WEBVTT
00:24.710 --> 00:33.340
Welcome to the lecture series on Time Value
of money-Concepts and Calculations. In this
00:33.340 --> 00:42.200
lecture I will teach valuation of bonds part
1 and valuation of bond part 2.
00:42.200 --> 00:50.210
Bond a debt instrument issued for a period
more than a year with the purpose of raising
00:50.210 --> 01:00.010
capital by borrowing. The government agencies
and business enterprises sell bonds generally
01:00.010 --> 01:07.600
a bond is a promise to repay the principal
on maturity along with interest.
01:07.600 --> 01:15.000
Now, let us see something which is related
to the bond. Par value it is the face value
01:15.000 --> 01:21.680
of the bond. Coupon rate is the specified
interest rate, interest payable to bond holder
01:21.680 --> 01:29.159
per year is equal to par value into coupon
rate. Maturity period it refers to the number
01:29.159 --> 01:35.659
of years after which the par value is payable
to the bond holder.
01:35.659 --> 01:45.420
Let us take an example. Example 1, a firm
as issued a 10 year bond with 1,000 par values
01:45.420 --> 01:52.640
with 10 percent coupon interest rate, with
interest paid semi-annually. The bond holder
01:52.640 --> 01:59.560
as a right to get rupees 100 and annual interest
which comes out from par value into coupon
01:59.560 --> 02:08.290
rate, which is 0.1 into 1,000 is equal to
100 paid as rupees 50, 100 divided by 2 at
02:08.290 --> 02:16.490
the end of every 6 months and rupees 1,000
par value at the end of 10'th year.
02:16.490 --> 02:24.600
Let us take an example. This is called example
2, a firm as issued a 10 year bond with 1,000
02:24.600 --> 02:30.959
par values with 10 percent coupon rate as
well as discounted; that means, discount rate
02:30.959 --> 02:38.290
and coupon rates are 10 percent with interest
paid annually. Compute the value of the bond.
02:38.290 --> 02:44.950
Now, if you analyze this, this bond is giving
100 rupees at the end of each year up to 10
02:44.950 --> 02:50.570
years. So, this is shown in the time line
at the end of the first year, you get 100
02:50.570 --> 02:55.860
rupees and end of the second year, you get
again 100 rupees so on so forth up to the
02:55.860 --> 03:01.549
end of 10'th year, you get 100 rupees. The
present values of these 100 rupees can be
03:01.549 --> 03:09.739
calculated using this formula, which is being
used to calculate the present value of annuity.
03:09.739 --> 03:16.900
This is P equal to A in brackets 1 plus i
to the power N minus 1 divided by i 1 plus
03:16.900 --> 03:24.510
i to the power N. And then this line shows
at the end of 10th year you will 1,000 rupees
03:24.510 --> 03:30.489
and the present value of these 1,000 rupees
can be calculated by bringing these values
03:30.489 --> 03:36.799
to time line t equal to 0.
So, here the present value is calculated in
03:36.799 --> 03:44.120
two groups. This first group shows the present
value of the annuity and second group is related
03:44.120 --> 03:49.669
to this figure; that is at the end of 10'th
year, you will get 1,000. So, when you put
03:49.669 --> 04:00.140
the values this becomes 100 into 6.14456 plus
1,000 into 0.38554 which comes out to be rupees
04:00.140 --> 04:08.790
999.996, which is almost equal to 1,000. Here
the 100 rupees interest paid annually for
04:08.790 --> 04:19.799
10 years is considered as annuity for 10 years
and discount rate is 10 by 100 is 0.1 conclusions.
04:19.799 --> 04:27.180
The bond value is equal to the par value when
coupon rate is equal to discount rate, then
04:27.180 --> 04:30.210
bond value is equal to par value.
04:30.210 --> 04:36.580
Now values of bond with different coupon and
discount rate. The values of the bond are
04:36.580 --> 04:44.159
the present value of the contractual payment.
It is issuer that is government or enterprises
04:44.159 --> 04:47.680
obliged to make from the beginning till maturity.
