Question

A perpetuity has a payment stream of Pt = 5t + 2 for t > 0 at an annual effective interest rate of r. Another
perpetuity pays $40 continuously for the first year, $45 continuously for the second year, and so on, forever
with an annual effective interest rate of 5%. The present values of the two perpetuities are equal. Determine
r.

Answer 1

Formula used is

Present value for perpituity = (A/r)

Where, A is amount of payments and r is rate of interest.

Perpituity 1

Annual payments = 5t + 2

Year 1: 7

Year 2: 12

Year 3: 17 and so on

Present Value = 7/(1+r) + 12/(1+r)² +17/(1+r)³ +22/(1+r)⁴ .... Infinity.

= 2( 1/(1+r) + 1/(1+r)² .... Infinity) + 5 * ( 1/(1+r) + 2/(1+r)² .... Infinity)

= {2/r + 5 * ( 1/(1+r) + 2/(1+r)² .... Infinity)}

Perpituity 2

Annuity Payments

Year 1: 40

Year 2 to infinity: 45

Present value of annuity=

40/1.05 + 45/(1.05)² + 45(1.05)³ ... infinity

= 40/1.05 - 45/(1.05) + { 45/(1.05) +45/(1.05)² + 45(1.05)³ ... infinity

= -5/1.05 + 45/.05

= - 4.76 + 900

= 895.24