Most online sportsbooks have a built-in option for allowing their bettors to "buy points".  Buying points allows the bettor to add to the spread for the game he is betting.  In exchange, he pays for the additional points by paying extra juice.  Typically bettors buy a half point at a time, paying 10 cents in juice for each half point.  In a standard juice arrangement, buying a half point moves the juice from -110 to -120.  Most books limit the bettor to purchasing 3 half points, or up to 1.5 points. Here is an example from CaribSports:

I have selected the bet "Chiefs +8" from the main board, leading to the above screen that I have captured after opening the drop down menu for buying points.  The default is always NOT BUYING ANY POINTS, but you have the option to buy up to 1.5 points in half point increments.  Consistent with the rest of the CaribSports quirky interface, they don't tell you its 10 cents more juice for each half point.  They tell you its $10 more per $100 wagered for each half point.  It means the same thing.  (Carib's a good book, and you get used to their quirks.)

Valuing the Half Point

If I buy a half point, then the terms of my bet change from +8 (-110) to +8.5 (-120).  Now all I need to know is which of those options is the better bet.  If I buy the half point and the final score is the Chiefs losing by exactly 8, I still win - but I have to put more money at risk to get the same payout as I would at -110.  If I don't buy the half-point and the Chiefs lose by exactly 8, it's a push.  Which is more valuable?

To resolve this we need to know the probability of a push happening when the spread is 8.  To know this we must consult a database.  I have access to a database of the last 23 NFL seasons.  From this database I can find that there is a push 2.14% of the time when there is a spread of 8.  (I will explain later how to access this database - it is online and free.)

The combined probability of a win, loss, or push will total 100%.  We know the push to be 2.14%, so assigning an equal chance of winning and losing, we get 48.93% for each.  

If I do NOT buy the half point, my chance of winning the bet is 48.93%.   The rest of the time I will lose or push.  If I buy the half point my chance of winning has increased by 2.14% to 51.07%.  Is that worth the extra 10 cents of juice?

King Yao gives us this following quick approximation method for evaluating the cost of the extra half point.  You find the implied value (in this instance) by dividing the win percentage by the lose percentage, multiply by 100 and subtract 100.  [(51.07/48.93)(100)] - 100 = 4.37 cents.  Buying a half point off the +8 here is worth about 4.37 cents.  As Carib is charging 10 cents for it, we should not buy the extra half point here.

When to Buy the Points

As a general rule of thumb, buying a half point to get on or off the spread of 3 or 7 is only time it is worth the extra 10 cents of juice in American style football.  For NFL football you could spend up to 20 cents to get on or off the spread of 3 and still be getting good value.  If you are using a reduced juice book that also sells points at reduced juice, then you will have to figure them out on a case by case basis.

Somewhat ironically, while buying a half point is rarely worth the extra juice, buying multiple half points frequently is. Each subsequent half point purchase becomes more valuable, as it becomes more likely you will win, but costs the same as the first half point.  Borrowing the example from Justin7 at SportsbookReview, imagine a book let you purchase 200 half points.  Your spread is now 100 points.  You're paying a very high level of juice, but since no NFL team has ever lost by 100, you are guaranteed a win.

Speaking of SportsbookReview - that is the home of the database I use for finding the value of points.  It is housed within their half point calculator.  Not only does it have 23 seasons of NFL data, but it also carries NCAAFB, NCAA Hoops, NBA, NHL and MLB.  It also employs a more sophisticated method of calculating the value of a half point (and multiple half points) than the King Yao method.  I recommend this tool, and will write a more detailed article on using it at a later date.