Constructing a successful investment portfolio and winning in professional

football share many common characteristics. Successful investing in the long-

term requires constructing a portfolio of assets that yield the highest expected

return given their levels of risk. Generally higher returns cannot be obtained

without taking on higher risk, so the typical investor faces a tradeoff between risk

and return. Much of financial economics has been devoted to determining how to

construct an optimal portfolio for an investor with a given amount of wealth to

invest and a given level of risk aversion1. The expected return of an asset is

typically measured by a historical average return and the risk is measured by a

historical variance of returns.

For a professional football coach (or offensive coordinator), the problem is

very similar. The success of a season could be determined by the coach

constructing an optimal portfolio of running and passing plays, each with an

expected return (measured by yardage) and a variance. The problem is to choose

the share of running and passing plays that yields the highest return for the

portfolio2. This is a very simplistic approach to the problem since the return and

variance to running and passing plays can change over the course of a single

season (and over several seasons). We assume that each coach knows what the

distribution of yardage from running and passing plays looks like for his team and

that this distribution is invariant over the course of a season. Portfolio models of

financial assets make the same assumptions so we see no great harm in making

the same assumption here. As we shall show, it is straightforward to derive the

optimal share of running and passing plays if the moments of their distributions

are known.

Alamar (2006) has noted that a passing premium appears to exist for NFL

teams, whereby teams do not appear to use as many passing plays as they should,

given the average yardage return advantage enjoyed by passing. This puzzle

persists despite rule changes in the late 1970’s that opened up the field for

receivers and made protecting the quarterback more stringent. Since 1960, Alamar

(2006) found that completion rates have increased significantly and interception

rates have fallen. Yards per completed pass have increased significantly since

1960, while yards per run have remained essentially unchanged. Alamar (2006)

argues that all this should convince a coach to pass more and run less, yet he notes

that passing rates have increased only slightly since 1960.

1 Many texts cover the basic theory. Some examples are LeRoy, Werner and Ross (2000) and

Bradfield (2007). 2 We do not consider the influence of defense and special teams to team success, although we

admit these are important factors which could be included in the optimal portfolio.

It is the purpose of this paper to examine whether a passing premium

exists using a more rigorous procedure than that used by Alamar (2006). Much of

the analysis in Alamar (2006) relies on subjective measures of passing and

running success and there is no notion suggested of what an optimal allocation of

passing and running plays in a team’s portfolio of plays is. The next section of the

paper outlines a basic portfolio selection model of passing and running plays. The

third section of the paper provides a statistical analysis for the 2006 NFL season.

The final section provides concluding remarks.

A Portfolio Model of Play Selection

Portfolio theory can be used to suggest how a football coach can try to determine

an optimal allocation of running and passing plays over the course of season. We

assume that there are N teams in a professional football league and that each team

has unique distributions for running yardage and passing yardage. The moments

of these distributions are assumed to be predetermined and known. Expected

return is measured by the mean of each distribution, for running and for

passing, and risk is measured by the variance of each distribution. This notion of

risk is quite standard, but it differs from the measure of risk used by Alamar

(2006), who assumes that risk is measured by pass incompletions and

interceptions. Formally, each of those is an uncertain event best captured by a

probability that reflects the magnitude of the uncertainty, not risk. An important

point is that neither running nor passing is risk free, passing has a variance of

and running has a variance of for one representative team.

Let be the share of plays in the coach’s portfolio that are running plays

( ) so that (1 - ) of the plays are passing plays. The expected return

from the portfolio, V, is then given by

( ) ( ) ( ) ( ) (1)

The terms XR and XP are the realized returns from a running play and a

passing play respectively. Each of these returns is assumed to be determined for

any single play by a simple process.