04:47.680 --> 04:56.170
Now, to find out what will happen if the bond
has got different coupon and discounted we
04:56.170 --> 05:03.669
have taken the example 3. The value of a bond
of SJ International matures in 5 years with
05:03.669 --> 05:10.669
a coupon rate of 7 percent and a maturity
value of 1,000. For simplicity, let us assume
05:10.669 --> 05:17.099
that the bond pays annually and the discount
rate is 5 percent. Here we see that coupon
05:17.099 --> 05:22.550
rate is 7 percent and discount rate is 5 percent
and hence discount rate is different than
05:22.550 --> 05:30.370
coupon rate. In earlier questions, the coupon
rate was equal to the discount rate. As the
05:30.370 --> 05:39.409
cash flow for each of the year is as follows;
after just 1 year this 70 rupees will get,
05:39.409 --> 05:45.990
as the interest that is par value 1,000 into
coupon rate divided by 100 coupon rate is
05:45.990 --> 05:50.699
7 percent. So, this 0.07 into 1,000 is 70
rupees.
05:50.699 --> 05:56.870
So, after just 2 years, 70 rupees, after the
3 years, 70 rupees, after the 4 years, 70
05:56.870 --> 06:05.229
rupees and after the 5 year, this is 70 plus
1,000. So, it becomes 1,070. This is par value
06:05.229 --> 06:10.220
plus interest. Now thus the PV the present
value of the cash flow is as follows.
06:10.220 --> 06:16.680
Present value for after just 1 year is rupees
70 divided by 1 plus discount rate divided
06:16.680 --> 06:25.520
by 100 to the power 1 is 70 divided by 1 plus
5 divided by 100, this becomes 66.67 rupees.
06:25.520 --> 06:35.009
Similarly, the present value after just 2
years is 70 divided by 1 plus 0.05 square
06:35.009 --> 06:45.080
which comes out to be 63.49. Similarly PV
after just third year is 60.47, PV after just
06:45.080 --> 06:55.400
fourth year is it is 57.59, PV after 5 years
is. This is 838.37. Here we will see that
06:55.400 --> 07:00.690
for discounting, we are using the discount
rate. While finding out the interest, we are
07:00.690 --> 07:04.770
using coupon rate.
Now, to find out the value of the bond all
07:04.770 --> 07:12.260
this present values are added up. So, the
bond value is 1086.59. The value of the bond
07:12.260 --> 07:19.189
when discount rate is same as the coupon rate
was 1,000, but in this case, it is different.
07:19.189 --> 07:25.469
We will illustrate the valuation bond with
reference to basic bond valuation, yield to
07:25.469 --> 07:30.199
maturity and semi-annual interest and bond
values.
07:30.199 --> 07:37.249
Now, let us take an example. Example 4, the
bond of SJ International matures in 5 years
07:37.249 --> 07:39.849
with a coupon rate of 7 percent.
.
07:39.849 --> 07:47.569
And the maturity value of 1,000. For simplicity,
let us assume the bond pays annually and the
07:47.569 --> 07:56.529
discount rate is 5 percent. Now here, we are
using formula to get these values. So, every
07:56.529 --> 08:02.350
end of the every year you will be getting
rupees 70, rupees 70, rupees 70, rupees 70
08:02.350 --> 08:10.499
and rupees 70 up to 5th year and at the end
of 5'th year rupees 1,000. So, interest for
08:10.499 --> 08:16.759
1,000 into coupon rate divided by 100 is equal
to 70 rupees. If interest per year is considered
08:16.759 --> 08:23.330
annually, annuity then the present value of
the series of annuities can be calculated
08:23.330 --> 08:29.569
as PV of the annuity is annuity into this
formula, which I had already told you in the
08:29.569 --> 08:37.450
last question and if you put just this, this
comes out to be rupees 303.0634. Where N is
08:37.450 --> 08:42.289
the maturity period, 5 years and the i is
discount rate is 5 percent.
08:42.289 --> 08:47.650
At the end of the maturity period the bearer
of the bond gets 1,000 rupees. So, if you
08:47.650 --> 08:54.570
want to find out the present value of 1,000
rupees this is equal to 1,000 divided by 1
08:54.570 --> 09:02.280
plus i to the power N. Where N is 5, this
comes out to be rupees 783.526 the present
09:02.280 --> 09:15.510
value of bond is therefore, this value 303.0634
plus this value 783.526 come out to be 1086.59.
09:15.510 --> 09:22.220
Now, you see the results are the same. Whether
I can go from the first principal to find
09:22.220 --> 09:28.720
out the present value or I use the formula
effect of discount rate on bonds value. So,
09:28.720 --> 09:34.810
what we saw here that we will find out the
same problem we solved for two different discount
09:34.810 --> 09:41.690
rates, 5 percent and 10 percent and then we
will find out the conclusion what the effect
09:41.690 --> 09:46.820
of discount rates on bonds value is. For this
we have take an example 5; the value of a
09:46.820 --> 09:52.140
bond with different discount rates. Bond of
SJ International matures in 5 years with a
09:52.140 --> 09:59.600
coupon rate of 7 percent and a maturity value
of rupees 1,000. For simplicity, let us assume
09:59.600 --> 10:05.740
that the bond pays annually compute the value
of the bond for two different discount rates
10:05.740 --> 10:09.980
5 percent and 10 percent.
