In finance, risk and return are considered to be two sides of the same coin – you can’t have one without the other. Return is pretty straightforward. We buy a stock, wait a year, and then check our brokerage account. It’s increased in value, giving us a decent return of 10%. Nice! So the return of an investment, the 10% that we earned, is something that we can directly observe.
What about the risk? That’s much less obvious. To most people, the risk of an investment is the probability that it will lose money. But ex-ante, that’s really hard to estimate. We can only see what actually happens. The other alternate realities where your investment did not perform as well (or performed even better) cannot be directly observed. We only see the set of events that led to us making 10% on our investment. Conversely, that means that we will never really know just how much risk we assumed in order to earn that 10%.
The Drivers Of Risk
There are two primary drivers of risk in investing:
- The inherent quality or lack of quality of a particular stock or bond. An investment in a company (this applies to both stocks and bonds) is a claim on that company’s future cashflows. Let’s focus on stocks for this post. If we invest in a company’s stock, then we are betting that the company will consistently earn more and more money making our share of the company (buying a few shares in a company grants us a tiny percentage ownership in that company) increasingly valuable. To do that, the company needs to both stay in Business and grow its business. Companies like Google that have strong brands, balance sheets, and business models are much more likely to do that compared to companies like WeWork that have weak business models and terrible management. So if the underlying business declines in quality, then that will increase the riskiness of the investment, all other things equal.
- Uncertainty around our analysis, a.k.a. the risk of being wrong. When we analyze a company, we can never be sure that we’ve uncovered everything, or that we have analyzed and understood everything correctly. If we incorrectly classify a business model that is actually flimsy as strong, then our investment performance will suffer.
Both of these are difficult and time consuming to estimate for one investment, let alone a portfolio of hundreds or even thousands of investments. So in quantitative Finance, the standard deviation of an investment’s return (often referred to as its volatility) is often used as an approximation for its risk.
Calculation Details: In order to estimate an investment’s risk (a.k.a. volatility), we would collect several years of say weekly returns data (each observation is the investment’s return over the specified week) and take the standard deviation of that. If we want to annualize it, then we would multiply the calculated value by the square root of 52.
Standard Deviation As An Approximation For Risk
Let’s first think about why standard deviation might be used as an approximation for risk. In statistics, the higher the standard deviation around our estimate of something, the less certain we are of its true value.
Take a look at the 2 bell curves below. They have different standard deviations, which show up on the plot as different distribution widths. When the distribution is narrow, that means the true value lies within a very small range – or in other words, we are pretty sure what the value is. When the distribution is very wide, we are very unsure what the value is and must entertain the possibility that it could be equal to any number of values.
So in statistics, a large standard deviation equates to high uncertainty. But is that truly risk? Let’s use an example to examine this more deeply. Say we believe that the expected annual return for a portfolio of stocks to be 7% with a standard deviation of 14%.
This means that while we expect stock returns to generally be positive, there is also a significant chance that they will end up negative (given our statistical assumptions, there is a 30% chance to be exact). So the higher the standard deviation of an investment, the higher the probability that its return might end up below zero, all other things equal. But volatility works symmetrically – while there might be a 30% chance of experiencing negative returns, there is also a 30% chance that we get returns of 14% or more:
Also, don’t forget that there are two key properties to any normally distributed random variable – the mean and the standard deviation (for a refresher on the normal distribution, check out my previous blog here). In this case, the mean is the expected return of our investment (7%). In finance, we typically ignore the mean when calculating the risk of an investment and look only at the standard deviation. If we step back and think about it though, that seems counter intuitive. Take a look at the following two histograms depicting the risk and return of 2 different investments? Which one looks riskier?
Finance theory would tell us that investment B is riskier since it has a wider distribution (higher standard deviation). But it also has a much higher return, so the chance of experiencing negative returns on an annual basis are lower than investment A’s. So in terms of trying not to lose Money, investment B is actually the safer (less risky) option. The key point is that if the investment’s expected return is high enough, even a very high standard deviation (relative to other investments) is not a big deal – though you better double and triple check how you estimated that expected return. Thus, if we think of risk as the probability of losing money, an investment with a low expected return relative to its standard deviation (and a high probability of producing negative returns) is the truly risky one. So it looks like the expected return of an investment is fundamentally tied to the risk of that investment.
That is, we need to consider both the mean and the standard deviation when we calculate risk.
So how does the finance industry get away with ignoring an investment’s expected return and using just its standard deviation to estimate risk?
Efficient Markets
It’s because the models assume that markets are generally efficient. Meaning that financial markets as a whole (all investors in aggregate) are assumed to do a good job of pricing assets.
The price you pay for an investment determines its return, all other things equal. If you pay $2 million for a McDonald’s restaurant, you will probably earn a decent return (over a long period of time). If you are able to buy that same restaurant for $1 million, you will earn a much higher return. And if you get suckered into paying $4 million, you may never earn back your initial investment.
So if markets do a good job of assimilating information and assessing risk, then it follows that they do a good job of pricing investments. And if they price investments correctly, then that means capital markets also assess expected returns correctly.
Now let’s think about what the implications of this are. In an ideal world where markets correctly assign price (and return), how would these prices be set? Prices would vary based on whatever factor investors cared about the most. Finance theory assumes that this all important factor is risk – the riskier an investment, the lower the price (relative to the cashflows it produces), and the higher the expected return. That is, there is assumed to be a roughly linear relationship between risk and return:
If you think about it, this is a tremendously simplifying shortcut. By assuming that markets are efficient, we no longer have to worry about how to figure out the expected return – rather, we rely on the fundamental belief that the risk of an investment determines its price and therefore its return. Or in other words, expected return is assumed to be a clearly defined function of volatility and nothing else. And if we run with the assumption that volatility determines return, we can now safely ignore the expected returns component when analyzing an investment’s risk.
