WEBVTT
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Welcome to the lecture series on Time value
of money-Concepts and Calculations.
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The topic of the present lecture is Perpetuity.
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Perpetuity is an annuity that provides an
infinite stream of equal cash flows received
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at regular intervals over time.
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Since this type of annuity is unending, it
is sum or future value cannot be estimated.
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However, the present value of perpetuity can
be calculated.
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Therefore, in perpetuity once the initial
fund has been paid the subsequent payments
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will flow from the fund indefinitely which
implies that these payments are nothing but
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the annual interest payments.
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One of the examples of perpetuity is the British-issued
bonds called Consoles.
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The present value of perpetuity is the perpetuities
cash flow has the end of the period 1 A is
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divided by the interest at i.
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So, PV is equal to A by i.
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Derivation of perpetuity; a perpetuity is
the series of equal payments over an infinite
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time period into the future.
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Consider the case of cash flow here have the
end of first year there is a payment A, end
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of second year there is another payment A,
end of third year there is another payment
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A and this goes on of to infinite.
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Now if you want to find out what is the PV
that is present value of this perpetuity this
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derivation is made.
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Present value is given by an infinite series.
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So, PV is equal to a divided 1 plus i.
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So, when I transfer this amount 2 present
value, it will be A divided by 1 plus i and
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when I transfer this amount to the present
value, this will be a divided by 1 plus i
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whole square.
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Similarly, it will be a plus i whole q up
to infinite.
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From this infinite series a usable present
value formula can be derived by first dividing
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each side by 1 plus i.
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So, when I divide PV by 1 plus i this is PV
1 plus i and this becomes a divided by 1 plus
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i whole square, this becomes A divided by
1 plus i whole square q and so on so forth.
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In order to eliminate most of the terms in
the series, subtract the second from the first.
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So, it becomes PV minus PV 1 plus i is equal
to A divided by 1 plus i and when we solve
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this equation we get PV is equal to a by i.
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Now, let us takes few problems.
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Example 1 what will be the value of a single
annual payment of a perpetuity cash flow beginning
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one year from today, if the interest rate
is 9 percent and the payment of 100000 is
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done today.
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So, PV is equal to a by i.
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So, PV is in this case is 1000000 i is nine
percent.
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So, A is PV into i.
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So, it is 1000000 into 0.09 is equal to 90000.
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Hence the annual perpetual payment A is 90000.
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Now, question number 2, what would you be
willing to pay for infinite stream of annual
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equal cash flows of Rupees 1000 each received
beginning 1 year from today if the interest
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rate is 10 percent.
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So, the formula is PV A by I, A is 1000, i
is 10 percent.
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So, the formula is PV is equal to A by i this
is 1000 divided by 0.1 is equal to 10000.
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Thus 1 has to deposit 10000.
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Today to get a infinite stream of cash flows
of 1000 per year.
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Let us take the example 3, while planning
is after a retirement life Ravi estimated
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that if he receives Rupees 20000 on retirement
date and also each month there after you will
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be comfortable.
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He also wishes to pass on this monthly payment
to his future generations after his demise
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as a gift as per the present scenario.
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He can easily earn interest of 9 percent compounded
annually.
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How much money he should set aside on the
date of retirement?
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So, that he starts getting his payment of
amount Rupees 20000 on his retirement date
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and on each month after it for an infinite
period.
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Now, if we see here the i per month is equal
to i per year divided by 12.
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So, this is 0.09 divided by 12 is 0.0075 and
PV is equal to A dived by i is 20000 divided
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by 0.0075 which comes out be 2666666.7.
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If Ravi wants to get Rupees 20000 on the date
of retirement, that is on the same date he
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deposited the money then he has to add Rupees
20000 with the computer amount of Rupees 2666666.7,
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which comes out be 2686666.7.
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Thus if he deposits this amount on the day
of retirement, then immediately he gets Rupees
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20000 as first payment and the rest of the
fund that is Rupees 2666666.7 will provide
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him future payments of Rupees 20000 per month
thereafter.
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The present value of PV of the perpetuity
is the perpetuities cash flow at the end of
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period 1 which is A divided by the interest
rate.
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Let us taken example Mr. Anshul Agarwal wants
to give a scholarship to IIT Roorkee under
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this scholarship every year, some student
will receive Rupees 1000 and you are paying
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for it even after you, your kids and your
grand kids are dead you are still paying for
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it forever, the question is how much money
will it cost you in today’s Rupees, in other
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words what is the present value of this perpetuity
when interest rate in your bank is 3 percent
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per annum.
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Now solutions every year the interest Mr.