Now, this 5 percent values and 10 percent
10:09.980 --> 10:16.560
values has been taken with a purpose 5 percent
is less than 7 percent and 10 percent is more
10:16.560 --> 10:23.390
than 7 percent. So, the value of the bond
when discount rate is 5 percent is 1086.59
10:23.390 --> 10:30.071
and the value of the bond when discount rate
is 10 percent is 886.52 and the value of the
10:30.071 --> 10:38.490
bond, when coupon discount rate is 7 percent
is 1,000. So, what we saw, that when the discount
10:38.490 --> 10:43.340
rate is 5 percent, the value is more than
1,000 and when the discount rate is 10 percent
10:43.340 --> 10:46.700
is more than 7 percent, the value is less
than 1,000.
10:46.700 --> 10:53.090
So, let us see here the present value of the
cash flow when discount rate is 10 percent.
10:53.090 --> 10:59.520
So, this can be calculated from the first
principal. The end of the first year interest
10:59.520 --> 11:08.581
70 is when present value is calculated by
dividing it with 1.10 to the power 1 here
11:08.581 --> 11:16.710
I am taking the interest discount rate to
be 10 percent. This is 63.63. Similarly, these
11:16.710 --> 11:24.250
values can be converted to the present value.
So, is comes out to be rupees 886.52 and this
11:24.250 --> 11:30.310
is for discount rate 10 percent. Now what
conclusion we make, when discount rate is
11:30.310 --> 11:36.360
different than coupon rate. The value of the
bond will differ from its par value. This
11:36.360 --> 11:37.360
is number 1 conclusion.
11:37.360 --> 11:47.550
The number conclusion 2; when discount rate
will be more than the coupon rate, the value
11:47.550 --> 11:56.000
of the bond will be less than the par value.
Now par value was 1,000 and will sell at a
11:56.000 --> 12:02.380
discount I know, when the coupon rate was
more that is 10 percent the value of the bond
12:02.380 --> 12:08.440
was 886.52 which was less than 1,000. So,
it will sell at a discount and discount will
12:08.440 --> 12:17.610
be rupees 1,000 minus rupees 886.52, which
comes out to be rupees 113.48. When discount
12:17.610 --> 12:23.660
rate will be less than coupon rate, the value
of the bond will be more than the par value
12:23.660 --> 12:30.800
and will sell at premium and this premium
is equal to 1086.59 minus 1,000 which is the
12:30.800 --> 12:38.760
par value is rupees 86.59. So, when my discount
rate was 5 percent, the bond will sell with
12:38.760 --> 12:41.320
the premium of 86.59.
12:41.320 --> 12:48.490
Now, let us take an example. In this example,
we have calculated the value of the bond with
12:48.490 --> 12:55.330
different discount rates 5, 7, 10 percent,
12 percent, 14 percent and 16 percent. So,
12:55.330 --> 13:03.200
here we see when it is 5 percent, the value
of the bond 1086.59. When it is 7 percent,
13:03.200 --> 13:16.450
it is 1,000. When it is 10 percent, it is
886.2764. When it is 12 percent, it is 819.7612.
13:16.450 --> 13:25.360
When it is 14 percent, it is 759.68 and when
it is 16 percent, it is 705.3136.
13:25.360 --> 13:31.301
So, when the discount rate is 7 percent, which
is equal to coupon rate, the value of the
13:31.301 --> 13:37.620
bond is equal to the par value and the discount
rate are more than 7 percent, the bond rate
13:37.620 --> 13:44.990
decreases. So, in this figure, it shows this
is the value of the bond that is par value
13:44.990 --> 13:50.610
of the bond when discount rate is 7 percent
and when discount rate is less than 7 percent,
13:50.610 --> 13:56.130
the value of the bond increases, but when
the discount rate is more than the coupon
13:56.130 --> 14:03.620
rate the value of the bond decreases. So,
this is discounting and this is premium. The
14:03.620 --> 14:06.280
green is premium, red is discounting
14:06.280 --> 14:11.130
To show the effect of maturity period of bond
value, we have takes an example 7. The value
14:11.130 --> 14:17.720
of a bond computed for different maturity
period when discount rates vary. M/s SJ International
14:17.720 --> 14:23.490
has floated bond with a coupon rate of 7 percent
and a maturity value of 1,000. For simplicity,
14:23.490 --> 14:29.770
let us assume that the bond pays annually.