(2)

(3)

The terms uP and uR are random shocks with expected values of zero,

variances and

and they share a covariance of zero. They are also serially

uncorrelated. The shocks to running and passing yardage on any play show no

persistent pattern to yardage on any previous plays. This assumption could be

violated in real games, however it is quite a common assumption in the sports

statistics literature to assume that outcomes of plays in professional sports follow

a Markov process so that they have no memory3. The problem is to choose the

return maximizing value for . Team coaches are assumed to maximize their

utility from the team portfolio. Utility is a measure of satisfaction that is simply a

monotonic transformation of (1). Different forms of utility functions can be used

to capture risk aversion (Grossman and Stiglitz (1980)). We assume a utility

function of the form

( ) (

)

(4)

This utility function maintains the principle of diminishing marginal

utility4 in the portfolio return V. The constant is the Arrow-Pratt measure of

absolute risk aversion5. Risk aversion is implied by > 0 while risk loving is

implied by < 0. To find the solution for the optimal , substitute (2) and (3) into

(1), then substitute the resulting expression into (4). After taking expected values

and some simplification, this gives

( ) ( ( )

(

( ) ( )

( ) ))

(5)

The last step is to take the derivative of (5) with respect to , set the

resulting expression equal to zero, and solve for the optimal . Without showing

all of the steps, the solution is given by

( )

(6)

The solution in (6) has some interesting properties. First, as the variance to

running increases, the optimal share of running plays decreases, ceteris paribus6.

3 Examples are Riddle (1988), Bukiet, Harold and Palacios (1998), and Hirotsu and Wright (2002). 4 Proper utility functions require that utility be increasing in V at a diminishing rate, or that

⁄ and ⁄ . 5 The Arrow-Pratt measure of absolute risk aversion is given by - ⁄ ⁄⁄ . It calculates

the degree of curvature of the utility function. 6 Differentiating (6) confirms these comparative static results.

Second, as the variance to passing increases, the effect on the optimal share of

running plays is positive the higher the degree of risk aversion, although if

, the effect is always positive. Little else can be said of (6) without

estimates of the parameters. That is what we do in the next section.

Data and Results

We propose to compute the optimal share of running plays given by (6) for each

NFL club over the 2006 season and compare the results with the actual running

and passing shares. Play-by-play net yardage was obtained for each team for the

regular season of 16 games from nfl.com. Penalty yardage was included in the

yardage figures if the penalty was a direct result of the play. Penalties such as

taunting the defense and others that occur well after the play is finished were not

included in the net yardage. We thought this was justified since coaches cannot

anticipate penalties that are more or less unrelated to the play. We followed

Alamar (2006) in the treatment of pass interceptions and fumbles by removing 45

yards from the yardage at the point of the interception or fumble. Some running

and passing plays were excluded altogether, mainly a quarterback “spiking” the

ball to stop the play clock and a quarterback kneeling to run out the play clock.

The total number of passing and running yardage observations was 31,159.

We use play-by-play data to compute the variances and

in (6).

Table 1 presents the main findings. Columns 2 and 3 provide estimates of R and

P. The lowest average yardage for rushing was 2.7 yards for the Cleveland

Browns, while the highest average yardage for rushing was 5.7 yards for the

Atlanta Falcons. Note that these figures cannot be compared to average yards per

carry provided by the NFL (3.6 yards and 5.5 yards respectively) due to the

inclusion of the 45 yard penalty for fumbles. Average yardage per carry for the

league was computed to be 4.14 yards. The lowest average yardage per pass

attempt (not completion) was 3.39 yards for the Cleveland Browns, while the

highest yardage per pass attempt was 7.3 yards for the New Orleans Saints.

Columns 4 and 5 of Table 1 provide estimates of R and P. The lowest

and highest variance runners were the Chicago Bears and the San Francisco 49ers

respectively. The lowest and highest variance passers were the Houston Texans

and the New Orleans Saints respectively. Financial assets usually display a

positive association between risk and return. Figures 1 and 2 plot this relationship

for running and passing plays, respectively. The risk-return tradeoff for running is

positive with a correlation coefficient of 0.322 (p-value = 0.073), but there does

not appear to be any significant risk-return tradeoff for passing (correlation

coefficient = 0.094). This implies that coaches might be risk-neutral with regard

to passing meaning that when considering choosing between two different passing

plays, no consideration is given to risk (high positive or negative net yardage).

