WEBVTT
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Welcome to the lecture series on Time value
of money-Concepts and Calculations. In this
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lecture we will deal multiple cash flows part
one and part two. The cash flow is the amount
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of fund that is flowing in and out of the
company. If a company is consistently generating
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more cash than it is using, the company will
be able to increase its dividend, reduce debt
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and acquire another company. The annual cash
flow of a company is the net profit it gets
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plus the depreciation charges for that year.
In general cash flow of a project runs for
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the life time of the project. It is necessary
to convert these to equivalent values either
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by discounting future cash flow values or
compounding earlier cash flow values. In the
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present unit use a present worth or future
worth of different cash flow patterns dealing
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with the equal and unequal cash flow patterns
and continuous cash flow patterns will be
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demonstrated.
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Now, let us see how to construct a cash flow
diagram. The horizontal time axis is marked
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up in equal increments 1 per period up to
the duration of the project. That means, duration
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of the project is from here to at this point.
Receipts are represented by arrows directed
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upward. This is the receipt 5,000, this is
the receipt 5,000, they have marked arrow
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upward. This is a receipt, this is the receipt
and there are two receipts here. Disbursements
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are represented by arrows directed downward.
This is the disbursement 20,000. The arrow
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length is approximately proportional to the
magnitude of the cash flow. Two or more transfers
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in the same period are placed end to end and
this may be combined after wards. Like, here
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5,000 is the receipt and 1,000 is the disbursement.
So, they are put end to end. Expenses include
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before t is equal to 0 are called sunk cost.
Sunk cost are not relevant to the problem
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unless they have tax consequences in an after
tax analysis.
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Now, based on this let us plot a cash flow
diagram. Now for this we are takes an example.
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For example, consider a mechanical device
that will cost 20,000 rupees. Now, this is
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20,000 rupees at disbursement because you
have to spend 20,000 rupees. So, this 20,000
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is put down ward, when purchased. Maintenance
will cost rupees 1,000 each year. So, maintenance
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is another disbursement. So, maintenance are
put like this each year. The device will generate
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revenue of 5,000 each year for 5 years. So,
this is revenue that is receipts so 5,000
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is put like this. After which this is salvage
value is expected to 7,000 rupees. So, after
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the useful light of the equipment it can be
sold and this salvage value is 7,000. So,
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this 7,000 is put above the 5,000. The cash
flow diagram is shown in a, and is simplified
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version is shown in b.
Now, this is the simplified version, this
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disbursement here it is 20,000 from 5,000,
1,000 deducted. So, it is 4,000, 4,000, 4,000,
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4,000 and 5,000 plus 7,000 it is 12,000 minus
1,000 so it is 11,000. These standard cash
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flows are single payment cash flows, uniform
series cash flows and radiant series cash
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flows. Let us explain these three types of
cash flows. Single payment cash flow; a single
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payment cash flow can occur at the beginning
of the time line, designated has t equal to
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0, at the end of the time line designated
t is equal to N or at any time in between.
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Here we see that, at equal to 1 there is a
cash flow, at t equal to 2 also there is a
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cash flow. So, this is a uniform series cash
flow, but this is not a single payment cash
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flow. In a single payment cash flow, any one
of this places there be a cash flow. A uniform
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series cash flow, this is a uniform series
cash flow illustrated in figure 1 consists
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of series of equal transactions staring at
t is equal to 1, there is a transaction, t
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is equal to2, there is a transaction, t is
equal to 3, there is a transaction and so
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on so forth up to t is equal to N. And ending
at t equal to N the symbol A, representing
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an annual amount is typically given to the
magnitude of each individual cash flow.
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So, we are representing here the magnitude
of the cash flow as A. Now gradient series
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cash flow; the gradient series cash flow is
from here. The gradient series cash flow illustrated
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in figure 2 starts with a cash flow typically
given the symbolic g at t equal to 2. So,
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at t equal to 2, there is a cash flow g and
increases by g each year until t equal to
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N. So, this cash flow increases up to N at
which time the final cash flow is N minus
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1, in brackets into g. The value of the gradient
at t equal to 1 is 0. Here t equal to 1 is
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0. Uniform series cash flow patterns
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This type of pattern a cash flow is called
constant end of the month cash flow for a
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year. So, this is a cash flow is shown every
month or end of the month for a duration of
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12 month. Here series of unequal end of the
month cash flow for 1 year. Here we will see
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that the cash flow is not uniform though it
is end of the month cash flow for a period
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of 1 year.
