On December 1, 2020, we acquired 100% of the shares of AGI, a premier provider of mission-simulation, modeling, testing and analysis software for aerospace, defense and intelligence applications. The acquisition expands the scope of our offerings, empowering users to solve challenges by simulating from the chip level all the way to a customer's entire mission. The transaction closed with a purchase price of $722.5 million.

On April 1, 2020, we acquired 100% of the shares of Lumerical, a leading developer of photonic design and simulation tools, for a purchase price of approximately $107.5 million, paid in cash. The acquisition adds best-in-class photonic products to our multiphysics portfolio, providing customers with a full set of solutions to solve their next-generation product challenges.

During the year ended December 31, 2020, we incurred $5.1 million in acquisition-related expenses, recognized as selling, general and administrative expense on the consolidated statements of income.

The assets and liabilities of AGI and Lumerical have been recorded based upon management's estimates of their fair market values as of each respective date of acquisition. The following tables summarize the fair value of consideration transferred and the fair values of identified assets acquired and liabilities assumed at each respective date of acquisition:


(in thousands)	AGI		Lumerical		Total
Cash	$	489,797	$	107,545	$	597,342
Ansys common stock(1)	218,348		—		218,348
Consideration not yet paid	14,342		—		14,342
Total consideration transferred at fair value	$	722,487	$	107,545	$	830,032

(1)We issued 0.6 million shares of our common stock in an unregistered offering to the prior owners of AGI with a fair value of $217.4 million, and we issued $0.9 million from shares held in treasury.


(in thousands)	AGI		Lumerical		Total
Cash	$	17,663	$	11,844	$	29,507
Accounts receivable and other tangible assets	29,456		4,244		33,700
Developed software and core technologies	184,100		31,614		215,714
Customer lists	25,400		1,616		27,016
Trade names	18,200		1,756		19,956
Accounts payable and other liabilities	(23,279)		(1,106)		(24,385)
Deferred revenue	(5,467)		(1,405)		(6,872)
Net deferred tax liabilities	(50,207)		(7,452)		(57,659)
Total identifiable net assets	$	195,866	$	41,111	$	236,977
Goodwill	$	526,621	$	66,434	$	593,055

The goodwill, which is generally not tax-deductible, is attributed to intangible assets that do not qualify for separate recognition, including the assembled workforce of the acquired business and the synergies expected to arise as a result of the acquisitions.

The fair values of the assets acquired and liabilities assumed are based on preliminary calculations. The estimates and assumptions for these items are subject to change as additional information about what was known and knowable at the acquisition date is obtained during the measurement period (up to one year from the acquisition date).

We determined the fair value of our intangible assets using various valuation techniques, including the relief-from-royalty method and the multi-period excess earnings method. These models utilize certain unobservable inputs classified as Level 3 measurements as defined by ASC 820, Fair Value Measurements and Disclosures. The determination of fair value requires considerable judgment and is sensitive to changes in underlying assumptions, estimates and market factors. Estimating fair value requires us to make assumptions and estimates regarding our future plans, as well as industry and economic conditions. These assumptions and estimates include, but are not limited to: selection of a valuation methodology, royalty rate, discount rate and attrition rate.

The weighted-average useful life, valuation method and assumptions used to determine the fair value of the intangible assets acquired with the AGI acquisition are as follows:

Intangible Asset

Weighted-Average Useful Life

Valuation Method

Assumptions

Developed software and core technologies

10 years

Relief-from-royalty

Royalty rate: 40.0%

Discount rate: 11.0%

Trade names

9 years

Relief-from-royalty

Royalty rate: 1.0% - 2.0%

Discount rate: 11.0%

Customer lists

15 years

Multi-period excess earnings

Attrition rate: 10.0%

Discount rate: 12.0%

The weighted-average useful life, valuation method and assumptions used to determine the fair value of the intangible assets acquired with the Lumerical acquisition are as follows:

Intangible Asset

Weighted-Average Useful Life

Valuation Method

Assumptions

Developed software and core technologies

10 years

Multi-period excess earnings

Discount rate: 16.5%

Trade names

6 years

Relief-from-royalty

Royalty rate: 2.0%

Discount rate: 16.5%

Customer lists

10 years

Multi-period excess earnings

Attrition rate: 10.0%

Discount rate: 12.5%

On November 1, 2019, we completed the acquisition of 100% of the shares of LST, the premier provider of explicit dynamics and other advanced finite element analysis technology, for a purchase price of $781.5 million, inclusive of final net working capital adjustments. The acquisition empowers our customers to solve a new class of engineering challenges, including developing safer automobiles, aircraft and trains while reducing or even eliminating the need for costly physical testing. The purchase price was paid with $472.8 million in cash and 1.4 million shares of our common stock valued at $308.7 million. We issued $307.2 million of common stock in an unregistered offering to the prior owners of LST and the remaining $1.5 million was issued from shares held in treasury.

On February 1, 2019, we completed the acquisition of 100% of the shares of Granta Design for a purchase price of $208.7 million, paid in cash and inclusive of final net working capital adjustments. The acquisition of Granta Design, the premier provider of materials information technology, expands our portfolio into this important area, giving customers access to materials intelligence, including data that is critical to successful simulations.

