Building the Perfect Portfolio: An Introduction to Empirical Asset Pricing and Hedge Fund Strategies
Introduction
My reasons for creating this series are threefold. Firstly, I genuinely enjoy thinking and writing about these topics. Secondly, I want to share my knowledge with others. Finally––and somewhat shamelessly––I am hoping to extend my reach as I embark on my upcoming job search. But before we dive in, here’s a little bit about myself.
I’m Aidan Vyas, a senior at Rice University studying Economics and Computer Science,\footnote{At the time of writing.} with a deep passion for empirical asset pricing. I have previously completed research on global macroeconomic investing, individual stock selection, and am currently working on understanding how prices and fundamentals affect stock returns in the long run.
This series aims to provide a suitable introduction to the economics of investing. We’ll start by establishing some common ground, walking through the relevant academic literature and industry best practices, and finally, conclude with my own thoughts and research.
Please feel to reach out on LinkedIn, X (neé Twitter), or via email at [email protected] if you have any questions, comments, or simply want to chat!
For a better viewing experience and the most up to date version of this series, please check out my website: https://aidanvyas.github.io/index.html.
The Assets
Principally, investors allocate their money between the risk-free asset and one of many risky assets––e.g., stocks, bonds, commodities, and various derivatives.
Common stocks, which represent an ownership stake in a publicly held company, are perhaps the most well-known group of risky assets. For example, owning 15 shares of Apple stock roughly equates to controlling 1 billionth of Apple.
As business owners, stockholders are entitled to a portion of the company’s profits (in the form of dividends) and can vote on key corporate matters (such as electing the board of directors, approving mergers and acquisitions, and setting executive compensation).
The value of the underlying company can be thought of as the sum of future cash flows discounted back to their present value, after accounting for outstanding debt obligations and current cash balances. Because there is uncertainty in future cash flows and money tomorrow is worth less than money today, the two main risks born by equity investors are cash flow risk and interest rate risk––i.e., will the company make as much as the market expects and how much less will those future cash flows be worth today?
Equity indices seek to diversify away idiosyncratic (firm-specific) risk by investing in a group of stocks––leaving only the systemic (market-wide) risk. Indices can cover specific industries, countries, or even the entire global equity market. Some common examples include the S&P 500 and the Dow Jones Industrial Average, but it is important to note that indices are not directly investable, as they simply just track a basket of stocks.
Exchange-traded funds (ETFs), on the other hand, are investment vehicles that pool money from multiple investors and are designed to track the performance of a specific index. ETFs trade on exchanges like individual stocks, so they can be bought and sold easily, and these instruments are often used to gain broad-based exposure to equity markets.
Beyond equities, investors can also purchase bonds, which are essentially loans to governments and businesses that need money to finance budget deficits or to invest in new projects. Bonds often pay periodic interest payments (known as the yield) and return the principal (the initial investment) at maturity (the bond’s expiration date).
For example, let’s imagine that you buy a $1,000 30-year Treasury bond yielding 5%. In this case, you give Uncle Sam $1,000 today, and in exchange, every year for the next 30 years, you receive $50 (that’s your 5%), and after 30 years, you get your original $1,000 back. Because governments and businesses might face bankruptcy and interest rates may dramatically change, fixed-income investors assume default risk and interest rate risk––i.e., will you get your money back and will it be worth as much today?
The major types of bonds include government bonds, corporate bonds, and convertible bonds. Government bonds are considered the safest of the bunch, but they can differ drastically in terms of issuer (which country sells them) and maturity (when they’ll return your money).
Corporate bonds––often issued by public companies to open a new factory, acquire a competitor, or invest in research and development––have higher yields than government bonds due to their higher default risk.
Moreover, convertible bonds combine characteristics of common stock and government bonds. They typically have a lower interest rate than typical corporate bonds and are more exposed to default risk as a result of their lower seniority, but they provide the option of being converted into a predetermined amount of common stock.
Apart from stocks and bonds, you can also invest in physical commodities like gold, oil, and wheat. However, given that the transportation and storage of large amounts of commodities presents a challenge, futures contracts are employed.
Simply put, a futures contract is an agreement to buy or sell a specific commodity in the future at a price determined today. At the settlement date (when the contract expires) either the physical commodity is delivered, or more commonly, the contract is settled in cash, where the difference between the previously agreed-upon price and the current market price is exchanged.
Futures contracts are mainly used by hedgers and speculators. Hedgers are typically businesses (think McDonald’s) that need to guarantee a specific price in the future (let’s say beef for a Big Mac). While speculators seek to profit on the movement in commodity prices.
