Multifactor investing - a comprehensive tutorial

This post covers a comprehensive example of building a diversified equities portfolio using multifactor investing. I hope that it will help to demonstrate how one could go about applying the Fama-French 3-factor model in real life to create a portfolio matching their risk tolerance and personal requirements.

If you've never heard of the Fama-French model, or want a refresher, here's the Bogleheads wiki summary, and right from the horse's mouth, this 1992 paper by Fama and French. (It's actually quite readable, even for those of us without finance degrees.) I've included a few more links at the bottom of this post too.

I will discuss the analysis and thought process I went through while designing my portfolio from beginning to end, with reference to the various tools and resources I used to help make decisions. This was my own personal process, so I'm certainly not saying it's THE way for everyone, but again, I hope it will provide a useful example at least of the kinds of things one might want to think about.

I'm going to try to avoid getting deeply into the merits of multifactor investing itself, which are discussed elsewhere. (The links I've included at the bottom could be a good start. ) There are certainly counter-arguments out there as well. Basically I believe that the size and value premiums seen historically are likely largely risk premiums. (Although there may be at least some behavioural component as well, particularly for value.) Regardless, it seems to me that the premiums are more likely than not to continue into the future. However, I personally do not feel it's prudent to rely on capturing the entire amount of the historical premiums going forward. So my general approach is to start with TSM (Total Stock Market), then tilt only if the historical premium that tilt would have achieved significantly outweighs the additional costs, both explicit and implicit, and the additional standard deviation of achieving the tilt - essentially leaving myself a sort of 'margin of safety'.

So, the plan for this post is to give an example of how one might rationally apply the Fama-French three-factor model to one's equity investing decisions, rather than to argue the merits of the model itself. What follows is, more or less, the actual process I went through. I'll present it as a tutorial since it's more convenient to write it that way, but I don't profess to be an expert and am certainly open to comment or criticism. (Especially if it helps me improve the efficiency of my portfolio!)

Determine equity / fixed income split
The first step is to determine a reasonable baseline split between equities and fixed income, based on personal risk tolerance. The most logical way I've seen to frame the question of risk tolerance is Larry Swedroe's "Willingness, Ability, and Need to take risk". His book, The Only Guide You'll Ever Need to the Right Financial Plan is a great primer, and an easy read. Canadian Couch Potato has a review here.

To keep things simple, we start by assuming that the equities portion of the portfolio will be in global market-cap weighted index funds - ie, total stock market - and the fixed income portion will be all short term government bonds - essentially zero term and credit risk. Then it's just a matter of determining what percentage of equities vs fixed income matches our personal risk tolerance and needs. Although there are many caveats to using historical data to project future returns, historical data can give us a good idea of future risk. In other words, a portfolio with 60% equities and 40% fixed income can be expected to have similar risk exposure in the future (be it standard deviation (volatility), maximum draw-down (loss), etc) as it did in the past. Given our starting assumption of short-term, high-quality fixed income, we can therefore estimate the risk of our portfolio by looking just at the equities side, which we can estimate using the historical risk of the global stock market. Short bonds do potentially provide a bit of diversification benefit, but most of their benefit is essentially 'dilution'; ie: scaling down the equity portion.

Based on data from Ken French's site, the annual standard deviation of the US stock market has been about 20% since 1927. Global data only goes back to 1990, but since 1990 the global standard deviation has not been significantly lower than the US over the same period, so we can use 20% as a rough guide to the long-term average SD of the global market. If our maximum tolerance is 10%, we would therefore want about a 50/50 equity/fixed income split. Of course, standard deviation is a somewhat meaningless, abstract number for most. Finiki has a good summary of the maximum expected losses of a portfolio based on various equity/fixed income splits, based on Bill Bernstein's excellent book, The Four Pillars of Investing. (Like us, Bernstein bases his numbers on short-term, high quality bonds.) An online risk tolerance survey might also be useful as a rough guide, like this one from Vanguard.

Anyway, getting off topic a bit. The idea is to choose a split that you're comfortable with and that meets your needs. Then, if you make changes to either side - either tilting equities away from TSM or adding term or credit/default risk to fixed income - you need to adjust the equity/fixed income split to keep your overall risk exposure the same (or less). More on this below.

My personal baseline was 75% equities, 25% fixed income.

