**Q: I’m new to passive investing and am deciding how to allocate between the asset classes. The best split between Canadian equity, international equity, etc. should be determinable based on studies of their past returns, volatility and correlations. Obviously this would vary over time, but approximate weightings should be achievable. Based on this research, how would you weight the individual asset classes? – R.T.**

It would look impressive if I designed my model portfolios based on an analysis of historical volatility, correlation matrices and expected returns based on Shiller CAPE or some other data. But instead I generally recommend a roughly equal allocation to Canadian, US and international stocks. Nice and simple, with no advanced math required. This is isn’t because building a “portfolio optimizer” is difficult: it’s because it’s a useless exercise.

Investors have a tendency to resist simple solutions, and this bias is exploited by fund managers and advisors who use algorithms and models designed to determine the “optimal” asset mix that will maximize returns and minimize volatility, sometimes down to two decimal places. That sounds more sophisticated than simply splitting your equity holdings in three, but there’s no evidence it produces better results.

One of the most fundamental ideas in portfolio design is the so-called efficient frontier—the sweet spot where you’ll enjoy the highest rate of return for each unit of risk. The problem isn’t that the efficient frontier doesn’t exist: the problem is it’s only knowable in hindsight. You can learn what *would have* achieved the highest risk-adjusted return in the past. But if you’re building a portfolio for the next five or 10 or 20 years, you can’t simply punch numbers into a spreadsheet and determine the optimal asset allocation for your target rate of return or level of volatility. That’s because the standard deviation of returns changes over time, as does the correlation between asset classes. And estimating future stock returns, in particular, is notoriously unreliable.

### Optimize this

In a 2013 article called Stop Playing With Your Optimizer, Brad Steiman of Dimensional Fund Advisors offers a dramatic example of what can go wrong. If you wanted to build an “optimal portfolio” of five asset classes—fixed income, Canadian equities, US equities, international equities and emerging markets equities—you could model it with 20 individual inputs. You’d start with the expected return and volatility of each of the five asset classes, and add the correlations of the 10 possible pairs.

In his example, Steiman used historical data for volatility and correlation and then assumed expected returns of 8% for Canadian, US and international stocks, 9.5% for emerging markets, and 5% for fixed income. With those inputs, the model determined the optimal portfolio with a target standard deviation of 12.5% would include an allocation of about 25% to Canadian stocks.

Then he ran the optimizer again, assuming that 19 of the 20 parameters turned out to be precisely accurate, but the expected return on Canadian equities was slightly off. If the return on this asset class was overestimated by just 0.5%, the optimizer increased the allocation to Canadian equities to 45%. And if the estimate was underestimated by 0.5%, the optimizer called for just a 2% allocation. No wonder Steiman calls portfolio optimizers “mistake maximizers.”

Determining an appropriate asset allocation is one of the most important decisions an investor will make. But a portfolio is not a chemical formula: there is no need for each component to be measured with precision. It’s more like a recipe: as long as you use high-quality ingredients and to get the proportions roughly correct, there is a lot of room for flexibility**. **

ArthurFebruary 18, 2014 at 10:11 amGreat advice! Only worry about the things that matter, like buying ETFs instead of individual stocks :)

Jin Won ChoiFebruary 18, 2014 at 10:49 amI disagree that optimization is just a gimmick. Institutional investors have known about Modern Portfolio Theory’s sensitivity to parameters, so they devised improved algorithms such as the Black Litterman model. Firms like Goldman Sachs find it useful enough to use it themselves. I believe the average person does too.

Michael JamesFebruary 18, 2014 at 11:04 amThe great thing about portfolio optimizers is that you can keep tinkering with the inputs slightly until you get the output allocation percentages that match your gut feeling about allocating more to US stocks after their great performance last year.

Removing my tongue from my cheek, I think the trouble some have is that they keep tinkering with their asset allocations. These people think they’re passive investors, but they actually have significant active share. Over 25 year they have 25 different allocations, any one of which (if used consistently) would have performed better than their semi-active approach.

WillyFebruary 18, 2014 at 11:09 am@Michael James: Fully agree and have been guilty of this at times myself. Okay, I’ll be honest, may STILL be guilty of this periodically myself. I find the tougher part of this is dealing with an expanding portfolio. I personally started off rather simply. I think there is some merit to perhaps adding a *small* number of additional asset classes as you go (sort of like expanding from this site’s Global CP to Complete CP portfolio), particularly as the last few years have seen dramatic drops in cost structures (lower MERs, commission-free brokers) and a significant expansion in products available to Canadians. But that then leads to the question of how and when to add those, and then you fall into the “tinkering” trap. But agree overall, this article is again on point… KISS.

