Calculus Derivation of Perpetuity Formula
The present value of perpetuity is given by
𝑃𝑉 =
𝐶1
𝐶2
𝐶3
+
+
+⋯
(1 + 𝑟) (1 + 𝑟)2 (1 + 𝑟)3
Now multiply both sides of this equation by (1 + 𝑟) to get:
𝑃𝑉 (1 + 𝑟) =
𝐶1
𝐶2
𝐶3
(1 + 𝑟) +
(1
+
𝑟)
+
(1 + 𝑟) + ⋯
(1 + 𝑟)
(1 + 𝑟)2
(1 + 𝑟)3
𝑃𝑉 (1 + 𝑟) = 𝐶 +
𝐶1
𝐶2
𝐶3
+
+
+⋯
(1 + 𝑟) (1 + 𝑟)2 (1 + 𝑟)3
Next subtract the first equation from the second one
𝑃𝑉 (1 + 𝑟) − 𝑃𝑉 = 𝐶 +
−
𝐶1
𝐶2
𝐶3
+
+
+⋯
(1 + 𝑟) (1 + 𝑟)2 (1 + 𝑟)3
𝐶1
𝐶2
𝐶3
−
−
−⋯
(1 + 𝑟) (1 + 𝑟)2 (1 + 𝑟)3
Simplifying provides our result
𝑃𝑉 (1 + 𝑟) − 𝑃𝑉 = 𝐶
𝑃𝑉 + 𝑃𝑉 ∗ 𝑟 − 𝑃𝑉 = 𝐶
𝑃𝑉 ∗ 𝑟 = 𝐶
𝑃𝑉 =
𝐶
𝑟