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evaluation of a debenture let's understand through it very simple practical example so this is the question the power value of a debenture is $1,000 it carries a coupon rate of 10% and is going to be redeemed in 6 years at a premium of 4% if the required rate of return is 12% calculate the value of this debenture so we have present value here that is the sorry we have the face value here that is the $1,000 coupon payment that is the 10% of 1,000 so we get $100 years are six years so any six years we have to find our B new face value because this debenture is going to be redeemed at an increase of 4% so we have to calculate the 4% of $1000 that is a $40 then it becomes 1000 $40 so in the formula we will use $1,050 not $1000 because this is going to be redeemed at an increase of 4% and now the required rate of return is 12% and find out the value of this debenture so this is the formula you see here face value 1,000 new face value is 4% of 1,000 that is the 1040 and coupon payment is $100 years and that is six years and the required rate of return is 12% find out the value so what is the formula for it this is coupon payment each each year after at the end of each year year you have to find out the present value for example if you get 100 at the end of the first year you have to find out the present value of this coupon payment so this is the formula coupon payment in the first year then second year third year fourth year then fifth year and six years so in after each year we have to find out the present value so this is the present value means $100 divided by one point one two and then here after two years that's why we have taken two here it is one but not written now it is - yeah after finding out the factor one point one two we have to divide 100 by it and in each case we do this thing so this is the after sorry after finding out the present value we have to add them together so present value of coupon payments here you have to add then present value of a single sum so this is a series of payment this that this is why this is the present value of annuity and this is the present value of single sum because we are going to receive this amount after after the after six years means at maturity we are going to receive this so this is the single payment that say we have used this method so present value of principal or the face value is this one and after adding all these present values we have found this thing's so he calculate all these present values so let's do it here in the first case means after the first year we received this $100 as a coupon payment so we have to find the present value of this $100 which is going to be received after first year we divided by 1.1 to this and we received eight nine point two eight so this is what we have written here now in the after the second year we again receive hundred-dollar so we have to in order to find out the present value of this hundred dollar we have to divide it by one point 1/2 raised to the power two because it is going to be received after two years that face we have written to here now one point one two raised to the power 2 so this is the value we have taken now I copied then I divided $100 by this factor so I received the present value of $100 which is going to be received after two years and now the present value of this $100 is seventy nine point seven two dollar so now in the case of in the case of this coupon payment which is going to be received after three years so again we have to write this and we have to now use the exponent three years and I copy it after copying I divided $100 by this factor and I find I find what I have got 71 point one seven dollar I have rounded it off so it becomes same seventy one point one $8 so this is the present value of $100 which is going to be received after three years so in each case what we do that firstly we find out this we solve this and then we divide hundred dollar by this factor and we get the present value after finding out the present value in each case we add them up and find out the present value of coupon payments so this is the present this is this is nothing but the present value of a series of payments or the present value of an 80 so here the second part that is the present value of a single sum why it is single sum because it is going to be received after six years at maturity so what formula I have used here I have used one point 1 2 raised to the power 6 then I get a factor I copy it and then I divide 1 0 4 0 not coupon payment because this is F face value and we have used new face value that is $1,040 not 1000 because this debenture is going to be redeemed at an increase of 4% at a premium of 4 person so that say we have taken $1,050 now I divided by the factor we have calculated here calculate now I divide I find out five hundred twenty six point eight nine that is five hundred twenty six point nine zero dollar so I have written here this thing now we add both of these present values together I find out the value of this debenture that is nine hundred thirty-eight point zero three dollar so as I told that this is a series of payment means $100 after each year so actually this is nothing but we are doing what we are doing that we are finding out the present value of her and any T's this is an annuity the and we are finding out present values so we can use this formula instead of writing the coupon payment for each year we can directly write this formula the formula for annuity this formula this formula will replace all these payments means all these payments through which we are going to calculate the present values so if we write this formula instead of that graph means writing after each year we can find out directly the present value of coupon payments I told that there are two parts in the first part here we are calculating the present value of coupon payment in the second part we are calculating the present value of face value present value of T face value or the par value there is a principal now if we write we use this formula we will find out we will get the same result but the calculation will become easier so let's do it this is nothing but for example 1 plus R raised to the power minus n so we can solve it by doing this 1 upon 1 plus R raised to the power n so here if we calculate for example firstly we have to find out this 1 point 1 2 raised to the power 6 so this is the factor now I copied after copying it what I do I have solved this portion now I have to divide 1 by this factor so 1 and the factor now I have found this thing so I have solved 1 + R raised to the power minus n I get this value I copy it now the second step is subtract this value from 1 so I subtract I subtract this value from of one and I get this value so I have solved now the numerator this numerator I have solved now I have to divide it by R naught 1 plus R R if it is our only then in this case we have to write only 0.12 now 0.12 here I have to write 0.12 now I have to divide it by 0.1 to 0.12 if I divide I get I get this factor present value factor now I multiply the coupon payment that is the $100 I write $100 and I get 400 11.1 $4 so this is the present value of all the coupon payments you see when we had already calculated the present ala coupon payments it came 411 point 1 $3 and in this case we have found 411 point one for a little increase of point zero one so now this is this is the same the method is same here and I I have done here this is the first part the first part has been solved this thing has been solved now now we have to solve this thing so now what I do that I write sorry I I have to solve this thing first I have to find out the factor that is the one point 1 2 raised to the power 6 so this is the value I have found now I have to divide this $140 by this factor the general factor now a present value in order to find out the present value I divided by this factor I find what is the problem 1.1 2 raised to the power 6 so this is the factor I have found now I divide the new face value by this factor so I get five hundred twenty six point eight nine not five hundred twenty six point nine zero dollar so here you see now I add them up both of these values I find out the present value of this debenture that is nine hundred thirty-eight point zero four here in this case we had nine hundred thirty-eight point zero three and here we had we have nine hundred thirty-eight point zero four the same the same value but this method is easier than this so we should better use this method use this because we directly calculate the present value of all the coupon payments through a single formula and what is the formula present value of an annuity so so thank you for watching this