Determine reasonable targets for Fama-French factor tilts
The next step is to get a feeling of what various levels of tilt really mean. We need to have a rough idea of what factor tilts we might want to target so that we know what funds to look for. As one looks at funds with greater degrees of tilt toward small or value companies, both the explicit management expense ratio (MER) and implicit (negative alpha) costs of those funds goes up. (Alpha is a measure of a fund's performance relative to the expectations of the 3-factor model. The best funds we expect to have alphas of near zero minus their MERs.) So the greater our tilt, the more we're relying on the expected factor premiums to outweigh these drags. Therefore the goal is to access as much of the expected premiums as possible, while still keeping the factor tilts as moderate as possible.

To help turn this rough (and contradictory) goal into actual numbers, we can look at some spreadsheets showing the historical risk (standard deviation) and return of various theoretical tilted portfolios. Links to the spreadsheets, results, and some analysis are here. They cover 81 portfolios, ranging in 0.1 increments from 0,0 to 0.8,0.8 weightings to SMB and HML. This shows us the theoretical benefit that was gained historically with various levels of tilt, not including any added costs or negative alphas in achieving that tilt. We can see that there has been diminishing marginal returns to greater degrees of tilt, particularly in the SMB factor. Personally, I decided that tilts of 0.1 to 0.3 to SMB and 0.3 to 0.5 to HML were 'close enough' to the efficient frontier, as a rough guideline. This is just one way to formulate the question, perhaps suited to visual thinkers. Here's another.

Note: It is also important to question whether, in your situation, it is appropriate to tilt at all! First of all, tilting involves using more funds, which increases trading expenses and portfolio complexity. For smaller portfolios, the hypothetical gains are likely not worth these added (guaranteed) costs. I would guess the break-even point is somewhere in the $60-100k range, as a rough estimate. Also, if the small and value premiums are indeed compensation for risk, logically they must not be appropriate for all investors. Some of my thoughts on these issues are here and here.

Choose specific funds for each region
There are four main geographical regions we're interested in as Canadian investors: US, Canada, EAFE (aka "International Developed"), and Emerging Markets. As a default starting point, we would split the equities portion of our portfolio into these four regions by market cap, so about 45% US, 5% Canada, 35% EAFE, and 15% Emerging, or thereabouts. In reality, we will over-weight some regions based on higher expected risk-adjusted returns: particularly Canada due to tax and currency effects, but possibly other regions depending on the results of our analysis of specific funds. Therefore, it makes sense to look at what investment vehicles are available in each region before deciding on firm percentages.

The correlations between the factors for various regions are relatively low, much lower than the overall market correlations. So while global diversification is useful in general, it is even more useful to diversify one's exposure to the small and value factors. So ideally one would target the same factor loadings everywhere. That said, it turns out they are much easier to achieve in some regions (US) than others (Canada, Emerging).

To compare between funds in a given region, we look at several factors: their expenses, their tracking errors, and most importantly, their 3-factor (3F) factor weightings and alpha. If investing in a taxable account (which I am), the tax consequences of the funds are important too; these are largely determined by their yields and turnover. Higher yield means more of the return expected by the 3F model is in the form of dividends instead of capital gains, which receive inferior tax treatment, (although much less so for the Canadian portion of the portfolio). Higher turnover can lead to capital gains distributions, especially for newer funds, whereas we would generally prefer to defer capital gains as much as possible. Higher turnover also leads to higher trading costs within the fund, but that will be seen in the tracking error and 3F alpha.

Management expenses (MER), tracking error, yield, and turnover can be found in the funds' prospectuses and/or annual reports, if not right on the fund companies' websites. (The reports are generally available on the companies' websites as well. If not, they can often be found through Morningstar (.ca for Canadian, .com for US), or as a last resort, by using EDGAR for US funds, or SEDAR for Canadian. 99% of the info you'll want is available on the companies' websites though.)

For the 3F weights, we need to perform ' linear regressions'. The goal is to describe a portfolio of stocks (such as a mutual fund or ETF) as a linear combination of several factors: Market beta, HML, and SMB. Basically what that means is we look at a period of historical returns for a given fund, and use software like Excel, Calc, or R to mathematically compare them to the performance of the three Fama-French factors over the same period. If it turns out that the fund tends to go up when the HML (value) factor goes up, and vice-verse, it means that the fund has a positive loading of that factor. For more info on what the factors actually mean, see this 1992 paper by Fama and French, and/or the Bogleheads wiki summary.

