Football is pretty amazing. A few short months ago, we weren’t even sure if we were even going to have a season. Now, we have 49ermania taking over the Bay Area, the possibility for a rematch of one of the greatest superbowls ever between the Giants and the Patriots, and Ray Lewis, well, he’s just being Ray Lewis.
For the rest of us, the end of yet another fantasy football season begins to sink in, we’re left to reflect on all triumphs and regrets of the last 5 months. Hopefully you’ve managed to stockpile enough bragging rights to last at least until next season. If not, there’s always next year.
If you’re like me, the book Moneyball by Michael Lewis (or maybe even the movie, if you’re a Brad Pitt fan) presented an intriguing idea that taking an objective statistical approach to evaluating talent allows one to identify the inefficiencies in assessing their value. Basically, using the right statistics, you can find the right players who will help you win at a discount and avoid the players who will cost too much for the amount they contribute to your team’s success.
Working for a finance company, it’s hard not to notice the parallels that fantasy sports present to investing (Paul DePodesta of Moneyball fame also graduated from Harvard with a degree in economics, which is not a coincidence). When we talk about making moves to acquire players who are due to break out or squeezing out a hefty premium for an overachieving player, we’re really just participating in a virtual market, trading players just as we’d trade stocks. Buy low and sell high is still the goal.
That being the case, we can apply many of the same techniques used by sophisticated investors to squeeze out a few basis points that could mean the difference between winning and losing. While there’s lots of debate about the efficiency of the public markets, I can guarantee that your fantasy league with your college roommates is not efficient. Let’s take a look at how we can use Modern Portfolio Theory to make the most strategic investments in our team to give us the best chance possible for success.
We recently put up a Slideshare presentation on how to invest in ETFs using Modern Portfolio Theory that explains some of the concepts in an investing context, but they’re effectively the same when applied to our fantasy sports domain. We’re well versed in understanding that every player has an expected return (i.e. the number of fantasy points they will generate over a given period), but as with investing, we commonly mischaracterize an investment in terms of the presented risk. Every investment has an expected return and an associated risk. In finance, the risk we try to assess is the probability that an investment will decrease in value. In this case, we’re more concerned with the opportunity cost that we’ve missed out on from the value attained by another alternative. For this exercise, we can simply use mean fantasy points generated by a player as our expected return and standard deviation of points as the risk. Then, using players to compose our portfolio (a.k.a. “team”), we can determine the combination of available players that presents the best risk/return characteristics for us to be successful.
Let’s look at a simple fantasy football example. We’ll create a 3player team consisting of a quarterback, a running back and a wide receiver pulling from a universe of 6 players. Calculating mean and standard deviation are pretty straightforward. (note: You can also mix in projected numbers as samples with the actual numbers if desired, this example just tries to keep it simple)
Aaron Rodgers (QB) 
Drew Brees (QB) 
Ray Rice (RB) 
LeSean McCoy (RB) 
Calvin Johnson (WR) 
Wes Welker (WR) 


Week1  35.01  40.04  33.19  30.61  26.00  34.55 
Week2  29.25  33.75  21.19  29.78  17.64  13.36 
Week3  33.86  39.88  22.67  26.18  28.82  50.10 
Week4  60.62  23.99  22.07  17.34  28.73  29.36 
Week5  30.92  30.76  23.08  22.45  22.82  16.27 
Mean  37.93  33.68  24.44  25.27  24.80  28.73 
Std.Dev.  12.89  6.73  4.94  5.49  4.70  14.85 
For our team, to get the mean we simply add the mean for our players together. For standard deviation of our team, we’ll make use of this equation:
To simplify the math, we’ll assume that players’ performances are independent of each other. We know this isn’t quite true, but we’ll leave this exercise as an advanced topic. When variables are independent, their covariance is 0, which allows us to simplify the equation to:
All we need to do is add the variance of each player on our team together and then take the square root.
Team  Mean  Std.Dev. 