Futures contracts are our first foray into the world of derivatives––financial instruments that derive their value from an underlying asset. Broadly, they all share the same hedger and speculator dynamic, where hedgers have a legitimate economic interest in locking in prices, while the speculators are looking to profit from price movements. Beyond futures contracts, other common types of derivatives include currency forwards, swaps, and options.
Currency forwards are agreements to exchange a specific amount of one currency for another at a future date and at a predetermined exchange rate. They are often employed by multinational corporations to hedge against changes in exchange rates.
An interest rate swap, as its name suggests, allows two parties to swap their interest rate payments. One party pays a fixed-rate (the current rate), while the other pays a floating-rate (based on the future rate). For instance, companies will often use interest rate swaps to convert their floating-rate debt into fixed-rate debt.
Credit default swaps (CDS) are a type of insurance contract that pays out in the event of a default on a bond or loan. You can think of them as fire insurance on your house. If your house burns down, or a company goes bankrupt, you get paid out. And the price of these contracts rise with the perceived default risk of the underlying assets.
Options are another important type of derivative that gives the holder the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a predetermined price (strike price) within a specific time frame (expiration date). After selling Broadcast.com to Yahoo, Mark Cuban famously purchased put options on the Yahoo stock that he was given, and when the bubble burst, Cuban made a fortune.
Finally, volatility futures are contracts that allow investors to speculate on the future volatility for a given asset (typically the American or European equity indices). These are almost exclusively used by speculators.
If an investor chooses not to take any risk, they can always invest in the risk-free asset––typically assumed to be short-term U.S. government bonds. These assets are considered to be risk-free because they eliminate the two risks associated with government bonds: default risk (the U.S. government can always print more money to pay off its debts) and interest rate risk (short-term bonds are relatively insensitive to changes in interest rates
And the return on said asset is the risk-free rate. You can think of this as the hurdle that all other risky assets ought to surpass. If you’re interested in seeing how the risk-free rate has changed over time, please check out the Ken French Data Library (French, 2024).
The assets that we’ve explored up to this point are publicly traded and relatively liquid. Their readily available prices make them ripe for empirical analysis and systematic investment strategies. However, there exists another realm of investments: private assets. These are characterized by their illiquid and idiosyncratic nature, setting them apart from their public counterparts. While real estate is the most familiar (and biggest) example, this group also includes venture capital, private equity, private credit, and even extends to more unconventional investments like fine art, rare wines, and professional sports teams.
The Jargon
The basic gist of the Modern Portfolio Theory (MPT) is that investors ought to maximize their returns per unit of risk––not just their total returns––so they get the most bang for their buck (Markowitz, 1952).
There are many different forms of risk, some of which were covered in the previous section, but for the purposes of the MPT, we focus on a measure of statistical risk––volatility (formally the standard deviation of an asset’s excess returns). Conceptually, volatility is a measure of how much an asset’s returns fluctuate over time. The higher the volatility, the less predictable the asset’s return, and the more risk it carries.
As previously stated, the risk-free rate is the interest rate on short-term government bonds. It’s how much an investor can receive if they don’t take any risk by lending to the U.S. government, and in academia, it is commonly presumed to be the cost of borrowing.
The total return of an asset is simply how much its price (in percentage terms) went up or down after adjusting for its yield (i.e., payments dispersed to the owner––e.g., dividends for stocks, interest payments for bonds, and rental income for real estate).
In asset pricing, we prefer to look at the excess return of an asset, which is defined as the total return minus the risk-free rate, as it allows us to compare "zero-dollar" portfolios. Essentially, an investor with zero dollars can borrow money at the risk-free rate to buy some asset. The value of their portfolio would simply equal the total return of said asset minus the risk-free rate.
Under this construction, returns become a lot less important because one could theoretically borrow more money (thereby taking on more risk) to achieve a higher return. This fact perfectly transitions us to our next piece of jargon––the Sharpe ratio––which is the excess return of an asset divided by its volatility (Sharpe, 1965). This equation essentially boils down to return over risk or the very thing that the MPT seeks to maximize.
Once an investor has determined the portfolio with the highest Sharpe ratio, the Modern Portfolio Theory dictates that they should simply hold the portfolio––leveraging it up (via borrowing money) or down (by holding more of the risk-free asset) to meet their individual risk preferences. The rest of this paper attempts to find said portfolio.
In practice, the MPT recommends that investors hold a group of uncorrelated assets with positive expected returns. Correlation refers to how much and in what direction two assets move in relation to each other. If two assets are positively (negatively) correlated, then they are likely to go up or down in the same (opposite) direction. Moreover, if they are uncorrelated, then the movement of one asset doesn’t affect the movement of the other.