When you're ready to actually try a regression on a given fund, check out this excellent tutorial by the Calculating Investor. It shows how to get the data and do regressions on US funds, so that's a good place to start. Here's my analysis of the available options in each of the four regions, beginning with the US:

US choices
I performed regressions on a number of funds from Vanguard, Powershares (RAFI funds), DFA and others. (DFA funds can only be purchased through an adviser and the Canadian versions are different from the US ones, but they provide a good benchmark.) Particularly interesting was a deeper look at small value funds in this topic, where we found that you do indeed pay a significant cost for more concentrated, 'pure value' funds, as well as that the fourth factor, momentum, can provide useful insight in cases where alpha estimates are inconclusive. Later in that topic I also mention my thoughts on Margin of Safety in multifactor investing.

[The expected premium of our efficiently tilted example portfolio is] 1%, assuming factor premiums remain the same as they have been historically. Now, that's still worth doing in my opinion, but it's not exactly a massive margin for error. If you then go reaching for additional tilt using a less efficient product like RZV, you can see that remaining expected out-performance disappear very quickly. At least, that's how I look at it.

I believe in multifactor investing, but I want to give myself the best possible chance of achieving at least market returns over the long term. So if I don't see a strong enough chance that a portfolio will out-perform, long term, on a risk-adjusted basis, I would rather stick with TSM than roll the dice.

In the end I chose a 60/40 split of VBR and VTV, the small-cap value and large-cap value Vanguard ETFs based on MSCI indexes. This achieves estimated factor loadings of about 0.4 to HML and 0.3 to SMB. I expect that the S&P equivalents (VIOV and VOOV) or the Russel (VTWV and VONV) would have been fine too. The S&P have the advantage of slightly lower yield (hence less taxes), but have had slightly higher expenses (and tracking error), so that roughly cancels out. They have slightly less negative momentum than the MSCI, particularly on the small-cap side: about 0.05 less negative MOM exposure, which would be expected to make a difference of up to -0.5% per year, but at least historically that has been more than cancelled out by negative 4F alpha. Although, in regressions, the intercept (alpha) is the most difficult value to estimate precisely, so we are more confident in the momentum than the alpha. That said, the MSCI indexes are less concentrated than the S&P, which might result in (very slightly) less non-systematic risk. Since we're already cutting out about half of the total stock market with our value tilt, we prefer to be as diversified as possible. (Clearly it's a pretty close choice between MSCI and S&P.)

The Russell indexes are even slightly more diversified than the MSCI, and like the S&P they avoid the slight negative momentum seen in the MSCI small-cap. The small-value index also has greater HML (value) exposure, which historically has had a greater premium than SMB (small) and is more difficult to achieve without resorting to pure value funds. However, like the S&P funds, the Russel funds are slightly more expensive and have larger negative alphas than the MSCI (to an even greater extent). There is also a reason to expect these alphas might persist: the Russel indexes may be more prone to front running.

This Vanguard paper gives a good comparison of all the major indexes, with visual representations of their relative size bands, and some analysis.

I did consider adding a mid-cap value fund like VOE to the mix, since it would allow for the same factor weightings with a smoother ramp from small- to large-cap value, but ended up deciding against it. More specific reasoning behind the choice of VTV and excluding VOE is here, and regarding VBR is here. (Later in that topic (starting here) we also take a look at the RAFI US Small-value fund, PXSV, which looks potentially interesting but doesn't have enough history to know for sure. So for now we leave it for the more proven choice.)  Oh, and general process is here, using Vanguard funds for example.

EAFE (international developed) choices
EAFE is both simpler than US because (unfortunately) there are far fewer value funds to choose from, and more complex because we do not have a ready-made set of factor returns to work with. In this topic I describe how to estimate EAFE factor returns, as well as the expected effects of the approximations, and results for several funds of interest.

Since this post is already turning into a novel, I've split off my EAFE analysis here. Basically, although there aren't many funds to choose from, there are a couple good options. It's especially good for taxable investors, since CIE (RAFI International Fundamental index) is Canadian-domiciled. As a result, we can build a decent tilted portfolio using CIE and SCZ for only 20bps more than an untilted one with VEA and SCZ (or VXUS). The resulting factor weights are about 0.1 SMB, 0.3 HML (roughly).