Rodgers/Rice/Johnson  87.17  14.58 
Rodgers/Rice/Welker  91.10  20.28 
Rodgers/McCoy/Johnson  88.01  14.77 
Rodgers/McCoy/Welker  91.93  20.42 
Brees/Rice/Johnson  82.93  9.58 
Brees/Rice/Welker  86.85  17.04 
Brees/McCoy/Johnson  83.76  9.88 
Brees/McCoy/Welker  87.68  17.21 
Unsurprisingly, we find in general that high risk yields a higher return and that lower risk yields a lower return. This intuitively makes sense and more on this later on, but also take notice of something else. We find that there are combinations that demonstrate a lower risk and a higher expected return than other combinations. Obviously any individual week can vary drastically, but in aggregate over the course of a season you will be better off* with the lower risk and higher return team. Think about it. You wouldn’t want more risk for the same number of expected fantasy points or conversely, less fantasy points for the same amount of risk. On a weekbyweek basis, other lineups might make more sense, but over a larger period of time, we expect more points from Rodgers/McCoy/Johnson with less risk than Brees/McCoy/Welker or Brees/Rice/Welker. It’s that kind of information that we hope will give us the edge we’re looking for.
*This is obviously highly dependent on the data you use. The more accurate you are with your predictions, the closer you’ll be in the outcome. In the finance world, we’re required to say “past performance is no guarantee of future results” in the disclaimers. We’re not fortune tellers, so the same applies here as well.
As you evaluate all the combinations of players, you’ll find a maximum to the number of fantasy points (a.k.a. the best possible team) at each level of risk, and together they form a curve. The finance world refers to this as the “Efficient Frontier.”
Now that we have this tool available, how do we use it? Let’s look at a few game formats to see where this information comes in handy.
Salary Cap Leagues
Salary Cap leagues assign each owner a pool of money to spend on players’ salaries. You can add any player (sometimes even if owned by another team) as long as adding the player’s salary doesn’t put your team’s total over its cap. At an individual level, we can use each player’s mean and standard deviation to determine if the player is performing above or below expectations, and standard deviation tells us how big of a swing (either up or down) is common. For each team combination, we can filter out teams that are over our salary cap and pick the most appropriate combination for a given point in the season. To know how much risk we should take on, we’ll look at Total Points Leagues.
Total Points Leagues
Total Points leagues have pretty simple scoring mechanism where each team adds up all of the points generated by its players. Team with the most points wins. In contrast to Salary Cap leagues, the most common format only allows one team to possess a player at any given time (Salary Cap leagues commonly allow a player to be owned by multiple teams). This player scarcity will limit your ability to assemble your ideal team to mostly what you are able to obtain in your initial draft.
Overall, you should approach your season with the same strategy as Target Date Funds do in the investing world. In a nutshell, Target Date Funds invest in securities that have a risk/expected return that’s in ratio with the distance to a target date in the future. The farther away from the target date, the higher expected return/risk the investments will be, and as the target date gets closer, the fund will transition into lower risk investments limiting the potential downside. You should effectively do the same.
Early in the season, you can afford to take on a little bit more risk knowing that there’s a good chance the player will perform well over the course of a season. Putting this team together is pretty straightforward, especially since the experts care almost exclusively about performance and a player’s expected return (even if they did care about risk, they wouldn’t know how to apply it to your team’s unique scenario). However, as the season progresses, you’ll either need to make up ground if you’re behind or try to lock in points if you get ahead. If you’re behind, you’ll need to make some moves to acquire higher risk / higher expected return players. (Note: This is absolutely not what you want to do with your retirement account. No one wants to crater their savings right before they retire. In fantasy football, it’s usually better to burn out in spectacular fashion than to fade away quietly.) Unfortunately, these highreward players have the most demand, so it’s usually difficult to pull off favorable trades.
In the case that you’re trying to maintain a lead, you have many more options. As you get closer to the end of the season, you should try to lower the risk of your portfolio of players. Perhaps that means trading a high reward/risk player and a scrub for two players who lower the volatility of your lineup. At the very least, you can stick to playing favorable player matchups and avoiding injuryprone guys to make sure your team consistently chugs along to the finish line.
