"You can't expect big rewards unless you're willing to take big risks."

—Guy who is about to walk into traffic while blindfolded

Financial markets make it very easy to earn more money over time through the simple expedient of taking more risk: cash has no volatility, so a risk/reward analysis of it gives nonsense answers unless you start with the premise that the return on a risk-free, infinitely liquid asset like cash in a bank account or a money-market fund is the benchmark against which all other assets' risk/reward relationship is measured. So that's what investors do: they think about total returns by looking at the excess return from allocating to a particular asset class or investing style. And, as it turns out, many different asset classes converge on roughly the same risk-reward tradeoff: for every 2.5 points you add to the annualized standard deviation of an asset, you pick up around a point of return. There's some noise in the data, in part because we don't really have a robust sample size going back far enough say this definitively, but that's a decent approximation.

Over time, asset allocation decisions that can be cheaply expressed through systematic rules should roughly approach this ratio, unless they happen to have an inverse correlation with the biggest asset classes.¹ And this applies not just to strategies but to flavors of strategies: momentum and value investing both work, historically, but if every investor knows this and acts accordingly, their excess return has to drop to the overall risk/reward tradeoff the rest of the market offers.²

The temptation is to think that if risk and reward are correlated—if you can reach for higher returns by selling your investment-grade bonds and replacing them with stocks, or selling your large-cap US stocks and replacing those with wilder emerging market stocks—that all forms of risk-taking are rewarded with higher returns. But this isn't true: the expected return of a single stock of average volatility is the same as the expected return of the market, but the volatility of one stock, compared to a diversified portfolio, is much higher. And thanks to the relentless math of "volatility drag," i.e. the observation that you don't offset a -50% loss until you've made a +100% return, the expected overall outcome from this particular form of risk-taking is negative.

And there are other forms of unfavorable risk/reward tradeoffs: paying higher fees for exposure to the same asset class, for example, is a straightforward cut to returns with no attendant reduction in risk. In the general risk/reward tradeoff, there are some extreme cases where things don't work as planned: the longer-term a bond is, the more it's a bet on interest rates, and the more it's excess return is compensation for taking interest rate risk. But 30-year bonds don't get returns quite commensurate with their risk—because when bond portfolio managers want to make a big bet on rates, that's what they have to bet on. There's a similar phenomenon where the best-rated high-yield bonds have better returns and the worst-rated investment-grade bonds have worse ones. Many bond fund managers have a mandate with that cutoff, and they'll pay up to take a bigger swing.³ And the smallest stocks and worst-rated bonds have worse risk-adjusted returns than average, too: there are buyers who want a lottery ticket, but they're hard to short. So in this particular instance, taking extra risk doesn't pay. Treat the market like a casino, and you should expect to pay a vig.

How can we square these observations with the fact that many investors have achieved great success through concentrated bets on a handful of positions? One obvious answer is that some of them are lucky; "lottery ticket" is an accurate pejorative term for assets and strategies whose risk isn't commensurate with reward, but lottery winners do exist.

However, there's a subtler answer. The Capital Asset Pricing Model, the broad term for all these risk-reward tradeoffs, assumes efficient markets. And it works fine as a description for the outcome in inefficient markets, too, as long as there are available index funds or if market participants can allocate in a diversified but truly random way. Those concentrated investors are really part of the process by which everyone else can make generalizations about asset classes. As Adam Smith might have said, it is not through the benevolence of the capital structure arbitrageur, the macro trader, the private equity acquirer⁴ , or the asset allocator that we expect sharpe ratios of different asset classes to converge, but from their regard to their own self-interest.

And this actually provides a framework for thinking about whether active management is worth it: there's some cost in time to coming up with ideas, and there's a tradeoff between a portfolio of, say, forty pretty decent opportunities, which will have volatility that's hard to distinguish from the market itself, and a portfolio of, say, five extremely high-conviction trades. Suppose we assume that the average stock has an annualized volatility of 40%, and that the correlation between stock movements is .5. A 1-stock portfolio has annualized volatility of 40%, 5 stocks gets you 26%, 25 stocks gets you 23%, and buying a 500-stock index gets annualized volatility of 22%. If you can beat the market by 1 point a year, a five-stock portfolio actually has a slightly lower Sharpe ratio than just buying an index fund. At half a point of outperformance each year, a 10-stock portfolio also does worse than a broad index.

