what was the percentage change in the expected volatility from 2012 to 2013
republic services , inc . notes to consolidated financial statements 2014 ( continued ) in december 2008 , the board of directors amended and restated the republic services , inc . 2006 incentive stock plan ( formerly known as the allied waste industries , inc . 2006 incentive stock plan ( the 2006 plan ) ) . allied 2019s shareholders approved the 2006 plan in may 2006 . the 2006 plan was amended and restated in december 2008 to reflect republic as the new sponsor of the plan , and that any references to shares of common stock are to shares of common stock of republic , and to adjust outstanding awards and the number of shares available under the plan to reflect the allied acquisition . the 2006 plan , as amended and restated , provided for the grant of non- qualified stock options , incentive stock options , shares of restricted stock , shares of phantom stock , stock bonuses , restricted stock units , stock appreciation rights , performance awards , dividend equivalents , cash awards , or other stock-based awards . awards granted under the 2006 plan prior to december 5 , 2008 became fully vested and nonforfeitable upon the closing of the allied acquisition . no further awards will be made under the 2006 stock options we use a lattice binomial option-pricing model to value our stock option grants . we recognize compensation expense on a straight-line basis over the requisite service period for each separately vesting portion of the award , or to the employee 2019s retirement eligible date , if earlier . expected volatility is based on the weighted average of the most recent one year volatility and a historical rolling average volatility of our stock over the expected life of the option . the risk-free interest rate is based on federal reserve rates in effect for bonds with maturity dates equal to the expected term of the option . we use historical data to estimate future option exercises , forfeitures ( at 3.0% ( 3.0 % ) for each of the periods presented ) and expected life of the options . when appropriate , separate groups of employees that have similar historical exercise behavior are considered separately for valuation purposes . the weighted-average estimated fair values of stock options granted during the years ended december 31 , 2014 , 2013 and 2012 were $ 5.74 , $ 5.27 and $ 4.77 per option , respectively , which were calculated using the following weighted-average assumptions: .

|  | 2014 | 2013 | 2012 |
| :--- | :--- | :--- | :--- |
| Expected volatility | 27.5% | 28.9% | 27.8% |
| Risk-free interest rate | 1.4% | 0.7% | 0.8% |
| Dividend yield | 3.2% | 3.2% | 3.2% |
| Expected life (in years) | 4.6 | 4.5 | 4.5 |
| Contractual life (in years) | 7.0 | 7.0 | 7.0 |

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expected_volatility_2012 = 28.9
expected_volatility_2011 = 27.8
change = expected_volatility_2012 - expected_volatility_2011
percent_change = change / expected_volatility_2011
answer = percent_change * 100