Compute the value of bond for three different
14:29.770 --> 14:36.720
discount rates; 5 percent, 7 percent and 10
percent and maturity periods starting from
14:36.720 --> 14:43.530
0 years to 10 years and this have been, the
results has been tabulated in this table.
14:43.530 --> 14:50.180
Now, if see did maturity period is 0 then
bond value is 1,000 for 5 percent discount
14:50.180 --> 14:55.480
rate 1,000 for 7 percent discount rate and
1,000 for 10 percent discount rate; that means
14:55.480 --> 15:01.070
that irrespective of the discount rate, if
the maturity period is 0, the bond value remains
15:01.070 --> 15:08.240
1,000. Now, if it is 1 year I find that the
bond value for 5 percent discount rate is
15:08.240 --> 15:18.530
1019.048 and for 7 percent it is 1,000 for
10 percent it is 972.7273; that means, bonds
15:18.530 --> 15:25.421
value will be more than 1,000, that is par
value for even when the period, maturity period
15:25.421 --> 15:31.600
is 1 year and the bond value will be less
than 1,000 when the discount rate is more
15:31.600 --> 15:36.230
than the coupon rate.
Now, here we see that when the bond discount
15:36.230 --> 15:41.830
rate is 7 percent which is equal to coupon
rate for all time to come, that is all maturity
15:41.830 --> 15:46.350
periods, the bond value remains same that
is 1,000; however, when the discount rate
15:46.350 --> 15:55.230
is 5 percent, the bond value increases with
maturity period. For 1 year it was 1019.048
15:55.230 --> 16:05.270
and for 10 years it is 1154.835; that means
the value is increasing with maturity period.
16:05.270 --> 16:13.560
Let us take see the scenario for 10 percent
discount rate. Here the value is decreasing
16:13.560 --> 16:23.860
and for the maturity period 1 year it was
972.7273 and for 10th year, this is 815.663.
16:23.860 --> 16:30.360
So, the value is decreasing, when I am increasing
the maturity period.
16:30.360 --> 16:36.930
So, I get a plot like this. This will be a
discount bond, this is the par value. So this
16:36.930 --> 16:41.030
will be obviously, a discounted bond this
is a premium bond, when the discount rate
16:41.030 --> 16:48.050
is 5 percent. Now the changes like this; so
when the discount rate is different than coupon
16:48.050 --> 16:54.180
rate the time to maturity would affect the
value of the bond. Even if the discount rate
16:54.180 --> 16:59.290
remains constant till maturity and what is
the conclusion? The shorter is the time to
16:59.290 --> 17:06.750
maturity, the smaller is the impact on bond
value caused by a given difference between
17:06.750 --> 17:09.220
the coupon and discount rate.
17:09.220 --> 17:17.600
Now, the other method for finding out to evaluate
the bond is yield to maturity. Which is called
17:17.600 --> 17:28.360
YTM? Yield to maturity, YTM is the rate of
return that an investor earns if he buys the
17:28.360 --> 17:36.370
bond at a specific price and holds it until
maturity it is presumed that the issuer of
17:36.370 --> 17:43.669
the bond makes all the interest payments and
payments of the principal as contracted. YTM
17:43.669 --> 17:50.769
is a bond whose current price equals to par
value, that is purchase price equals to maturity
17:50.769 --> 17:55.419
value, will be always equal to it is coupon
rate.
17:55.419 --> 18:03.490
In case, the bond value differs from the par
value YTM value will be different than the
18:03.490 --> 18:09.559
coupon rate. To show this how to calculate
YTM, let us take an example which is example
18:09.559 --> 18:19.100
number 8 to compute YTM. SJ International
is currently selling a bond for rupees 1,900
18:19.100 --> 18:26.419
with a coupon rate of 10 percent and a maturity
value of 10,000. The maturity period is 10
18:26.419 --> 18:33.830
years. The bond pays interest annually computes
YTM. Now let us see the solution of this problem.
