Price the Perpetuity

Question

A contract pays 1 one year from now, 1 the year after, and so on forever. The discount factor per year is v = 0.96 (a dollar next year is worth 0.96 today). What is the contract worth, and derive, don't recite, the series formula you use.

Accepted Answer

The value is the geometric series S = v + v^2 + v^3 + cdots with |v| < 1. Derivation by self-similarity (the trick to know cold): vS = v^2 + v^3 + cdots = S - v implies S(1 - v) = v implies S = fracv1-v. With v = 0.96: S = frac0.960.04 = mathbf24. A sanity check that builds intuition: with discount factor v = frac11+r, the perpetuity value is fracv1-v = frac1r, here r approx 4.17% and indeed frac10.0417 approx 24. "A perpetuity is worth one over the rate" is the fastest mental anchor in fixed income, and the same self-similarity trick (S reappearing inside itself) is the engine behind first-step analysis in every expected-value recursion on this site, geometric series and Markov recursions are the same move.

Price the Perpetuity

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