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Visualizing How Market Volatility Impacts Risk and Returns

S&P500 Performance for 2020 (Yahoo Finance), pulled Aug 9, 2020

2020 has seen the greatest market volatility in history for American stocks. The roller-coaster ride investors have experienced over the last 6 months included a steep ~33% single-month drop followed by a four-month bull market run taking the S&P500 back roughly to where it started.

While not usually so dramatic, volatility is a fact of life for investors. In researching how to create a long-term investment strategy that can cope with volatility, I found a lot of the writing on the subject unsatisfying for two reasons:

First, much of the writing on investment approaches leans heavily on historical comparisons (or “backtesting”). While it’s important to understand how a particular approach would play out in the past, it is dangerous to assume that volatility will always play out in the same way. For example, take a series of coin tosses. It’s possible that during the most recent 100 flips, the coin came up heads 10 times in a row. Relying mainly on backtesting this particular sequence of coin tosses could lead to conclusions that rely on a long sequences of heads always coming up. In a similar way, investment strategies that lean heavily on backtesting recent history may be well-situated for handling the 2008 crash and the 2010-2019 bull market but fall apart if the next boom or bust happens in a different way.

Second, much of the analysis on investment allocation is overly focused on arithmetic mean returns rather than geometric means. This sounds like a minor technical distinction, but to illustrate why it’s significant, imagine that you’ve invested $1,000 in a stock that doubled in the first year (annual return: 100%) and then halved the following year (annual return: -50%). Simple math shows that, since you’re back where you started, you experienced a return over those two years (in this case, the geometric mean return) of 0%. The arithmetic mean, on the other hand, comes back with a market-beating 25% return [1/2 x (100% + -50%)]! One of these numbers suggests this is an amazing investment and the other correctly calls it as a terrible one! Yet despite the fact that the arithmetic mean always overestimates the (geometric mean) return that an investor experiences, much of the practice of asset allocation and portfolio theory is still focused on arithmetic mean returns because they are easier to calculate and build precise analytical solutions around.

Visualizing a 40-Year Investment in the S&P500

To overcome these limitations, I used Monte Carlo simulations to visualize what volatility means for investment returns and risk. For simplicity, I assumed an investment in the S&P500 would see annual returns that look like a normal distribution based on how the S&P500 has performed from 1928 – 2019. I ran 100,000 simulations of 40 years of returns and looked at what sorts of (geometric mean) returns an investor would see.

This first chart below is a heatmap showing the likelihood that an investor will earn a certain return in each year (the darker the shade of blue, the more simulations wound up with that geometric return in that year).

Density Map of 40-Year Returns for Investment in S&P500
Densities are log (base 10)-adjusted; Assumes S&P500 returns are normally distributed (clipped from -90% to +100%) based on 1928-2019 annual returns; Years go from 0-39 (rather than 1-40)

This second chart below is a different view of the same data, calling out what the median return (the light blue-green line in the middle; where you have a 50-50 shot at doing better or worse) looks like. Going “outward” from the median line are lines representing the lower and upper bounds of the middle 50%, 70%, and 90% of returns.

Confidence Interval Map of 40-Year Return for Investment in S&P500
(from outside to middle) 90%, 70%, and 50% confidence interval + median investment returns. Assumes S&P500 returns are normally distributed (clipped from -90% to +100%) based on 1928-2019 annual returns

Finally, the third chart below captures the probability that an investment in the S&P500 over 40 years will result not in a loss (the darkest blue line at the top), will beat 5% (the second line), will beat 10% (the third line), and will beat 15% (the lightest blue line at the bottom) returns.

Probability 40-Year Investment in S&P500 will Exceed 0%, 5%, 10%, and 15% Returns
(from top to bottom/darkest to lightest) Probability that 40-year S&P500 returns simulation beat 0%, 5%, 10%, and 15% geometric mean return. Assumes S&P500 returns are normally distributed (clipped from -90% to +100%) based on 1928-2019 annual returns

The charts are a nice visual representation of what uncertainty / volatility mean for an investor and show two things.