Two assets are said to be perfectly correlated if their returns can be completely explained by a positive linear function. For instance, if the returns of asset X are always equal to, double, or one-half of asset Y, then asset X and Y are perfectly correlated. Likewise, two assets are perfectly negatively correlated if their returns can be completely explained by a negative linear function, and two assets are perfectly uncorrelated if a linear function explains none of the relationship between the returns of the two assets.
Perfect positive (negative) correlations are represented by a correlation coefficient of 1 (-1), while perfectly uncorrelated assets are represented by a coefficient of 0. Between 0 and 1 (-1), increases (decreases) in the correlation coefficient represent a stronger positive (negative) linear relationship.
Correlation should not be confused with beta, which represents an asset’s sensitivity to the market portfolio (or any portfolio for that matter). Beta is measured as the coefficient in a linear regression of the asset’s excess returns on the portfolio’s excess returns. If a stock has a market beta of 1, then it is expected to match the return of the market, but if it has a market beta of 0.5 or 2, then it is expected return one-half or double the market return.
Alpha, the intercept of that very same linear regression, is the 'holy grail' of finance, and it refers to returns that cannot be explained by the market or other controls. It is important to note that while there is only one alpha, there can be multiple beta (e.g., a stock market beta, a bond market beta, and a gold beta), whereby each beta represents the coefficient on each control portfolio.
The final pieces of jargon are skew and kurtosis. Skewness informs us where the outliers are. If a distribution has negative skew, then it has a longer left tail (i.e., more extreme negative returns), while if a distribution has positive skew, then it has a longer right tail and more extreme positive returns.
Kurtosis refers to how observations cluster around the mean. A distribution is said to have positive excess kurtosis, or be leptokurtic, if there are more outliers relative to a normal distribution. In the opposing case, distributions with negative excess kurtosis are called platykurtic and have fewer outliers relative to a normal distribution. The specific calculations for both skew and kurtosis are relatively involved and will generally be handled by common statistical packages.
Apart from the aforementioned terms, other crucial asset pricing mathematics includes various regressions––simple linear regressions, Fama-MacBeth regressions, panel regressions, and spanning tests––in addition to the basic t-statistics, p-values, and R^2 (Fama & MacBeth, 1973). Moreover, the Black-Scholes model is a crucial tool for pricing options (Black & Scholes, 1973). However, these exact derivations are beyond the scope of the main text.
The Players
Broadly and crudely, there are two types of investors––individuals and institutions––and this distinction leaves no shortage of confusion in the financial industry. Individual investors are mainly focused on maximizing their real returns––the total return minus the rate of inflation. While institutional investors should be concerned with their Sharpe ratios.
The reason being is that institutional investors have the ability to employ leverage and participate in short sales. The Modern Portfolio Theory assumes that all investors have access to leverage, but this is only true for institutional investors. For example, a strategy that returns 2% annually in excess returns with 1% volatility would be an excellent investment for institutional investors. After all, it has a Sharpe ratio of 2 (commonly thought to be the ceiling for strategies at scale) and could be easily leveraged up to meet the investor’s risk tolerance, but it would be relatively useless for individual investors, especially over the long run.
And shorting is simply making a bet that an asset will fall in price. In practice, you borrow an asset from another market participant, sell it immediately, and then buy and return the asset at a later date (hopefully at a lower price). The return of any short sale is simply the final price divided by the initial price.
Moreover, there are also differences in which products and markets each type of investor has access to. Individual investors trade common stocks and ETFs over exchanges like the NYSE or NASDAQ, while institutional investors have the ability to trade any asset––including those traded over-the-counter (OTC) like corporate bonds or currencies.
Most of the academic literature focuses on institutional investors given their outsized influence (both in terms of the money they manage and the amount of trades they execute), as will the vast majority of this paper. However, when we cover my own research, we will touch on the optimal equity portfolio for long-term individual investors.
Diversification
Harry Markowitz, the Nobel Prize-winning economist and the father of the Modern Portfolio Theory, once said that "diversification is the only free lunch" in investing.
To illustrate this fact, let’s take the example of n perfectly uncorrelated assets––each with the same excess return of 5% and volatility of 10%. When n = 1, the Sharpe ratio is trivially 0.5, and as n increases, the portfolio’s volatility decreases, while the portfolio’s Sharpe ratio increases, as shown by Figure 1 below.
Once you have 10 uncorrelated assets, you reach a portfolio volatility of nearly 3% and a Sharpe ratio of almost 1.6––a far cry from the 10% volatility and 0.5 Sharpe ratio with just 1 asset.
At this point, we have only considered perfectly uncorrelated assets, but in reality, assets will often have non-perfect correlations with each other. Let's explore a simple portfolio consisting of two assets held at equal weight––each with the same 5% excess return and 10% volatility––but this time with different correlations in Figure 2 below.