18:33.830 --> 18:40.080
Now, for the present problem N is equal 10
years. Interest paid per year is 10,000 into
18:40.080 --> 18:46.809
coupon rate by 100 is equal to 1,000 and at
the end of 10 years; the firm pays 10,000
18:46.809 --> 18:53.240
as maturity value. Now if you see the interest
that is 1,000. So, interest is paid for 10
18:53.240 --> 18:59.309
years. So, interest is annuity and the present
value of this annuity is this, where this
18:59.309 --> 19:05.600
A is the annuity 1,000 rupees and this formula
I was used to find out the P, this is present
19:05.600 --> 19:13.130
value. And the end of 10 years one gets 10,000
and the present value of this 10,000 are 10,000
19:13.130 --> 19:18.340
divided by 1 plus i to the power of N.
So, when we add the present values of these
19:18.340 --> 19:25.170
two, then this is equal to 10,900. So, the
problem is that, what should be the value
19:25.170 --> 19:32.830
of i, which will give us this value and that
value of i is required. Thus one has to compute
19:32.830 --> 19:39.130
the value of i discount rate, which will make
the present value of the above investment
19:39.130 --> 19:46.120
as 10,900. As the maturity value is more than
10,000 i should be less than the coupon value.
19:46.120 --> 19:49.980
The value of i for the present problem will
be the YTM.
19:49.980 --> 19:59.230
Now, let us see the computation. As a first
estimate, let us consider discount rate in
19:59.230 --> 20:06.539
this case YTM as 9 percent. The present value
of this series above investment for 9 percent
20:06.539 --> 20:14.150
discount rate is rupees 10,641.77 which is
less than 10,900.
20:14.150 --> 20:21.499
Hence a lower value of discount rate has to
be considered and let us take it as 8 percent.
20:21.499 --> 20:30.529
The present value of the series of above investment
for 8 percent discount rate is rupees 11,342.02,
20:30.529 --> 20:37.289
which is more than 10,900. Hence the actual
value of the discount rate should be in between
20:37.289 --> 20:41.970
8 percent and 9 percent.
The approximate value of discount rate which
20:41.970 --> 20:50.720
will offer present value of 10,900 can now
be computed using interpolation or by trial
20:50.720 --> 20:54.730
and error. The difference between the bond
values for 8 percent.
20:54.730 --> 21:05.059
And 9 percent discount value is rupees 11,342.02
minus rupees 10,641.77, which comes out be
21:05.059 --> 21:12.720
700.25. That is for 1 percent difference in
discount rate, there is a difference of rupees
21:12.720 --> 21:18.320
700.25 in the present value of the bond.
Difference between desired value and the bond
21:18.320 --> 21:26.619
value for lower discount rate 8 percent is
442 which comes out as rupees 11,342.02 minus
21:26.619 --> 21:38.220
rupees 10,900. So, actual discount rate will
be 8 plus 442.02 divided by 700.25 is equal
21:38.220 --> 21:49.310
to 8.631232. Now for the discount rate of
8.631232 percent, the present value of the
21:49.310 --> 21:58.580
bond is 10892.87 which are almost equal to
10,900. Now the actual value of discounts
21:58.580 --> 22:05.960
rate by trial and error is about 8.621 for
which the present value of bond is 10899.95.
22:05.960 --> 22:13.740
So, the YTM rate is 8.621 which is less than
the coupon rate of 10 percent.
22:13.740 --> 22:20.139
Now, let us see a third type of evaluation
that is semi-annual interest and bond value.
22:20.139 --> 22:26.389
The procedure to value bonds paying interest
semi-annually, that is half yearly is similar
22:26.389 --> 22:32.720
in computation to the situation, where compound
interest rate is charged more frequently then
22:32.720 --> 22:40.149
annually. Following steps are involved in
the computation. Convert annual discount rate
22:40.149 --> 22:47.490
to semi-annual discount rate by dividing it
by 2. Convert the maturity period given in
22:47.490 --> 22:56.470
years N to number of semi-annual period by
multiplying it with 2. Convert the required
22:56.470 --> 23:01.570
interest paid annually to semi-annually by
dividing it by 2.
23:01.570 --> 23:07.480
So, let us see an example to demonstrate this,
example 9. Compute the value of the bond when
23:07.480 --> 23:14.440
interest is paid semi-annually. M/s SJ International
has floated a bond with a coupon rate of 7
23:14.440 --> 23:22.159
percent and the maturity value of 1,000. For
simplicity, let us assume that the bond pays
23:22.159 --> 23:27.560
semi-annually for two required discount rates
10 percent and 5 percent and maturity period
23:27.560 --> 23:33.169
is 5 years. Compare the value of the bond
under similar condition, when the interest
23:33.169 --> 23:38.169
rate is paid annually. So, there are two discount
rates 10 percent and 5 percent.
