Shire's (NASDAQ: SHPG) c-suite thinks the company's future is pretty bright. The company's board dismissed a $46 billion offer to be acquired by drug giant AbbVie (ABBV 1.05%) in May, even though that offer represented a 23% premium to its pre-offer price.

According to AbbVie, the company made three attempts to win over Shire's board, but AbbVie came up short each time. Given that Shire's management says the AbbVie offer "fundamentally undervalued the company and its prospects," let's take a closer look at what Shire has to offer.

Source: Shire.

First, a bit of background
Shire's built a $5 billion-a-year business around treating ADHD and rare diseases including hereditary angioedema, or HAE, a condition characterized by spontaneous and recurring swelling that affects just 18,000 in Europe and the United States.

Although Shire has made moves to diversify its business away from ADHD therapies, the majority of the company's revenue still comes from the sale of ADHD drugs Vyvanse, Aderral XR, and Intuniv.

However, that business has come under pressure in the past year as Adderal XR lost patent protection in 2012, forcing its Adderall sales down 12% to $375 million in 2013. Shire's ADHD franchise will suffer another blow in December when Intuniv loses exclusivity, putting another $335 million in annual sales in jeopardy.

To offset some of its patent risk, Shire acquired ViroPharma for $4.2 billion in January. The ViroPharma deal gives Shire ViroPharma's HAE drug Cinryze, an ultra-high-priced drug that costs about $2,300 per vial, or $4,600 per two-vial dose. Since patients receive that dose every few days, Cinryze generates hundreds of millions in sales annually. 

Cinryze is the second of Shire's two HAE drugs. The first is Firazyr, a drug developed by Shire that was approved for use in HAE patients in 2011. Sales of Firazyr doubled to nearly $235 million last year, and Firazyr's first-quarter sales jumped 80% to $75 million. That means that Shire's two HAE drugs are on pace to produce more than $700 million in sales this year.

While that's impressive, Shire's HAE success is dwarfed by its mega-blockbuster ADHD drug, Vyvanse. That drug is one of the most widely used ADHD medications in patients under 18 years old, and Vyvanse has captured 15% market share in adults too. Last year, Shire sold $1.2 billion worth of Vyvanse, up 19% from 2012.

In addition to HAE and ADHD drugs, Shire also markets other successful drugs for rare diseases, including Elaprase for Hunter syndrome, Lialda for ulcerative colitis, and Replagel for Fabry disease. Those three drugs generated a combined $1.5 billion in sales last year.

Source: Shire Plc

Potential for growth
In addition to the stable of high-market-share ADHD drugs and high priced drugs for orphan disease, suitors like AbbVie are also likely interested in Shire's pipeline.

In Shire's Q1 investor presentation, the company said it thinks its neuroscience sales could double by 2020. That's a bold goal, given the Adderal and Intuniv patent headwinds.

SHP 465 is one reason behind Shire's optimism. SHP 465 is a long-lasting ADHD drug that Shire thinks has a shot at becoming a market share leader in adults. The adult segment is the fastest-growing segment within the ADHD market and Shire projects that growing awareness of adult ADHD will help total ADHD prescriptions climb to 70 million per year by 2020, up from 60 million today. If Shire is right, SHP 465 has a very good chance of capturing a significant piece of that market, especially since Adderal and Vyvanse already claim 73% market share.

Shire hopes to launch SHP 465 in 2015, but that isn't the only promising program in Shire's pipeline. The company also plans to file for a Vyvanse label expansion to include use as a treatment for binge eating, and it's working on other treatments for rare diseases.

Fool-worthy final thoughts
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AbbVie would benefit from adding the ADHD and orphan drugs to its product lineup, given that the company is facing its own patent hurdle in a few years for its mega-blockbuster autoimmune drug Humira. But the real reason behind AbbVie's interest in Shire is Shire's tax-friendly Irish address.

If Shire were to accept an offer from AbbVie, AbbVie would do the deal as an inversion that could significantly lower AbbVie's tax rate. Given that AbbVie's trailing-12-month tax rate is 23% and Shire's is just 14.55%, that could mean hundreds of millions in shareholder-friendly tax savings per year.