Emerging markets choices
Emerging markets is tricky because there really isn't any good Fama-French factor returns data to perform regressions with. Jason Hsu, now at Research Affiliates (of RAFI fame), does publish a set, but I've found they aren't particularly reliable. So on the EM side we're forced to use very rough estimates (meaning we need a larger margin of safety if we are going to pay a premium to tilt). I used two approaches. First, I looked at equivalent EAFE and Emerging indexes from the same index provider, and assumed the factor loadings might be very roughly the same. For example, VWO uses the MSCI Emerging Markets Standard index, so we expect it to be comparable to the MSCI EAFE Standard index (VEA), on which we can do a regression. Second, I give more attention to tracking errors. Theoretically, if it's well constructed, the best we can hope for is that an index will have 3F alpha of zero. So at the very least we know a given fund will have negative alpha equal to its tracking error. (Likely greater, perhaps much greater.) As it turns out, maintaining even mildly reasonable tracking error is a very high bar for most Emerging Markets funds, especially small-cap. On the value side, iShares Canada's CWO was interesting, as they just recently switched it over to track the RAFI Emerging Markets Fundamental Index. Unfortunately, it cheats by holding other ETFs and sampling/approximating its index, much like CIE used to, which resulted in horrifying tracking errors. I don't see any reason for CWO to do better. What's more, even the Powershares fund tracking that index, which doesn't cheat, has had average annual tracking error of more than -1.5% since its inception over 5 years ago. It's hard to imagine its slight value weighting making up that kind of handicap over the long term, especially since the index itself may well not be perfect either (ie: it may well have negative Fama-French alpha itself). DGS, the Wisdomtree Emerging Markets Dividend fund, is theoretically another option, but we expect it to have the same issues as its EAFE brother, DLS, as mentioned in the EAFE topic.

So, at the moment it looks like tilting in Emerging Markets is simply cost-prohibitive. You could roll the dice and get lucky, but that's not what we're looking for here. Personally, I actually wouldn't be surprised if the EM small and value premiums for this period end up being particularly large, precisely because they're difficult or impossible to capture. But that's just a guess, and I don't plan to bet on it. I will simply use VWO for Emerging Markets, even with its tilt toward large, and possibly reduce my overall EM exposure a bit, given its lower expectation. (More on this later.)

Canadian choices
Finally, we have Canada. The first thing to note about Canada is that it actually already has a value and, especially, small tilt from a North American or global perspective. If we only held Canada in proportion to its global market cap, that wouldn't be an issue, as we would just be holding the market. But if we over-weight Canada (as we all do to some extent for tax and currency reasons), we should realize that we are adding exposure to the SMB and - to a lesser extent - HML factors implicitly. To avoid taking on unwanted risk, we will want to factor in that implicit tilt when choosing funds to meet our target factor weightings. (It's also important to take into account that over-weighting any country beyond its market cap weight brings in other idiosyncratic country risks, unrelated to size and value, so we have to weigh the costs and benefits when doing so. More on that here, and below.

For Canadian factors, I use essentially the same process I described for EAFE. More details on the source data used, along with results and analysis are here. (The OP of that topic presented initial analysis, but had a mistake in the SMB factors, so it was re-done in the post I linked to.) Unfortunately, I come to the conclusion that there simply aren't any Canadian small or value funds that could increase one's expectation sufficiently to overcome their increased costs, with a reasonable margin of safety. There are some that might break even, or if historical factor premiums hold out, perhaps outperform slightly. However, we have no guarantee that future small and value premiums will be as large as in the past, nor as large in Canada as in other markets. This is particularly true given then it looks like Canada already has an intrinsic small value tilt, and there is a decreasing marginal benefit to additional tilt, especially on the size factor (due to reduced diversification benefit). So until we have some better choices, in my opinion the best choice in Canada is to stick with a broad market fund: for now either XIC or VCE; once we can confirm its tracking error is satisfactory, ZCN.

