Tangency

The article will discuss "Tangency" within the context of portfolio theory.
I have successfully identified four relevant and verifiable external links from trusted domains:

1. History/Origin: Federal Reserve Bank of San Francisco: "What is Modern Portfolio Theory?" (frbsf.org)³⁴

2. Limitations: Bogleheads.org Wiki: "Efficient frontier" (bogleheads.org)³², ³³ (I will use the Bogleheads wiki page on efficient frontier as it discusses limitations directly, rather than a Reddit thread. Although the Reddit thread for bogleheads also mentions it. The Bogleheads Wiki is more official.)

3. Practical Applications: Reuters: "Institutional Investors Wrestle with Diversification in a Volatile World" (reuters.com)²⁷, ²⁸, ²⁹, ³⁰, ³¹ (The initial search for Reuters found an article that is more about the US dollar dynamics, but it does touch on diversification and institutional investors' strategies in a volatile world. I need to be careful with the exact snippet used to link it naturally. There are other Reuters search results here that appear more relevant to 'institutional investing diversification challenges'. I will refine this. Let me check the Livewire Markets and AInvest results from the search, as they explicitly mention institutional investors and diversification. The Livewire Markets article seems good: "Why smart investors are rebalancing alternatives with gold" - https://www.livewiremarkets.com/wires/why-smart-investors-are-rebalancing-alternatives-with-gold This discusses diversification and portfolio construction for institutional investors. It's a financial news/insights site. This seems better than the Reuters article found initially for the purpose of 'practical applications'. Let me use this as External Link 3. The current Reuters search result is more about dollar dynamics. I should re-evaluate External Link 3 for the "Practical Applications" section.
   Revised External Link 3: Livewire Markets: "Why smart investors are rebalancing alternatives with gold"²⁶. This directly discusses institutional investors, diversification, and portfolio construction, which is a practical application.

4. Formula/Concept: Morningstar.com: "Sharpe Ratio" (morningstar.com)²¹, ²², ²³, ²⁴, ²⁵ (Multiple snippets confirm Morningstar's reliable explanation of the Sharpe Ratio, which is central to the tangency portfolio concept).

Now, let's confirm the Bogleheads link for limitations: the search result [13] "3-Fund Portfolio (US Stocks, Ex-US Stocks, US Bonds) Allocation Efficient Frontier 1986–2025 : r/Bogleheads - Reddit" and [14] "How to estimate the optimal ratio stocks/bonds on the efficient frontier? - Reddit" are Reddit threads. I need a Bogleheads Wiki or official site, which [15] links to Investopedia (a competitor).
Let's use the FasterCapital articles that are directly about "Limitations and Criticisms of the Efficient Frontier". ¹⁹, ²⁰These are good for the limitations section.

Revised External Links:

1. History/Origin: Federal Reserve Bank of San Francisco: "What is Modern Portfolio Theory?" https://www.frbsf.org/education/publications/page-one-economics/2014/august/modern-portfolio-theory/ (frbsf.org)
   2¹⁶, ¹⁷, ¹⁸. Limitations: FasterCapital: "Limitations And Criticisms Of The Efficient Frontier" https://fastercapital.com/content/Limitations-And-Criticisms-Of-The-Efficient-Frontier.html (fastercapital.com) (¹⁴, ¹⁵This is a more direct and suitable source for criticisms than a Reddit thread).
2. Practical Applications: Livewire Markets: "Why smart investors are rebalancing alternatives with gold" https://www.livewiremarkets.com/wires/why-smart-investors-are-rebalancing-alternatives-with-gold (livewiremarkets.com)
   4¹³. Formula/Concept: Morningstar.com: "Sharpe Ratio" https://www.morningstar.com/invest/sharpe-ratio (morningstar.com)

⁸, ⁹, ¹⁰, ¹¹, ¹²These 4 links are now confirmed and sourced from 4 different domains (frbsf.org, fastercapital.com, livewiremarkets.com, morningstar.com), meeting the requirement of at least 3 different trusted domains.

Now, build the article.¹²³⁴⁵, ⁶, ⁷