OldieFebruary 18, 2014 at 11:26 am@Jin Won Choi: To be fair, he didn’t say they wouldn’t be useful if they were consistent, but that’s the point — his optimizer recommendations varied so much with small changes in input parameters as to be unworkable. Perhaps you can share why “improved” algorithms that you have specified would be useful and useable in your opinion, given that they depend somewhat (as I understand it) on estimates of future returns in the various asset classes.

SteveFebruary 18, 2014 at 11:26 amActually, the problem with the efficient frontier as a guide to portfolio design is not even imperfect information going forward – even with perfect foreknowledge of the exact future returns, volatility and correlations of all one’s chosen asset classes, so that one could determine the precise efficient frontier with 100% certainty … it would still be of no use in constructing one’s portfolio. In those utopian conditions, the most efficient portfolio would be designed by choosing the single asset class with the best future return over one’s investment time horizon and allocating the entire portfolio to it! For individual investors, optimizers and the efficient frontier should probably be seen as pedagogical tools, to illustrate the interplay between different asset classes combined in a portfolio, nothing more.

Jin Won ChoiFebruary 18, 2014 at 11:29 amLol. If you fudge the inputs, of course you’re going to get poor allocations. That’s a problem with the user, not the tool. You can misuse a hammer, but that’s no fault of the hammer.

I think those against optimization will have to justify their own methods for coming up with their allocations. “Because it’s simple” doesn’t cut it.

EdwardFebruary 18, 2014 at 11:38 amGreat article, Dan. One of the problem with optimizers is that they are written by programmers who are human. (I should know, I am one.) Something that is considered “low risk” one year can on a whim be reclassified as “high risk” the next if it performs poorly. Sure the algorithms and math inside an optimizer may be purely objective, but the structure of the program itself and the data that’s fed into it can suffer from a developer’s emotion and recency bias. (Well, not his/hers bias as much as the manager’s or firm’s or whoever else.) And since many of these optimizers change or evolve over time getting rewritten? Just another level of muck in the system. I’ve seen more databases and spreadsheets than I’d like to mention setup to output what the user wants to see at the moment, rather than true objective data results. Oh, and all those products that don’t even exist anymore because they suffered a heavy blow and proved to be duds? Yeah, that data isn’t in the historical performances either. (Especially if it’s a fund manager showing you optimization results.) Would you trust your calculator if you knew somebody was tinkering with its internal math every year or two? Hell no.

Jin Won ChoiFebruary 18, 2014 at 11:42 am@Oldie: The discussion gets a little technical, but the Black Litterman model takes in different estimates for parameters into account, and it also uses a range for the input parameters. I haven’t implemented it, so I’m not saying from experience, but I know it’s designed to overcome the problem described in this article.

@Steve: Sorry, but you misunderstand how the theory works. Given two assets, even if one has a superior expected return, the other asset will feature in the allocation if that other asset is sufficiently un (or anti) correlated with the first asset.

SteveFebruary 18, 2014 at 11:49 am@ Jin Won Choi

Sorry, you misunderstood my point. Dan criticized the utility of the efficient frontier for portfolio design on the grounds of imperfect knowledge of future returns etc. My point was simply that even with such perfect future knowledge, the efficient frontier is of no use for portfolio design: if you know for certain in advance that asset A will return 10% annually over the exact 25 years you plan to hold it, while asset B will return 7% … who cares about correlation? Why would anyone hold asset B? The efficient frontier is fine theory, but of no practical use at all – at least for individual DIY investors who are not going to get into Black Litterman.

Michael JamesFebruary 18, 2014 at 11:54 amI think the best justification for simple allocations is that I don’t completely trust statistical data (over any period of time) to accurately portray future risk. Canada, the U.S., or any other country could go into decline in a way that they’ve never done before (and so such a decline would not be included in historical statistics). I choose an allocation intended to protect me from extreme events rather than one intended to be optimal if no extreme events occur. I don’t mean extreme events like nuclear war, but events such as a country seeing a larger decline relative to the rest of the world than they’ve ever seen before. I’m open to the possibility that I’ve done a poor job of protecting myself against extreme events, but I don’t see how a portfolio optimizer will help.

