compensate for difficult (and, occasionally, poorly written) questions.

Pros
Barron's guides tend not to change much from one edition to the next. For example, except for question 18, Model Test 1 is basically the same in the 10th and 11th editions. You can buy a used 10th edition for five dollars if you want.

Because Barron's is a major publisher, you can find its books at the public library. That's not the case for self-published books like Steve Warner's and John Chung's.

Cons
The pro listed above is also a con: errors tend not to get fixed from one edition to the next.

You have to be proactive in order to check your work, as the answer explanations are short, and it may be hard to tell whether a question is badly written or if you've simply answered it incorrectly.



Ivy Global's Online Math Level 2 Practice Test and Answer Explanations
Ivy Global, which has published fairly accurate SAT practice tests, has recently released a Math Level 2 practice test.

Their material is pretty challenging and includes topics that the real Math Level 2 test doesn't cover. That said, the answer key is accurate, so go ahead and take their test if you need the extra practice.



Dr. John Chung's Mathematics Level 2

This book has problems that are much harder than the real thing and covers some obscure topics. The problems are all doable in 30 seconds or less, though, so they're hard in a way that may be helpful if you're already scoring 800 and want some extra practice.

Pros
Only two official practice tests have been released for Math Level 2, so you may need the extra practice in this book if you absolutely must get an 800 on your test.

Though the book tests some obscure concepts that rarely show up on official tests, you could see those concepts in your precalculus class. The difficult problems in this book might be fun if you're obsessed with math.

Cons
The practice tests are very difficult, so don't treat your scores from this book as accurate diagnostics.

Books to Avoid

The Arco and McGraw-Hill books contain inaccurate questions and answer key errors.

Going for a Perfect Score

A raw score of 44/50 will usually get you a perfect Math Level 2 score. Even after the test deducts a quarter of a point for every question you get wrong, you can afford to miss five of the fifty problems. That's like getting an A-minus on an advanced high school math test.

The books above contain everything you need to get an awesome score, but if you'd like personalized help, you can sign up for in-home or online tutoring.

July 7, 2020

Timing Rules for the SAT and ACT

Update: I've added additional comments to the Math section.

The easiest way to avoid making "stupid mistakes" on the SAT and ACT is to slow down. Skipping steps and not reading questions carefully tend to lead to small mistakes. You can slow down if you have a way to keep track of time.

For example, on SAT Critical Reading, you have 13 minutes per passage (65 minutes for a total of 5 passages).

That means that if you know the multiples of 13, you can check the clock and make sure that every passage is finished by its appropriate multiple of 13 minutes. That means you need to have read the passage, done the questions, double-checked, and bubbled within the 13 minutes that you’re allowed.

Passage 1: 13 minutes
Passage 2: 26 minutes
Passage 3: 39 minutes
Passage 4: 52 minutes
Passage 5: 65 minutes

If you find that you can't finish the Reading section even with the timing rule given above, you can choose to skip the passage type that's hardest for you. For example, it's okay to skip the 1800's-era passage and to spend 65/4=16 minutes per passage instead of 13 minutes. If you do this, choose "D" for all of the questions in the passage you're skipping and don't even bother to read that passage.

Whenever you do a problem, whether it's math, grammar, and reading, read the question carefully, but don’t stare at it for more than 10 seconds. If a method for solving the problem doesn’t become apparent within 10 seconds, look at the answer choices and try to eliminate some. For about a quarter of the questions, you can eliminate three of the four choices immediately.

Math

Show all your work on math problems. Speed increases come from using fewer steps, not from skipping steps. (For example, solving a 3-4-5 triangle using ratios is faster than using the Pythogorean theorem. We don’t skip any steps, but we do solve the problem in one step instead of five.) Small mistakes come from either mis-reading the problem or by skipping steps. If you show all your work, you’ll know immediately if you re-read everything that you solved for the right variable and plugged the numbers in correctly.

When you’re done, read the problem carefully a second time. Doing this catches about 90% of small mistakes, including ones that involve setting up equations incorrectly, solving for the wrong variable, or forgetting to convert units (for example, hours to minutes). Double-check to make sure the problem was set up correctly, that you plugged the right variables in the right places, and (importantly) that you solved for the correct variable.

If you have a lot of time, do each question two or three different ways if it looks like there’s an easy opportunity to do so. Doing this will eliminate almost all small mistakes and nearly guarantee an 800 on the Math section. (One way to teach yourself creative ways to solve problems is to work through a practice book without using any paper: this will force you to look for simple solutions. I started trying this in May 2017 and have been doing it during actual tutoring since September 2018.)