First, the level of uncertainty around what an investor will earn declines the longer they can go without touching the investment. In the early years, there is a much greater spread in returns because of the high level of volatility in any given year’s stock market returns. From 1928 – 2019, stock markets saw returns ranging from a 53% increase to a 44% drop. Over time, however, reversion to the mean (a fancy way of saying a good or bad year is more likely to be followed by more normal looking years) narrows the variation an investor is likely to see. As a result, while the median return stays fairly constant over time (starting at ~11.6% in year 1 — in line with the historical arithmetic mean return of the market — but dropping slowly to ~10% by year 10 and to ~9.8% starting in year 30), the “spread” of returns narrows. In year 1, you would expect a return between -21% and 44% around 90% of the time. But by year 5, this narrows to -5% to 25%. By year 12, this narrows further to just above 0% to 19.4% (put another way, the middle 90% of returns does not include a loss). And at year 40, this narrows to 4.6% to 15%.

Secondly, the risk an investor faces depends on the return threshold they “need”. As the probability chart shows, if the main concern is about losing money over the long haul, then the risk of that happening starts relatively low (~28% in year 1) and drops rapidly (~10% in year 7, ~1% in year 23). If the main concern is about getting at least a 5% return, this too drops from ~37% in year 1 to ~10% by year 28. However, if one needs to achieve a return greater than the median (~9.8%), then the probability gets worse over time and gets worse the greater the return threshold needed. To beat a 15% return, in year 1, there is a ~43% chance that this will happen. But this rapidly shrinks to ~20% by year 11, ~10% by year 24, and ~5% by year 40.

The Impact of Increasing Average Annual Return

These simulations are a useful way to explore how long-term returns vary. Let’s see what happens if we increase the (arithmetic) average annual return by 1% from the S&P500 historical average.

As one might expect, the heatmap for returns (below) generally looks about the same:

Density Map of 40-Year Returns for Higher Average Annual Return Investment
Densities are log (base 10)-adjusted; Assumes an asset with normally distributed annual returns (clipped from -90% to +100%) based on 1928-2019 S&P500 annual returns but with 1% higher mean. Years go from 0-39 (rather than 1-40)

Looking more closely at the contour lines and overlaying them with the contour lines of the original S&P500 distribution (below, green is the new, blue the old), it looks like all the lines have roughly the same shape and spread, but have just been shifted upward by ~1%.

Confidence Interval Map of 40-Year Return for Higher Average Return Investment (Green) vs. S&P500 (Blue)
(from outside to middle/darkest to lightest) 90%, 50% confidence interval, and median investment returns for S&P500 (blue lines; assuming normal distribution clipped from -90% to +100% based on 1928-2019 annual returns) and hypothetical investment with identical variance but 1% higher mean (green lines)

This is reflected in the shifts in the probability chart (below). The different levels of movement correspond to the impact an incremental 1% in returns makes to each scenario. For fairly low returns (i.e. the probability of a loss), the probability will not change much as it was low to begin with. Similarly, for fairly high returns (i.e., 15%), adding an extra 1% is unlikely to make you earn vastly above the median. On the other hand, for returns that are much closer to the median return, the extra 1% will have a much larger relative impact on an investment’s ability to beat those moderate return thresholds.

Probability Higher Average Return Investment (Green) and S&P500 (Blue) will Exceed 0%, 5%, 10%, and 15% Returns
(from top to bottom/darkest to lightest) Probability that 40-year S&P500 returns simulation beat 0%, 5%, 10%, and 15% geometric mean return. Assumes S&P500 returns are normally distributed (clipped from -90% to +100%) based on 1928-2019 annual returns. Higher average return investment is a hypothetical asset with identical variance but 1% higher mean

Overall, there isn’t much of a surprise from increasing the mean: returns go up roughly in line with the change and the probability that you beat different thresholds goes up overall but more so for moderate returns closer to the median than the extremes.

What about volatility?

The Impact of Decreasing Volatility

Having completed the prior analysis, I expected that tweaking volatility (in the form of adjusting the variance of the distribution) would result in preserving the basic distribution shape and position but narrowing or expanding it’s “spread”. However, I was surprised to find that adjusting the volatility didn’t just impact the “spread” of the distribution, it impacted the median returns as well!

Below is the returns heatmap for an investment that has the same mean as the S&P500 from 1928-2019 but 2% lower variance. A quick comparison with the first heat/density map shows that, as expected, the overall shape looks similar but is clearly narrower.

