Radian Group Inc. Notes to Consolidated Financial Statements - (Continued) The following table summarizes information concerning outstanding and exercisable options at  December 31, 2012: In 2012, we used the Monte Carlo valuation model in determining the grant date fair value of stock options issued to  executives and non-executives. Prior to 2012, we used the Black-Scholes valuation model in determining the grant date fair  value of stock options issued to executives, non-executives and non-employee directors. Each of these was determined using  the assumptions noted in the following table:  ______________ (1)  In 2011 and 2010, the expected life of stock options granted was estimated to be less than the full term of the options  awarded. The expected life is estimated using historical data. (2)  In 2012, we began using a Monte Carlo valuation, which assumes a derived service period of between 3.14 years and 4  years instead of an expected life assumption. (3)  The risk-free interest rate is based on the U.S.  Treasury yield curve in effect at the time of grant. (4)  Volatility is determined at the date of grant using historical share price volatility and expected life of each award. For stock option awards granted in 2012 and 2011, in addition to the time-based vesting requirements, the options contain  a performance hurdle whereby the options will vest if the closing price of our common stock on the NYSE exceeds  $4.90  (200% of the option exercise price) and $4.48 (125% of the option exercise price), respectively , for ten consecutive trading  days ending on or after the third anniversary of the date of grant. We elected to apply the short-cut method in accounting for the windfall tax benefits under the accounting standard  regarding share-based payment. Should future of fsets to the windfall resulting from cancellations, expirations or exercise  shortfalls exceed the balance of  $18.3 million at December 31, 2012, the excess would be reflected in the consolidated  statements of operations. 254 Year Ended December 31, 2012 2011 2010 Expected life (years) (1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2) 6 years 6 years Risk-free interest rate (3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.66% 1.88% 2.85% Volatility (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.97% 114.51% 110.83% Dividend yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.41% 0.28% 0.10% Options Outstanding Options Exercisable Range of Exercise Prices Number Outstanding Weighted Average Remaining Contractual Life (Years) Weighted Average Exercise Price Number Exercisable Weighted Average Exercise Price $2.45 - $3.58 . . . . . . . . . . . . . . . 2,288,360 5.45 $ 2.72 444,800 $ 2.48 $5.76 - $7.06 . . . . . . . . . . . . . . . 76,827 5.18 6.89 — — $10.42 . . . . . . . . . . . . . . . . . . . . 214,100 4.36 10.42 — — $20.34 . . . . . . . . . . . . . . . . . . . . 946,400 1.70 20.34 946,400 20.34 $35.79 - $53.17 . . . . . . . . . . . . . 494,580 0.04 40.48 494,580 40.48 $56.03 . . . . . . . . . . . . . . . . . . . . 382,077 0.10 56.03 382,077 56.03 4,402,344 3.51 2,267,857