 Hello and welcome to the session. In this session we will discuss Present Value or Present Worth of an NUT. Now the Present Value or Present Worth of an NUT is the sum of the present values of its different instruments that is the total worth of all the variables of NUT under consideration at the beginning of the NUT. It can also be understood as the lump sum is invested at compound interest with the objective of obtaining a series of payments over some future period of time. The lump sum invested is called the present value of an ordinary NUT. Now consider an ordinary NUT of $10,000 per year, money or present worth for a percent is it in three equal installments $1000. The present worth so we have taken that Mr. X takes a loan and repays it in three equal installments of $10,000. Now by using the compound interest formula that is the amount a is equal to p into 1 plus i whole raised to power n is the number of periods i is equal to r upon 100 where r percent is the rate of interest per period. Now here the rate of interest per period is given as 4% that is the rate of interest per year is given as 4%. So here r upon 100 which is equal to 4 upon 100 which is equal to 0.04. Now using this formula for the compound interest for the first installment $10,000 will be equal to p1 into 1 plus i the whole. For the first interest is equal to 1 year let p is equal to p1 and a is equal to this implies p1 is equal to $10,000 upon 1 plus i that is 1 plus 0.04 which is equal to $10,000 upon 1.04. Now for the second installment p is equal to p2 so this is equal to $10,000 upon 1 plus i whole raised to power as here n is equal to 2 years so this is equal to $10,000 upon 1.04 whole square. Then for third installment let p is equal to p3 which is equal to $10,000 upon 1 plus i whole raised to power 3 that is here n is 3 years this is equal to $10,000 upon 1.04 whole raised to power 3. Now we know that the present value or present worth of a given NUT is the sum of the present values of its different installments so we have the value of present worth of the given NUT with p so p will be equal to p1 plus p2 plus p3. Therefore p is equal to p1 plus p2 plus p3 which is equal to $10,000 upon 1.04 plus $10,000 upon 1.04 whole square plus $10,000 upon 1.04 which is further equal to $10,000 upon 1.04 into 1 plus 1 upon 1.04 plus 1 upon 1.04 whole square on this complete hold. Now this is a geometric progression with A is equal to 1 and R is equal to 1 upon 1.04 which is less than 1 so when R is less than 1 then in a geometric progression we can find out the sum at 2 N terms that is Sn by using the formula which is A into 1 minus R raised to power N level where upon 1 minus R so this is equal to $10,000 upon 1.04 into now A is 1 so this will be 1 minus R raised to power N that is 1 upon 1.04 whole raised to power 3 whole upon 1 minus R that is 1 minus 1 upon 1.04 whole and this complete hold is equal to $10,000 upon 1.04 into 1 minus 1 upon now 1.04 whole cube is 1.124864 and the denominator it will be 1.04 minus 1 whole upon 1.04 and this complete hold now this is equal to $10,000 upon 1.04 into 1 minus 0.888996 the whole into 1.04 upon 0.04 further on solving this is equal to $10,000 into 0.11004 into 100 whole upon 4 which is equal to $27,751. So one important result that is present value of an annuity is equal to amount into 1 plus i whole raised to power minus N. Now for the present value of an annuity it is discussed in case one that is to find the present value of an annuity immediate. Now in the case of annuity immediate the present value E will be equal to A upon 1 plus i plus A upon 1 plus i whole square plus 0 and up to plus A upon 1 plus i whole raised to power N which is equal to A upon 1 plus i whole raised to power N into 1 plus i whole raised to power N minus 1 plus 1 plus i whole raised to power N minus 2 plus 0 and up to plus 1 this complete hold further this can be written as A upon 1 plus i whole raised to power N into 1 plus 1 plus i the whole plus 1 plus i whole square plus so on up to plus 1 plus i whole raised to power N minus 1 and this complete hold. Now this is the symmetric progression where the r is equal to 1 plus i and A is 1 by the whole that means r is greater than 1 and when r is greater than 1 then in a geometric progression we can find out the sum of two n terms by using the formula that is Sn is equal to minus 1 the whole whole upon r minus 1 1 plus i whole raised to power N into 1 plus r N minus 1 i the whole minus 1 and this complete hold this is equal to A upon i into 1 minus upon 1 plus i raised to power N and this complete hold equal to A upon a whole raised to power minus N the present value which is denoted by capital H of annuity immediate. First the case tool and that is to find the present value of an annuity due. In the present annuity due we have A into 1 plus i the whole whole upon i whole raised to power N minus 1 and this complete hold annual payment of each installment of an annuity the present value of an annuity so the present value of an annuity that is B is equal to 0 that is A into 1 plus i the whole whole upon i into 1 plus i whole raised to power N minus 1 and this complete hold into 1 plus i whole raised to power minus N further this is equal to A upon i into 1 plus i the whole into 1 raised to power N into 1 plus i whole raised to power minus N will be equal to 1 plus i whole raised to power N minus N which is equal to whole raised to power 0 minus 1 into 1 plus i whole raised to power minus N will be 1 plus i whole raised to power minus N and this complete hold is equal to A upon i into 1 plus i the whole into 1 plus i whole raised to power 0 will be 1 minus 1 plus i whole raised to power minus N and this complete hold. So this is the present value B of an annuity due. So in this session you have learnt about present value of present world of an annuity and this completes our session. Hope you all have enjoyed this session.