I personally like to circle the variable I'm supposed to be solving for when I first read the question, solve the problem, and put a vertical bar next to the question when I've read it a second time. This allows me to mentally be finished so that I can move on to the next question. (If I've solved a problem two different ways, I also place a check mark next to the question to remind myself not to double-check it if I have time at the end of the test.)


Doing the Math section very quickly and going back to double-check very quickly doesn’t work. That’s because the test is purposefully designed to be difficult to interpret, and if you try to go through the problems too quickly, you’ll mis-read them. Even though you’re double-checking, you’re likely to mis-read twice if you’re in a rush. Again, it makes more sense to slow down and be careful than to rush through and double-check later.

Bubbling

Work on two pages at a time and don’t bubble until both pages are finished. This will (1) minimize the distraction of moving from the test page to bubble sheet and back again and (2) reduce errors from accidentally bubbling in a question that was skipped and then being off by one.

If you want, you can print each practice test out onto double-sided pages and put the pages in a binder. Your test will then have two pages that face each other, just like the booklets you'll be using on test day.

Timing Rules

The table below has timing rules you can use for each section of the SAT and ACT. If necessary, you can modify the rules if you plan to skip a passage or to go faster in order to spend more time on difficult material.

Section Total Time Number of Questions Time for each set of 10 questions Comments
SAT Critical Reading 65 minutes 52 questions 13 minutes Instead of checking the clock every ten questions, give yourself 13 minutes per passage. If there's a passage type that's particularly hard for you, try finishing the others in 12 minutes so you can spend 17 minutes on the difficult one.
SAT Grammar 35 minutes 44 questions 8 minutes Since this section of the test has four passages (11 questions each), you can give yourself 8 minutes per passage with 3 minutes left at the end of the test to check your bubbling.
SAT Math (no calculator) 25 minutes 20 questions 12 minutes

SAT Math (calculator) 55 minutes 38 questions 14 minutes

ACT English 45 minutes 75 questions 6 minutes

ACT Math 60 minutes 60 questions 10 minutes You have one minute per question on this section.

ACT Reading 35 minutes 40 questions 8 minutes Give yourself 8 minutes per passage. The science-related passage at the end is hardest for most people, but note that 8*4=32 and that you have 35 minutes to finish the test, so you actually have 11 minutes, not eight, to finish the final passage.
ACT Science 35 minutes 40 questions 8 minutes


In order to use these rules effectively, you need to have your own timing device. Most test centers don't have digital clocks, and even if you're good at reading analog clocks, there's no guarantee that the clock on the wall is going to to be in sync with the proctor's timing device, which will probably be his phone.

Finally, remember that these timing rules are designed to make the test less stressful for you. If you forget your watch or find it inconvenient to check the time, you can still do very well if you've practiced a lot and know that you can work quickly enough.

June 27, 2020

SAT: The Best Prep Books

Update: I've revised the Errata section of John Chung's SAT Math book.

This is a current list of the books I'm using with my own students. I'll revise it as new books are released.


This new edition eight official practice tests with answer explanations, though the College Board is no longer recommending that students take tests 2 and 4. The 2020 edition of this book has two replacement tests that are already available online.

Pros
Official material is a true confidence builder. Every question you get wrong contains skills you need to practice.

Most prep books have poorly written questions, answer key errors, and questions that are unrealistically easy, difficult, or off-topic. If you get questions wrong or run out of time on unofficial tests, you'll have trouble figuring out whether the fault lies with you or with the book you're using.

Not all unofficial prep guides have a good feel for the SAT yet, so it's doubly important to practice with official material.

Cons
The practice tests and answer explanations are already available for free online. However, it is cheaper to buy a $16 book than to print 1280 pages. Even with a laser printer at two cents a page, you'd have to pay $25.60 to print the book out yourself.

The book itself doesn't contain the conversion tables that you need to figure out your scaled (out of 1600) scores. You can download those online by visiting the SAT practice test page and clicking on the Scoring Chart link underneath each test.

The first four tests in this book were released as practice material, and the raw/scaled score conversion tables may not be accurate. (The College Board is now recommending that students skip Tests 2 and 4.) Tests 5-8 were taken by real students in 2016 and 2017.


SAT Prep Black Book, Second Edition

If you don't have a tutor, this is the book to start with. Barrett's answer explanations are very detailed, and he uses only official SAT practice tests.

