Task: Table-QA (source dataset: FinQA)

Task: 
You are given a table containing structured data and a natural language question referring to the table. Your goal is to analyze the table and provide an accurate and concise answer based on the given data.

Instructions:
1. Understand the Table: Examine the structure, headers, and data in the table. Identify relevant columns and rows related to the question.
2. Interpret the Question: Determine whether the question requires direct retrieval, aggregation, comparison, sorting, filtering, or inference.
3. Extract the Answer: Use logical reasoning and numerical operations (if necessary) to derive the correct response from the table.
4. Format the Answer: No explanation, return the answer in a structured JSON format: {"answer": <answer>}. If the question cannot be answered due to missing or ambiguous data, return a JSON response with {"answer": null}.

Table:
| Unnamed: 0              | 2012             | 2011             | 2010             |
|:------------------------|:-----------------|:-----------------|:-----------------|
| expected volatility     | 24.0% ( 24.0 % ) | 28.8% ( 28.8 % ) | 41.7% ( 41.7 % ) |
| risk-free interest rate | 0.8% ( 0.8 % )   | 2.1% ( 2.1 % )   | 2.1% ( 2.1 % )   |
| dividend yield          | 2.2% ( 2.2 % )   | 0.0% ( 0.0 % )   | 0.0% ( 0.0 % )   |
| expected life ( years ) | 6.1              | 6.0              | 6.1              |

Question:
what was the average expected volatility of the weighted-average estimated fair value of employee stock options from 2010 to 2012


pre_text: defined contribution plan the company and certain subsidiaries have various defined contribution plans , in which all eligible employees may participate .
in the u.s. , the 401 ( k ) plan is a contributory plan .
matching contributions are based upon the amount of the employees 2019 contributions .
after temporarily suspending all matching contributions , effective july 1 , 2010 , the company reinstated matching contributions and provides a dollar for dollar ( 100% ( 100 % ) ) match on the first 4% ( 4 % ) of employee contributions .
the maximum matching contribution for 2010 was pro-rated to account for the number of months remaining after the reinstatement .
the company 2019s expenses for material defined contribution plans for the years ended december 31 , 2012 , 2011 and 2010 were $ 42 million , $ 48 million and $ 23 million , respectively .
beginning january 1 , 2012 , the company may make an additional discretionary 401 ( k ) plan matching contribution to eligible employees .
for the year ended december 31 , 2012 , the company made no discretionary matching contributions .
8 .
share-based compensation plans and other incentive plans stock options , stock appreciation rights and employee stock purchase plan the company grants options to acquire shares of common stock to certain employees and to existing option holders of acquired companies in connection with the merging of option plans following an acquisition .
each option granted and stock appreciation right has an exercise price of no less than 100% ( 100 % ) of the fair market value of the common stock on the date of the grant .
the awards have a contractual life of five to ten years and vest over two to four years .
stock options and stock appreciation rights assumed or replaced with comparable stock options or stock appreciation rights in conjunction with a change in control of the company only become exercisable if the holder is also involuntarily terminated ( for a reason other than cause ) or quits for good reason within 24 months of a change in control .
the employee stock purchase plan allows eligible participants to purchase shares of the company 2019s common stock through payroll deductions of up to 20% ( 20 % ) of eligible compensation on an after-tax basis .
plan participants cannot purchase more than $ 25000 of stock in any calendar year .
the price an employee pays per share is 85% ( 85 % ) of the lower of the fair market value of the company 2019s stock on the close of the first trading day or last trading day of the purchase period .
the plan has two purchase periods , the first one from october 1 through march 31 and the second one from april 1 through september 30 .
for the years ended december 31 , 2012 , 2011 and 2010 , employees purchased 1.4 million , 2.2 million and 2.7 million shares , respectively , at purchase prices of $ 34.52 and $ 42.96 , $ 30.56 and $ 35.61 , and $ 41.79 and $ 42.00 , respectively .
the company calculates the value of each employee stock option , estimated on the date of grant , using the black-scholes option pricing model .
the weighted-average estimated fair value of employee stock options granted during 2012 , 2011 and 2010 was $ 9.60 , $ 13.25 and $ 21.43 , respectively , using the following weighted-average assumptions: .
post_text: the company uses the implied volatility for traded options on the company 2019s stock as the expected volatility assumption required in the black-scholes model .
the selection of the implied volatility approach was based upon the availability of actively traded options on the company 2019s stock and the company 2019s assessment that implied volatility is more representative of future stock price trends than historical volatility .
the risk-free interest rate assumption is based upon the average daily closing rates during the year for u.s .
treasury notes that have a life which approximates the expected life of the option .
the dividend yield assumption is based on the company 2019s future expectation of dividend payouts .
the expected life of employee stock options represents the average of the contractual term of the options and the weighted-average vesting period for all option tranches .
the company has applied forfeiture rates , estimated based on historical data , of 13%-50% ( 13%-50 % ) to the option fair values calculated by the black-scholes option pricing model .
these estimated forfeiture rates are applied to grants based on their remaining vesting term and may be revised in subsequent periods if actual forfeitures differ from these estimates. .
label: 31.5
qa: answer: 31.5%
ann_table_rows: 1
steps: op: add1-1
arg1: 24.0
arg2: 28.8
res: 52.8
op: add1-2
arg1: 41.7
arg2: #0
res: 94.5
op: divide1-3
arg1: #1
arg2: const_3
res: 31.5%
program: add(24.0, 28.8), add(41.7, #0), divide(#1, const_3)
gold_inds: table_1: the expected volatility of 2012 is 24.0% ( 24.0 % ) ; the expected volatility of 2011 is 28.8% ( 28.8 % ) ; the expected volatility of 2010 is 41.7% ( 41.7 % ) ;
exe_ans: 31.5
tfidftopn: text_19: the weighted-average estimated fair value of employee stock options granted during 2012 , 2011 and 2010 was $ 9.60 , $ 13.25 and $ 21.43 , respectively , using the following weighted-average assumptions: .
text_25: the expected life of employee stock options represents the average of the contractual term of the options and the weighted-average vesting period for all option tranches .
program_re: divide(add(41.7, add(24.0, 28.8)), const_3)
model_input: text_17
for the years ended december 31 , 2012 , 2011 and 2010 , employees purchased 1.4 million , 2.2 million and 2.7 million shares , respectively , at purchase prices of $ 34.52 and $ 42.96 , $ 30.56 and $ 35.61 , and $ 41.79 and $ 42.00 , respectively .
text_19
the weighted-average estimated fair value of employee stock options granted during 2012 , 2011 and 2010 was $ 9.60 , $ 13.25 and $ 21.43 , respectively , using the following weighted-average assumptions: .
table_1
the expected volatility of 2012 is 24.0% ( 24.0 % ) ; the expected volatility of 2011 is 28.8% ( 28.8 % ) ; the expected volatility of 2010 is 41.7% ( 41.7 % ) ;