Transcription
Hey guys, welcome to this video. In this video, we're going to be talking about the sharp ratio. The sharp ratio is a really simple metric. It's just a ratio, but what we'll see is, is it is at the same time really deep and really useful. Sharp ratios are definitely the number one metric used by quants on Wall Street. And just by trying to talk about the sharp ratio, just trying to even introduce it, we'll be forced to touch on many of the key topics in Quant research and trading. We'll talk about portfolio construction, diversification, correlation, leverage, hedging, and risk. So, so if you're interested in Quant Finance, this is a great place to get started.
Now, to motivate the sharp ratio, we're going to first have to talk about why investing is not only about returns. Why it can't just be about percent returns, which might be counterintuitive. In order to illustrate this point, we're going to start with an example. Here are two investments, green and black. They both returned exactly the same amount. They both returned 40%. The question is, which is better?
Now, in order to answer this, obviously we can't just look at percent returns because they're the same between the two. So we have to look at something that's different. One thing that's different between the two is the path that each took to reach that 40%. If you look at green, the path is pretty steady. In comparison, black wiggles around and moves around a lot more, right? And in particular, has this steep up and down right in the middle here. You probably already intuitively know that these wiggles that black has are bad. Here are a few reasons why.
The first reason the wiggles are bad is that it makes black a lot more painful to hold. Sure, sometimes it is beating green, right? But other times it underperforms by a lot. And in particular, it would be incredibly scary to be all the way down here, not knowing if and when you'll ever recover your investment. From basic trading psychology, we know that these losses are going to loom larger than any gains and make the holding experience a lot more painful for black.
The second reason the wiggles are bad is it gives us less confidence in black going forward, which makes us not really want to invest in it, right? After wiggling around so much, doesn't it seem like black could really do anything? Whereas green seems like it would make steady gains.
The third reason the wiggles are bad is because of liquidity. Liquidity. What happens if we suddenly needed cash? Black wiggles around so much that we could be down when we need the money, and we would be forced to close the position at a loss.
By now, many will recognize that the wiggles here actually represent risk. And we can answer our initial question a lot more directly. Black is a worse investment than green, despite having returned the same, because it took on a lot more risk to get that return. We need a performance metric that reflects this, because simple percentage returns clearly fails.
The first thing we have to do in order to derive this new metric is to quantify risk. Our risk metric should capture this intuition that if the returns graph wiggles a lot like black, there's a lot more uncertainty, pain, or luck involved, and therefore, uh, it's a riskier investment. One of the simplest and most common ways of doing this is by calculating the standard deviation of returns, also known as volatility, denoted by sigma. The formula for volatility is given here. Uh, in words, it's, it computes the average of the square deviations from the mean return. That's quite a mouthful. I think it's a lot easier to understand volatility visually. The graph here represents the average return in the formula, or average, by this orange trend line, and it represents the RT in the formula by, uh, the blue curve, which is the actual investment, of course. But volatility measures is deviations from this trend line, represented by these red arrows. In other words, it's precisely trying to measure the wiggles in the graph that we want to capture.
All right, now that we have our risk metric, we're ready to introduce the sharp ratio, which is a risk-adjusted measure of performance. The sharp ratio can be calculated pretty simply by taking your average daily returns, putting that in the numerator, and dividing that by your vault. So, um, when we just look at this formula, it, it makes a lot of sense. The higher your average returns are, the higher the sharp will be, which is exactly what we want. But also, it's penalized in the denominator by volatility. So the higher your risk, more wiggles there are in your graph, the lower your sharp will be, which is exactly what we want.
Now, one detail I left out here is that typically we multiply this in practice by a factor of 252. And that's because that will annualize the sharp ratio. And, and that will annualize the sharp ratio. And typically, just to keep things apples to apples, we look at things on an annual time frame. Uh, the factor of 252, we won't derive it here, but it comes about because we're using daily data, and there are 252 trading days in a calendar year, excluding weekends and holidays. If you're using monthly data, you would have a square of 12 here instead, because there are 12 months in a year.