You have to use College Board practice tests 1-4 from Official SAT Study Guide, since Barrett's book has answer explanations for those tests but doesn't include the questions themselves. Fortunately, you can also download the tests online for free..

Pros
A top tutor can explain any officially released ACT question to you in as much detail as you want. Barrett's answer explanations are almost as good. At $27, it's a lot more affordable than hiring a real tutor.

His book really shines in its strategy suggestions for the Critical Reading section, where the right approach to the test is more important than reviewing content.

The answer explanations in this book are more organized and easy to read than the ones from the (older) ACT Prep Black Book. Unlike his ACT book, in which he skips some problems but explains others, Barrett includes explanations for every single SAT practice problem.

Cons
Since Barrett chooses to use only official SAT practice questions, he doesn't include any practice questions in the content review chapters. You have to read the entire book and then take an official practice test. For this reason, I suggest treating your first few tests as untimed practice.

Barrett's content review for the Math section is limited, and he emphasizes the use of guessing strategies and calculator tricks over doing problems the "right" way. If you're shooting for a perfect score on the Math section, you really need to know both calculator shortcuts and the "correct" methods in order to decide which approach is the fastest. His English grammar content review is limited and written in a confusing way.


IvyGlobal's New SAT Guide

IvyGlobal's SAT 6 Practice Tests

IvyGlobal's 3 New PSAT Practice Tests

Answer Explanations for IvyGlobal's Practice Tests

These are the best unofficial SAT practice tests I've seen. Even better, IvyGlobal's Web site has detailed answer explanations that are much more helpful that the ones the College Board provides for its tests. I've been impressed enough to include two IvyGlobal tests alongside the College Board tests on my SAT practice test page.

My evaluation is based on four practice tests in a 2015 edition IvyGlobal book. One of my students took all four Critical Reading sections and one of the Writing/Grammar sections and checked IvyGlobal's answer explanations for the questions he got wrong. He made a list of the questions he still couldn't figure out and brought them to me. After spending an hour going through the list with him, I couldn't find any problems that felt ambiguous or inauthentic. Overall, I've been impressed with both the tests and my student's performance on them.

(Update: He ended up getting a 1520 on the real SAT, largely due to practicing with Ivy Global's tests.)

I've taken Ivy Global online practice test #1 on a timed basis and was impressed. The questions are tricky without being unfair, and the answer explanations are concise but accurate.

Pros
IvyGlobal continually revises its books to correct mistakes. It publishes the corrections alongside its answer explanations online for students who don't own the revised books.

Ivy Global's Web site automatically scores practice tests and generates score reports. Select the test you want to score from Ivy Global's menu and enter the answers from your bubble sheet into the online interface.

Cons
It would have been convenient for the answer explanations to have been published inside the books instead of online. IvyGlobal's Amazon book reviews reveal that not everyone is aware that the answer explanations exist.

Errata
Ivy Global's "Six Practice Tests book" doesn't give you enough information to definitively solve #38 on page 300 (practice test #3, section 4). The third column of the table isn't labeled clearly enough.

#41 on page 476 (practice test #6, section 2) can't be solved unless you replace the phrase "no dollar signs" with "numerals only."

#20 on page 483 (practice test #6, section 3) can't be solved unless you assume that segment CG is perpendicular to segment AE.


Marks Prep: Four Realistic SAT Practice Tests
Marks Prep: Four Realistic SAT Practice Tests

These four high-quality practice tests are about as good as Ivy Global's (see the link above) and also come with helpful answer explanations at the back of the book.

I've had several students work through these tests with no issues.

Pros
The answer explanations for the tests are at the back of the book (unlike Ivy Global's, which you have to find online).

Cons
Unlike Ivy Global, Marks Prep doesn't offer online scoring. It can also be a bit of a pain to score and review the tests, as the four tests begin on page 7, the scoring instructions are on page 228, the answer keys are on pages 229-237, and the answer explanations start on page 237. You'll be flipping back and forth a lot.

There's also no bubble sheet in the book, so you'll have to find and print out your own.


Khan Academy's SAT program

Khan Academy provides an online SAT practice program that's great if you don't have time to buy and read prep books.

Pros
The practice tests are the same as the ones in the May edition of The Official SAT Study Guide, but you can print them from Khan Academy for free without having to wait for the book to arrive. Note that Khan Academy doesn't have links to the answer explanations. You can find the links at my SAT practice test page.

I consider the questions in the Web site's drills to be semi-official. They've been approved by the College Board but haven't been tested on students on actual SATs. That may change in the future if the College Board decides to add some of its real SAT questions to Khan Academy's practice pool.