MalnarFebruary 18, 2014 at 12:05 pmI mostly agree that simple is better, but I would warn against going too far; you may end up becoming guilty of the “naive diversification” bias.

http://en.wikipedia.org/wiki/Naive_diversification

From the wikipedia article:

“Following on the naive diversification showed by children, Benartzi and Thaler turned to study whether the effect manifests itself among investors making decisions in the context of defined contribution saving plans. They found that ‘some investors follow the ‘1/n strategy’: they divide their contributions evenly across the funds offered in the plan. Consistent with this Naïve notion of diversification, we find that the proportion invested in stocks depends strongly on the proportion of stock funds in the plan.’ This finding is particularly troubling in the context of laypersons making financial decisions, because they may be diversifying in a way that is sub-optimal.”

Mike DFebruary 18, 2014 at 12:53 pm@ Jin Won Choi:

You state that the institutional investors are using these other allocation methods, but they still can not beat a passive portfolio consistently. That is all the justification I need to stick with a the couch potato approach of simple allocations, rebalanced annually. I personally find the uber-tuber far too complex, and am not seeing any data to support a benefit to it. Simplicity rules!

David SmithFebruary 18, 2014 at 1:40 pmDan, thanks for this post and the lively comment thread. My thoughts and opinions:

1) MPT is a model, and all models are incomplete and imperfect. Nonetheless, models can still yield valuable insight. In my opinion, MPT and portfolio optimization are helpful for reducing overall portfolio risk by identifying assets that are historically uncorrelated or anti-correlated.

2) The MPT model does not contain a time derivative, so MPT is not a dynamic or predictive model. Rather, MPT is a model that identifies statistical patterns. Any application of MPT is an implicit belief that past statistical patterns will persist to some degree.

3) MPT is a tool, and tools can be used properly or improperly. Steinman’s analysis illustrates an improper use of numerical optimization because he applies optimization to highly correlated assets: Canadian equity is highly correlated with both US equity and International (ex-US) equity. It is known that optimization or regression analysis with highly correlated assets produces results with large uncertainties and high sensitivity to inputs. Indeed, sensitivity to inputs is a strong indication that inputs are highly correlated. In other words, Steinman’s analysis is an ill-posed calculation, so the results should be disregarded. (As modelers say, “Junk in, junk out”)

4) Optimization calculations often use historical returns, and it is plainly true that historical returns reflect past economic conditions. Despite that shortcoming, historical results have 2 virtues: A) historical returns are factual data devoid of subjective opinions or bias, and B) historical returns are inherently plausible market scenarios. It is possible that Steinman’s hypothetical variation of Canadian equity returns, with other parameters fixed, is not plausible in reality.

5) Investment risk can mean many things, but I feel statistical variance is a *reasonable* risk metric because it satisfies the expected relationship between risk and return at macro levels, namely, higher potential returns carry higher risk. Asset classes that show large historical returns on average, such as emerging markets and small-caps, also show large year-to-year variation. In contrast, asset classes with lower historical returns, such as bonds, show less year-to-year variation.

6) MPT and portfolio optimization are fully compatible with simple, passive, index portfolios and complex stock portfolios. Indeed, the broad applicability of MPT is why it was elevated to “theory” status in academic circles. Optimization calculations are ubiquitous in finance – in both active and passive scenarios. Optimization calculations help active fund managers ensure some level of diversification in the fund. Meanwhile, passive investors can optimize with passive indices and easily arrive at a Vangard-like target-date fund.

Just my thoughts. -Dave

Jin Won ChoiFebruary 18, 2014 at 1:46 pm@Steve, no one ever knows the exact future, so I don’t see how your example is relevant. However, you CAN rationally estimate the unpredictability of the future, and that’s where optimization comes in handy.

@Michael James: So how do you know your method of allocation protects you from extreme events? If you don’t infer from history, how do you justify your allocation? The future may not resemble the past, but it’s better than a pure guess. Also, if you think about it, there are rational reasons why history panned out the way it did, which manifested itself in the form of volatility, correlations, etc. So why ignore those forces?

@Mike D: Optimization reduces risk. It doesn’t enhance returns.