As the correlation between the two assets rises (falls), the volatility of the portfolio increases (decreases), while the Sharpe ratio decreases (increases). At the extrema of -0.9 and 0.9, the Sharpe ratios are 2.236 and 0.513, respectively. Even with more moderate correlations of -0.5 and 0.5, the difference in Sharpe ratios is still meaningful––at 1.000 and 0.577. Simply put, diversification is only mildly useful when the underlying assets are highly and positively correlated, but can become particularly useful in the cases where the assets are negatively correlated.
Risk Parity
Putting all of these facts together, we could summarize the investment advice provided so far as "maximize risk-adjusted returns via the power of diversification". However, we have not provided any concrete advice on what this actually looks like––after all, there is no investment store that you can walk into to purchase a group of uncorrelated assets with modest Sharpe ratios.
To remedy this situation, let’s imagine that you are an investor whose portfolio solely consists of the S&P 500––an index of approximate the 500 largest companies in the United States weighted by their market capitalization––and while this portfolio has certainly performed well over the past century, we believe that you could do better.
For a start, adding smaller American stocks would provide some––albeit minimal–– diversification benefits. Moreover, broadening beyond the American markets to hold a global portfolio of all stocks would reduce country-specific risk, thereby raising the overall Sharpe ratio of the portfolio.
Furthermore, diversifying into other asset classes––e.g., bonds and commodities––can be beneficial as well. Bonds, in particular, have a relatively low correlation to stocks, which makes them great additions to an equity portfolio. In fact, the global 60/40 portfolio––which consists of 60% stocks and 40% bonds, held at their respective market weights within each asset class––is a mainstay in the wealth management industry.
Finally, adding commodities (e.g., gold, oil, wheat, etc.) can additionally enhance a portfolio. While commodities have a lower (but still positive) excess returns, they tend to perform well in inflationary environments when stocks and bonds typically suffer, which reduces the portfolio volatility.
However, this addition begs the question of how to properly weigh commodities in a portfolio. With stocks and bonds, we can easily employ a market-weighted approach––albeit the 60/40 asset class split is arbitrary. But for commodities, we have to decide how to assign weights within the asset call, as well as between assets in the same asset class. Because we do not have market values for commodities (i.e., we typically don’t know the total value of all the cattle or oranges in the world), the typical approach taken by commodity indices involved weighting by the total annual production value or liquidity (how much of that asset was traded).
A more elegant solution to this problem is risk parity. It was initially pioneered by Ray Dalio and Bridgewater Associate’s in the 1990’s, and essentially, it aims to balance risk between asset classes regardless of the macroeconomic environment (i.e., whether growth and inflation are rising or falling) (Bridgewater Associates, 2012).
To build a simplified version of such a portfolio, one would first estimate the risk of each asset (for instance, by measuring the realized volatility over the past 36 months), and then by setting the weight of each asset in the overall portfolio to the normalized inverse of the volatilities.
This portfolio now balances the risk between asset classes, and the overall portfolio can be leveraged up or down to meet an investor’s individual risk tolerance. Not only does risk parity present a clean theoretical model for weighting different assets and asset classes, it also leads to an empirical outperformance relative to other weighting schemes. As with all strategies that provide alpha (i.e., positive performance that cannot be explained by other factors), it is crucial to determine the reason behind the outperformance and who is on the other side of the trade.
As previously mentioned, many investors are leverage constrained or leverage averse (Frazzini & Pedersen, 2014). Retail investors, for instance, face exceeding high costs to access leverage via margin loans, and many mutual funds have explicit rules against employing leverage. Given that these investors are focused on maximizing their returns, they’ll choose to invest in riskier assets (i.e., stocks as opposed to bonds) if those assets provide higher unleveraged returns. This demand mechanically increases the price of stocks relative to other asset classes, thereby decreasing their expected returns and risk-adjusted returns.
To capitalize on this mispricing, investors with access to leverage can overweight bonds (and low-risk assets in general) and underweight stocks (given their lower risk-adjusted returns) to achieve a higher Sharpe ratio and higher returns per unit of risk.
Furthermore, this simplified risk parity implementation can be improved to mitigate macroeconomic shocks, not just the risk of asset class concentration. Let’s take the example of inflation––a period in which stocks and bonds suffer, but commodities thrive. The question now becomes how do we weigh stocks, bonds, and commodities in our portfolio such that we are neutral to changes in the inflation rate.
One approach might be to calculate an inflation beta––or how sensitive each asset class is to inflation. Practically, we would regress the returns of each asset class on changes to inflation expectations, and over a sufficiently long period of time, we’ll be able to estimate the durable relationships between inflation shocks and the different asset classes.
Once we've determined the inflation beta for each asset class, we could then scale the weights of each asset class up or down such that the overall portfolio has an inflation beta of zero. While this example neutralizes the impact of inflation shocks, this approach can be generalized to eliminate growth, interest rates, volatility, and liquidity shocks.