23:38.169 --> 23:44.039
So, let us see solution; the value of the
bond when interest is paid annually. Now we
23:44.039 --> 23:49.320
will start the calculation with discount rate
of 10 percent. So, my first calculation is
23:49.320 --> 23:55.990
with discount rate of 10 percent. Where N
is the maturity equal to 5 years, i is the
23:55.990 --> 24:01.230
discount rate 10 percent, which is equal to
10 divided by 100 is equal to 0.1. Present
24:01.230 --> 24:08.080
value of the bond is 70. This is the interest
per year and this interest per year is taken
24:08.080 --> 24:13.010
as annuity and is converted into the present
value and this 1,000 which we are getting
24:13.010 --> 24:17.900
after 5 years is also converted into the present
value. So, when we add the two present values
24:17.900 --> 24:28.919
it comes out to be 886.27634.
Now, the same problem we will evaluate with
24:28.919 --> 24:34.649
semi-annual interest rate and we will see;
what is the value of the bond for semi-annual
24:34.649 --> 24:40.659
interest rate, the value of the bond when
interest is paid semi-annually with discount
24:40.659 --> 24:41.780
rate of 10 percent.
24:41.780 --> 24:47.480
Now, here the value of the interest, the bond
value into coupon rate is 1,000 into 0.07
24:47.480 --> 24:55.970
is 70 rupees and this 70 rupees is paid semi-annually
and thus it is equal to 35 because 70 divided
24:55.970 --> 25:03.260
by 2. So, it will be paid over to 6 months
that is, why it is rupees 35 paid every 6
25:03.260 --> 25:10.440
months up to the end of 5 years. This means
that in 5 years 10 payments each 35 will be
25:10.440 --> 25:19.200
paid every 6 months. Thus annually it can
be taken has 35 for 10 consecutive payments
25:19.200 --> 25:25.639
when gap between the two payments are 6 months.
Please note that the present value of annually
25:25.639 --> 25:32.510
formula has been changed to accommodate compounding
semi-annually in this case. When compounding
25:32.510 --> 25:39.190
is semi-annually, compounding every 6 months
and twice annually and hence N is equal to
25:39.190 --> 25:45.549
2. For this problem N is equal to 5 years
and i equal to r equal to 10 divided by 100
25:45.549 --> 25:47.750
is equal to 01.
25:47.750 --> 25:54.419
So, for compounding semi-annually this formula
is used. Which is a discrete compounding formula
25:54.419 --> 26:00.239
and this is the P, that is present value and
this is the A which is the annuity value.
26:00.239 --> 26:08.549
The present value of the annuity is equal
to annuity that is 35, 100 into this group
26:08.549 --> 26:18.350
which comes out to be, when we plays the value
it comes out to be rupees 270.261. Now again
26:18.350 --> 26:25.059
the PV; the present value of the 1,000 paid
at the end of 5 years, when compounding semi-annually
26:25.059 --> 26:32.809
is this 1,000 divided by in brackets 1 plus
i by m to the power mN which comes out to
26:32.809 --> 26:42.070
be rupees 613.9133. So, thus the present value
of the bond is this 270.261 plus this one,
26:42.070 --> 26:50.929
which is 613.9133 which comes out to be 884.1743.
The value of bond when interest rate is paid
26:50.929 --> 26:57.000
annually with discount rate 5 percent, so
we have calculated for 10 percent discount
26:57.000 --> 27:03.280
rate. Now we will calculate for 5 percent
discount rate and here, the present value
27:03.280 --> 27:11.059
of the bond is 70 into this formula, which
is annuity formula.
27:11.059 --> 27:20.590
And compounding is annually and this is 1,000
is transferred to present value by dividing
27:20.590 --> 27:27.139
it with 1 plus i to the power N. So, this
is annually. So, it comes out to be rupees
27:27.139 --> 27:33.510
1,086.5896.
So, when discount rate is 5 percent and the
27:33.510 --> 27:41.020
compounding is annually, the value of the
bond is 1,086.5896; the value of the bond,
27:41.020 --> 27:48.400
when interest is paid semi-annually with discount
rate 5 percent and compounding semi-annually.
27:48.400 --> 27:54.889
So, the semi-annually compounding formula
is used and this comes out to be 1,087.521.