Cons
Khan Academy provides good SAT practice, but its answer explanations aren't always helpful. SAT Math problems always have more than one solution, and the fastest solution takes 30 seconds or less. Khan Academy's explanations only include one solution per problem, and it may not be the one that is fastest or easiest for everyone.

Its answer explanations for the SAT's Critical Reading section can be particularly incomplete and confusing. To be fair, I've looked at most of the prep books for the SAT, and all of them have this problem to some extent.

If you miss questions, Khan Academy doesn't offer related questions for targeted practice. Khan Academy's normal math courses do do this, so I hope these features are added to its SAT program in the next few years.



Critical Reading is the hardest section to write good practice questions for. Erica does a very good job of emulating the trickiness of the real test. 

She includes great strategies for managing time, skimming, taking notes, and answering tricky questions.

Pros
Erica's practice questions and strategies are several levels better than anything else that's out there. She has detailed strategy descriptions and plenty of practice material.

The Suggested Reading list has a good sweep of different types of writing, from financial journalism to historical United States documents.

Erica's Web site offers several reading quizzes.

Cons
This is a big one: Critical Reading questions are very tricky, and Erica's answer explanations aren't long enough to fully explain why each question has three objectively wrong answer choices and one objectively correct choice.

Supporting Evidence questions on the SAT are hard, and some of them seem to have two correct answers unless you read very carefully. She mentions this in Chapter 5, but there's not enough information there for most people to apply her strategies to the toughest one or two questions on the test. To fill the gap, I've written detailed instructions for conquering Supporting Evidence questions.




IES SAT Reading: World Literature (Advanced Practice Series)

IES SAT Reading: Vice and Virtue in the Exploration of Democracy (Advanced Practice Series)

I used to use AP English Language and Literature tests to over-prep students for SAT Reading - at least until I ran across these two IES books.

Since each book focuses on a specific passage type (either Literature/Fiction or History), you can choose the one that best matches what you need to work on.

The SAT is designed to reward the use of background knowledge from AP Euro, AP U.S. History, and AP English Language/Literature classes. (Did you know that Abraham Lincoln was not an abolitionist and that early feminism resembles modern Christianity more than it does modern feminism?) Keep that in mind as you work through these books, especially when you read history passages.

Pros
The practice tests are very accurate in their passage selections and practice questions. The answer explanations are detailed enough that most students haven't needed my help when reviewing their work.

Once you graduate from these IES books, you should be able to move on to AP English Language practice tests if you still need to over-prep. The AP Lang released exams don't have any answer explanations, so I strongly suggest either doing both IES books first or getting a tutor to help you.

Cons
These books are more about practice than strategy, so I suggest starting with the SAT Black Book unless you've taken AP English.

The science-based passage at the end of the Reading section can also be challenging, but IES hasn't produced a dedicated book to help students with that yet.

Errata
#7 on page 105 of the World Literature book has four answer choices that are all incorrect. The answer key indicates that (D) is the answer, but the passage doesn't provide evidence that the author progressively withdrew from social events during the holidays.




This two-book set is similar to Erica's ACT English book in its comprehensive grammar rules, practice questions, and answer explanations. Unlike the ACT version, the SAT version has a second volume called the Workbook that contains six extra practice tests.

Pros
If you understand every grammar rule Erica teaches, SAT Writing questions become objective, and you can tackle them like math questions. This doesn't mean they're easy, but it does mean that you can figure out why the wrong choices are actually wrong.

Since the first volume has a practice test and the Workbook has six, you get a total of seven practice tests with answer explanations.

Erica's Web site has a complete list of grammar rules and and reading/grammar quizzes.

Cons
Of the six tests in the Workbook, only tests 1, 3, 5, and 6 are useful for timed practice. Tests 2 and 4 contain a few confusing questions and answer key errors. It only takes one confusing question to throw your timing off for the whole test, so it's better to use tests 2 and 4 for light practice.

Erica's strategies are great for all of the question types except those that involve adding and deleting sentences. For those, read the Relevance and Purpose article on the College Panda's blog.

Her books are like textbooks (long and possibly boring). Decide now that you're going to be dedicated enough to read everything, including the answer explanations for the practice questions.



This unusual book addresses vocabulary skills that can help with both the Critical Reading and Grammar sections.

Chapter 5 is devoted entirely to the SAT Essay and includes two student essays that earned perfect scores.