So efficient markets would tell us that investment B is not possible. Either we are calculating its return to be way too high or its standard deviation to be way too low . Compared to investment A (and the other investments in the world), the risk and return characteristics of investment B are "out of whack with reality" (violates the function) and therefore impossible. After all, as adherents to market efficiency like to say, "there is no free lunch".
But Does It Make Sense?
The premises of efficient markets are powerful. Believing in market efficiency is basically a belief in the wisdom of crowds. That millions of professionals have pored over the data for countless hours – so any insight you think you might have on a stock is not actually an insight, but rather information that has already been incorporated into the stock’s price. Moreover, as long as there is a diversity of opinions, the errors that the individual constituents of the market make will diversify (similar to a random forest) and the whole will produce an output that is on average correct.
At a high level this does sound sensible, but here are some counterpoints as to perhaps why markets might not always be the best price-setters:
- The madness of crowds also applies. When there is a diversity of opinions, errors diversify. But when there is no diversity and everyone more or less believes the same thing, errors magnify. This is how investment bubbles like Bitcoin can occur.
- Not every investment exists in a market with enough buyers and sellers (or data) to efficiently set prices. For example, I would characterize the market for startup equity as extremely illiquid, opaque, and inefficient (anyone unlucky enough to be paid in WeWork stock in 2018 or 2019 probably doesn’t think that their strike price was an efficient one). It is also one with an extreme lack of diversity in terms of viewpoints and opinions.
- During boom times, markets tend to rise gradually (accompanied by a gradual decline in volatility). Thus, as the memory of the last crash fades, and the economic expansion becomes a relatively larger portion of our sample, our models will start to understate risk. While you could argue that we care more about the relative risk (how the volatility of one investment compares to that of another or to that of the overall market), the finance industry does use absolute volatility and metrics derived from it to risk budget (i.e figure out how much money to put in riskier investments and how much to keep as risk free cash). So there is the danger investment managers taking on more and more risk as volatility declines (by adding money to riskier investments like stocks) and the next recession looms.
- Market crashes happen very quickly and are generally measured in weeks if not days. The large negative returns of a crash usually cause volatility to spike, tripping risk alarms all over the place. All of a sudden, investment managers everywhere deem their portfolios to be extremely over-risked (because the volatility of investments and correlation between investments have increased so much), and there is a mad dash to sell risky investments as fast as possible.
- The impact of a rise in correlations on total portfolio risk can be massive. Those Thai stocks that proved diversifying against your US stocks during the good times will almost certainly drop in lockstep along with all of your other stock investments in a market crash. In a crisis, correlations go to 1.
Notice something odd about the behavior I described in bullets 3 and 4? It’s classic buy high sell low. In retrospect, that’s not very rational or profitable behavior. But it’s a byproduct of fear and the belief that volatility equals risk. Another reason that it occurs is that as managers scramble to revise their risk estimates upwards, they hold their expected return estimates constant. This makes the average risky investment look a lot less attractive due to it suddenly having a much higher volatility while still having the same expected return.
There is actually a more sensible way to go about things that doesn’t require us to break the link between volatility and risk (and throw away all its simplifying benefits). Volatility is a dynamic value, constantly changing as the prices of investments fluctuate.
Rather than view volatility as an unbiased and accurate estimate of the risk of an asset, we should view volatility like an insurance quote. The price of, say, hurricane insurance is primarily a function of two things – the perceived riskiness of the disaster (which affects the number of people looking to buy it, a.k.a. demand) and the number of companies willing to underwrite (supply). Similarly, at any point in time, the volatility of an investment is a function of two things – the perceived riskiness of the investment and the number of people willing to bear that risk.
Right after a hurricane, everyone becomes massively aware of how destructive the storm can be, thus demand for insurance is very high. At the same time, insurance companies that wrote a few too many contracts for too little in premium during sunny times are reeling, so the supply of insurance is very low. If you are able to step in at this time to provide insurance, you would probably reap significant financial rewards, due to the extremely high premiums you could earn.
It works the same way in finance. When the volatility of an investment changes, it’s because the perceived riskiness of the investment is changing along with the amount of investors willing to own that risk. During good times, strong growth and profits reduce the perceived riskiness of investments like stocks, and strong returns make more and more investors willing to own stocks. The net effect of all this is a sustained decline in volatility for as long as the good times persist. But just like with insurance, if no one is concerned with the risk of stocks and everyone is willing to own them, that must mean that stock prices are abnormally high, and that their expected returns in the short to medium term going forward will be abnormally low. And when markets crash and companies’ profits crater, the opposite will occur. Now stocks will be perceived as extremely risky at the same time that the pool of willing owners shrinks dramatically (due to fear, job losses, etc.) – making expected stock returns abnormally high.
If we truly believe that there is a link between risk and return, then as volatility declines, we should be reducing our estimates of expected returns at the same time. This reduction in expected return would offset the decrease in volatility and neutralize the urge to buy more stocks (it might even make us want to reduce our stock holdings). And when volatility shoots up, we should revise upwards our estimates of expected return, which would get rid of the urge to sell low and may even motivate us to buy low instead.
Conclusion
Investment risk is an extremely complicated subject, and one that is debated over constantly. In this post, I did my best to give you the 10,000 foot view of how things work. There are parts of investment risk theory that I don’t agree with, but overall it is a really useful tool that touches so many aspects of modern finance. But like any other tool that we decide to start using, we should first learn its limitations. I hope this was helpful. Cheers!
