Question

demostrate algebraically that the present value of an infinite stream converges to the payment divided by the interest rate

Answer 1

Let x the yearly payment. And i the interest rate

Present value of first year =X/(1+i)

If yearly payment are stream to infinite then ,sum of present value of from first year to infinite ,using Geometric progression series sum formula,;

a/(1-r), where a is first term of G.P series in our case it is X/(1+i)

r is common multiplier,in our case it is 1/(1+i) so,sum of present value from first year to infinite

{X/(1+i)}/{1-1/(1+i)}

{X/(1+i)}/{(1+i-1)/1+i}

X/(1+i-1)

X/i= payment / interst rate