Staying Sharpe

It’s hard to make decisions in the face of uncertainty, but often we must – whether in the case of our investments or our broader lives.  I find it useful to apply concepts from finance as I reflect on my decisions and prepare to make decisions in the future.  In part, this is because I’m lazy: I get paid to think this way already.  But, unromantic though this may sound, aren’t we all investment managers for ourselves?  We invest our resources (our time, attention, energy, skills) in the hope of achieving some sort of returns (perhaps happiness, comfort, companionship, pleasure) that are in all cases subject to fundamental uncertainties and disruptions.  If we accept this metaphor, perhaps it’s reasonable to think that we could learn something by reflecting on how an investment manager makes her decisions, and how her skill would be measured by others.

Roughly two-thirds of this post will be basic finance, in which I’ll take some shortcuts with your permission.  If you’d like to fast-forward to the punchline (punchparagraphs?), search for ‘dolphins’.

Obviously one important question to us as investors is: what rate of return do we expect to get on the investment?  Mechanically this analysis is pretty simple – it’s a question of cash flows in versus cash flows out, and their respective timing, which a computer can calculate in tiny fractions of a second.  Bear in mind that by treating the future cash flows, the probability of realizing those cash flows, and the time horizon associated with their realization, as discrete and knowable in the examples that follow, I’m glossing over 95% of the art and science of investing.  Let’s just assume for now that we can.  Do not try this at home.

Suppose we buy a piece of paper for $10 today.  We believe there’s an equal probability that the paper will be worth $30, $15, $10, and $5 in one year’s time, and that no other outcome is possible.  We could take a weighted average of those outcomes (i.e., 25% * $30 + 25% * $15 … ) and conclude that the expected (i.e., probability-weighted) value of the paper in one year is $15.  Our expected return would be $5, since we bought it for $10, and so we could think of our return as 50%.  If we wanted extra credit, we’d also look at what return we could have earned on a risk-free investment over the same time horizon.  The yield on a one-year Treasury bill is about 0.23% as of this writing, so we could have earned about a whopping two pennies with our cash if we had not chosen to buy the paper instead.  So we’d really think of our expected ‘excess’ return as being the $4.98 difference between the expected return on our investment and return on a risk-free asset (i.e., $5.00 – $0.02).

A second important question is: how risky is the investment?  Suppose we have two pieces of paper, Paper X and Paper Y.  Paper X will be worth $11 or $9 in one year, with equal probability (i.e., the expected value is $10).  Paper Y will be worth $20 or $0 in one year, also with equal probability, so the expected value is also $10.  Even though each piece of paper has the same expected value, the difference between them is clear.  Paper X has a very tight range of outcomes, but Paper Y is either going to be a home run or an easy fly ball for the outfielder.  Intuivitely, Paper Y seems much riskier, even if we don’t have a well-defined notion of risk.  Fortunately for our intuition, one measure of the risk of an investment is the standard deviation of its expected values (or returns), which essentially computes how wide is the dispersion or range of possible outcomes with respect to the expected outcome; the wider the dispersion, the higher the standard deviation.  The standard deviation of the possible values of Paper Y will clearly be higher than that of Paper X, which accords with our intuition that it has more risk.

The marriage of these concepts – expected return and the risk associated with that expected return – is the core of investment management.  One powerful intuition is that if we have two pieces of paper with the same expected return, we generally prefer the one that gets us there with less risk.  (If you want more extra credit, though, think about situations where that might not be the case.)  Similarly, if we have two pieces of paper with different levels of risk, we would generally demand a higher expected return on the riskier piece of paper in order to prefer it over the less-risky one.  If you would like to see me twitch uncontrollably, say something like, “higher risk means higher return” – a tragically common colloquial bastardization of the preceding concept.  The kernel of truth is that you should demand to be compensated for the risk you take, so a higher-risk investment should have a commensurately higher expected return.  But to see the distinction between the kernel and the bastardization, compare picking up a dollar from the sidewalk with picking one up in front of an oncoming train.

We can use these concepts to characterize the historical performance of investments, and investment managers by extension.  We could look at the manager’s performance over time, e.g., the annual return of her portfolio of investments.  For extra credit, we’d subtract the return she could have earned if the portfolio had been entirely in risk-free investments, which would produce the annual ‘excess’ return of her portfolio.  We could derive a notion of the expected return of her strategy just by taking the average of her annual excess returns.  We could derive a notion of the risk of her strategy by taking the standard deviation of her annual excess returns.  We could even calibrate the returns of her strategy relative to the risk of her strategy by dividing the first quantity (return) by the second (risk), since we’re expressing both quantities as annual percentages.

The result is known as the Sharpe Ratio, which essentially captures how much excess return an asset, strategy, or portfolio generated (or is expected to generate) per unit of risk.  It gives a way of normalizing investments that may have very different return expectations.  An investment with an expected return of 2% and standard deviation 1% would have the same Sharpe Ratio as one with an expected return of 10% and a standard deviation of 5%.  A higher Sharpe Ratio suggests that you are getting more “bang for your buck” when you take risk.  Going back a few paragraphs, assuming Paper X and Paper Y sold for the same price, an investment in Paper X would have a much higher Sharpe Ratio than one in Paper Y.  The Sharpe Ratio doesn’t tell you which of a set of investments is better, but it helps you assess how well compensated you are for the risk you have taken or expect to take.

Which brings me to dolphins.  They actually have nothing to do with what follows, but I needed to pick a word that wouldn’t show up anywhere else in this post.  When I reflect on how I’m managing my portfolio of investments in myself, I find the Sharpe Ratio to be a useful framework.  I think I’ve managed my life to an extraordinarily high Sharpe Ratio.  I’m generally very happy and comfortable, and I’ve been growing my personal capital (i.e., not just money; my rich memories, experiences, network of relationships, professional skills, opportunities, etc.) at a solid rate.  At the same time, I’ve taken very few significant risks, and I think I’m very well insured against downside – again, not just literal insurance; I think my strong relationships with family and friends are like ‘insurance’ against negative experiences like loneliness and fear.  Of course I have rough periods, like everyone, but my choices have been remarkable for how little volatility they’ve introduced for me.

I’m mindful, though, that a higher Sharpe Ratio is not necessarily better.  Some people are perfectly happy to live their lives pursuing incredibly low Sharpe Ratio ambitions – the aspiring actress, the serial entrepreneur.  The likelihood of failure is so much higher than the likelihood of success, and there’s a wide dispersion of outcomes under each of those headings.  These are lifestyles that will have a high standard deviatio in their actual past and likely future returns, but perhaps for them the pursuit of the highest possible ‘highs’ is subjectively worth the risk in a way that something as reductive as the Sharpe Ratio can’t capture.  (Actually, to reinforce my metaphor, it’s well known that the Sharpe Ratio is less useful when applied to non-Gaussian returns…)

The conventional wisdom is that one’s life migrates to a higher Sharpe Ratio as one becomes more of an adult.  A family, a mortgage, a professional reputation – as one acquires them, they (sensibly) constrain the amount of risk that one is willing to take.  If I look at how my own decision-making has changed, particularly in the last year, the remarkable thing to me is that I’ve gone in the opposite direction.  I’ve accepted a lot more volatility than I’m accustomed to (although, objectively, still not a lot!) in pursuit of uncertain higher-highs: relationships that might grow deeper, career trajectories that might be more rewarding, investments in people and causes that might make a big difference – or, in all cases, might not.  The outcomes all remain to be seen, but now that I’ve proven to myself that my return on risk has historically been pretty good, I’m ready to fly a little closer to the sun.


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