27:54.889 --> 28:03.580
So, what are the conclusion we draw out of
this the conclusion is, the value of a bond
28:03.580 --> 28:12.429
selling at discount is lower, when semi-annual
compounding is used in comparison to annual
28:12.429 --> 28:16.350
compounding.
And number 2 conclusion is for bonds selling
28:16.350 --> 28:24.289
at premium, the present value with semi-annual
compounding is greater than the annual compounding.
28:24.289 --> 28:30.289
And we already know that what is the discounting
bond and what is a premium bond. In the discounting
28:30.289 --> 28:37.989
bond the discount rate is more than the coupon
rate and in a premium bond the discount rate
28:37.989 --> 28:40.130
is less than the coupon rate.
28:40.130 --> 28:44.759
Now, let us take another example. Example
number of 10; S J International is selling
28:44.759 --> 28:53.580
bond of 100000 dated January 1, 2016 for coupon
rate of 9 percent and having interest payment
28:53.580 --> 29:01.259
dates of 30'th June and 31'st December each
year for consecutively 5 years and will following
29:01.259 --> 29:07.340
semi-annual interest payments and one time
principal payment of rupees 100000 at the
29:07.340 --> 29:12.581
end of the maturity period.
And discount rate is 8 percent. This time
29:12.581 --> 29:21.730
line very clearly shows this numerical. So,
here in the first period which is 6 month
29:21.730 --> 29:29.870
away from the t equal to 0 time line, that
is on 30 6 2016, 4,500 will be paid and then
29:29.870 --> 29:38.710
after 6 months again 4,500 will be paid and
this will move till the 10'th period comes
29:38.710 --> 29:41.220
where 4,500 plus 100000 will be paid.
29:41.220 --> 29:48.749
Let us see the solution. As indicated in the
time line diagram ever the bond issuing from
29:48.749 --> 29:56.529
SJ International will pay to it is bond holder
10 identical interest payments of rupees 4,500.
29:56.529 --> 30:04.669
From where it has come, Rupees 100000 into
9 percent into 6 by 12 of a year that is equal
30:04.669 --> 30:12.509
to 4,500; at the end of each 10 semi-annual
periods along with a single payment at the
30:12.509 --> 30:19.619
principal value of 100000 at the end of 10th
period. The present value of this bond depends
30:19.619 --> 30:25.519
on market interest rate that is discount rate.
At the time of computation the discount rate
30:25.519 --> 30:30.610
was 8 percent. So, the present value of the
bond compounded semi-annually and by given
30:30.610 --> 30:36.129
by this. This is the annuity of 4,500 for
10 consecutive payments and this is converted
30:36.129 --> 30:43.150
into the present value using a discounting
method, which is compounding semi-annually
30:43.150 --> 30:49.840
and this is 100000 is converted into the present
value using a discounting method, which is
30:49.840 --> 30:55.200
compounded semi-annually. This comes out to
be rupees 104055.4
30:55.200 --> 31:04.519
Where i is the discount rate, 8 percent, N
is the maturity period, which is 5 years and
31:04.519 --> 31:13.259
m equal to 2, that is compounding twice per
year; the present value of 36,499.03 of the
31:13.259 --> 31:23.019
annuity. Now from where this 36,499.03 has
come, you see this is the 36,499.03 and this
31:23.019 --> 31:28.090
comes from the annuity payment of the annuity,
the present value of the annuity, this group,
31:28.090 --> 31:40.869
so the present value rupees 36,499.03 of the
annuity 4,500 for 10 periods. So, that if
31:40.869 --> 31:52.029
an investor invests rupees thirty six 36,499.03
on January 1, 2016 will in return will get
31:52.029 --> 32:02.940
10 semi-annual payments of rupees 4,500 with
first payment on 30'th June, 2016 when discount
32:02.940 --> 32:10.859
rate is 8 percent. The difference between
the 10 semi-annual future payments of 4,500
32:10.859 --> 32:13.739
is 450000.
32:13.739 --> 32:26.570
And the present value of rupees 36,499.03
equals 8,500.97. This return that is 8,500.97
32:26.570 --> 32:34.559
the investor gets on an investment of rupees
36,499.03 provides the investor an annual
32:34.559 --> 32:40.899
return of 8 percent compounded semi-annually.
It is reasonable to accept that a bond promising
32:40.899 --> 32:50.989
to pay 9 percent coupon rate, will sell for
more value. That is rupees 104055.4 when the
32:50.989 --> 32:56.239
market is expecting to earn only 8 percent,
that is discount rate.