Additionally, during the year ended December 31, 2019, we acquired Dynardo, Helic and DfR Solutions to combine the acquired technologies with our existing comprehensive multiphysics portfolio. These acquisitions were not individually significant. The combined purchase price of these other acquisitions was $138.5 million, paid in cash and inclusive of final net working capital adjustments.

During the year ended December 31, 2019, we incurred $6.6 million in acquisition-related expenses, recognized as selling, general and administrative expense on the consolidated statements of income.

The assets and liabilities of the acquisitions have been recorded based upon management's estimates of their fair market values as of each respective date of acquisition. The following tables summarize the fair values of consideration transferred and the fair values of identified assets acquired and liabilities assumed at each respective date of acquisition:


(in thousands)	LST		Granta Design		Other Acquisitions		Total
Cash	$	472,800	$	208,736	$	138,548	$	820,084
Ansys common stock	308,704		—		—		308,704
Total consideration transferred at fair value	$	781,504	$	208,736	$	138,548	$	1,128,788


(in thousands)	LST		Granta Design		Other Acquisitions		Total
Cash	$	8,520	$	13,644	$	6,231	$	28,395
Accounts receivable and other tangible assets	20,569		6,941		10,734		38,244
Developed software and core technologies (10-year weighted-average life)	167,700		32,445		25,018		225,163
Customer lists (15-year weighted-average life)	25,900		20,016		15,743		61,659
Trade names (10-year weighted-average life)	10,600		4,579		2,051		17,230
Accounts payable and other liabilities	(3,887)		(5,164)		(5,946)		(14,997)
Deferred revenue	(3,565)		(1,426)		(1,889)		(6,880)
Uncertain tax positions	—		—		(257)		(257)
Net deferred tax liabilities	(47,412)		(8,976)		(8,151)		(64,539)
Total identifiable net assets	$	178,425	$	62,059	$	43,534	$	284,018
Goodwill	$	603,079	$	146,677	$	95,014	$	844,770

At the closing of the transaction, LST had uncertain tax positions inclusive of interest and penalties of $34.0 million and a corresponding indemnification asset. The uncertain tax positions reflected potential federal and state tax liabilities associated with tax years 2016 to 2019. As all tax positions have been resolved, the uncertain tax positions and the corresponding indemnification asset recorded at closing have been removed.

Intangible Asset

Valuation Method

LST Assumptions

Granta Design Assumptions

Developed software and core technologies

Relief-from-royalty

Royalty rate: 50.0%

Discount rate: 10.0%

Royalty rate: 8.0% - 10.0%

Discount rate: 12.5%

Trade names

Relief-from-royalty

Royalty rate: 2.0%

Discount rate: 10.0%

Royalty rate: 2.0%

Discount rate: 14.0%

Customer lists

Multi-period excess earnings

Attrition rate: 10.0%

Discount rate: 11.0%

Attrition rate: 10.0%

Discount rate: 12.5%

The operating results of each acquisition have been included in our consolidated financial statements since each respective date of acquisition. The effects of the business combinations were not material to our consolidated results of operations individually. The table presented below reflects the aggregate impact on our results of operations of the 2019 acquisitions from the date of acquisition to December 31, 2019. The operating income does not include integration costs borne directly by us and our non-acquired subsidiaries as a result of the acquisitions.

On May 2, 2018, we completed the acquisition of 100% of the shares of OPTIS, a premier provider of software for scientific simulation of light, human vision and physics-based visualization, for a purchase price of $291.0 million, paid in cash. The acquisition extends our portfolio into the area of optical simulation to provide comprehensive sensor solutions, covering visible and infrared light, electromagnetics and acoustics for camera, radar and lidar.

During the year ended December 31, 2018, we incurred $3.5 million in acquisition-related expenses, recognized as selling, general and administrative expense on the consolidated statements of income.

The assets and liabilities of OPTIS have been recorded based upon management's estimates of their fair market values as of the acquisition date. The following tables summarize the fair value of consideration transferred and the fair values of identified assets acquired and liabilities assumed at the acquisition date:


(in thousands)	OPTIS
Cash	$	7,957
Accounts receivable and other tangible assets	15,910
Developed software and core technologies (10-year weighted-average life)	47,597
Customer lists (12-year weighted-average life)	41,303
Trade names (9-year weighted-average life)	10,749
Accounts payable and other liabilities	(11,941)
Deferred revenue	(2,470)
Net deferred tax liabilities	(23,438)
Total identifiable net assets	$	85,667
Goodwill	$	205,316

During the one-year measurement period since the OPTIS acquisition date, we adjusted the fair values of the assets acquired and liabilities assumed, with the offset recorded as a $2.6 million increase to goodwill. These adjustments were made as we obtained new information about facts and circumstances that existed as of the acquisition date that, if known, would have affected the measurement of the amounts recognized as of that date.

Full pro forma results of operations have not been presented as the effects of the OPTIS business combination were not material to our consolidated results of operations. The table presented below reflects the impact of OPTIS from the date of acquisition to December 31, 2018. The operating loss does not include integration costs borne directly by us and our non-OPTIS subsidiaries as a result of the acquisition.