So, welcome to the second lecture. And we
have proceeding on our journey to learn some
mathematical finance, but I promised yesterday
that I will talk to you about something related
to the internal rate of return or internal
interest rate. I will give your result which
you can try to prove using basic facts about
equations and at the end of this talk so today
we have going to talk about a very important
class of financial instruments called a fixed
income security.
Now there are 2 parts here that we here have
to look into. That what is fixed what you
mean by fixed income and do you mean by security.
You see there is something called a financial
instrument. We use financial instruments to
generate a cash flow so financial instruments
will lead to something called a cash flow.
And fixed income financial instruments are
the ones where you are guaranteed a certain
amount of income if you put in certain amount
of money now. And by the word security we
used very often financial markets, but the
real meaning of security is any financial
instrument that can be traded in the market
freely is called a security so very important
class of fixed income security that you would
all know is called say fixed deposit in a
bank. In sometimes also called certificate
of deposits in the US so in our country it
is called fixed deposits so fixed deposits
mean, you give the bank sum amount of money
and it holds a money for a 3-year period or
4-year period or 10-year period, and then
at the end of that period which is called
the maturity date, they will provide you some
money. In between they can keep on paying
you a little bit of interest or a percentage
of the final thing that you are supposed to
get.
So, what you generate is essentially a cash
flow. And fixed deposits guarantee you that
at the end of the maturity you will get this
amount of money so that is what we do in banks
and most people would like to keep the money
in fixed deposits this a certain growth of
money. Well like me show you that this a certain
growth is not at the same that you have when
you actually buy a share buy shares in a stock
market so fixed deposits or may fixed income
securities fixed deposit is one of the fixed
income securities allows your money to grow
and so therefore, there is an interest rate
associated with that growth. So the banks
take your money and invest it somewhere else
and as a result of holding your money it gives
you some more money at the end wherein another
important class of fixed income financial
instrument is mortgages. So if you want to
buy say big house when you want to buy a house
usually have to mortgage sum amount of money
or say you have to mortgage all your fixed
deposits to the bank, so though the bank will
it own all your fixed deposits. For example,
an on that basis they will give you some amount
of money as a loan and which you are going
to pay at a monthly rate of installment. So
the bank gets a cash flow so banks get some
money from this sort of home loans or mortgage
loans.
Another type of fixed income deposit is a
pension. Which is also called annuity, means
a pension holder starting from a time 0 at
time 1 he receives a fixed amount of payment
and time 2 he receives another fixed amount
of payment. So the fixed payments every month
he receives that fixed payment over a certain
period of time and that is called annuity.
You will your own an annual period of time,
so this called annuity. I am just giving you
some examples of fixed income securities.
Very important fixed income securities in
the financial market are bonds. So bonds are
essentially like fixed deposit where, but
fixed deposits are you know certificate of
deposits which the banks give you and you
give your money to the bank. Bonds can be
issued by a company, so bonds can be issued
by any government organization, bonds can
be issued by say even what to say even a financial
firm can issue it is own bonds.
So, bonds are slightly different from fixed
deposit. Though the nature is almost a same,
but their initial intrinsic character is quite
different. And both in case in fixed deposit
if you are depositing with the government
bank you always sure that you will get back
your money after at the time of maturity,
but while for bonds your very not sure. Because
bonds if issued by a private company and the
company goes broke a goes bankrupt they may
not be able to pay you your required amount
of money. So in that case bonds are not really
completely secured you know items, so one
has to worry that bonds are fixed income,
but not such a guarantee of income like the
fixed deposit in the government bank. So the
2 types of bonds one is called coupon paying,
and another is called zero coupons. So suppose
I had a bond which will say pay me I will
buy the bond at 100 rupees now, after one
year get 1 10 rupees, get 1 10 rupees. So
I had 10 percent interest rate basically.
Say is given a 10 percent interest rate now.
Another might be a bond might be that I will
pay you 10 percent interest rate. 10 percent
of the face value as coupon means in whole
year I will not only pay you 1 10 rupees I
will also pay you additionally 10 percent
of the total money. Right so 10 percent of
the face value. So this is called a face value
of bond though money that you get at maturity
is called the face value. So it will say that
I will pay, you 10 percent of the face value
as a coupon so and I will pay it every 6 months.
So the first 6 months will pay you 10 percent
of 1 10. Which is how much which is 1 1 rupees.