32:56.239 --> 33:04.070
Thus the 9 percent coupon bond value will
pay 500 more semi-annually, then the bond
33:04.070 --> 33:10.850
market is expecting. If a bond pays 4,000
interest semi-annually for 10 periods and
33:10.850 --> 33:18.149
the principal is 100000 at the end of 10'th
period having a maturity of 5 years, the present
33:18.149 --> 33:26.229
value of the bond taking 8 percent discount
rate is 10000. That is rupees 4,500 minus
33:26.229 --> 33:31.770
rupees 4,000 is equal to 500.
Now, the investor will be receiving an additional
33:31.770 --> 33:42.009
500 semi-annually for 10 periods; if the investor
is willing to pay rupees 4,055.4 more than
33:42.009 --> 33:50.860
the bonds par value of rupees 100000. The
rupees 4,055.4 more than the par value amount
33:50.860 --> 33:59.869
is referred to as premium on Bonds Payable,
Bond Premium, Unamortized Bond Premium or
33:59.869 --> 34:09.500
Premium. The present value of bond, when compounded
semi-annually is equal to 4,000, that is the
34:09.500 --> 34:16.480
annuity and then we use a formula for semi-annual
compounding plus 100000 is converted into
34:16.480 --> 34:26.760
the present value comes out to be 100000.
What is a perpetual bond? A perpetual bond
34:26.760 --> 34:33.080
is a bond with no maturity date. Perpetual
bonds are not redeemable.
34:33.080 --> 34:43.980
But pay a steady stream of interest forever.
Some of the only notable perpetual bonds in
34:43.980 --> 34:52.599
existence are those that were issued by the
British Treasury to pay off smaller issues
34:52.599 --> 35:02.780
used to finance the Napoleonic War in 1814.
Some of the US believes it would be more efficient
35:02.780 --> 35:09.609
for the government to issue perpetual bonds
which may help it avoid the refinancing costs
35:09.609 --> 35:17.840
associated with bond issues that have maturity
dates. A perpetual bond is a bond that never
35:17.840 --> 35:26.440
matures. It has an infinite life. So, perpetual
bonds present value PV is equal to AI divided
35:26.440 --> 35:34.180
by 1 plus i to the power 1 plus AI divided
by 1 plus i to the power 2 like this up to
35:34.180 --> 35:40.569
AI divided by 1 plus i to the power infinite.
So, it is summation of t equal to infinite
35:40.569 --> 35:49.260
AI 1 plus i to the power t. So, AI is per
period interest, which the bond is paying
35:49.260 --> 35:55.119
and i is the discount rate.
Now, you can divide the PV b by 1 plus i.
35:55.119 --> 36:01.930
So, it becomes AI 1 plus i whole square plus
AI by 1 plus i whole cube so on so forth plus
36:01.930 --> 36:10.210
AI 1 plus i to the power infinite. Now if
I deduct Pb minus Pb divided by 1 plus i it
36:10.210 --> 36:20.420
is AI divided by 1 plus i to the power 1 or
I can write down PV b is equal to AI divided
36:20.420 --> 36:27.750
by i. Where, AI is the annual interest, i
is the annual discount rate and in case maturity
36:27.750 --> 36:35.920
period N is infinite. Example 11, valuation
of perpetual bond; A bond P has a rupees 1,000
36:35.920 --> 36:42.280
par value having a annual coupon rate of 8
percent, if the discount rate is 10 percent.
36:42.280 --> 36:43.650
Find the value of the perpetual bond solution.
36:43.650 --> 36:51.910
The annual interest is rupees 1,000 into 0.08
is equal to 80 rupees, discount rate is 10
36:51.910 --> 36:56.990
percent, that is i is equal to 0.1. Hence
the present value of the perpetual bond is
36:56.990 --> 37:03.900
equal to rupees 80 divided by i is equal to
rupees 800.
37:03.900 --> 37:12.799
Summary, what we have had read in this lecture,
number 1; bond valuation with a reference
37:12.799 --> 37:21.530
to basic bond valuation. Second, yield to
maturity and third semiannual interest and
37:21.530 --> 37:28.570
bond value has been explained. Number 2, two
methods for basic evaluation has been explained.
37:28.570 --> 37:35.710
Number 3, effect of time of maturity, coupon
rate and discount rate on bond valuation has
37:35.710 --> 37:42.780
been explained. Number 4, premium and discount
bonds and their dependency on discount rate
37:42.780 --> 37:53.480
have been explained. Number 5, perpetual bond
and it is present value has been explained.
37:53.480 --> 38:03.369
Thank you.