Coaches expect a higher return for riskier running plays but not for passing plays.

This is suggestive evidence that coaches may choose to pass too frequently since

no regard is given to risk.

The calculated optimal share of running plays () is contained in column 6.

This is computed by inserting the means and variances from columns 2, 3, 4 and 5

into (6). The choice of the value for the absolute risk aversion coefficient, , is

subjective and rather ad hoc. This does not affect the ranking of the teams from

most efficient to least, but it does affect the gap between and the actual share of

running plays in column 7. We used a logical approach to the value of by

setting it equal to the value that just equals the actual share of running plays for

the team with the best regular season record, the San Diego Chargers (14 wins

and 2 losses). This provided = 0.0285. Our reasoning is that such a successful

record is, at least partly, due to choosing the correct portfolio of offensive plays.

Figure 3 graphically reproduces the results in Table 1 using three different values

for . Increasing from 0.0285 to 0.0365 increases the concavity of the utility

function, making coaches more risk averse and thus more likely to prefer a greater

share of running plays to passing plays (since running plays have a smaller

variance of yardage gained). For given shares of running and passing plays, this

makes the efficiency measure in column 7 of Table 1 larger, as can be seen in

Figure 3. Decreasing the value of implies more risk loving coaches and will

have the opposite effect.

The evidence suggests that the majority of teams choose more passing

plays and fewer running plays than the portfolio model predicts as optimal for the

2006 season. The exceptions are the New England Patriots, the New Orleans

Saints, the San Diego Chargers and the Indianapolis Colts, whose inefficiencies

are arbitrarily close to zero. The New Orleans Saints passed for more yardage

than any other team in the NFL (281.4 yards per game), however the portfolio

model suggests that the Saints were a very efficient passing team in the sense of

return versus risk, allowing for fewer running plays. The worst offenders were the

Pittsburgh Steelers and the Oakland Raiders who passed the ball far too often

given their relatively poor efficiency in passing. This could be because their

coaches were poor portfolio managers, but it also could be because they had poor

defenses who let the team fall behind early games very often. In this situation, the

higher return to passing might outweigh the higher risk to passing if a large

number of points are needed to catch up in a game.

Table 1

Team R

(1)

P

(2)

R

(3)

P

(4)

(5)

Running

share (6)

Inefficiency

(5)-(6)