In one sense, these are modest goals. On the other hand, there are plenty of well-paid professionals with a mandate to get exposure to equity markets who spend an entire career trying to beat the market's average return by a point. And many of them end up looking at more volatile situations in order to do so. Crank up the average volatility of selected stocks to 60% from 40%, and stick with a concentrated five-stock portfolio, and the threshold where outperformance on a per-stock-pick basis translates into risk-adjusted outperformance for the whole portfolio rises to five points.

So another way to frame this is that there are many mediocre tradeoffs between risk and reward, where you can get the same mix of them in different amounts. And there are some bad tradeoffs available, with incremental-reward-free risk. To get a good tradeoff, you need a good reason to expect it: some justified belief that there's a particular asset that's mispriced, and a good meta reason to think that nobody else has considered it. In a competitive market, you need not just a theory about the asset in question, but at least some kind of theory of why your competitors haven't spotted it. Yes, "I'm a better investor than they are" is a theory—but do any of your competitors pick stocks despite consciously thinking they're bad at it? Especially in an environment where cheap index funds are available, you should assume that the average market participant is about as self-confident as you are. And one fun implication of this is that the best returns accrue to people who think they're less skilled than the average stock picker, but skilled enough that they shouldn't leave the whole process up to Standard & Poors.

Which is not to say that investing by picking a small set of smallish, high-variance companies, watching them like a hawk, and enjoying long-term outperformance with some painful dips is a bad trade. It can be a very rewarding hobby! But the opportunity cost is defined by the average performance of index investing. So stock picking ends up being like many other hobbies: it takes time and effort, and if you're very good at it and somewhat lucky, you might break even.

Read More in The Diff

The capital asset pricing model is a subtext to several Diff pieces, including:

• "Passive" as a factor, exploring whether the rise of passive investing creates a new kind of risk.

• We covered why systematic value investing has been so challenged since the financial crisis ($).

• And we covered what happens when you buy at an all-time high.

1. Why?  Because an asset allocator will tend to rebalance into whichever asset class has the best risk-reward and rebalance out of the worst ones. Since new asset classes are almost always smaller than the legacy ones—the only real exception is single-family real estate, but the proportion owned by institutions remains miniscule—this can lead to wild price swings as people crowd into assets that seem to offer better returns and diversification. For example, this paper made a strong case for investing in commodity futures based on historical data, which was immediately followed by that strategy absolutely tanking: in six of the next ten years it was the worst broad category of assets, including a half-decade stretch from 2011-15 in which it was always at the bottom of the list.

2. What's the mechanism by which this could happen? "Value" stocks are statistically cheap, i.e. they trade at a low price to book value or price to some measure of earnings. They're either cheap for a bad reason or cheap for a good one. If everyone over-allocates to value as a category, cheap-for-a-good-reason companies will have more access to capital, and will be less likely to go under and more likely to grow. Cheap-for-a-bad-reason companies also don't stay cheap. So the more investors systematically bet on cheap stocks, as opposed to identifying individual companies that are undervalued, the worse a business the typical value stock will be.

3. This, incidentally, is one reason that so many companies will lever up as much as they can subject to the constraint that they keep their investment-grade rating. It's also part of why AAA-rated companies are so rare—from 60 in 1980 to two today. A high rating leaves money on the table! It would be amusing for someone to start a bond fund whose mandate was to invest in bonds rated BB to single-A, i.e. a narrow slice of the ratings spectrum that overlapped with a visible underpriced/overpriced split.

4. Who is partly betting that stocks are too cheap and bonds too expensive, and whose execution of this bet entails removing a company from the set of public companies while typically issuing some new bonds

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