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Now, derivation for gradient series cash flow;
uniform cash flow derivations we have already
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seen because they are basically annuities.
So, in the section of annuity we have seen
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the derivative. Here we will derive the derivation
for gradient series cash flow. Now the arithmetic
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gradient is a series of increasing cash flow
as follows: The value of F which is the some
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of the all cash flow is equal to G 1 plus
i to the power N minus 2. Now if i am finding
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the final value and this is t 2, then here
is the final value at t equal to N. Then if
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I find out the value of G at this point then,
it will be G into 1 plus i to the power N
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minus 2. The second t equal to 3, 2 G it will
be 2 G into 1 plus i N minus 3 and so on so
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forth up to N minus 1 G because here there
is no compounding period and that is why it
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will be N minus 1 G and here this would be
compounded up to this time period.
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This would be compounded up to this, this
would be compounded up to this and this would
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be compounded up to this. So, you can write
down this series for future value. Now here
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we can take G common. So, this is the series
we get. Now multiplying equation 1, this is
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equation 1 multiplying equation by 1 plus
i this is 1 plus i F equal to G, this is N
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minus 2. So, it becomes N minus 1 when you
multiply with 1 plus i and so on so forth
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up to here N minus 1 into 1 plus i.
Now, subtracting equation 1 from equation
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2; so this is equation 1, up to this and this
is the value of equation 2 up to this then,
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we are taking G common. So, this is i F because
this F and this F cancels out i F is equal
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to G into this whole series minus N G. We
know that, the some of the series below is
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equal to the summation of this series is equal
to n brackets 1 plus i to the power N minus
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1 divided by i. So, when we substitute the
value of series by this then, it becomes i
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F is equal to G into 1 plus i to the power
N minus 1 divided by i minus N G or F is equal
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to G by i in brackets 1 plus i to the power
N minus 1 divided by i minus N. So, this is
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equation series. So, this gives you the future
value of a gradient series of cash flow.
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Now, if you want to find out the present value
of this gradient cash flow then, P is equal
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to G by i into one plus i to the power N minus
1 divided by i minus N brackets close into
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1 by 1 plus i to the power N. Now this multiplied
by P is the F that is final value and hence,
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we can calculate P is equal to G this 1 plus
i to the power N minus i N minus 1 divided
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by i square 1 plus i whole to the power N.
What it has been done it has been solved i
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N here divided by i then this is multiplied
this i multiplied i square and this is 1 plus
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i to the power N.
So, you can write down the factor P by G i
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N is equal to this. Equation 5 is the arithmetic
gradient present worth factor, multiplying
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equation 5 by sinking fund factor we get.
Now if you want to find out the value of A,
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that is the annuity then, we can multiply
this by the sinking fund factor, this is sinking
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fund factor to give you this value. So, we
can write down the arithmetic gradient uniform
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series factor as A by G i N is equal to 1
by i minus N 1 plus i to the power N minus
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1. So, these equations will be used to solve
the numerical.
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Now, let us start taking numerical. Assume
that you will have no need for money during
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the next 2 years and any money you receive
will immediately go in to your account and
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earn a 10 percent effective annual interest
rate. Which of the following options would
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be more desirable for you? Now one option
is that, at t is equal to 0, you get 100 rupees,
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receive rupees 100 down or we receive 110
rupees in 1 year; that means, at the end of
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the first year you get 110 rupees or second
is receive 121 in 2 years.
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So, at the end of the second year, you get
121. So, which one of this offers you will
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take. The solution none of the option is superior
under the situation given. If one chooses
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the first option, he will immediately place
rupees 1 into a 10 percent account and in
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20 years the account will have grown to rupees
121. In fact, the account will contain 121
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at the end of the 2 years, regardless of which
options you chose. Therefore, these alternatives
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are set to be equal.
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Now, you see the multiple cash flow problem
matrixes. So, we have five type of problems.
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One is calculate present worth of annual cash
flow with annual compounding, when annual
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interest rate and cash flow at the end of
the year is given and will call this type
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of problem; problem type A. Calculate future
worth of annual cash flow with annual compounding
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when annual interest rate and cash flow at
the end of the year is given, such type of
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problem will be called problem type B.
