 Hello and welcome to the session. In this session, first of all, let us find the present value of perpetual annuity. Now, our perpetuity is an annuity whose payments being on a fixed date and continue forever. So, we can find out the present value of a perpetuity, but we cannot determine the future amount as the future extends to infinity. Now, in case of a present value of an annuity in date, p is equal to a upon i into 1 minus 1 plus i by 2 minus 2 power minus n and this complete whole. Now, if n tends to infinity, then we have a perpetuity that is forever to infinity. 1 plus i whole raised to power minus n tends to 0. So, the present value is equal to a upon i into 1 minus 0 by whole, which is equal to a upon i. The present worth for perpetual annuity is equal to a upon i. Now, let us illustrate this with the help of an example. Now, suppose an amount is deposited at 5 percent per annum, so that it gives a return of 100 dollars per annum in perpetuity. The present value is for a perpetual annuity, that is, p is equal to a upon i. Now, p is equal to a upon i implies, now a is also given to us, but i is equal to r upon 100. So, using these values, that is, the rate of interest is given as 5 percent per annum. So, p will be equal to 500 upon i, which is equal to r upon 100, that is, 5 upon 100. So, this will be 500 over 5 into 100. This is minus 0, 500 dollars for the first year after giving back this interest to the individual. Again, 10,000 dollars remain as it is, 10,000 dollars, but the present of 10,000 dollars will remain intact for the first annuity. Now, suppose the preferred annuity begins for Q years, the annuity begins at the end of from 0 to p is called the interval of definite at the end of p is du, is du at the end of p plus 2 years and so on. Now, here the present value is upon 1 plus p plus 1, that is, by using, we have called the present value of a at the end of p plus 1 the whole years as a upon 1 plus i whole raised to power p plus 1. Now, the present value and of p plus 2 raised to power p plus 2 is equal to a upon 1 plus i whole raised to power p plus 1 plus a upon 1 plus i whole raised to power p plus 2 i whole raised to power p plus q a geometric progression. Now, in a geometric progression, the sum of p n terms, that is, S n is given by the formula n a whole over 1 minus i less than 1 and for the geometric progression is equal to 1 plus i whole raised to power p plus 1 into 1 minus 1 upon 1 minus 1 upon 1 plus i the whole and this definite whole is equal to a upon 1 plus i whole raised to power p plus 1 into 1 plus i into 1 minus 1 upon 1 plus i whole raised to power q and this complete whole. Now, this is equal to a upon i into 1 plus i the whole whole upon 1 plus i whole raised to power p plus 1 into 1 plus i whole raised to power q minus 1 whole upon 1 plus i whole raised to power q and this complete whole. Now this is equal to a upon i into 1 plus i whole raise to power 1 minus p minus 1 whole upon 1 plus i whole raise to power q into 1 plus i whole raise to power q minus 1 and this complete whole and also I mean this is equal to a upon i into 1 plus i whole raise to power q minus 1 and this complete whole 1 plus i whole raise to power p plus q p is equal to a upon i into 1 upon 1 plus i whole raise to power p minus 1 upon 1 plus i whole raise to power p plus q which is further equal to a upon i into 1 minus 1 upon 1 plus i p plus q this complete whole minus 1 upon 1 plus the whole that is the present with instalments is equal to the present value of an immediate annuity for p minus the present value is equal to a into 1 plus i power q minus 1 plus a into 1 plus i whole raise to power q minus 2 plus 1 plus a into 1 plus i the whole plus a on summing up this geometric progression we get to a upon i into 1 plus q minus 1 and this complete whole now this one for the amount of a definite annuity is same as that for ordinary annuity you have learnt about the present value of perpetual annuity and the present value of definite annuity hope you all have enjoyed the session.