This simplified risk parity strategy will serve as the base of our perfect portfolio. The following sections of this paper will explore how we can take advantage of core concepts from academia and industry to further maximize risk-adjusted returns.
Academia
It is generally a good strategy in life to follow the consensus of expert opinion. In fact, the entire field of management consulting exists to operationalize industry best practices.
Thankfully, there is a field of economics––empirical asset pricing––that seeks to understand the prices of returns of assets. We have already discussed the Modern Portfolio Theory, but it would be helpful to go over a brief review of the relevant advances in asset pricing since then to better understand how to construct the perfect portfolio.
The MPT states that one should maximize their risk-adjusted returns through the power of diversification. If the returns of all stocks were independently risky and on average had a positive expected return, then it would be trivial to construct a portfolio with a near-infinite Sharpe ratio.
Obviously, such a portfolio does not exist because stocks are not independently risky. There is of course some portfolio of idiosyncratic risk in each stock (i.e., whether the company’s sales are growing, whether they beat earnings, etc.), but there is also a portfolio of systemic risk, determined by whether the overall market goes up or down (typically due to changes in interest rates or other macroeconomic factors).
The nation of systemic risk was first introduced by the Capital Asset Pricing Model (CAPM), which expresses the excess return of an asset as a function of its market beta (Treynor, 1961); (Treynor, 1962); (Sharpe, 1964); (Lintner, 1965); (Mossin, 1966). In a CAPM world, all idiosyncratic risk can be diversified away by holding a market portfolio, so the only risk that you are exposed to is the systemic (or market) risk.
Closely connected to the CAPM is the Efficient Market Hypothesis (EMH), which states that asset prices reflect all available information, so consistently beating the market on a risk-adjusted basis becomes nearly impossible (Fama, 1970). Given that investors ought to maximize their risk-adjusted returns and they can’t beat the market, they will simply hold the market portfolio and leverage it up or down to meet their risk preferences. Moreover, on average, all the returns of all actively managed portfolios ought to equal the market portfolio, minus trading costs (and fees if they are paying another manager to invest on their behalf) (Sharpe, 1991).
Since the introduction of the CAPM and EMH, academics have discovered a variety of anomalies (i.e., strategies that beat the market without taking additional market risk––also known as factors). Most notably, the fact that stocks with low price-to-earnings ratios, positive earnings surprises, and low betas generate alpha relative to the CAPM (i.e., when regressing the returns of these stocks against the market, the intercept is positive) (Basu, 1977); (Ball & Brown, 1968); (Black et al., 1972).
These strategies are usually constructed as long-short portfolios, where an investor borrows a group of assets with unfavorable characteristics (e.g., stocks with high price-to-earnings ratios), sells said assets, and then invests the proceeds to buy a group of assets with favorable characteristics (e.g., stocks with low price-to-earnings ratios). At the end of a given time period, they will see the assets they bought, and then repurchase and return the assets that they initially borrowed. The value of their portfolio will simply equal the difference in returns between the two portfolios. Because these portfolios (theoretically) don’t require any initial capital, they are directly comparable to the "zero-dollar" portfolios that we discussed earlier.
There are four generally accepted explanations for the existence of market anomalies. Either they a) are compensation for non-market risk (e.g., anomalies could be profitable for taking on illiquidity risk which is consistent with the EMH), b) are a result of mispricing which directly contradicts the EMH, c) run into the limits to arbitrage (i.e., the costs to trade the long-short portfolio are greater than the underlying anomaly), or d) are a result of data mining––essentially, a random finding generated by parsing through heaves of past data that is unlikely to persist.
In order to summarize the then-current literature and provide an explanation for the anomalies of the time, Eugene Fama and Ken French created the Fama-French three-factor model, which explained the popular anomalies of the time as a function of the excess return of the market, a size factor, and a value factor (Fama & French, 1993). As time progressed and more anomalies were discovered, the three-factor model has expanded to include the profitability, investment, and momentum factors (Fama & French, 2015); (Fama & French, 2018).
These factors are long-short portfolios that represent the difference in returns between small vs. big companies, cheap vs. expensive stocks, profitable vs. less profitable companies, firms with low vs. high investment rates, and stocks that are outperforming vs. underperforming, respectively.
The returns of said portfolios are on the aforementioned Ken French Data Library, and as a quick example, let’s construct a portfolio that simply holds the five long-short factors. The portfolio would return 1.66% a month with a volatility of 6.59%, yielding an annual Sharpe ratio of 0.87 from July of 1963 to December of 2023.