So, does the sharp ratio actually work? In other words, is it able to identify that green is the better investment when returns could not? Coming back to our original graph, we calculated the sharp of green versus black as two versus 1/2. So yes, the sharp works. It reflects this intuition that we have that green is the better investment, well, returns alone would say the two are the same. It does this by incorporating volatility to tell you how much return you actually earn per unit of risk you took.
To get an even better sense of what the sharp ratio is measuring, we've plotted hypothetical investments with increasing sharps here: two, five, and 20 sharps. As the sharps increase, the wiggles in the curve get smaller. At a 20 sharp, you basically have a straight line going up and to the right. A 20 sharp would be extremely difficult to capture at scale and represents a near-perfect arbitrage.
So, if a 20 sharp is a bit unrealistic, what's an actual realistic good sharp ratio? Well, the trailing 20-year sharp of the S&P 500, as measured by SPY, an ETF that tracks the S&P 500, is around 0.45. Warren Buffett, one of the most famous investors in the world, has been able to beat the market for a long time and has a sharp ratio of around 0.75. Good hedge funds, however, have sharps in the 2-plus range, and that should really be our goal.
So, we've seen that sharps are useful for evaluating standalone investments. However, just as importantly, they let you evaluate combinations of investments. Here's a quick stylized example to illustrate this point. Red and blue here are both equal sharp investments. They both have a sharp ratio of 2.0. They also have the same volatility and return. So, which of these should we go with? Maybe we're indifferent since they have the same return and sharp, so we'll just flip a coin. Heads, go with red, and tails, with blue.
But actually, there is a best investment that can be carefully chosen here. It can be found by observing that when red zigs, blue zags, and vice versa. In other words, their wiggles are opposite. They are negatively correlated. So, if we combine the two investments, couldn't we cancel the wiggles, reduce risk, and obtain a higher sharp portfolio with a smoother return profile?
In this graph, perp does exactly this. It's a 50/50 portfolio of red and blue. It looks a lot smoother, right? Indeed, while the sharp of red and blue were each two, perp has a gigantic sharp ratio of five, and this is all while maintaining the same solid return. If we had only looked at returns as a performance metric, we wouldn't even be able to tell that perp is any better than red or blue, and the concept of creating perp as a better investment would never even be on our radar. However, when you think in terms of sharps, this becomes too obvious. When people say they're trying to diversify, diversify, or hedge their portfolios, this is exactly what they're referring to. They're trying to create a smoother return profile or investment by combining investments that move in opposing ways, and sharp ratios are one way to tell how effectively we've done that. With returns, you wouldn't even be able to gauge how well you've hedged or diversified your portfolio.
I now want to switch gears and address one of the initial concerns people often have when dealing with sharps. In this graph, I've plotted two investments: one that has a higher sharp (High SR) and one that has a higher return (High R). Which is better? While the smoothness of High SR is nice, in the end, we can't eat sharp ratios, and we'll have more dollars in our pocket investing in the blue line (High R). So, perhaps High R is better. Does that, that somehow make sharps less meaningful to look at? The answer, in the large majority of cases, is no. But to understand this more clearly, we have to understand leverage.
All right, guys, here is a quick primer on how leverage works. Leverage is pretty simply borrowing money to invest more than you own. That's it. It's what you do when you take out a mortgage to invest in a house. Here's a concrete example that applies to finance. Let's say you initially have $100 of cash to invest. This is what we call your equity capital. If you then borrow another $100 to invest, the total amount you invested is $200, and your leverage is 2x. For every 1% gain in the original investment, the stock that you invested in, or you know, crypto, or whatever, you're actually going to earn $2, not $1. You're going to earn $2 because you get a dollar on the cash that you invested, your equity, and then you get another dollar on the $100 you borrowed, right, your leverage. So you get a 2% return on your actual equity, the original $100 that you invested. In other words, dollar P&L swings are doubled, and so returns are doubled. You earn 2% rather than 1%. It's not too hard to show that volatility is also doubled, and that's because if the investment increases by 1%, you get $2. But on the flip side, if it decreases by 1%, you also lose double, or you lose $2. So the key point here is that leverage amplifies both returns and volatility. In a financial or investment context, and so sharp ratios remain constant. So if you took 2x leverage, you'd have 2x returns, 2x volatility, but the sharp would remain the same. Same applies for 3x leverage, 4x leverage, or whatever other leverage that you took.
