Difference between continuously payable annuity and perpetuity

[bij_30]

Hello,

What is the basic difference between continuously payable annuity and perpetuity? I understand that for perpetuity n (number of years) tends to infinity. But why is continuously payable annuity a factor of force of interest (small delta) and perpetuity a factor of effective interest (i)?

Thanks for your help in advance,

Regards,
Bijal

 

[DevonMatthews]

Hello,

What is the basic difference between continuously payable annuity and perpetuity? I understand that for perpetuity n (number of years) tends to infinity. But why is continuously payable annuity a factor of force of interest (small delta) and perpetuity a factor of effective interest (i)?

Thanks for your help in advance,

Regards,
Bijal

Hi there,

This is because a perpituity (ie. a_infinity) is the present value of a payment at the end of years 1,2,3, up until infinity. So as a function of v (which is a function of i), this is v + v^2 + .... + v^infinity. Summing this as a geometric series you are left with v/(1-v) = 1/i

A continuously payable perpetuity however is abar_infinity. This is the "same" as abar1 + v*abar1 + ... (v^2)*abar1 + ... + (v^inf)abar1, summing this also as a geometric series one attains abar1(1/(1-v)) = ((1-v)/delta)*(1-v) = 1/delta.

Basically, whenever payments are at the end of the year, we use "i", when they are at the start of the year, we use "d", and when they are continuous, we use "delta". You could also however make this a function of i by substituting delta=Log(1+i) if you wanted.

Please excuse my unorthadox method of demonstrating this, but i hope this answers your question.

 

[bij_30]

Thanks a lot Devon!!