Density Map of 40-Year Returns for Low Volatility Investment
Densities are log (base 10)-adjusted; Assumes S&P500 returns are normally distributed (clipped from -90% to +100%) based on 1928-2019 annual returns but with 2% lower variance. Years go from 0-39 (rather than 1-40)

Looking more closely at the contour lines (below) of the new distribution (in red) and comparing with the original S&P500 distribution (in blue) reveals, however, that the difference is more than just in the “spread” of returns, but in their relative position as well! The red lines are all shifted upward and the upward shift seems to increase over time. It turns out a ~2% decrease in variance appears to buy a 1% increase in the median return and a 1.5% increase in the lower bound of the 50% confidence interval at year 40!

The probability comparison (below) makes the impact of this clear. With lower volatility, not only is an investor better able to avoid a loss / beat a moderate 5% return (the first two red lines having been meaningfully shifted upwards from the first two blue lines), but by raising the median return, the probability of beating a median-like return (10%) gets better over time as well! The one area the lower volatility distribution under-performs the original is in the probability of beating a high return (15%). This too makes sense — because the hypothetical investment experiences lower volatility, it becomes less likely to get the string of high returns needed to consistently beat the median over the long term.

Probability Low Volatility Investment (Red) and S&P500 (Blue) will Exceed 0%, 5%, 10%, and 15% Returns
(from top to bottom/darkest to lightest) Probability that 40-year S&P500 returns simulation beat 0%, 5%, 10%, and 15% geometric mean return. Assumes S&P500 returns are normally distributed (clipped from -90% to +100%) based on 1928-2019 annual returns. Low volatility investment is a hypothetical asset with identical mean but 2% lower variance

The Risk-Reward Tradeoff

Unfortunately, it’s not easy to find a “S&P500 but less volatile” or a “S&P500 but higher return”. In general, higher returns tend to go with greater volatility and vice versa.

While the exact nature of the tradeoff will depend on the specific numbers, to see what happens when you combine the two effects, I charted out the contours and probability curves for two distributions with roughly the same median return (below): one investment with a higher return (+1%) and higher volatility (+2% variance) than the S&P500 and another with a lower return (-1%) and lower volatility (-2% variance) than the S&P500:

Probability Low Volatility/Low Return (Purple) vs. High Volatility/High Return (Gray) Exceed 0%, 5%, 10%, and 15% Returns
(from top to bottom/darkest to lightest) Probability that 40-year returns simulation for hypothetical investment with 1% higher mean and 2% higher variance than S&P500 (gray) and one with 1% lower mean and 2% lower variance than S&P500 (purple) beat 0%, 5%, 10%, and 15% geometric mean return. Both returns assume normal distribution clipped from -90% to +100% with mean/variance based on 1928-2019 annual returns for S&P500.

The results show how two different ways of targeting the same long-run median return compare. The lower volatility investment, despite the lower (arithmetic) average annual return, still sees a much improved chance of avoiding loss and clearing the 5% return threshold. On the other hand, the higher return investment has a distinct advantage at outperforming the median over the long term and even provides a consistent advantage in beating the 10% return threshold close to the median.

Takeaways

The simulations above made it easy to profile unconventional metrics (geometric mean returns and the probability to beat different threshold returns) across time without doing a massive amount of hairy, symbolic math. By charting out the results, they also helped provide a richer, visual understanding of investment risk that goes beyond the overly simple and widely held belief that “volatility is the same thing as risk”:

  • Time horizon matters as uncertainty in returns decreases with time: As the charts above showed, “reversion to the mean” reduces the uncertainty (or “spread”) in returns over time. What this means is that the same level of volatility can be viewed wildly differently by two different investors with two different time horizons. An investor who needs the money in 2 years could find one level of variance unbearably bumpy while the investor saving for a goal 20 years away may see it very differently.
  • The investment return “needed” is key to assessing risk: An investor who needs to avoid a loss at all costs should have very different preferences and assessments of risk level than an investor who must generate higher returns in order to retire comfortably, even at the same time. The first investor should prioritize lower volatility investments and longer holding periods, while the latter should prioritize higher volatility investments and shorter holding periods. It’s not just a question of personal preferences about gambling & risk, as much of the discussion on risk tolerance seems to suggest, because the same level of volatility should rationally be viewed differently by different investors with different financial needs.
  • Volatility impacts long-run returns: Higher volatility decreases long-term median returns, and lower volatility increases long-term returns. From some of my own testing, this seems to happen at roughly a 2:1 ratio (where a 2% increase in variance decreases median returns by 1% and vice versa — at least for values of return / variance near the historical values for S&P500). The result is that understanding volatility is key to formulating the right investment approach, and it creates an interesting framework with which to evaluate how much to hold of lower risk/”riskless” things like cash and government bonds.