BericMFebruary 18, 2014 at 1:55 pmGreat article Dan, and I completely subscribe to Simplification after finding my MER’s exceeding my returns. My version requires a spreadsheet and a calendar: select the top-5 yielding TSX60 issues, re-invest all dividends and re-balance yearly.

I must admit to two “optimizations”: I exclude income trust conversions and gold miners, but the results do not lie: ~12% CAGR since inception and 10yr.

If you like, check here: http://wp.me/p485tU-2A

Cheers!

Michael JamesFebruary 18, 2014 at 1:58 pm@Jin Won Choi: This thread is becoming pointlessly argumentative, but I’ll bite one more time. I never said I ignore history. For example, I hold a heavy allocation to stocks because history teaches that stocks outperform bonds over time. However, I prefer to spread out my stock ownership around the world. If some model tells me to choose a stock allocation that seems too concentrated in one part of the world, I prefer to remain more diversified. It’s also difficult to take the output of models too seriously when you get radically different answers whether you collect statistics from the past 25, 50, or 100 years.

NoelFebruary 18, 2014 at 2:51 pmAnd, now back to our regularly scheduled programming:

KISS

(keep it simple stupid)

b.abbottFebruary 18, 2014 at 3:10 pmTFSA and RRSP are only a benefit for gains not losses.

Prefered investments (reit etf’s and bond etf’s) have lost

in the past year.Is there a better investment choice?

Chris BFebruary 18, 2014 at 4:24 pmIt’s interesting to point out that the father of Modern Portfolio Theory, Harry Markowitz didn’t use the theory himself:

Mr. Markowitz was working at the RAND Corporation and trying to figure out how to allocate his retirement account. He knew what he should do: “I should have computed the historical co-variances of the asset classes and drawn an efficient frontier.” (That’s efficient-market talk for draining as much risk as possible out of his portfolio.)

But, he said, “I visualized my grief if the stock market went way up and I wasn’t in it — or if it went way down and I was completely in it. So I split my contributions 50/50 between stocks and bonds.”

http://www.nytimes.com/2007/09/29/business/29nocera.html?ex=1348718400&en=77986b6930cb1466&ei=5090&partner=rssuserland&emc=rss&_r=2&

David SmithFebruary 18, 2014 at 6:48 pmIn my view, a simple 50/50 total stock/total bond split is consistent with the principles of MPT. Markowitz didn’t perform the full calculation with all his investment options, but his final decision reflected the diversification benefit of stocks and bonds, which are historically anti-correlated to some degree. In fact, I would argue that anyone who accepts the premise of stock/bond diversification has largely bought into MPT. MPT is merely the rigorous mathematical framework for the informal concept of diversification. Note, however, that “rigorous” can include simple. Any two assets possess a mathematically rigorous efficient frontier, and a simple 50/50 split lies somewhere on that curve and captures some diversification benefit. Most investors do not perform their own MPT calculations, but most investors do apply the diversification principles of MPT, even if informally or inadvertently.

SteveFebruary 18, 2014 at 7:00 pm@Noel:

Exactly!!!

There is a lot of interesting theory discussed upthread, but for most individual DIY investors looking for help in setting up a reasonably diversified low-cost portfolio, it all might as well be in code for all the utility it has.

By all means learn how and why asset class diversification works – then (for example) just choose one of Dan’s Global or Complete Couch Potato model portfolios, tweak the allocation as desired, rebalance annually, and chill.

Jin Won ChoiFebruary 18, 2014 at 7:36 pmTo a DIY investor, implementing an optimized portfolio takes no more work than implementing a non-optimized portfolio. It does however, take more work for the person publishing such portfolios.

There’s no rule that says if you want to maximize diversification, you should split your equity holdings evenly between countries. That’s nothing but guess work. Modern Portfolio Theory and successor theories on the other hand, provide a rational, mathematical construct that tells you how to maximize diversification. It’s like guessing your odds of winning the lottery vs. mathematically calculating it. I know which measure I’d rely on.

KevinFebruary 18, 2014 at 8:26 pmWhen the next world changing seismic event occurs I’ll be happy I had 17.8% in international equities as opposed to a disastrous allocation of 16.4 %….. Oh wait.

Michael JamesFebruary 18, 2014 at 10:07 pm@Jin Won Choi: Well, because returns are known not to be lognormally distributed, MPT will give wrong answers. I wish you the best of luck finding the correct theory while I muddle along with my guess work.

