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Firstly let me apologise if this has been asked on on here before - I went through some of the factor modelling post on here but I've struggled to find the answers. I am new to the quant world and I am fascinated on how I can introduce science to my equity portfolio

I want to learn more about factor modelling and how can it be used to potentially help improve my current portfolio? When you run a regression on the 3 factor F&F model on your current portfolio it will tell me what exactly? So as the value of the independent variable increases (assuming the coefficient is positive), the mean of the dependent variable also tends to increase. Am I correct to assume that the purpose of it is to understand what factors are driving the returns in my portfolio - be it momentum, value, quality, etc. How can I use this to find out what stocks I should have to improve my portfolio? So let's assume I have a portfolio of 50 stocks from FTSE AIM All share index. I run the regression, learn that a bunch of my stocks are influenced by momentum.. what happens next? how do I use this information to improve my portfolio? How is does factor modelling used at a portfolio level? what are the typical regression and data pitfalls in this sort of exercise?

I really appreciate your time and thoughts, and again I apologise if this has been answered before. If you have links, directions that could help me on my path I would be extremely grateful.

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    $\begingroup$ Did you try searching on portfolio factor models here? $\endgroup$ – kurtosis Aug 4 at 20:04
  • $\begingroup$ Sorry: I see you did search but are asking more about the meaning of these factors. That topic is not so well addressed in many places. I have posted an answer which is (hopefully) helpful on that front. $\endgroup$ – kurtosis Aug 5 at 2:57
  • $\begingroup$ You might also do well to focus your question more and change the title. Some of your questions are asked elsewhere (how factor models can help improve your understanding of a portfolio; typical regression pitfalls; what risk factor betas mean) and some of your questions are specific to implementation and maybe even to you (what stocks to hold). The questions you ask which are not asked often enough are those related to what these factors mean -- which helps you answer the "what next" when you find portfolio exposures to (say) Carhart's momentum. $\endgroup$ – kurtosis Aug 5 at 3:06
  • $\begingroup$ Many Thanks. I will take this on board. $\endgroup$ – Newbie56809 Aug 5 at 6:21
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Congratulations on trying to manage your portfolio better. That is wise.

If you lose money, you may at least understand why you lost money. Ideally, you could also protect your portfolio against risk factors you do not want to take. However, this gets a little tricky since some factors are not clearly risk factors. (This is especially true for the Fama-French factors.) Therefore, you should think a bit about what these factors mean.

Note that more detail can be found in chapters 14 and 15 of A Quantitative Primer on Investments with $R$.

Returns versus Holdings

In general, we assess risk by analyzing returns. There may be value that can be added by looking at holdings; however, we should not look at holdings alone.

We analyze returns because we want to see if the company's performance varies with factors we might not realize are important. For example, many small-cap firm returns are highly correlated with credit spreads. That might seem sensible with some thinking, but not enough equity portfolio managers use fixed income risk measures on their portfolios to see that connection.

Limits of Holdings: An Example

A simple example can show the limits of looking only at holdings. In the late 1980s, the Japanese stock market had risen dramatically. In 1989, it was estimated that the land under the Imperial Palace in Tokyo was worth as much as all of the real estate in California. A portfolio of Japanese equities (say, mirroring the cap-weighted TOPIX index) would have shown holdings across a variety of industries.

The Japanese market peaked in 1991 and then declined 40% through mid-1992; after that, the market rose and fall but has not yet recovered to the 1991 peak. Declines in stock prices were linked to a few factors including cross-holdings and cross-lending (so most stocks in a keiretsu were correlated with other stocks in the same keiretsu) and real estate having inflated stock valuations (since the Japanese bubble was real-estate driven).

Unfortunately, a holdings-based approach would not have revealed the cross-holding problem and would definitely not have revealed the real estate exposure. The only hope for uncovering the cross correlations among stocks and stocks' exposure to real estate was by analyzing returns.

Which Factors to Consider?

If you are managing an equity portfolio, there are a number of factor models to consider.

CAPM: What is the "Market?"

The base model is the CAPM and is generally a one-factor model: you explain your portfolio's excess returns (returns beyond the risk-free rate $r_f$) as a linear multiple of excess returns on some broad-based market index: $$ \underbrace{R_i}_{r_i - r_f} = \alpha_i + \beta_i \underbrace{R_M}_{r_M - r_f} + \epsilon_i. $$

Lots of people will tell you "the market" is the S&P 500. That is absurd since the S&P 500 only covers large-cap US stocks. US-focused equity investors should also consider small-cap and mid-cap stocks; and, other equity investors should consider other indices, use a global index, or combine many indices. (US investors should also do that.)