What’s Next

Having demonstrated how simulations can be applied to get a visual understanding of investment decisions and returns, I want to apply this analysis to other problems. I’d love to hear requests for other questions of interest, but for now, I plan to look into:

  • Diversification
  • Rebalancing
  • Withdrawal levels
  • Dollar cost averaging
  • Asset allocation
  • Alternative investment return distributions
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Searching for a Narrative

I love the webcomic XKCD. Not only is it incredibly nerdy, its surprisingly on-point in terms of its take on reality. I found the comic below to be a good example of this:

sports

But whereas I find sports commentary to be somewhat plausible (because its about a specific person or small group of individuals that you might be able to interrogate and make inferences about), I think this is especially true on press describing the stock market.

Take the recent massive market downturn which occurred on Thursday, Aug 4th. Almost immediately, every press outlet had to have an explanation – people talked about fears of a Eurozone crisis, fears that the US and Chinese stimulus which have propped up global demand would vanish, fears that the US would be downgraded, and even talks that this was the media’s fault or the role of greedy banks using flawed computer systems.

The question that you never hear the press answer but which may be more relevant than all these narratives: is it even possible to know? You can’t ask the market what its thinking in the way that you might be able to ask a sports player or even a sports team, and its hard to run controlled experiments in the way a scientist might. And, while the psychology of the buyers and sellers certainly plays a big role, I think the simple truth is this: there is no real way to know, and its not only pointless to speculate but possibly counterproductive to try to explain the market’s movements. We’re all  hardwired to want a reason for something which is insightful and reveals something – but the fact of the matter is that trying to find reasons that aren’t necessarily there or even possible to validate pushes people into investing time and energy trying to control or understand things they can’t.

In my mind, its far better to take Warren Buffett’s approach: don’t waste your time on things you can’t predict or control or understand, take what you can get (the price of a stock or an asset) and make a decision based on that. Who cares why someone is offering to sell you something for $100 that is worth $200 – just make the right choice.

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The Making of an American Capitalist

buffettIf you’re interested in investing, regardless of your investment philosophy/strategy, you should be interested in Warren Buffett, potentially the most successful investor of our time. And, if you’re interested in Warren Buffett and investing, you should read Roger Lowenstein’s biography: Buffett: The Making of an American Capitalist.

Lowenstein’s biography is not “official” in the sense that Buffett did not go out of his way to authorize or support it. But, despite this (and maybe even because of this), it does a great job of delving into the psychology of the “Oracle of Omaha.” It doesn’t pretend to fully grasp the inner workings of Buffett’s mind (otherwise, I believe Mr. Lowenstein would be better suited investing full time in the stock market rather than writing biographies! :-)), but by diving into Buffett’s childhood and by closely examining what the individuals/companies who interacted with Buffett had to say, the book does a great job at painting with broad strokes a hint at Buffett’s character and thinking style.

Rather than the image of an entitled baron or a megalomaniac with a god complex that one might expect from someone with Buffett’s investment record, the biography portrays an insecure and, at times, painfully awkward person who lived in the same house for decades, seemed to subsist entirely on Cherry Coke, hamburgers, and steaks, was (and probably still is) a remarkable cheapskate, and found it difficult to communicate with his family. And yet, what is particularly inspiring is that these very human, “Main Street” traits seemed to have been at the core of his success. Where “more sophisticated” investors chased new gimmicks involving leverage and options and forecasts, Buffett stuck with very simple ideas, like making sure you have a “margin of safety” before making an investment, being patient, and not venturing out into things he did not understand. And, while those other “more sophisticated” investors came and went, Buffett has stuck through year after year delivering impressive results.

More so than that, while Lowenstein doesn’t paint Buffett out to be a saint – he is interested in making money after all – I was amazed by the degree to which Buffett has seemed to retain a moral character that so many financiers have not. Case in point: the end of the book covers Buffett’s takeover of Salomon Brothers after it was embroiled in a Treasury bond scandal that could have seen the firm collapse. Buffett played a major role in helping to save the company, instituting practices which many in the industry (and even at the firm) viewed as naïve – such as bonuses tied to actual performance, a willingness to be truly open book with regulators, and a desire to not just “not cross the line” by employing huge legal teams, but to genuinely stay “way, way away from the line.”

