Independent solution

How to solve this Continuous Perpetuity question

Setup

Setup

Split the continuous payment stream at time 10, where its growth pattern changes. The first component is level; the second grows continuously at 3%.

PV1=∫010(1.06)−tdt=7.58PV_1=\int_0^{10}(1.06)^{-t}dt=7.58

Model

Model

Discount both integrals at the 6% annual-effective accumulation convention. For the deferred tail, first discount ten years and then integrate the net growth-discount exponent.

PV2=(1.06)−10∫0∞(1.03)s(1.06)−sds=19.45PV_2=(1.06)^{-10}\int_0^\infty(1.03)^s(1.06)^{-s}ds=19.45

Compute

Compute

The first component is 7.58 and the tail is 19.45. Their sum is 27.03.

PV=27.03PV=27.03

Answer

Answer

The calculation gives 27.03 for continuous perpetuity, matching published choice A.

PV=27.03(A)\boxed{PV=27.03\quad\text{(A)}}