In private wealth planning, when do you use the finite-$N$ annuity formula vs the infinite-perpetuity formula?

Both have real-world use cases, but they apply to different vehicles. Here's the practical guide.

Use finite-$N$ when:

• Annual gifting plans during a grantor's lifetime. $N$ = remaining life expectancy.
• GRATs (Grantor-Retained Annuity Trusts) with a fixed term, typically 2-10 years. The annuity stream is finite by design.
• CRTs (Charitable Remainder Trusts) with a term certain. $N$ is the stated trust term.
• Estate planning illustrations with a defined planning horizon.

Use infinite (perpetuity) when:

• Dynasty trusts designed to last forever or until perpetuity laws void them. Some states (Florida, South Dakota, Alaska) allow effectively perpetual trusts. Math approximation uses $S_\infty$.
• Charitable endowments (university foundations, religious institutions) that intend to spend only the income and preserve corpus forever.
• Inheritance income streams modeled as "the family will own this forever" — useful for very-long-horizon planning where $N > 100$ years effectively equals infinity.
• Quick-cap valuation of high-quality dividend stocks for which the Gordon growth model is appropriate.

A subtle case — long but finite $N$:

When $N$ is large enough (say, 40+ years), the difference between the finite-$N$ formula and the infinite formula becomes small. Specifically:

For $r = 0.952$ ($5\%$ discount) and $n = 40$, $r^{40} \approx 0.142$. So $S_{40}$ captures about $86\%$ of $S_\infty$. For $n = 80$, capture is $98\%$.

This means in practice, a 40-year planning horizon already approximates "forever" for most calculation purposes. Some planners use the infinite formula as a conservative upper bound when $N$ is uncertain.

Tax considerations also pick one or the other:

• Annual gifts (finite-$N$) leverage the annual gift exclusion — $19,000/recipient/year in 2025.
• Lump-sum trust funding (infinite-projection) consumes the lifetime exemption — $13.6M in 2025.

The choice between finite gifting and lump-sum funding is often driven by tax considerations as much as math.

Exam framing:

CFA Level III vignettes specify the planning horizon. If the vignette says "the client wants to plan for the next 30 years," use finite-$N$. If it says "the family wants to preserve wealth across generations indefinitely," use perpetuity. Read carefully.