These basic theories are all that are required for the purposes of this paper, but if you are interested in learning more about asset pricing, there are a variety of excellent textbooks that one could peruse, namely from Dr. Kerry Back, Dr. John Y. Campbell, Dr. John H. Cochrane, and Dr. Wayne E. Ferson (Back, 2017); (Campbell, 2018); (Cochrane, 2009); (Ferson, 2019).
Industry
While academics have certainly contributed an invaluable body of work to the field of empirical asset pricing, given the large financial windfalls reaped by the hedge fund industry, it stands to reason that there are similar innovations on the other side of the fence. In particular, practitioners have pioneered a wide variety of arbitrage-esque strategies.
Arbitrage refers to a riskless profit, often when the exact same asset is trading at two different prices in two different places––e.g., if Bob wants to sell an ounce of gold for $95 and Sally wants to buy an ounce of gold for $105, then an arbitrageur could buy from Bob and sell to Sally for a profit of $10.
Given that these scenarios are rare in actuality, the industry typically focuses on similar assets trading at different prices. Perhaps the first example of this type of strategy was introduced by Benjamin Graham and David Dodd, Columbia Business School professors and investors who proposed buying stocks trading below their net current asset value (i.e., the value of their current assets minus total liabilities, per share) in their magnum opus: Security Analysis––a textbook that serves as the intellectual foundation for value investing (Graham & Frank, 1934).
Net current asset value is tantamount to the liquidation value of a company, or what an investor could receive if they paid off the debts, pocketed the cash, and shut down the company’s operations. While this strategy is not a perfect arbitrage because management could still squander away their cash reserves, more often than not, it is a profitable trade.
Moreover, merger arbitrage––a strategy that bets on mergers going through––has long been a hallmark of the hedge fund industry. There are two types of mergers: cash mergers and stock mergers.
In cash mergers, the acquirer buys the target company’s stock for a set amount of cash per share, and when the merger is announced, the price of the target company rises close to (but not perfectly at) the acquisition price. At this point, the spread between the two prices serves as a proxy for the probability of the merger being consummated and represents the arbitrageur’s profit margin.
This strategy makes money when the merger goes through, as a small premium is collected, while it loses money when the merger fails and the target company’s stock craters to its pre-acquisition price. Because most mergers go through, merger arbitrage makes a little bit of money most of the time, and loses a lot of money some of the time.
Because of this distribution of returns, merger arbitrage has a negative skew (given the more extreme negative returns), and in particular, it does poorly when the overall stock market suffers and more mergers and acquisitions fall through. Given the prospects of large and untimely losses, why would any investor pursue this trade? Well, because, over the decades, it has proven to be reliably profitable (Mitchell & Pulvino, 2001).
Let’s for a second imagine the investor on the other side of this trade. They just experienced a large jump in the value of their holdings, and they now face the option to sell and lock in that gain, or hold on for some period of time (typically a couple of months), for a smaller additional bump with the chance of a large loss.
In a sense, when a hedge fund manager buys a stock after the merger announcement, they are providing insurance to the investors that sell their shares, whereby the investors forgo a small bump in the stock price in return for escaping the possibility of a large drop.
For a stock merger, instead of purchasing the target company with cash, the acquire converts the target company’s shares into their own. In response to this added wrinkle, the hedge fund shorts the acquirer's stock––making a bet that the two prices will converge when the merger occurs. The basic insurance principle, however, remains the same.
Moving on, convertible bond arbitrage is another strategy that is popular in the hedge fund industry. Convertible bonds are traditionally issued by companies with lower credit ratings that need quick access to capital and are incentivized by the prospect of lower interest payments. Given that these bonds are often issued at a discount to their fair value to encourage investors to provide quick liquidity, an arbitrage opportunity presents itself. Specifically, arbitrageurs can purchase the convertible bond, short the stock, and buy credit default swaps and interest rate swaps to hedge out the equity, default, and interest rate risks (Asness et al., 2010).
At the end, investors are (hopefully) left over with a clean arbitrage opportunity. However, similar to merger arbitrage, convertible bond arbitrage tends to suffer when the overall market drops. In times of market stress, investors shun less liquid assets like convertible bonds, which causes their prices to drop. Moreover, because of the leverage employed by convertible bond arbitrageurs, this drop in prices often leads prime brokerages to force these hedge funds to unwind the trades: selling the convertible bonds and causing their prices to fall even more.
Furthermore, a perhaps misleadingly named strategy is statistical arbitrage. The basis behind this strategy is that there are a variety of market participants who engage in noise trading––meaning that they are buying or selling a given asset without having an informed opinion on the future price of that asset. Some examples include investors selling to access liquidity or retail traders purchasing meme stocks.
