Question

an increasing perpetuity immediate makes annual payments. the first payment is 100 and each subsequent payment is larger than the preceding payment by an amount X. based on an annual effective interest rate of 10%, the present value of the perpetuity at time 0 is one half of its present value at time 20. what is rhe value of x?

Answer 1

Given, First Payment = 100, and subsequent payment increases by X

Interest rate = r = 10%

NPV = Σ CFn/(1+r)ⁿ, where CFn is the cash flow in year n

Present Value at Time 0

=> NPV = 100/(1+0.10) + (100+X)/(1+0.10)² + (100+2X)/(1+0.10)³ + .....

NPV/(1+0.10) = 100/(1+0.1)² + (100+X)/(1+0.10)³ + (100+2X)/(1+0.10)⁴ + .....

NPV - NPV/(1+0.10) = 100/(1+0.10) + X/(1+0.10)² + X/(1+0.10)³ + .....

=> 0.10NPV/(1+0.10) = 100/(1+0.10) + X/(1+0.10)² / (1 - 1/(1+0.10))

=> NPV = (1.10/0.10)[100/1.10 + X/(0.10*1.10) ] = 1000 + 100X

Present Value at time 20

=> NPV = (100+19X) + (100+20X)/(1+0.10)¹ + (100+20X)/(1+0.10)² + .....

NPV/(1+0.10) = (100+19X)/(1+0.1)¹ + (100+20X)/(1+0.10)² + (100+20X)/(1+0.10)³ + .....

NPV - NPV/(1+0.10) = (100+19X) + X/(1+0.10)¹ + X/(1+0.10)² + .....

=> 0.10NPV/(1+0.10) = (100+19X)/(1+0.10) + X/(1+0.10) / (1 - 1/(1+0.10))

=> NPV = (1.10/0.10)[(100+90X)/1.10 + X/0.10 ] = 1000 + 900X + 110X = 1000 + 1010X

Present Vale at Time 0 = Present Value at Time 20 / 2

=> 1000 + 100X = (1000 + 1010X)/2

=> 1000 + 100X = 500 + 505X

=> 500 = 405X

=> X = 1.234