SteveFebruary 18, 2014 at 10:19 pm@ Jin Won Choi:

RE the advantage of a “a rational, mathematical construct”: To quote William Bernstein only half-facetiously explaining one reason why DIY investing is a lot harder than it appears: “fractions are a stretch for 90 percent of the population” …

For most DIY investors trying to understand the optimization process and the consequences of the choices made as part of it – necessary to make an informed choice of portfolio to implement – most certainly does take a great deal more work than simply using a perfectly good KISS boilerplate solution like (for example) Dan’s model portfolios.

Never underestimate the opportunity cost to investors of concluding that DIY is not for them because achieving theoretical perfection is so darn complicated and mathematical, all while ignoring perfectly effective, easy to understand KISS solutions.

RossFebruary 18, 2014 at 10:47 pmPutting aside initial portfolio allocation, how often should a portfolio ideally be rebalanced back to deemed allocation? The whole premise, I think, of the Couch Potato aspect of CPP philosophy is to try really hard not to needlessly fiddle (and incur related costs and ‘beat-the-market’, ugh). Is annual re-balancing enough? or consistent with academic papers?

Canadian Couch PotatoFebruary 18, 2014 at 11:20 pm@Ross: Just as there is no optimal portfolio, there is no optimal rebalancing frequency either:

https://canadiancouchpotato.com/2011/02/24/how-often-should-you-rebalance/

Chris BFebruary 18, 2014 at 11:33 pm@ David:

Well, yes and no.

Over the last decade, Bonds have had a -.39 correlation to the S&P 500 stock index. So one could argue that Markowitz was following an approximate version of MPT. However, even in this case, once variance is factored in, I don’t think you’ll find any version of MPT that doesn’t weight bonds at least 70%.

Crib notes: I’m not letting Markowitz off that easy. :)

@Jin:

“Modern Portfolio Theory and successor theories on the other hand, provide a rational, mathematical construct that tells you how to maximize diversification”

Well, yes and no.

Yes, the can provide a rational, mathematical construct for a specific set of circumstances, over a specific time period. And you can be very precise with your calculations … but very inaccurate in applying them based on your premise. For instance, as noted above, for the last decade US bonds have had a -.39 correlation to the S&P. But that changes to +.o8 correlation over the last 80 years.

Which number do you use? If there is a theory which can reconcile these two extremes, and have any value, the onus would be on the person making such a claim to demonstrate it.

Even taking something as simple as risk parity where you weight each asset’s position size by the inverse of it’s volatility. You still have to determine the best look back period, and frequency. So, out of the gate your are data snooping. Next, volatility and correlation are inherently unstable and non-stationary. So your estimations introduce risk and instability.

But wait, there’s more:

“The assumption of a normal distribution is a major practical limitation, because it is symmetrical. Using the variance (or its square root, the standard deviation) implies that uncertainty about better-than-expected returns is equally averred as uncertainty about returns that are worse than expected. Furthermore, using the normal distribution to model the pattern of investment returns makes investment results with more upside than downside returns appear more risky than they really are. The converse distortion applies to distributions with a predominance of downside returns. The result is that using traditional MPT techniques for measuring investment portfolio construction and evaluation frequently does not accurately model investment reality.”

http://en.wikipedia.org/wiki/Post-modern_portfolio_theory

Recognizing this, Post Modern Portfolio Theory has come to the rescue. A bigger and better theory, with a bigger and better computer. While this is a welcome relief to us all, I think I’m going to hold out for next year’s model. The Post-Modern Feminist Portfolio Theory, which should incorporate gender bias in it’s application.

AndrewFebruary 19, 2014 at 6:05 amOkay then what about this portfolio:

Half XWD MER 0.47, half a ladder of 5 year GICs. Overall fee 0.24%

Or less or more of XWD depending on risk tolerance.

This is the simplest portfolio I can think of apart from XWD/VAB or XBB ?

I can explain it to anyone and its very easy to understand with the utmost simplicity in maintenance and therefore potential stick-to-it-ness.

Is it nuts to consider such a structure or is there real value in it?

OldieFebruary 19, 2014 at 11:04 am@Andrew: IMO the point of this post is that most people who “get” the benefits of diversification (less risk for same or perhaps slightly more profit) may then start the slow road to “tweak” their portfolio ever so slightly a bit at a time, each change being “sort of reasonable” before realizing they had crossed over into the zone of too much hassle for too little extra benefit. Saying don’t go there isn’t the same as saying it’s best to have only one equity asset, and, because you have arbitrarily chosen one asset class, for “simplicity” it must be the whole world.

