Read the following texts and table with financial data from an S&P 500 earnings report carefully.Based on the question-answer history (if provided), answer the last question. The answer may require mathematical calculation based on the data provided.


republic services , inc . notes to consolidated financial statements 2014 ( continued ) 11 . employee benefit plans stock-based compensation in february 2007 , our board of directors approved the 2007 stock incentive plan ( 2007 plan ) , and in may 2007 our shareholders ratified the 2007 plan . in march 2011 , our board of directors approved the amended and restated 2007 stock incentive plan , and in may 2011 our shareholders ratified the amended and restated 2007 stock incentive plan . in march 2013 , our board of directors approved the republic services , inc . amended and restated 2007 stock incentive plan ( the amended and restated plan ) , and in may 2013 our shareholders ratified the amended and restated plan . we currently have approximately 15.6 million shares of common stock reserved for future grants under the amended and restated plan . options granted under the 2007 plan and the amended and restated plan are non-qualified and are granted at a price equal to the fair market value of our common stock at the date of grant . generally , options granted have a term of seven to ten years from the date of grant , and vest in increments of 25% ( 25 % ) per year over a period of four years beginning on the first anniversary date of the grant . options granted to non-employee directors have a term of ten years and are fully vested at the grant date . 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 , to reflect 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 2014 and 2013 ) and expected life of the options . when appropriate , separate groups of employees that have similar historical exercise behavior are considered separately for valuation purposes . we did not grant stock options during the year ended december 31 , 2015 . the weighted-average estimated fair values of stock options granted during the years ended december 31 , 2014 and 2013 were $ 5.74 and $ 5.27 per option , respectively , which were calculated using the following weighted-average assumptions: . <table class='wikitable'><tr><td>1</td><td>-</td><td>2014</td><td>2013</td></tr><tr><td>2</td><td>expected volatility</td><td>27.5% ( 27.5 % )</td><td>28.9% ( 28.9 % )</td></tr><tr><td>3</td><td>risk-free interest rate</td><td>1.4% ( 1.4 % )</td><td>0.7% ( 0.7 % )</td></tr><tr><td>4</td><td>dividend yield</td><td>3.2% ( 3.2 % )</td><td>3.2% ( 3.2 % )</td></tr><tr><td>5</td><td>expected life ( in years )</td><td>4.6</td><td>4.5</td></tr><tr><td>6</td><td>contractual life ( in years )</td><td>7.0</td><td>7.0</td></tr></table> .
Question: what was the change in value of weighted-average estimated fair values of stock options from 2013 to 2014?

0.47