Anshul Agarwal earns he is used to pay for
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this scholarship.
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The principle in the bank account does not
really change year to year.
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So, PV of perpetuity is called payment divided
by interest rate.
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So, PV is equal to Rupees 1000 divided by
0.03 is comes are to be 33333.
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So, if Mr. Anshul Agarwal deposits Rupees
33333 into the bank.
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Each year the money will earn Rupees is 1000
interest and that interest becomes the scholarship.
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Another example, Hari is thinking of buying
some stock in a company listed on the Bombay
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stock exchange BSE.
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Before Hari buys any stock he should compute
a price based on the dividends that he accepts
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a stock to pay in the future.
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This company has paid day Rupees 2.50 dividend
every quarter for the past 12 years and thus
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it can be a excepted that this trend to continue
for infinity.
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What should be the excepted price of this
stock?
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If Hari excepts return of 15 percent quarterly
compounded, the solution is giving that A
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is equal to Rupees 2.50, rate of return is
15 percent, quarterly rate of return is 15
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percent divided by 4, which is 0.0375.
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So, PV is equal to 2.50 divided by 0.0375
which comes off to be 66.67, with the exception
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that the company will continued to pay Rupees
2.50 dividend every quarter indefinitely and
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a 15 percent required rate of return he believes
the stock price should be 66.67.
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Now, we have to see the derivation of another
type of perpetuity, which is called growing
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perpetuity.
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Sometimes the payment in perpetuity is not
constant, but rather increases at a certain
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growth rate G, as depicted in the following
time line.
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If you see this time line at the end of 1
year we are getting an amount A, but end of
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second year we are getting an amount A into
1 plus G and third year we are getting an
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amount A into 1 plus G whole square and so
on and so forth up to infinite period this
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is a time line for a growing perpetuity.
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Let us try to find out what is the PV that
is present value of this type of perpetuity.
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So, because this cash flow continues for ever,
the present value is given by an infinite
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series PV is equal to A divided by 1 plus
i.
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So, if this A is converted to present value.
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So, the present value of this A will be A
divided by 1 plus i the present value of this
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A into 1 plus g will be A into 1 plus G divided
by 1 plus i whole square and so on and so
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forth up to infinite.
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From this infinite series a usable present
value formula can be derived by first multiplying
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each side by 1 plus G divided by 1 plus i.
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So if I multiplied this P with one plus G
divided by 1 plus i, so this is PV 1 plus
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G divided by 1 plus i.
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So, this becomes a factor becomes A into 1
plus G divided by 1 plus i whole square and
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so on and so forth.
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In order to eliminate most of the terms in
the series subtract the second equation from
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the first.
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So, it is PV minus PV 1 plus G divided by
1 plus i is equal to A divided by 1 plus i.
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So, this gives PV is equal to A divided by
i minus G, where i is greater than G.
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So, this is the formula for growing perpetuity
to find out present value.
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Now, let us take some numericals.
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If you investing stock that will pay a dividend
of Rupees 20 next years and grows at 6 percent
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per year and you require a 15 percent rate
of return how much is the stock worth to you
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today.
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So, here A is twenty g is six percent i is
15 percent; so PV is equal to a divided by
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in brackets i minus G, which comes out to
be Rupees 222.22.
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Let us take other example, what would you
pay for a share of stock given a required
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annual rate of return of 15 percent, compounded
quarterly and the following information the
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next quarterly dividend will be Rupees 2.25
and it is the companies policy is increase
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the dividend by 3 percent each quarter, also
calculate how much you will pay for constant
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perpetuity of Rupees 2.25.
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The solution is recognized than this is a
constant growth perpetuity problem, because
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the dividends continue forever and grow at
a constant rate of 3 percent.
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You can use this equation to decide how much
you would pay for this stock.
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So, PV is equal to a divided by i minus G
and what are given is A is 2.25, G is 3 percent
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each quarter and i is could 15 percent annually
return and PV is what is.
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So, I want to find out what is the value of
the PV.
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Now, quarterly rate of return will be i is
equal to 15 percent.
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So, 0.15 divided by 4 it should be 0.0375.
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So, PV is equal to 2.25 divided by in brackets
0.0375 minus 0.03 which comes out to be Rupees
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300.
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Rarely a stream of payments that gets larger
every quarter is worth more than a stream
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of constant payments, one pays Rupees 300
for this constant growth perpetuity and how
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much would you pay for it if he did not grow
is 20 divided by 0.0375 which comes out to
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be 60.
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So, if we see that for a constant perpetuity
you have to pay 60 Rupees and for a growing
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perpetuity we have to pay 300 Rupees.