These uninformed trades can move the market, so statistical arbitrage seeks to profit from this fact by buying stocks that have gone down recently, while sorting stocks that have gone up––making the bet that stock prices will revert to their mean. There is substantial evidence that this short-term reversal effect exists at the hourly, daily, and even the monthly level (Heston et al., 2010); (Keloharju et al., 2016); (Fama, 1965).
An important aspect of implementing this strategy is disentangling informed and uninformed trades. Crudely speaking, there are sharks and minnows in the world of investing. You don’t want to trade with sharks (e.g., the Warren Buffett’s and Ken Griffin’s of the world), but you do want to trade with the minnows. In fact, that’s why retail order flow, especially from Robinhood, is so valuable. For instance, market makers––i.e., firms that run extremely sophisticated versions of these statistical arbitrage strategies––like Citadel Securities will purchase order flow from retail brokerages, in order to take the opposite side of these trades.
Broadly speaking, statistical arbitrage strategies perform well when trading with uninformed investors and suffer when trading against informed investors. Moreover, these strategies are particularly valuable in times of market distress, where most arbitrage strategies unravel, as they provide liquidity to other investors (Nagel, 2012).
Additionally, volatility arbitrage is another hedge fund strategy that has grown increasingly popular in recent years. There are two main implementations that fall under the umbrella of volatility arbitrage. Firstly, hedge funds can sell put options, or provide insurance to the rest of the market. Other investors regularly overrate the probability of a recession or a bear market. Over the years, innumerable market commentators have made brash projections promising a downturn to no avail.
Because of this well-known behavioral bias, investors bid up the prices of put options (derivatives that pay out in the case of a stock falling in value), lowering their expected return. Selling puts, or in essence, selling insurance, is profitable, as these hedge funds provide protection from crash risk in exchange for a steady stream of income in non-turbulent times (Ang et al., 2018).
Secondly, hedge funds will often create their own forecasts of future volatility that might differ from what the market has currently priced in. For example, imagine that you anticipate the market to be more volatile than expected. To monetize this view, you would either purchase a volatility future––making a direct bet that volatility will go up––or by purchasing both a call and a put option––making a bet on large price movements in stocks, as you would get paid if they went up or down more than expected. The obvious risk of this strategy is that it’s difficult to create volatility forecasts that are more accurate than the market.
Another arbitrage-esque strategy pursued by hedge funds is ETF arbitrage, and again, there are two main implementations. The first is index reconstitution arbitrage. Many indices, like the S&P 500 and the Dow Jones Industrial Average rebalance their constituents periodically, as to properly represent the American economy. These events are always announced beforehand, and the ETFs that track these indices are required to buy the stocks that are added into the index and sell the stocks that are dropped from the index (Paleologo, 2024).
This forced buying and selling creates an inefficiency in the market that arbitrageurs can exploit. Let’s imagine that Tesla has been added to the S&P 500. By purchasing Tesla stock after the announcement and then selling it once the stock has been added to the index, you could profit from the mechanical trading rules of index ETFs. This profit, however, is not risk-free, as there is some degree of idiosyncratic risk borne by buying Tesla or any other stock.
The generalized form of ETF arbitrage is to take advantage of the fact that ETFs and their underlying securities trade independently, and there might be a shock in one market (e.g., Elon Musk selling his Tesla shares) that is not immediately reflected in the other market (e.g., a NASDAQ ETF).
Authorized participants are special actors in the marketplace that are allowed to create or redeem ETF shares at any point during the trading day in order to correct these discrepancies. Given this power, they can exchange shares of ETFs for their net asset value (NAV)––i.e., essentially, the value of the underlying assets.
For instance, if the stock price of Tesla drops significantly, and if this is not immediately reflected in the NASDAQ ETF, then arbitrageurs can purchase Tesla (and the rest of the NASDAQ), exchange them for ETF shares, and then sell those shares on the market for profit. The challenge with this form of ETF arbitrage is that it requires incredible speed and execution to implement effectively, and if you’re too slow, you might be left holding the bad.
Moving beyond trading individual stocks, there are a variety of hedge fund strategies that involve trading macro assets––like government bonds, currencies, commodities, and equity indices. One common example is fixed-income relative-value, which was made popular by Long-Term Capital Management.
Governments regularly issue bonds to finance their deficits, with the most recently issued bonds, known as on the run, typically trade at a premium due to their liquidity and status as a benchmark for various derivatives. In contrast, previously issued bonds of the same maturity, referred to as off the run, often trade at a discount relative to their on the run counterparts.
A classic fixed-income relative-value trade involves purchasing the most recent off the run bond, (usually the second most recently issued bond), while shorting the on the run bond. These bonds are essentially the same product, and once a new bond is issued, the current on the run bond loses its premier status, dropping in value to trade in line with the other off the run bonds.