So I will pay 10 percent of 1 10 means 1 1
rupees for the whole year. So at 6 months
you will pay you 5 and half rupees and then
the next 6 months at the end you will pay
you 5 and half rupees. So basically what will
now have 1 10 plus 1 1 rupees as your face
value total income, face value plus the coupon
value.
Total cash flow is this. So this is called
coupon paying bond is called coupon because
earlier days in the bond when you are the
bond certificates this coupons where attached
to it, so you have to send the coupon to the
issuer of bond and the guy would send you
the that particular amount of money. And many
government bonds are 0 coupon bonds. They
do not care it is like a fixed deposit it
does not care anything in between and then
they will be paid this amount. So bonds can
default either result to which there had been
credit rating agencies like moodys and standards
and poors, who rates the quality of bonds.
They do not get government bond because government
bonds are highly secured bonds. Government
has obligation to pay, but for private bonds
you afford some of ICICI bank in India, you
will see the issue a lot of bonds. So in those
case you honest to really check the moody’s
or standard in poor setting before buying
such bonds. So what we will go to move into
the mathematical structure of the whole thing.
Now, let us talk about something called perpetual
annuity. So we have also spoken about annuity.
Annuity means every month say or a given time.
So let us talk about months 0 1 2. So it is
the first month, second month, third month,
four months so the government a gives a pension
to somebody. I receive A amount of money same.
So this is called perpetual annuity. Suppose
I do it for forever. The perpetual annuity
is need not a feasible thing, but usually
it given till the death of the person or something
arrived, but perpetual annuity in some cases
is the made still be retained, but perpetual
annuity essentially means till the death of
the person. So from a mathematical point of
view what is the present value?
So, if I am going to make if I am a government
organization, I am going to pay a pension
to an employee what is the present worth of
these payments that I am going to make. So
if that is the money I now invest somewhere,
I would be able to generate that particular
money. So the present value of the annuity
present value of the annuity 
just means this theoretically infinite who
means give them for large times. Theoretically
it is infinite. So you know this end infinite
geometric series and that would sum up to
A by r where r is the simple rate of interest
which is quoted by the current financial market
is quoting and on an we have expected for
example, for just for the record, let it be
fixed over given time just for like a modeling.
So one might ask why if this formula how it
is useful. What is this meaning? How can you
pay somebody for infinite time? So the present
value you see is nothing, but A by r if I
want to pay A. So this is the money I have
to really invest now. That is coming use of
this geometric infinite geometric series.
Now, let us look at finite life steam finite
annuities. When you look at finite annuities
then off course I will write this as p annuity
up to n period. So I am writing it as n the
so what is the principle value. So how much
is the worth, how much I really into invest
to give somebody of fixed payment say every
month or n months? So that would be exactly
we are not taking compounding then also bring
compounding and all those things. So what
we get up is a finite GP series. If you want
to look into the thing as a finite GP series
and you know how do sum of finite GP series
then formula is, so these after summing the
finite GP series. In fact, you can if you
do not want to bother to look at the finite
GP series is like this, then you can look
at 2 perpetual annuities 2 perpetual annuities
one that starts at time 1 and one that starts
at time n plus 1.
So if you are going to pay so if you look
at 2 perpetual annuities. One that starts
at time 1 a keeps on paying and another that
starts at time n plus 1, so if I will take
the difference then I am looking at this.
So you really do not have to bother with summing
the GP. So then I can look at 
so p n annuity as some difference of the 2
perpetual annuities. One starting at n plus
1 and another starting at the time 1 so take
this whole minus this, so this would give
me a by r is the amount of present value that
is what I am supposed to invest, minus the
present value of the one starting at n plus
1, but I have to look at the present value
at this time right time 0.
So, I have to not only A by r is the A by
r is the money have to invested n plus 1.
Right in order to make a payment from n plus
1 till infinity, but then what is if I want
to make an investment of A by r at time n
plus 1 what should be the investment now at
0. It should be A by r the A by r divided
by 1 plus r to the power you have to come
back n steps. So that should be the formula
right? So if you look at it if you look at
it that is exactly now you have to and you
to come back n steps, right so you really
have to come back n steps, so you have to
start your discounting. Because here A by
r is the investment you make at time n plus
1 up to infinity.