Patriots 3.988 5.974 8.214 11.610 0.373 0.463 -0.089

Saints 3.125 7.305 9.026 15.651 0.375 0.391 -0.016

Chargers 5.720 6.549 9.408 11.452 0.488 0.488 0.000

Colts 4.454 6.604 5.938 12.423 0.479 0.445 0.034

Ravens 3.775 5.186 6.229 12.229 0.564 0.449 0.115

Cowboys 4.547 6.275 5.839 13.676 0.612 0.487 0.125

Chiefs 4.472 5.574 7.001 13.487 0.644 0.517 0.127

49ers 4.415 4.503 10.894 14.213 0.622 0.477 0.144

Rams 4.135 5.509 7.063 12.010 0.528 0.379 0.149

Bengals 4.070 6.492 6.013 15.047 0.585 0.426 0.159

Jets 3.574 4.918 5.922 13.770 0.658 0.496 0.162

Dolphins 3.591 4.306 8.901 11.961 0.543 0.373 0.170

Jaguars 4.768 5.177 6.947 11.860 0.679 0.507 0.172

Vikings 3.863 4.381 8.881 12.686 0.604 0.423 0.181

Texans 3.584 4.136 7.676 10.903 0.572 0.401 0.171

Redskins 4.770 5.518 6.195 12.804 0.699 0.493 0.206

Falcons 5.721 4.174 10.491 12.399 0.757 0.540 0.217

Eagles 4.875 6.149 9.485 15.576 0.616 0.389 0.227

Giants 4.870 5.008 7.967 12.264 0.684 0.447 0.237

Lions 3.936 4.798 9.123 12.982 0.564 0.318 0.246

Browns 2.699 3.394 8.046 13.801 0.659 0.402 0.257

Panthers 4.168 4.679 7.641 13.129 0.679 0.419 0.260

Cardinals 2.691 4.122 6.463 14.210 0.641 0.380 0.261

Titans 4.276 3.675 8.637 12.742 0.764 0.484 0.279

Packers 4.154 4.763 8.085 13.861 0.674 0.383 0.290

Broncos 4.807 4.702 8.372 13.451 0.734 0.437 0.297

Bears 3.912 4.800 5.314 15.098 0.782 0.483 0.299

Buccaneers 3.633 3.648 7.651 11.942 0.707 0.392 0.314

Seahawks 3.825 4.017 6.743 13.883 0.784 0.433 0.351

Bills 3.848 4.183 5.994 15.056 0.823 0.462 0.361

Raiders 3.601 2.469 8.392 13.685 0.866 0.437 0.429

Steelers 4.715 3.624 9.487 14.720 0.815 0.358 0.458

Further evidence is provided in support of the portfolio approach if play-

calling inefficiency is negatively associated with team success. We test this by

computing a least squares regression7 between winning percentage for the 2006

regular season and the inefficiency measure in column (6) of Table 1 for all 32

NFL teams. We compute White’s (1980) heteroskedasticity consistent covariance

7 If winning percentage is an accurate estimate of the probability of winning a contest, a linear

probability, with its noted shortcomings, is the result of a least squares regression. We also

estimated a logit regression model, but we found no qualitative differences between the two

methods.

matrix estimator to test the slope and intercept of the regression model8. The

results are given below (Student’s t statistics appear in parantheses).

Winning % = 0.6643 – 0.7948 Inefficiency

(14.82)**

(3.56)**

R2 = 0.283

** Indicates statistical significance at 95% confidence.

Figure 4 provides a graph of the association between winning percentage

and running inefficiency. The association is negative and statistically significant

at 95% confidence.

Conclusions

This paper began by reconsidering the passing premium puzzle posed by Alamar

(2006). The puzzle is that NFL teams do not appear to use passing plays enough,

despite rule changes over the last three decades that have increased the expected

return to passing. The difficulty is in determining just what the optimal amount of

passing is, given the characteristics of the team, its opponent and the rules. We

borrow from portfolio economics and construct a simple model of a risk-averse

coach that needs to choose utility maximizing shares of passing and running plays

over the course of a season. Utility is gained from net yardage of each play. The

solution for the optimal shares includes the mean yardage and the variance of

yardage for both running and passing plays, as well as an Arrow-Pratt coefficient

of absolute risk-aversion. We assume that each team coach knows the moments of

the team’s passing and running net yardage distributions at any time in the season.

Data collected for the 2006 NFL regular season were used to compute the

optimal share of running plays in the team portfolio of plays. These were

compared to the actual share of running plays for all 32 NFL teams. The results

suggest that most NFL teams pass the ball too often and that a running premium

exists. In addition, evidence was found for a tradeoff between expected net

yardage from a running or passing play and the risk (variance) of each type of

play. As in financial investments, higher returns generally came at the expense of

higher risk. Finally, the size of the gap between the optimal running share and the

actual running share was found to be negatively associated with winning

8 Heteroskedasticity is evident in the data if the variance of winning percentages is associated with

the inefficiency measure. White’s (1980) method corrects the standard errors of the slope and

intercept for an unknown form of heteroskedasticity.