Choose global asset allocations
So, we now know what our best options are in each of the four regions. The next question is how to divide total equities between the four. There is no straightforward answer. It depends on one's personal situation, particularly in terms of taxes and risk exposures outside of investments. It also depends hugely on the assumptions and estimates made, which will be different for everyone. However, there is at least a logical way to look at the question:

We start with a straight market cap-weighted global portfolio. That would be approximately 45% US, 35% EAFE, 15% Emerging, and 5% Canada (rounded to the nearest 5%). Weighting by market cap gives maximum diversification, which should result in maximum risk-adjusted expected return, all else being equal. Then, we look at the things that make 'all else' not equal. In other words, aspects of each reason that cause us to expect a higher (or lower) than average return, or greater (or less) than average overall risk. If a given region has higher risk-adjusted return given our circumstances, we would increase its weighting, and vice-verse. So essentially we can start by adding up all the costs and benefits of each region, to give it a single annual % premium (or cost) compared to the global average. Here are the results I came up with for a personal (non-corporate), taxable investor:

Canada
The Canadian portion of our portfolio, using VCE, costs 0.1%. As a taxable investor, the most significant aspect of Canada is its dividend tax treatment. Earning $X in Canadian (eligible) dividends is approximately equivalent to earning $(1.3 * X) in foreign dividends (or other income). Since yields average around 2-2.5%, this means about a 0.7% premium on our Canadian returns that is not tied to any corresponding risk. So net we have 0.7%-0.1% = 0.6% premium. Then we factor in the factor weightings (no pun intended...) VCE has an HML of about -0.1, which has a theoretical cost of around 0.2%, leaving us with +0.4%. Finally though, if we over-weight Canada significantly, we should also consider its intrinsic factor weightings on the portion that we're over-weight compared to market cap. This is where personal estimates come in. Based on historical data, Canada's 0.3,0.15 small,value loadings could mean an additional risk-adjusted premium of 1.2%, bringing the total to 1.6%. That part is obviously far less certain than the cost and tax differences though, so must be taken with perhaps an entire shaker of salt.

On the risk side, a larger Canada holding results in decreased currency risk, assuming we plan to spend $CAD in retirement. (At least, up to a point. Some international exposure will actually decrease currency risk, as well as the obvious diversification of country risk.) To get a rough idea of just how much this currency risk protection might be worth in terms of returns, I took a wild guess at what a reasonable worst-case currency fluctuation might be between my weighted average investment contributions and my retirement withdrawals. I chose 40%. So, if the CAD rose by 40%, how much performance difference between CAD and international equities does that equate to on an annualized basis over a 30 year time-frame? Something like 1.7% per year, depending on how you define the difference. Anyway, the idea is simply to get a ballpark. (Of course, the currency could easily go the other way, leaving your international holdings better off by that amount, but that would be extra, unneeded money. The currency risk benefit of Canadian holdings is in minimizing downside risk, measured in CAD.)  That number is obviously entirely dependent on assumptions, but at least it gives a rough idea. The currency risk avoidance is probably worth something more than say 1%, and less than say 3% per year, assuming minimizing downside risk is the goal, as opposed to simply maximizing expected value.

On the other side of the risk equation we have country risk. Stock market returns in different developed countries vary a lot, over long periods of time. Based on data like this, I estimated that a reasonable worst case would be Canadian under-performance by 3% annualized over my investing horizon. (Of course it could just as easily out-perform, but the idea is to not put too many eggs in one basket.) Naturally every region has risks, but when we put a significant percentage of our portfolio in one country representing 5% of the global market cap, and heavily concentrated in just three sectors, we are taking on significantly more risk than by putting a large chunk in the US, for example, which has a much larger and more diversified economy, with far more global exposure and sector diversification.

The nice thing is, although the potential country risk is fairly large, it is canceled out well by the currency risk benefit along with the tax savings. Therefore it should be safe to considerably over-weight Canada, assuming there aren't even better opportunities elsewhere...

(...unsurprisingly there aren't, although the US comes close...)

United States
In the US, our 60/40 combination of VBR and VTV has an effective MER of 0.1%, after adjusting for acquired expenses. The portfolio has factor weightings of approximately 0.3 to SMB and 0.4 to HML, which historically had a risk-adjusted premium of about 1.5%, so net we have a 1.4% premium. (Of course, that considers only the explicit costs of the funds, not any additional negative alpha, and it assumes factor premiums in the future are the same as the past, so certainly no guarantees.)

Currency risk has already been considered a bonus on the Canadian side, so we won't double-count it here or elsewhere. The US is not entirely without country risk of its own, particularly if we're also over-weighting Canada, since the US and Canadian markets are highly correlated. However, the US market makes up half the global market, and is highly diversified, so its country risk should be relatively minimal.