Pros
Instead of drilling vocab words using flash cards, Erica groups words by their function in the English language and provides practice questions for those functions. This is a practical approach that will pay dividends in college later: words like hypothesis, tentative, and analogous often occur in science-related passages, so it makes sense to group them together.

Because Erica's focus is on practical reading, her book is equally helpful for ACT English and Reading passages. The section on passage-based vocabulary for science passages (pages 31-35) will even help you on the ACT's Science section.

Cons
If you already score higher than 700 on SAT Verbal and 30 on ACT English/Reading/Science, this book might be too easy. You'd benefit more from taking practice tests and reading 10th-to-12th grade level books to build an advanced vocabulary.


A Guide to SAT Math (Richard Corn)

This is the closest thing to an SAT Math textbook I've seen. It organizes content review and practice drills by topic. If you don't feel comfortable with high school math, start your prep with this book.

Students who get A's in school math often struggle with the SAT. A school test focuses on one chapter of your book at a time, and a good teacher tells you exactly what's going to be on that test. The SAT, on the other hand, tests knowledge that ranges from 7th grade to trigonometry and includes Common Core material that not all students have seen yet.

Disclaimer: I received a free copy of this book from the author. I get free copies of books on a regular basis, but this one is unique enough to earn a spot on my SAT book review page.

Pros
Richard Corn's book is enough like a textbook to get you comfortable with the topics that are tested on the SAT, but it's not long enough to be truly intimidating.

Cons
The main strength of Corn's book, its textbook-style organization, is also its weakness. The real SAT won't tell you whether a particular problem is testing the standard-form equation of a circle, the area of a circle, or right triangles within the unit circle. It could potentially test all three topics at the same time!

To truly be ready for the SAT, you have to learn how to think on your feet. Corn's book is great if you need content review as a primer, but you'll want to graduate quickly to more advanced practice materials.

This is the first edition of Corn's SAT Math book, so you may notice some typos and ambiguously written questions.


28 SAT Math Lessons - ADVANCED Course: For Students Currently Scoring Above 600 and Want to Score 800
Each of these books is a strategy guide combined with a large set of problems and answer explanations.

Pros
The practice material is very similar to real SAT Math tests.

Dr. Warner presents strategy, followed by practice problems, followed by more strategy. The cycle repeats with increasingly difficult problems.

If you do one lesson a day, you can finish each book in a month, although many people will need more time than that.

The answer explanations provide more than one way to do each problem, and the fastest method is marked with a star.

Cons
You can use this book without a tutor, but the problems near the end of the advanced book get so hard that you might throw your pencil at the wall and give up. Dr. Warner is trying to help you get an 800 on the Math section, so you'll be seriously challenged.



If you absolutely must get a perfect score in SAT Math, you'll want Dr. Chung's 626-page tome for its extremely hard practice problems. Be warned: this is not a book for wimps.

Note that Dr. Chung's book is great for practicing difficult algebra and geometry problems but that it doesn't have nearly as many word problems as the SAT itself does. If word problems are your weakness, you're better off practicing with Dr. Warner's books and Ivy Global's practice tests.

The most recent version of this book is the fourth edition, which is basically the third edition with different page numbers and an additional practice test. (The page numbers are different because the fourth edition uses less empty space in between its answer explanations. The questions are identical; even the errors in the Third and Fourth editions are the same.)

The answers to Tips 1-60 start on page 113.

Pros
If you want a perfect score, I'd normally suggest doing a trillion official practice tests. The SAT doesn't have that many, so the next best option is to practice with questions that are difficult while still being accurate in other ways.

Every hard problem on the SAT has an easy solution that take 30 seconds or less; the difficulty involves finding that solution quickly. What you want is a set of accurate problems that looks like SAT Math but has twice as many hard problems and almost none of the easy ones. That way, you can challenge yourself without distorting the timing of the test too much.

Dr. Chung's problems are hard, but they're still doable within the 30-second limit.

There are other books that are unhelpfully difficult: you can get Barron's New SAT if you want problems that are time-soakers or that have to be done by plugging twenty different numbers into a calculator. In case you're wondering, that's not an approach I recommend.

Cons
Dr. Chung's book will frustrate you unless you already understand everything in Dr. Warner's advanced course. The answer explanations tend to skip several steps in a row (exactly the sort of thing your math teacher tells you not to do), and he doesn't always approach problems in the fastest or most intuitive way possible. The explanations also contain numerous typos. Use his book for the excellent practice problems and ignore the answer explanations.