Figure 1

Figure 2

2.000

2.500

3.000

3.500

4.000

4.500

5.000

5.500

6.000

20 40 60 80 100 120 140

Ave

rage

pe

r R

ush

Att

em

pt

Variance of Rushing Yardage

Risk-Return Relationship for Rushing

2.000

3.000

4.000

5.000

6.000

7.000

8.000

120 140 160 180 200 220 240 260

Ave

rage

pe

r P

ass

Att

em

pt

Variance of Passing Yardage

Risk-Return Relationship for Passing

Figure 3

Figure 4

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Effi

cie

ncy

Ind

ex

Sensitivity of Efficiency Index to Choice of

=.0285

=.0365

=.0445

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Win

%

Inefficiency

Winning Percentage vs. Running Inefficiency

percentage so that, in general, poor portfolio managers (coaches) suffered lower

success.

Some caveats should be mentioned before accepting these results as

definitive evidence. Coaches are assumed to have full knowledge of the moments

of their running and passing net yardage distributions before the season starts.

They can construct these moments since they are assumed to know their own

team level of talent, as well as the talent of all the other teams in the league, in

particular, the teams they must play during the regular season. They also know

how the same plays have worked in the past, perhaps even with different teams.

This is the benefit of coaching experience. However, they cannot have knowledge

of injuries, weather, suspensions, and any other factors that might affect these

distributions. This is not much different from the assumption in financial

economics that the future returns of an asset will behave the same as the

distribution of past returns. Business cycles, company reorganizations, new

products, new government regulations, and so on, cannot be fully anticipated

either. Nevertheless, assuming coaches know the net yardage distributions of their

plays might be pulling the apple cart before the horse.

The assumption that coaches maximize the expected return from offensive

plays in order to win games may not be appropriate for some teams. The Chicago

Bears did not fair well in offensive play calling based on their high inefficiency

rating in Table 1, yet they made it to the Super Bowl. This result was most likely

due to a coaching emphasis on minimizing points given up on defense, then

taking whatever offensive points that can be obtained. This points out that the

inefficiency ratings in Table 1 must be considered in combination with defense

and special teams performance, although most of the teams that scored poorly on

offensive play efficiency also performed poorly on defense and special teams for

the 2006 season.

To fully assess the claim of a “passing premium”, more than one season of

data is necessary. Alamar (2006) considered only the 2005 NFL regular season in

detail – we consider only the 2006 NFL regular season. It would be insightful to

obtain data for a season prior to the passing rule changes that came in the 1978

season. It could be that teams had a running premium prior to 1978, but that the

running premium was smaller. This would provide evidence in favor of the

argument that teams do not appear to pass frequently enough today, given the

passing rule changes, even though they passed too often in the distant past.

Unfortunately we have not found a source for play-by-play net yardage data prior

to the 1978 season.

References

Alamar, B.C. 2006. The Passing Premium Puzzle. Journal of Quantitative

Analysis in Sports. 2(4): 1-8.

Bradfield, J. 2007. Introduction to the Economics of Financial Markets. Oxford

University Press, USA.

Bukiet, B., Harold, E., Palacios, J. 1997. A Markov Chain Approach to Baseball.

Operations Research. 45: 14-23.

Grossman G., Stiglitz, J. 1980, On the Impossibility of Informationally Efficient

Markets, American Economic Review, 70(3): 393-408.

Hirotsu, N., Wright, M. 2002. Using a Markov Process Model of an Association

Football Match to Determine the Optimal Timing of Substitution and Tactical

Decisions. Journal of the Operational Research Society. 53(1): 88-96.

LeRoy, S.F., Werner, J., Ross, S.A. 2000. Principles of Financial Economics.

Cambridge University Press, Cambridge MA.

Riddle, L. 1988. Probability Models for Tennis Scoring Systems. Applied

Statistics. 37(1): 63-75.

White, H. 1980. A Heteroskedasticity-consistent Covariance Matrix Estimator

and a Direct Test for Heteroskedasticity. Econometrica. 48: 817-838.