So what is that how much to make that in start
that is investment how much investment have
to make at time now, that is that is a by
1 plus r to the power n, A by r divided by
1. Discounting this A by r itself, so then
you get simply the formula which is exactly
that what you get by summing finite GP series.
As you see so this perpetual annuity can be
used to understand this finite steam annuity.
Now how much do I need to pay you know this
is very important. Which among this 2 terms
what I have express, interest rate is known
to me so I am basically linking p with A.
Right so I can also link A with p suppose
I know that this is the amount of funds I
have available which really you know about
a comp in a company or in a government organization
that is the fund which is available, then
how do I set my annuity. How do I give pension
to a person?
So basically then you can also write down
a relationship between annuity and the principle
value that is, the money that is available.
Now so annuities, so from here you can calculate
out the annuity as r into p, divided by r,
to the 1 minus not 1 plus 1 plus r to the
power n minus 1.
So, this is what is really used. You so in
real organizations you will find that list
of principle values list of interest rates
list of annuity. Tables are made you cannot
be it is not so easy to compute. This it is
and periods are given so tables are already
made. So when you really sell those annuities
you cannot just be sitting down and computing
every time. You really have to have table
based on this formula, it looks out to a very
simple formula, but based on this formula
huge numbers of tables are created and used
in the financial market daily. So what we
are now learning is, what we are learning
at this movement, at this movement is essentially
something related to the practical techniques
used in the financial market. And for as the
financial instrument that we use go more and
more complex and you will see when we really
take into answered into account all the things
that we did in part one of the course we all
come back on the board, which includes bavani
and bushani to integral and everything else.
Now, what this is A or what is this A? It
is telling you if I know what is how much
money I have I know how much payment that
I have to made every month. So this process
is called amortization. So you have to pay
a loan, so your current obligation is to pay
a loan of this amount say x amount for a car,
and you say that this is the current interest
rate and if I say choose a period of 5 years,
I will be able to pay this amount so that
is this is exactly the way, your payment that
every month that you make your payment to
the banks. So those low only loan repayments
that you make it the installment that you
give every month to the bank say for buying
a car that is actually calculated by this
formula that is called an amortization loan.
So suppose I that this is my worth of the
car the loan I take from the bank, this is
the interest rate and this is the amount I
am ready to pay every month, then what is
my period. So this formula is a lot of things
it is very powerful formula little thing very
simple, but is a lot of information. So if
I know p if I know the interest rate the bank
is charging if I know what amount I want to
pay, then I know what amount what are what
is the total number of months or periods I
have to pay and suppose n terms of to be finally,
that I have to pay overall 5-month period
60s any 60. Then I will go to the bank and
ask for a 5-year-old, 5-year loan on automobile.
So this process is called amortization.
Now, we will study of very important class
of financial instrument called bonds which
we have already mentioned. Bonds as I again
want to clarify a different from fixed deposits.
A fixed deposit is usually issued by banks
and largely by government banks also a private
bank. And fixed deposits and private banks
are quite like bonds because of a chance of
defaulting by government bank there is no
chance of defaulting you have to pay back
that money. But bonds are something on which
people can default so let us now try to compute
the value of the bond, the current value of
the bond means why this current value is so
important, mean see I am all these instruments
are paying some money in the future, for which
I now have to pay some money. So this present
value is what I have to determine in his whole
story of finance is essentially story of determining
present value. There is hardly anything else
all the technology or techniques so mathematical
tools that you would see would actually depend
on that particular thing that computing the
present value. There is nothing else much
then computing present values.
So, if you now look at bonds. So bonds are
that if I if some money p now, and I buy I
become the bond holder. So the bond holder
pays the money p now, to get an amount f which
is called a face value of the bond. And this
will be paid by the bond issuer. Maybe intermediately
suppose I am holding it for a year, I will
get a payment say c 1 or if is the coupon
paying bond and I will get a payment say c2,
then what I total income from the bond is
f plus c 1 plus c2.
Now just like this notion of internal data
of return all this stories at we are trying
to say you is depending on whether all these
stories at you are trying to tell you, now
is actually depending on the interest rate
which is prevailing in the market which is
guaranteed by the federal banks or guaranteed
by the reserve bank of India all these things.