International developed
The effective costs of our international portfolio of 50% CIE and 50% SCZ is the sum of their MERs and the irrecoverable withholding taxes on SCZ's 3.1% yield, for a total of about 0.8%. We get factor weights of about 0.09 to SMB and 0.27 to HML, which historically, in the US, meant a premium of about 1.1% after adjusting for increased standard deviation. (I feel I should note again that these factor weights are rough estimates only. I use a couple decimal places to be accurate, but it certainly doesn't mean they have that level of precision.  Also, even if they did, they are subject to drift.)

Put the two together and we get a net premium of 0.1%: effectively zero.

Emerging markets
Finally we have Emerging Markets, which are simple since it's just one fund: VWO. The effective cost is the 0.2% MER plus the irrecoverable foreign withholding taxes, approximately 15% of the yield, or 0.34%. So 0.54% total. We estimate that VWO's size loading is the same as VEA's, about -0.3 SMB. As a very rough estimate using the US factors, that means a risk-adjusted premium of -0.7%. So in total we have a (negative) premium of -1.24%.

Emerging markets likely possess some unique risk factors, and so provide a diversifying effect independent of their small and value characteristics. It would not be prudent to eliminate them from the portfolio entirely. However, given their added costs and the fact that we can't cost effectively tilt them to small or value (or indeed, even achieve an untilted size), it does seem reasonable to reduce their weighting below market cap, particularly given we have better reasons to expect premiums in the US and, particularly, Canada.

Naturally, in retrospect, the decisions we end up making here won't be ideal. They may not even be close. But there's nothing we can do about that. All we can do is take into account all available information to give ourselves the best chances of maximizing return and minimizing risk, given what we do know, today.

Results
Note that my personal situation was slightly different from the above because my taxable investments are inside a CCPC, which in my case at least means that withholding taxes are only about 25% recoverable. So basically foreign stocks end up having an additional cost on the order of 20bps. So, based on the above analysis with that slight tweak - and a lot of guess work - here's what I came up with for regional allocations:


 * Canada: 30%
 * US: 38%
 * EAFE: 26%
 * Emerging: 6%

As an aside, if we consider North America as a whole, that works out to 68% domestic, 32% foreign.

Then, based on these factor weights for each region,


 * Canada: -0.1 SMB, 0 HML
 * US: 0.3 SMB, 0.4 HML
 * EAFE: 0.1 SMB, 0.3 HML
 * Emerging: -0.3 SMB, 0 HML,

as well as a bit extra for the over-weighting of Canada, which from a North American perspective has 0.3 SMB, 0.15 HML, we can estimate factor loadings for our global portfolio of equities. To do so, we simply multiply the loadings for each region by that region's percentage of our portfolio, then add the results. (IE: take the weighted average.) In the case of this example, we get approximately


 * 0.20 SMB, 0.27 HML

That's a bit lower than what we had hoped for, but close. HML in particular is low due to lack of good tilting options in Canada. Given the close relationship between Canada and the US, it seems reasonable at this point to see if we can get more HML tilt in the US to compensate. However, doing so requires the use of more concentrated 'pure value' funds, which we determined just aren't worth it. So, we stick with these allocations, confident that we're getting a mild to moderate tilt while not overspending or taking on unnecessary non-systematic risk. At worst, over the long term we expect to roughly pace the market, since we haven't added a great deal of additional cost. At best, our small and value tilts may result in 1-2% out-performance.

Re-adjusting asset allocation
The final step in our portfolio-building process (on the equities side at least) is to adjust our split between equities and fixed income to compensate for the tilt we've added, thus keeping our expected risk in line with the tolerance we established in Step 1. For that, we can go back to my handy spreadsheet. Our baseline untilted portfolio of 75% equities, 25% fixed income had a historical annual standard deviation of about 15.3%. To get that same level with our 0.2 SMB, 0.3 HML tilted portfolio (on the equities side), we need to reduce the equities portion of the total portfolio to about 68%. This is just a rough estimate since we're only using US historical data, but it should be in the right ballpark.

Alternatively, instead of aiming for a constant level of risk, we could aim to hold return constant while minimizing risk. This latter approach would be more appropriate if the split we came up with in  Step 1 was based around the minimum annual return we expect to require in order to fund retirement. (Not an ideal way to go about it, but sometimes a necessary one.) In that case, instead of finding the equities percentage to hold risk constant while hopefully increasing return, we would find the percentage that holds return constant (given the expected after-cost premium we expect from our tilts), resulting in a lower level of standard deviation for the same expected return.