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Now another example since graduating for college,
we have grown rich as a gesture of goodwill
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you have decided to endow your alma mater
with a research grant in finance.
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The endowment calls for the grant to be a
constant growth amount paid once in a year
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in perpetuity the first payment will be Rupees
10000 and is to be made in exactly 1 year,
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which subsequent payments growing at the rate
of 6 percent annually.
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If you are able to secure a 8 percent annual
rate of return on the endowment fund, how
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much should you put into the endowment fund
today.
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So, we have a growing perpetuity at the end
of 1 year the perpetuity is a then at the
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end of second year this is a into 1 plus G
and so on and so forth up to infinite period.
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So, in our case a one is 10000, G is 6 percent,
i is 8 percent and what is the PV that is
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present value which I should invest to get
this growing perpetuity.
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So, PV is equal to 10000 divided by 0.08 minus
0.06 which comes out to be 500000.
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To set up the endowment today assuming 8 percent
rate of return it will cost you Rupees 500000.
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Now, let us take another complex problem a
chemical manufacturer wants to issue some
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new equity to it is stock holders.
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You have been hired to price it is stocks
based on it is current dividend policy and
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it is share holders requires rate of return.
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The company has to told you that over the
last 3 years it paid stock dividends of Rupees
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1.10 Rupees 1.13 and Rupees 1.16 respectively.
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The company has also assured you that this
growth rate in it is dividends will continue
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indefinitely.
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If the company will pay it is next dividend
in exactly 1 year and believes that it is
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stockholders require a 12.85 percents rate
of return what should be the price of it is
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stock, use an average of the growth rate of
dividend payment over the last 3 years and
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the formula for the present value of a constant
dividend to compute the stock price.
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Now, in the last 3 years, if you see here
in the last 3 years it has to paid 1.10,1.13,1.16
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and 1 year hence from this point it will pay
a one which we do not know.
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So, the solution is PV equal to a one divided
by i minus G, PV is the present value of the
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growth annuity is the stock price and a one
is the payment received at the end of the
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first period, i is the periodic discount rate
which is 12.85 annually and G is periodic
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growth rate annual growth for this problem.
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The problem does not provide the dividend
at the end of the first period that is A1
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or the periodic growth rate G it does; however,
give the dividend for the last 3 years, Therefore,
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estimate the growth rate given the last 3
years dividends to find the value of G.
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Then use this to compute the payment at the
end of the first period that is 1 year’s
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growth.
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There are several techniques that can be used
to solve for the growth rate of which one
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come up with 2.62 percent, by using simple
average compute the percentage change year
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to year and take the average if G1,2 is the
growth rate from year 1 to year G23 the growth
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from year 2 to year 3 and G1,2 is equal to
1.13 divided by 1.10 in brackets minus 1 which
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comes out to be 2.73 percent.
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Similarly G23 comes out to be 2.655 percent.
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So, average is 2.73 plus 2.655 divided by
2 comes out to it 2.6925 percent.
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Now if a take a geometric mean of this percent
change, this is 0.23 into 2.65 and root of
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this 2 multiplications comes out to be 2.6922
percent almost same as a arithmetic mean.
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Now, the compound growth rate from the previous
section on the present value of a single cash
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flow, the internal rate of return can be computed
as this, i is equal to AN divided by PV to
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the power 1 by N minus 1 on this, if i use
the last one and the first one that is 1.16
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and 1.10.
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So, it is 1.16 divided by 1.10 to the power
half minus 1, it is comes out to 2.691, based
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on the estimated growth rate up to 2.62 percent
compute the dividend at the end of the first
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year that is A1 using a 0 is equal to 1.16
as the current dividend.
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This produces A1 is equal to 1.16 into 1 plus
0.0262, that is 1 plus G is equal to Rupees
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1.1904.
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The dividend to be paid in 1 year at t equal
to 1 is Rupees 1.16 and brackets 1 plus 0.26
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is comes out to be Rupees 1.1904.
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The dividend to be paid in the 2 years at
t equal to 2 is 1.16 in brackets 1.1 plus
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0.026 whole square comes out to be 1.222 and
so on up to infinity.
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So, PV is equal to 1.1904 which is the value
at t equal to 1 divided by i in brackets 0.1285
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which is the rate of interest minus the growth
is 0.0262 which comes out to be Rupees 11636.
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So, you can inform the company that with the
implied growth rate of their dividends and
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their stock holders requires rate of return,
they should expect to sell the new equity
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at a price of Rupees 11.636 per sale.
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Thank you.