Heck, I’m still shocked Buffett does all of his own taxes!

But, coming back to what I took away from the book as the reasons for Buffett’s success, I saw four overarching themes:

  1. Keep it simple. Buffett’s entire investment philosophy seems to break down into three fairly simple ideas: (1) companies have an intrinsic value that their stock prices will ultimately reflect, (2) the market can be thought of as a manic-depressive named Mr. Market – on his manic days, he quotes very optimistic prices, but on his depressed days, he quotes very pessimistic prices – who doesn’t take offense if you ignore him, and (3) its hard to precisely determine a company’s intrinsic worth, but when that number and the number Mr. Market quotes are far apart, you have a “margin of safety” to make a decision. None of these ideas are particularly complicated or require extremely sophisticated analysis, and the allegory of “Mr. Market” feels more folksy than Ivy League-MBA-quant-heavy-investment-guru. Yes, there is certainly sophisticated thinking and math behind evaluating intrinsic worth, but in reading the book and even in reading Buffett’s essays, you find that his thinking often boils down to other folksy tenets like when he explained one of the main reasons he invested in Coca-Cola, “if you gave me $100 billion and said take away the soft drink leadership of Coca-Cola in the world, I’d give it back to you and say it can’t be done.”
  2. Be consistently honest and moral. I think the book, and the general investing community, focus too little on this. One reason Buffett was able to buy (or at least buy large stakes) in so many companies and inspire loyalty from those who worked with him was that people liked him because they knew he was consistently honest and moral. Its how he was invited on the board of the Washington Post despite the CEO’s initial fear of him as an investor who wanted to take over the company. Its how, in the 1980s, he was able to get into companies who wanted to avoid the aggressive and hostile overtures of the corporate raiders of that generation. And its how he was able to maneuver through the handful of times he or one of his acquisitions had to deal with legal trouble. Hard to dig up skeletons when there aren’t any, and its hard to mistrust someone who will probably always be straight with you.
  3. Be patient. If there’s one thing I’ve observed its that Buffett is remarkably patient. Many of the assets he buys/invests in, he holds for years (and in some cases, he has never let go of or sold). He is also willing to take the time to thoroughly understand a business before jumping in rather than simply jump in on a “hot stock tip” or because the stock price seems to be on an upward trend. But, and what is potentially the most important, he is willing to wait until all the conditions are right before jumping in. In the case of Buffett’s Coca Cola investment, although the stock price had been cheaper before Buffett bought in, Buffett was waiting for the right management with the right focus on growing shareholder value to be set in to invest.
  4. Read everything. One thing which Lowenstein referred to over and over again was the degree at which the CEOs Buffett invested in were surprised at how well Buffett understood the underlying business and the issues facing it. Granted, Buffett seemed to read financial reports and statements for fun, but I think the key takeaway there is that nothing can replace solid fundamental research.

If you’re interested in investing or a biography of one of the world’s most fascinating and successful people or just want a cool look at the inside of a few decades of business, I’d highly recommend the book.

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Making Macro Manageable

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On the advice of one of the members of the little investment club I am a part of, I picked up David Moss’s book A Concise Guide to Macroeconomics. I wasn’t expecting much, having taken a few introductory economics courses in college and being a casual economics aficionado, but I gave it a shot.

And, I think that the subtitle “What Managers, Executives, and Students Need to Know” is simultaneously very appropriate and a dramatic underselling of the book. Moss’s writing style and his very direct, conclusion-oriented (as opposed to “scholarly”) overview of basic macroeconomics makes the book not only accessible to people who need a working understanding of economics but not the extra academic theory, but also a great reference.

Now, if you’re an economic genius, or have even just taken basic economics, then you’re not going to learn anything earthshattering from the book, but what you may get out of it which could be just as valuable is a different way of thinking through or of explaining macroeconomic concepts.