Professional money managers often hedge exposure or unfilled positions with equity index futures. Therefore, indices which underlie futures are preferred for hedging.

This all neglects other asset classes like fixed income, commodities, real estate, and foreign exchange. Ideally, the broad-based index used should match the universe of potential investments.

Macroeconomic Risk Factors

Some factor models are based on macroeconomic risk factors. Many of these models look at factors beyond just equity markets. Since some of these factors come from other markets, that allows us to use different markets' forward-looking inferences about the macroeconomy.

The Chen, Roll, and Ross (1986) model considers five factors:

  • the percent change in industrial production;
  • the change in expected inflation;
  • the surprise in realized inflation;
  • the slope of the yield curve; and,
  • a credit spread between intermediate (approximately 10-year) corporate bonds vs US Treasuries.

The Litterman and Scheinkman (1991) model considers exposures to yield curve movements:

  • a rise or fall across the yield curve (related to duration);
  • a change in the yield curve slope; and,
  • a change in the yield curve curvature (bowing, perhaps related to interest rate volatility).

Other models consider:

  • an equity market index, a corporate bond index, a currency index, credit spread, implied equity index volatility, and a commodity index (Hasanhodzic and Lo, 2007);
  • equity indices, 10-year US Treasuries, a credit spread, and trend factors for bonds, foreign exchange, and commodities (Fung and Hsieh, 2004); and,
  • equity indices, government bond indices, short-term interbank lending rates, gold, and trade (Fung and Hsieh, 1997).

Microeconomic Risk Factors

We tend to believe investments should compensate us for risk. Therefore, if a stock becomes more volatile, that looks less appealing to current holders while new investors would demand more return from the stock. Therefore, the price falls to yield a forward-looking higher return. We can test this with an ARCH-in-mean or GARCH-in-mean model which handles predicting volatility and incorporating that as a risk factor (Engle, Lilien, and Robins, 1987; Bollerslev, Engle, and Wooldridge, 1998).

There are also liquidity risk factor models such as Amihud and Mendleson (1986) which supplements the CAPM with divergences of a stock's bid-ask spreads from the average; and, Acharya and Pedersen (2005) which considers divergences of bid-ask spreads from their average for an equity market index (similar to the CAPM).

Other researchers have looked at idiosyncratic volatility, cashflow volatility, and the volatility of assets versus liabilities.

Style Factors

You mentioned the Fama and French (1992) three-factor model which has factors they say represent the outperformance of small stocks (SMB=small-minus-big) and value stocks (HML=high-minus-low book-to-market): $$ R_i = \alpha_i + \beta_i R_M + \beta_{i,SMB} SMB + \beta_{i,HML} HML + \epsilon_i. $$ Many people also consider the Carhart (1997) model which adds a cross-sectional momentum factor (WML=winners-minus-losers): $$ R_i = \alpha_i + \beta_i R_M + \beta_{i,SMB} SMB + \beta_{i,HML} HML + \beta_{i,WML} WML + \epsilon_i. $$

More recently, Fama and French have admitted their three-factor model missed some factors, They therefore came up with a five-factor model (Fama and French, 2016) which adds profitability (RMW=robust-minus-weak) and investment (CMA=conservative-minus-aggressive) factors.

What are the Style Factors?

The problem is that unlike all of the macro and micro factor models, we do not really know what the Fama and French factors represent in terms of risks. This has led to many people referring to these factors as "style" factors. The lack of clarity about if style factors are risk factors is not mere opinion: Some findings challenge the interpretation (and even usefulness) of these factor models.

The existence of an SMB factor suggests that maybe the S&P 500 is not a broad enough index. Sure enough, Palacios-Huerta (2003) finds that using a more representative market index (including accounting for human capital) does better at explaining stock returns than the Fama-French three-factor mmodel or the Chen-Roll-Ross model.

Chan, Chen, and Hsieh (1985) show HML is related to credit spreads. Bassett and Chen (2014) show that even very-large-cap portfolios may have an exposure to SMB -- which calls into question how SMB could represent small-minus-big stocks.

Das, King, and Sinha (2012) show that the value premium is only positive for a few days per year for a given stock and thus seems to be related to earnings announcements and not the hard assets underlying a firm's market valuation.

As for Carhart's momentum, Sinha (2014) shows that news sentiment dominates WML momentum (and perhaps explains how cross-sectional momentum arises).