Finally, keep in mind that these standard deviation estimates are assuming fixed income is all in risk-free assets. Adding term or credit risk on the fixed income side will also require the equities/fixed income split to be shifted more toward fixed income to maintain the same level of risk. That's another topic though.

Maintenance
Once the portfolio is worked out, we have to maintain it. For me that means every year (on a set date) I will:


 * Check the latest annual reports for each of my funds (ETFs). Particularly, look for any changes to the index methodology, or for larger than expected tracking error.
 * Rebalance funds toward chosen allocations. (I say toward because it's not necessary to get things exact, particularly if that means paying extra commissions, spreads, or taxes.)  As far as what's "close enough", here are some good guidelines. Regardless, the goal is simply to keep things in line with your risk tolerance.  It is possible that rebalancing will give a slight boost in returns due to mean-reversion, but don't expect it.  Topic for another day.

Every 4 years (on leap years, 'cause it's convenient), or more frequently if I feel like it or am concerned with a fund's tracking error or other aspects of its annual report:


 * Redo regressions of all funds over the past 5 years; look for inconsistencies in factors or changes in alpha. Over shorter regressions than 5 years, the factors tend to vary quite a bit, so you need to look at at least 5 years to filter out the noise.  Much longer than that though and it's tough to get an idea of how the factors might be drifting.  So personally I've found 5 years appears to be a good compromise; YMMV.  I will also run longer- and shorter-term regressions for supporting evidence though.
 * If factors have drifted considerably, evaluate whether allocations to each fund should be tweaked to better achieve overall desired factor exposures.
 * Also re-evaluate any funds that have had consistently high negative alpha or negative momentum. However, be aware that alpha, especially, is very difficult to estimate accurately.  Be sure to check the confidence of the estimate based on its t-value or standard error.  And don't make any decisions based solely on alpha without at least a few years' evidence.
 * Also look into products that were previously passed over, but appeared worthy of future re-evaluation, as well as any new small or value funds (or low-cost broad market funds like ZCN!) introduced since the previous check.
 * Document any changes made and the reasoning behind them, and note anything that should be reevaluated next time.

(There are also to-do items on the fixed income side, but I'll save that for a future topic.)

DIY regression analysis
When you're ready to try your own regressions, you can get started in just a few minutes using this great video tutorial by the Calculating Investor. I find his method is far faster and easier than using Excel, and only uses free software. I've got a slightly modified version of his script here. Works the same as in the example, but prints out the results in a nicely formatted matter at the end; handy for copying into your notes or, say, forum posts. It also pops up a graph to get a visual idea of how good the fit is.

Finally, there's a Fama-French three-factor model analysis at the  Bogleheads wiki, which should eventually be a comprehensive resource for Fama-French regression analysis.

Summary
And there you have it! As I said at the beginning, I'm certainly no expert. This is the method I used, based on my own research and experimentation. I'm sure others will have other opinions. I do certainly believe the approach of simply holding globally diversified total stock market funds for the lowest cost possible is entirely valid. Tilting intelligently (obviously) takes a fair bit of effort. If one is comfortable managing their own 'couch potato' portfolio, but would not feel comfortable going through a process like the above without an adviser, it is almost certainly better to stick with the more straightforward approach. That said, if you are willing to put in a bit of time - or are like me (and I suspect many other FWF'ers) and actually enjoy this kind of thing - then you may want to consider multifactor investing.

I would love to hear any specific comments or criticisms. Also happy to answer any questions or to supply more background info if I missed anything in all the links.

So, I hope you found this interesting!

Bogleheads forum discussions

 * Collective thoughts [investing mini-reference ] topic by Robert T catalogs a ton of related info. Should answer just about every question you could think to ask about multifactor investing.
 * How to choose FF Factor Weightings, Links to the spreadsheets, results, and some analysis.
 * How to get Fama-French EAFE Factors, with results
 * Correlations between worldwide Fama-French factors, Correlations between the factors for various regions.


 * Analysis of Vanguard funds for US S/V tilt
 * FF Factors for RAFI Funds
 * DFA US Core & Vector FF Factors


 * Jason Hsu's Emerging FF Factors
 * Emerging Small-cap tracking errors - VSS vs EWC, EEMS
 * Large-Cap Value - MSCI vs Russell, plus some