But there is something called internal thing
internal rate of return of bond, means suppose
the prevailing rate of interest is 8 percent
and it is I will pay you 100 rupees if you
pay me 100 rupees after 1 year I will give
you 110 rupees. And I am actually paying you
10 percent, so that is not same as the current
given rate of interest so which means that
this is something internal to the bond structure
itself and that is called a yield, yield of
a bond. So we are going to now compute what
is the meaning of the yield of the bond. And
with the computation of the yield of a bond
we will stop our discussion today.
So, what is the yield of a bond? So let is
let me now do that calculation. It is not
difficult to calculate and here we will you
compounding the way that is really used in
the financial market.
Suppose a bond has n periods to go; so one
maturity will come after n periods of payment
suppose I am taking a coupon paying bond;
suppose, I am talking of a coupon paying bond.
Now because I am talking about a coupon paying
bond, and let me divide the whole thing into
m periods and their n periods too, now which
means that if the internal rate of interest
of the bond which I called yield, which is
the intern internal rate of interest suppose
that is lambda, suppose that is lambda, lambda
is an internal rate of interest so if this
is my internal rate of interest, how much
I actually have to pay if I want to by the
bond.
And the bond pays a coupon value c total coupon
worth c and that they pay me in m periods
each worth in c by m, at every period they
pay me c by m. So the face value the, to buy
the bond the money that I have to pay is a
face value of the bond. So I do not charge
because a m periods where coupon is given,
I do not charge a lambda every time and at
the what the what lambda m at each period.
So and I discount face value over each period,
plus over the n periods I am making a coupon
payment. So what is the coupon payment? The
coupon payment is c by m and that is also
discounted by the same interest internal interest
rate of the bond. That is if have to you have
to discount everything to get the current
price. So this is the price that you have
to pay for a coupon paying bond. If c is equal
to 0 that is a 0 coupon bond. Then you are
actually paying, so if you know, how many
times the coupon is paid. So if the coupon
is paid twice means an n is 1, sorry n is
the number of periods or number of n is the
number of period remaining, and m is the number
of times the coupons are paid. Suppose number
of times is coupons are paid is 2 and n is
1 year 0 1 year. So you will put 2 here you
will put one here.
So, one plus lambda by 2 by f, that would
be if this is a 100 and this is 110, this
110 and this 100, then the yield is so yield
even calculate out if this is I pay 100 to
a 110 rupees, so I pay a coupon twice at the
interest rate, and I sorry I do not pay any
coupon so because I do not pay any coupon
so I pay everything at the end. So you see
so m also becomes 1 and n also becomes 1.
So this becomes so this lambda is same as
the given interest rate, but now if I am paying
come coupon say for first 6 months and then
next 6 months, so it is c and then this becomes
2 and then this becomes 1 this becomes 2 and
this becomes 2 this becomes n becomes 1 so
if say if your m is equal to 2 and n is equal
to 1 and say this is 5.5. Then your yield,
your f is 1 10 this is a p, then this lambda
is different from the market interest rate.
It is not 10 percent interest rate right.
So, with this formula we end our discussion
today, and in the next class we will talk
about the term structure of interest rates
which will start discussing yield cause and
their behavior. So just let me give you the
problem I will just give you the problem which
I had said I will in the last class. So here
is the problem which I want to you to solve
which is independent of the assignments. So
do not consider it as an assignment problem,
but I would like you to solve or even check
it out in the book or try to learn. This is
the interesting little thing and then that
we you that that we end it. So it is a theorem
on the interest rate internal interest rate.
So, suppose x n is the cash flow or n periods,
is the cash flow. And 1 has x naught strictly
lessons you means you are paying the money
and after that you are getting the money just
like a 0 coupon bond just like a 0 coupon
bond. Or it could be that you have may a huge
deposit in the bank and then you are gradually
taking of every month right that, so for all
k equal to 1 2 such that, x k is strictly
bigger than 0 for at least 1 k. So you could
have that for every month you are supposed
to take some money every month, you are not
taking money may be 1 month you took some
amount of money is positive. Then there is
a unique positive root of the equation.
Now what is this c? C is related to the interest
rate you know if you look at the internal
cash flow internal rate of return that we
discussed yesterday. It was like this 
so c is actually equal to 1 by r. So which
means if this is what happening then as the
unique internal rate of return that is what
you really have to prove everything do it
in the class was it can be thought about the
problem.
So thank you and we will start discussing
term structure of interest rate we will talk
about bonds the what is the issue of duration
and what is the how duration effects the bond
prices and all those things. So we will discuss
it over 2 classes.
Thank you very much.