Case in point: I had never thought of “the job of a pension system is to divide national output between active workers and retirees.” While this is a simple and obviously true statement, Moss uses that underlying “framework” to explain why moving existing Social Security/pension plans to an IRA (stock-based) retirement system is unlikely to fundamentally solve anything:

Although all of us are accustomed to thinking that we can sell our financial assets for cash at a moment’s notice and then use the cash to buy goods and services, this obviously wouldn’t work if everyone tried to do it at once. If a large number of senior citizens liquidated their financial assets at the same time, in order to buy needed goods and services, they would soon find that the proceeds were much smaller than they had expected. Simply giving the elderly more pieces of paper – more stocks and bonds – does not guarantee that there will be more output for them to consume in the future …

The key question from a macroeconomic standpoint, therefore, is not whether the senior citizens of tomorrow have IRAs or traditional Social Security benefits, but whether they (or others) reduced their consumption to prepare for their eventual retirement. Unless savings are increased today, the division of output between active workers and retirees will be no less onerous tomorrow, regardless of whether we have a fully funded pension system based on individual accounts or a traditional pay-as-you-go system based on payroll taxes …

The amount of output a country produces is its ultimate budget constraint, regardless of how many stocks or bonds or Social Security cards may be floating around. Unless its output grows, a country cannot give more to its retirees without giving less to its workers.

Maybe you didn’t hear anything new there – and if so, pat yourself on the back as you are far smarter than I am – but I was blown away by the simplicity of Moss’s explanation of what is a very complicated problem. Mind you, he doesn’t have an answer to out-of-control entitlement programs like the one the US has, but being able to break this down only pages after explaining the different things that make up and affect GDP (national economic output) was impressive to me. And the cool thing is that Moss does this several times, explaining, for instance, why boosting monetary supply (i.e. when the Federal Reserve cuts interest rates) may have a certain effect on the exchange rate in the short-term but a different one in the long-term and how an “unsustainable current account deficit” (i.e. huge trade deficits) might look like a “high degree of investor confidence” at first.

If you’re interested in macroeconomics casually or as a business-person who needs a better grasp of it in his or her job, I’d highly recommend the book.

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2011 Goals

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I made a list of New Year’s resolutions last year which I am happy to say I did fairly well at achieving.

But, when I sat down again to think up what I wanted to publically commit to, I realized something. The list borrows heavily from last year, with a few slight modifications:

    • Read and blog one scientific paper a month: It’s been a staple of this blog, a lot of fun to read/write, and has succeeded in keeping me at least partially grounded in the science world that I once came from and, to some extent, still feel a part of.
    • Instead of reading Checkmate and Pawn in Frankincense from Dorothy Dunnett’s Lymond Chronicles, this year I’m aiming to read Queen’s Play – the last book in the series which my girlfriend and good friend recommend I read.
    • Continue to expand my network – I think I’ve graduated from super-shy wallflower, but I’m still not completely comfortable with the whole networking thing. The good thing about being in venture capital is that networking not only comes with the job but comes a lot easier when people know that your company has money and connections, so I will continue to work on breaking out of my introvert shell
    • Finish a rev 1 of Benchside – I failed last year, but gosh-darn-it, I will succeed this year!

But, in addition, I want to add a few other items:

    • Build out a public version of Iggregate – In addition to Benchside, I’m also working on a project I call Iggregate which I am hopeful I will be able to take the wraps off in this new year. It’ll be tough, especially with the Benchside goal, work, and my general programming incompetence, but you gotta aim high to make it anywhere big!
    • Improve my Chinese – The venture fund I work for is unique in its strong presence in US, Japan, and China. And, while my job and interests are focused on venture opportunities in the US, our recent company offsite has convinced me that I should improve my Chinese speaking and comprehension. I don’t intend (not that I ever could get) to be as fluent as a native speaker, but my goal is to be able to understand enough business Chinese that I can participate in my fund’s China team meetings without falling back on English.
    • Build out investment portfolio with more than just index funds – My traditional investment strategy has been heavily reliant on index funds due to lack of time, lack of training/practice, and fear of conflicts of interest from my consulting career. Now that I’m no longer a consultant, employed in an investment industry, and have a good friend from college who’s very interested in deploying his capital effectively and has the time to think about this non-stop (because he’s in business school), I think its about time I graduate from the low-risk, low-reward world of index funds and reallocate so my investment portfolio is 1/4 to 1/3 in specific stocks/commodities

Truth be told I’m starting to feel a little nervous about committing to all of this – but I’m also a little excited to get started. Happy New Year everybody! And good luck with those resolutions!

(Image credit)

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