As mentioned earlier, Dr. Chung's book doesn't have as many word problems as the SAT does. If you're better at math than at reading, you may find that your scores on his practice tests are higher than they would be on real College Board exam.

SAT math diagrams are always drawn to scale unless the test indicates otherwise, but the drawings in Dr. Chung's book are generally intentionally distorted. In addition, the SAT always states its assumptions (segment AB is the diameter of a circle, lines CD and EF are perpendicular, lines GH and IF are parallel), but Dr. Chung sometimes doesn't do this. Don't get in the habit of making assumptions that are not explicitly stated in the problem, as doing so can hurt you on the real SAT.

I suggest starting with an official practice test, then Dr. Warner's books, followed by at least one additional official practice test. If you score 750 or higher and want to get to 800, then (and only then) should you use Dr. Chung's encyclopedic volume.

Errata
Page 22: Tip #8, problem #4 has two answers (B and C). To fix the problem, assume that a ≠ 1 and that answer choice B is not an option.

Page 30: Tip #13, problem #1 should state that points P and R are the roots of the parabola. If you don't make that assumption, the problem is unsolvable.

Page 53: Tip #27, problem #1 should state that you are solving for k.

Page 107: Tip #58, problem #1 asks for the correlation coefficient, which is not actually tested on the SAT.

Page 272: Practice Test 4, section 4, problem #22: All four answers are incorrect.

Page 490: Practice Test 12, section 4, problem #35: The answer is 91, not 39.

Page 491: Practice Test 12. section 4, problem #38: The answer is 21.9, not 21.

Page 519: Practice Test 13, section 4, problem #31: The answer is 16, not 6. (The book's answer explanation mistakenly multiplies the trapezoid bases instead of adding them.)

Page 564: Practice Test 15, section 3, problem #15: The problem is un-solvable without a clearer definition of what "does not support the candidate" means. (Does it include both the "Against" and "No Opinion" responses?)

Page 594: Practice Test 16, section 3, problem #20: The answer is 41°, not 39°.

Page 605: Practice Test 16, section 4, problem #30: In the last sentence, replace the phrase "number of algebra workbooks" with "price of each algebra workbook." If you don't do this, all the answer choices will be wrong.

Books to Avoid

I strongly suggest staying away from Peterson's Master the New SAT 2016. Two of students came to me after taking practice tests from this book. There were so many poorly written questions and typos that the students' scores went up 100 points in the Math section alone after I adjusted the answer key.

Peterson's has a 2018 edition now, but you may want to use the excellent free practice tests provided by the College Board and Ivy Global rather than risk your sanity on a new edition of a poorly written book.

Working with Official Practice Tests

If you find unofficial questions to be inaccurate or confusing, you can still prep for the SAT using only official practice tests:
  1. Take a practice test under timed conditions.
  2. Score your test and clearly mark the questions you missed. Use a different symbol to mark the questions that you got right by guessing.
  3. Review the questions you missed or had to guesss on. Spend at least ten minutes on each one. You have to prove to yourself that each question objectively has one right answer and three incorrect ones.
  4. Make a list of the questions you're not able to figure out on your own. Look up the questions on the Internet or use a tutor's help to get to the point where you can clearly identify one correct answer and three incorrect ones for every single question. Your job isn't done until you can do this.
  5. Repeat the process using a new practice test. Keep the cycle going until you're happy with your scores.
This method can work really well if you have a tutor. Do a practice test and hire a tutor to go over some of the answers with you, then spend time at home going over the remaining answers on your own. In your next session, you can go over any questions that are still confusing. When you and your tutor are satisfied that you fully understand the first practice test, repeat the process with a new test.

Going for a Perfect Score

Getting a 1600 on the SAT isn't easy.

On the ACT, you can get two 35's and two 36's, and the four scores will round to a composite of 36. The SAT doesn't offer that luxury: the section scores are added, not averaged. The SAT's Critical Reading passages are tough, and the Math sections contain a large proportion of word problems. You also face the challenge of finding enough practice tests to search for and eliminate the careless mistakes we all have a tendency to make.

You'll really have to go above and beyond in your understanding of the test. Work on improving your vocabulary. Read college-level books for practice and, preferably, for fun. Go through Dr. Chung's SAT Math practice. If you really want to challenge yourself, prep for AP English Language and SAT Literature; SAT Critical Reading will seem like child's play in comparison.

Most of all, go easy on yourself if you don't make it. Colleges want students who will bring them glory, and the difference between a 1550 and a 1600 doesn't say much about that potential in the long run.