The risk faced by this strategy is that the spread between the two bonds is small, often just a couple of basis points, so to make these trades worthwhile, ample leverage needs to be applied. An excessive use of leverage, coupled with improper risk controls, is what led to the downfall of the aforementioned Long-Term Capital Management.
Another macro strategy revolves around exploiting the interest rate differentials between different currencies––known as carry. A classic example entails borrowing in a low-interest rate currency (e.g., the Japanese yen) and investing in a higher interest rate currency (e.g., the Australian dollar), where the investor’s return equals the difference between interest rates.
The risk posed by this strategy is that the value of each currency can change. Higher interest rates currencies typically have a reason––namely, these countries face higher inflation which devalues their currency or their bonds might be at risk of defaulting.
In particular, the carry trade suffers in global recessions when other investors shun risky assets in favor of safe assets––the apt-named "flight to safety". This tendency makes it similar to most other arbitrage-esque strategies that also suffer when the economy at-large declines and liquidity dries up, limiting the diversifying benefits of adding a carry strategy to the portfolio––granted, the returns are still positive and consistent in good times, which far outnumber the bad times (Koijen et al., 2018).
Trend following, on the other hand, is a strategy that performs exceptionally well in most market crashes (Hurst et al., 2017). Practically, a trend follower believes that other investors systematically underreact to new information, so to profit off of this behavioral bias, they simply buy assets that are going up in the time-series (i.e., assets that have gone up or down on an absolute basis) and short their counterparts.
The returns from a trend following strategy tend to dry up in high-correlation environments––or when the number of distinct trends to bet on dwindles. Overall, trend following has proven to be remarkably consistent across more than a century in time. Carry traders and trend followers are considered to be commodity traded advisors (CTAs), because they frequently trade commodities and other futures contracts.
Global macro is the final type of hedge fund strategy that trades these broad macro assets. Systematic implementations include making directional bets on equity indices, government bonds, commodities, currencies, and interest rate swaps based on the relationships between the macroeconomic variables––e.g., GDP growth, CPI growth, interest rates, and exchange rates––and the relevant asset classess (Brooks et al., 2024).
George Soros’ bet against the British pound is the quintessential example of a global macro trade. In the months leading up to Black Wednesday, as economic conditions in Great Britain deteriorated relative to the rest of Europe, Soros built up his large short position that ultimately paid off handsomely when the pound catered. And similar to volatility arbitrage, the risk faced by global macro investors is simply the challenge of beating the market.
And finally, the last hedge fund strategy that we’ll cover is implementing the academic research that has already been published: equity market neutral. Beyond the basic Fama-French factors, there are over 200 known predictors of returns that have passed peer-review journals (Chen & Zimmermann, 2022).
The vast majority of these relate to individual stock selection, while others focus on corporate bonds and macro market. The specific reasons why these strategies outperform vary, but collectively they can be thought of as some combination of bearing a variety of risks and identifying a multitude of mispricings.
Conclusion
So far, this paper has laid out the merits of a risk parity portfolio as well as various academic factors and arbitrages. These additional strategies are largely long-short and "zero-dollar"––i.e., they are virtually uncorrelated from our base portfolio and their returns can simply be stacked on top of each other (Israel et al., 2013).
A portfolio that holds all of the aforementioned strategies––risk parity, the Fama-French factors, the various arbitrages and macro trades––would surely greatly expand the efficient frontier relative to what most investors hold currently, but it is tantamount to a simple literature review of best practices in the asset management industry.
To add further alpha, we should look beyond these traditional strategies. For instance, Lauren Cohen has shown that there are a variety of individuals who are sophisticated actors––namely, insiders, other investors, analysts, and politicians––and that copying their trades is often profitable (Cohen et al., 2012); (Cohen & Frazzini, 2008); (Cohen et al., 2010); (Cohen et al., 2011). Moreover, ranking these actors by their connection to the company and its management––via geographical, educational, and nomenclatural ties––could be an effective method of processing these unique signals into a single return forecast.
Additionally, while data mining is a common concern in academia, Andrew Chen has embarked on an intriguing endeavor to show that datamined strategies actually perform as well as published findings, out-of-sample (Chen & Dim, 2024). This innovation greatly increases the number of anomalies that exist, from hundreds to thousands, which additionally helps with diversification. This same principle could be applied in other markets––e.g., the corporate bond market. Furthermore, machine learning has been recently on the forefront of asset pricing, and if you are interested in learning more, Bryan Kelly has a detailed review of the relevant literature (Kelly & Xiu, 2023).
In conclusion, the perfect portfolio includes risk parity, plus the exploitations of various anomalies and arbitrages. The base model can be extended to encompass the cutting-edge of asset pricing, namely strategies built on less traditional data sources and the use of rigorous statistics and machine learning techniques.