HeadtoHead Leagues
HeadtoHead leagues match your team up against another team each week. The team scoring the most points receives a win, and the more wins you have, the better your place in the standings. As we already know from Moneyball, the number of wins a team gets is highly correlated with scoring a higher ratio of runs than its opponents. For us, it’s the same formula, but using the fantasy points scored by our team. Let’s take a look at the Pythagorean Expectation formula invented by Bill James.
In fantasy sports, although you can argue that there is some level of correlation between players, one notable difference is that your score and your opponent’s score are mostly independent. This means that when you look at the Pythagorean Expectation formula, your points scored will be consistent with your average projected out for the season, but it also means that your points allowed should approach the league average as the season goes on. It’s likely that you will lose weeks where your opponents get lucky and score above average amounts of points. Don’t panic. Over the course of the season, your opponents will hopefully revert to the mean and you’ll get a few favorable breaks yourself.
Some HeadtoHead leagues will declare a winner at the end of the regular season, but it’s more common to have a playoff among the best regular season teams to determine the winner. Hopefully by the time the playoffs get closer, you’ll have locked up your spot and you have the flexibility to retune your roster to have the best chances against the other teams in the playoffs. You use the same techniques as you’d use in the Total Points leagues to lock in the best possible seeding for your playoff tournament, but remember to keep enough horsepower on your roster to compete against the other playoff teams. If you lock up your playoff spot and you’re not playing for other prop bets like highest points scored during a season, you might consider losing games to inferior teams that might make the playoffs over unlucky teams with higher variance or higher expected returns than your team. The last thing you need is for a high variance team to sneak into the playoffs and get hot in the last weeks. The playoffs are mostly about luck and the less chances you take playing with fire, the more likely you’ll be holding that championship trophy in the end. In the immortal words of Ricky Bobby, if you ain’t first, you’re last.
Now, let’s take a look at the results from my league this past season. First, this chart shows the mean and standard deviation of the weekly points generated by each team during the season. The league average for mean and standard deviation of scores is in red. As you might suspect, the numbers next to each represent which place the team finished in after the playoffs. The numbers in this charts are completely intuitive. Teams in the upper left that demonstrated above average points and lower than average risk performed the best. Teams that scored less than average points did not fare as well.
Behind all data is a story. In this case, the 4th place team scored slightly above average, but also showed significant volatility. This team absolutely crushed the #6 team in the first round of the playoffs, but was subsequently crushed twice in a row by the #3 and #2 teams due to team 4’s huge swings in point production, eventually settling in 4th place. You’ll also notice how the eventual 7th place team appeared among the top teams in the league. As it turns out, this team was in fact one of the better teams in the league, but unfortunately very unlucky.
This second chart shows the actual winning percentage of teams charted against their expected winning percentage calculated using Bill James’ Pythagorean Expectation formula:
As you can see, the line of best fit matches pretty well and means that using the Pythagorean Expectation formula has some merits. Teams that scored more points compared to their opponents also continued to perform well in the playoffs. What you’ll also notice is that our #7 team is well below the line here, despite scoring an aboveaverage amount of points in the previous chart. This likely means that the team was scoring enough points that they should have had a better winloss record, but probably lost several close matches by a small amount and won a few matches by a landslide. This was, in fact, the case. Even more to the point, Team 7 lost the last regular season week to the eventual 3rd place team by .87 points (remember the mean was about 141, so that’s a very small amount). Had Team 7 scored just a single additional point that week, it would have made the playoffs in the place of the eventual 3rd place team and taken 3rd place. Instead, Team 7 won all of the consolation games and finished on a 2game win streak, but could do no better than 7th place. It just goes to show that luck and great timing mean as much to winning as having a good team.
So, we’ve covered a number of ways that concepts in finance are completely analogous and applicable to fantasy sports. In this case, we’ve explored fantasy football, but the concepts translate to other sports as well. Remember that it’s important to keep in mind the risk characteristics of your players and team as well as their performance, just like you should for your investments. Good luck!
By the way, here’s a list of some of my favorite sites doing some really interesting advanced statistics work:
Football
 http://www.profootballreference.com/
 http://footballoutsiders.com/
 http://www.advancednflstats.com/
Baseball
Basketball