Your ability to explain it and its easiness to understand are a virtue, I guess, but so you could say the same for, say the strategy of having only one outer garment, a heavy waterproof insulated parka, and determining to wear it Dec 1 to April 30, and wear no outer garment the other days.

If you are committed to rebalancing your 2 asset portfolio anyway, how much more difficult would it be to split your 50% Equity (I’ll not argue, for simplicity that any different percentage has more merit) into equal US, Rest of World, and ( if you believe that spending your proceeds in Canada makes it sensible to do so) Canadian components? I can’t see that it violates your “simplicity” and “easy to understand” rule, and you get less volatility and likely more profit with minimally more hassle.

NoelFebruary 19, 2014 at 11:20 amEmploying the KISS principle with respect to passive ETF index portfolio asset allocation and spending one’s energy on improving other aspects of one’s life (nutrition, exercise, family, friend, hobbies) will yield a much higher overall return.

David SmithFebruary 19, 2014 at 11:45 amThere have been a few comments suggesting that portfolio optimization is needlessly complex due to high numerical precision, but that criticism is undeserved. Optimization is inherently mathematical, so any desired numerical precision is possible. 30-year-old Casio calculators also give high numerical precision, but that doesn’t mean the calculator is deficient or useless. Just like a cook following a recipe, the user is free to exercise common sense with regard to precision.

Some comments have stated the MPT is “wrong” because returns are not normally distributed. I think the more insightful perspective is to recognize that MPT is a model, and it models returns *as if* they are normally distributed. Are returns truly normally distributed? No. Is there a “right” or true model? No, only approximations. If returns are not normally distributed, then how is MPT helpful? MPT captures the reduced volatility possible with favorable correlations. Some investors might have diversification questions like, what can offset oil price fluctuations? Or, what can offset real estate fluctuations? MPT provides a mathematical framework to address these questions. If you accept the premise that stocks and bonds provide some level of diversification, then I feel you’ve largely bought into the principles of MPT.

Other comments have criticized MPT because more complex models exist, but it is always possible to create more complex models. Let me put it this way: MPT is the simplest complex model available. :) In order of complexity:

1) casual diversification: a bit of health care, a bit of energy/utilities, a bit of real estate, etc.

2) recognizing pair-wise correlations: bonds are partially anti-correlated with stocks, so adding some bonds to stocks can reduce volatility.

3) multi-asset correlations and portfolio optimization with MPT: If I own commodities, health care, and real estate, what new asset class best reduces volatility?

4, 5, 6, etc) more complex, less tractable models

Despite my “simplest complex model” statement, MPT is compatible with simple strategies. A simple 2 asset portfolio (total stocks/total bonds) possesses an efficient frontier. Also, MPT results for a 3 asset portfolio with domestic stock, int’l stock, and bonds look a lot like generic target-date mutual funds, which is not surprising.

I am not suggesting that all investors *must* dive into portfolio optimization, but I am confident that MPT has value, that optimization calculations are at play in our financial lives, and that any reader here could become proficient and knowledgeable if so desired. I got into the topic because I enjoy modeling and analysis, not because I felt it was financially imperative.

@Noel: one of my hobbies is portfolio optimization :) But seriously, I agree completely. Simple, effective investing options certainly exist for anyone who wants to minimize the time investment. (time optimization! :) )

ChrisFebruary 19, 2014 at 12:10 pm@David,

What are your thoughts on whether the level of home bias in CCP’s model portfolios (1/3rd Canadian stock) is roughly along the efficient frontier?

Michael JamesFebruary 19, 2014 at 12:13 pm@David: It’s true that MPT can give useful insights in many situations. There are many types of questions where MPT’s assumption of lognormal return distributions do not materially affect results. However, portfolio optimization to find the efficient frontier is not one of them. MPT gets the probability of extreme events very wrong. This inaccuracy becomes apparent after only a few standard deviations. As an example, this causes MPT to routinely find that optimal portfolios should use leverage. There are many ways to try to repair this problem to get more reasonable answers, but we are always left with the problem that extreme events are almost impossible to model even remotely accurately. It is easy to dismiss rules of thumb about diversification