Ball and Brown (1968) found post-earnings announcement drift (PEAD) helped explain stock returns; and, Sadka (2006) found WML and PEAD were likely related. Thus news, cross-sectional momentum, and earnings surprise momentum may all be related.

Summary: Which Factors to Consider?

This is a lot to wade through. However, it is important to realize that researchers have looked at many risk factors; some factors go away when we just look at a broader market index; and, we really do not know what the style indices represent.

If style factors do not represent risk factors, then we should not expect to be compensated (on average) for holding those risks. For example, the HML ("value") factor has spent years yielding no positive return for HML exposure. That could mean that HML is related to a rare risk or it could be that HML is just a proxy for some risk we could measure better with another factor.

What would I do? I would consider the macroeconomic and microeconomic risk factors: the level of interest rates, slope of the yield curve, and curvature of the yield curve; inflation (expected and unexpected); credit spreads; indices for commodities, foreign exchange, real estate, and bonds; volatility of equities and maybe a few other asset classes; and, well... just a better equity market index than the S&P 500.

That is a lot to look at; however, I think the arguments for why these are risks or relate to risks are clear. You will be a better portfolio manager if you can understand the risks your portfolio is exposed to instead of just factors it is correlated with which may (or may not) represent risks.

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There are multiple ways to model portfolio factor exposure, for instance:

1- Return based approach: regressing returns of a portfolio vs several factors (similar to what your doing).

2- Holdings based approach (more accurate): essentially you would want to get the factor exposure of all individual holdings, then, derive the portfolio factor exposure. In practice, portfolio factor exposure is compared with the benchmark (Search for MSCI Factor Exposure; they explain it very well).

Now, usually such analysis is used mainly to get a sense of what factors your portfolio is exposed to.

Example 1) Assume you do some market research and believe that Value stocks will outperform other factors and Low Vol will underperform. Then you would want to position your portfolio so that you have an overweight exposure to the Value factor and an underweight exposure to the Low Vol facto (relative to the benchmark). This is factor timing, which in practice is very difficult to time correctly.

Example 2) Some believe that you would want your portfolio to be diversified (i.e. does not have a bias toward a specific factor). Hence running such analysis (through time) would give you an idea how your portfolio is positioned.

Example 3) How much of your portfolio returns are explained by such factors? Is 90% of your portfolio performance explained by let’s say MSCI UK Value? Is it cheaper for you to buy MSCI UK Value rather than buy individual stocks and all the transaction costs/commissions associated with it.

Apologies for the long answer, but there are lots of way to use such analysis. Good luck!

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    $\begingroup$ The holdings-based approach has major problems. According to that approach, holding a diversified portfolio of Japanese stocks in 1994 would have seemingly had no exposure to real estate. Fast forward a few years and I would have lost a lot of money due to real estate exposure. $\endgroup$ – kurtosis Aug 5 at 0:06
  • $\begingroup$ I do not see the relationship between equity based factors (i.e. value, growth, quality, momentum, ... etc) and Real Estate. I assume exposure to Real Estate is an asset allocation problem and not related to this question. $\endgroup$ – user28909 Aug 5 at 10:12
  • $\begingroup$ Real estate may or may not be exposed to these factors depending on the concentration of tenant industries or types. Equity can be exposed to real estate through holdings or leases (which are rarely reported in "holdings" reports) or even customer exposure (which would not appear in any holdings report). For more, read up on the Japanese real estate bubble and how it affected share prices. These might be unusual situations, but detecting unusual exposures is a major reason to look at risk factors. $\endgroup$ – kurtosis Aug 5 at 12:16
  • $\begingroup$ It is actually totally unrelated to the subject matter. The only relationship I can see is with REITS and the yield factor, given that REITS somewhat pay an attractive dividend yield. Real Estate is an Asset Class and does not fall under equities. Assume you run an asset allocation optimizer, which allocates 0% of the portfolio in real estate and 30% of the portfolio in equities. You would then use the holding based factor model to analyze the equities portion of your overall portfolio. Hence, the real estate issue you are referring to falls under the asset allocation problem. $\endgroup$ – user28909 Aug 6 at 9:59
  • $\begingroup$ Why not actually read up on the Japanese bubble and how the real estate market was a hidden factor for all equities? The real estate exposure is absolutely related to the question at hand. Talk to any portfolio manager who had Japanese equities at that time and they all scrambled to ascertain their real estate exposure. $\endgroup$ – kurtosis Aug 6 at 10:17

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