# What is the intuition of a spread portfolio and how exactly is it constructed?

In a lot of papers spread portfolios are constructed, like in Harvey and Siddique (1999), Table IV, or in Fama and French (2005 from SSRN), page 15.

First, why is it important to construct such portfolios and what's their meaning in an analysis. Second, how exactly are they constructed? (is there some bibliography, somewhere?) Say I have a total of 20 stocks:

• I create one portfolio (P-high) with the highest value 10 stocks and calculate it's daily return; then I do the same for the rest of the stocks creating the P-low portfolio.
• The spread portfolio's return is then calculated subtracting the return of the P-low from P-high?

## 2 Answers

### Does variable $x$ forecast returns?

Let's say you have some variable $x$ that you think forecasts returns, and you want to conduct statistical tests of a null hypothesis that $x$ has nothing to do with expected returns.

# Why a long-short portfolio? (quick answer)

### 1. It gives you a reasonable shot at detecting an effect.

Imagine you have a stereo system and you want to see if dial $x$ does anything. You could turn the dial all the way to the right and compare sound with the dial turned all the way to the left.

Similarly, comparing returns of a portfolio of firms with high $x$ with returns of a portfolio of firms with low $x$ gives you a reasonable shot at detecting whether $x$ matters.

### 2. It's simple and intuitive.

Sorting into portfolios on some signal is classic, low-brow statistics, 1980s style finance.I'm not saying this to be derogatory. There's much to be said for simple, robust techniques done well.

With classic techniques, many quantitative finance types can instantly recognize and understand the broad outlines of what you're doing.

### 3. Forming portfolios naturally corrects test-statistics for cross-sectional correlation.

If you worked at the level of individual companies, your statistical tests need to be robust to the immense cross-sectional correlation in returns. Firms in the same industry will go up and down together. Firms with similar accounting characteristics go up and down together.

While a reasonably skilled statistician can handle this properly by clustering standard errors by time, a long-short portfolio is much easier to explain to those less skilled in statistics.

# Recipe overview

A standard thing to do since at least the 1980s is to form portfolios on signal $x_t$ and test if portfolios with high $x$ have statistically different returns than portfolios with low $x$.

1. Sort firms $i=1, \ldots, n$ into $k$ portfolios at time $t$ based upon which quantile $x_{it}$ falls into at time $t$.
• Eg. if $k=5$, companies in the bottom quintile of $x$ (i.e. lowest 20% of firms) would go portfolio 1 and companies in the top quintile (i.e. highest 20%) would go in portfolio 5.
• You may want to exclude micro-cap firms, etc...
• No cheating! Be extremely careful to not form portfolios based upon information unavailable at time $t$.
2. Compute monthly returns for your portfolios.
• The more frequently you rebalance, the more issues there are with trading costs, bid-ask bounce etc... and whether these portfolio returns are achievable in practice.
3. Compute average returns, risk adjusted returns (i.e. Jensen's alpha), or characteristic adjusted returns (i.e. excess returns over some benchmark portfolio return).
• Eg. For each portfolio $1, \ldots, k$, regress returns against the factors of the Fama-French five factor model and compute the alpha, t-stat for the alpha etc... (Note: the spread portfolio return is already an excess return and so you don't subtract the risk free rate from that portfolio return when estimating alpha).

If $x$ really delivers a signal, you'd want to see alpha increasing in $x$ across the portfolios and an economically and statistically significant number for the alpha of the long-short, spread portfolio. You'd ideally like to see portfolio 5 has higher returns than portfolio 4 has higher returns than portfolio 3, etc....

The first step is to divide a very large number of stocks into deciles (groups having 10% of the stocks) based on some ranked measure (for example book to market or liquidity etc.), then you construct a spread of the highest decile vs the lowest decile returns by subtracting average D1 returns from average D10 returns. The idea is simply to understand how long term returns differ between stocks high and low on this measure (including a test of statistical significance of the difference). It is also a way of approximating the returns of a portfolio long the first decile and short the 10th decile, which is a common quant approach to investing. HTH