All right, guys, so given this result around leverage, how do we actually use it? This is the key takeaway. You can use leverage with high sharp strategies to target even higher returns while maintaining the identical sharp. Right? So, and this is, this is going to be why, um, in many cases, high sharp strategies are better than lower sharp strategies, even if the lower sharp strategy had somewhat higher returns, because you can simply just like lever up the higher sharp strategy to obtain higher returns and maintain the same high sharp.
So, back to our original question here of whether High R or High SR is better. Given these results around leverage, we could just lever up the high sharp, this orange line, the High SR portfolio, to obtain equal or higher returns in the high red portfolio while maintaining that same nice high sharp. The graph shows High SR, the original orange line, levered up to 2x and 3x leverage. As you can see, we can maintain the smoothness, nice smooth return profile of the original orange line. Um, actually, the profiles are identical in terms of their wiggles, and at the same time, obtain, you know, the same or even higher returns than, than High R, the blue line, right? So you really do get the best of both worlds. And to answer our initial question, the best investment here is going to be a likely a leveraged version of the High R portfolio. Leverage, it does bring with it some risks and costs, and users should manage those carefully. If you take an excessive amount of leverage, you could potentially blow up, and we'll discuss this in another video. But for now, the key takeaway is that leverage makes high sharp strategies very powerful, even if they have lower returns initially. You can just target higher returns while maintaining the nice high sharp and smooth return profile.
All right, guys, and I just want to briefly touch on some theory. So, we've seen that higher sharp equals fewer wiggles. Wiggles isn't exactly a technical term, but there are a lot, some technical justifications for using the sharp ratio. Not going to spend too much time on this, but I just want to briefly mention a couple. A simple stat. So, the first is from statistics. The simple a simple stats approach to evaluating a return series, I might seek to determine if the returns are statistically greater than zero. Going back to just stat 101, we can compute a t-stat for this. If you never heard of a t-stat, don't worry, it's fine for now. It's just, just understand that it's a number that tells you how likely it is your investment actually will have a positive return going forward. The interesting thing about t-stats and sharps is that it's, it's not too hard to show that the t-stat is actually, you know, it's proportional to the sharp ratio, right? And if you're interested, you could actually back this out based on the formula for the t-stat. So here's the formula for the t-stat. Here are the, uh, values for the variables that go into the t-stat. If you actually plug these values into the formula, what you get is this guy, and you can see already in this formula that you basically have the sharp ratio right there. And if you just rearrange things, what you get is that this t-stat is equal to this guy, and the sharp ratio is right there. So the t-stat is directly proportional to the sharp ratio. If you're evaluating multiple investments, sharp ratio is already rank-ordering them by statistical significance, and you're just one scalar away from actually getting the full t-stat. And I think this is perhaps why intuitively we also feel less confident going forward in investments with more wiggles and lower sharps.
And finally, there is an academic justification for the sharp ratio, which I'll just briefly mention. We don't like getting too academic here, but there is some financial theory which says that rational investors who like returns but dislike risk, as measured by volatility, should only hold a single portfolio to maximize their happiness or utility, and that portfolio happens to be the highest sharp portfolio, which is also known as the tangency portfolio in, uh, academic finance. So we should always be striving to maximize the sharp ratio of our investments.
All right, guys, that wraps it up for sharp ratios. If any of this seemed fast, don't worry, it was meant to be an introduction to sharp ratios, and we'll go over everything in more detail in other videos. Here are some key takeaways for now. Returns matter in investing, but so does risk. Sharp ratios are one way to incorporate risk and to, to create a risk-adjusted measure of investment performance. They can be used to evaluate standalone investments, but also tell you how well investments mix together. In fact, it's impossible to do this using just returns. If your high sharp strategy ever doesn't have a high enough return, you can use leverage to lever up high sharp strategies to target higher returns and maintain that nice smooth return profile. Finally, a solid goal for us should be to achieve a two-plus sharp. If you can consistently achieve a two-plus sharp, you'll be doing better than 99% of investors out there. All right, hope you enjoyed this video and see you next time.