Calculate present worth and future worth of
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cash flow, with compounding other than annual,
when nominal interest rate and cash flow at
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the end of the period is given will call this
type of problem type C. Compare two cash flow
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patterns A and B in terms of their present
worth as well as future worth with compounding
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other than annually, when nominal interest
rate and cash flow for both patterns A and
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B at the end of the period is given. We will
call this type of problem; problem D. Find
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the present worth and future worth of a arithmetic
gradient cash flow will call it problem type
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E.
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Now, take another example, example 2. A cash
flow consisting of rupees 5,000 per year is
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received in one discreet amount at the end
of each year for 10 years. Interest will be
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9 percent per year compounded annually determining
the present worth of the cash flow. This is
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equal cash flow annually. This is problem
type A. So, given is R is equal to or say
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A is equal to 5,000 here we are representing
A with R, i equal to 0.09 N equal to 10 years
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because this is annuity problem. So, present
worth will be R in the brackets 1 plus i to
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the power N minus 1 divided by i into 1 plus
i to the power of n.
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So, this is the formula for finding all the
present worth. When you put values into this
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formulas R is 5,000. This is 1 plus i is 1.09
to the power 10 because it is for 10 years
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divided by i 0.09 and 1 plus i is 1.09 to
the power 10. This comes out to be 32,088.25;
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that means, the present worth of the equal
cash flow annually for 10 years will be equal
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to 32088.25.
Let us take another example. An unequal end
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of the year cash flow consisting of 5,000,
9,000 and 12,000 at the end of first year,
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second year and third year respectively has
been received. Interest rate it is 10.5 present
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per year compounding annually. Determine the
present worth of the total amount of this
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start of the first year.
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Now such type of problem which is a unequal
end of the year cash flow cannot be solved
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by formula and hence it has to be solved by
first principle. Now if you see the time line
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here at the end of first year, we have invested
5,000 rupees, at the end of second year we
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have invested 9,000 rupees and end of the
third year we have invested 12,000 rupees.
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Now, these 12,000 rupees has to be brought
to its present value. This 9,000 has to be
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brought to its present value and this 5,000
has to be brought to its present value and
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if the 5,000 is brought to the present value,
it is 4,524.886. If now, 9,000 are brought
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to its present value, it is 7,370.856 and
if it is 12,000 is brought to the present
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value, it is 8,893.994 and when we add all
these present values it is 20,789.686.
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Now, how this 5,000 will be brought to its
present value? Here we will see. Present worth
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of the money 5,000 invested at the end of
the first year is equal to 5,000 divided by
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1 plus the interest rate. This is 10.5 so
expressed in ratio it is 0.105 to the power
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1 is equal to 4,525.886 and the present value
of this will be 9,000 divided by 1 plus i
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to the power 2, which comes out to be 7,370.856
and this is the present value of this is 12,000
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divided by 1 plus i and i is 0.105 and to
the power 3 because 1 year, 2 year and 3 year
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and that is why to the power of 3. So, it
is 8,893.944. So, when we sum it up, it becomes
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20,789.686.
So, this is how the present worth of a unequal
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end of the year cash flow can be calculated
from the first principle. Example 4; an graduate
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plans to buy a home he has been advised that
his monthly house and property tax payment
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should not exceed 40 percent of his disposable
monthly income after researching the market.
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He determines that he can obtain a 30 year
home loan for 7 percent annual interest per
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year compounded monthly. His monthly property
tax, will be approximately rupees 200. What
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is the maximum amount he can pay for a house,
if his disposable monthly income is 3,000
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rupees?
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Let us see the solution. Here in the time
line, from t equal to 1, the person is able
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to pay rupees A amount A at t equal to 2,
also t equal to 3, t equal to 4, up to t equal
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to 360, is able to pay rupees A that is amount
A. Now, let us see how much you can pay, which
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is equal to amount A. So, the calculation
is here, the amount available for monthly
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house installment payment A is rupees 3,000
into 0.4, this is 40 percent of 3,000 minus
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the rent which is paying 200. So, this comes
out to be 1,000. So, at the max he can pay
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rupees 1,000 per month up to 360 months.
So, the present worth of this annuity of 1,000
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for 360 month is equal to A into P by A 7
percent and 360 and the formula for this is
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this. This is a discrete compounding formula,
where r is equal to 7 percent m equal to 12.
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So, r by m is equal to 0.07 divided by 12
equal to 0.0058333 m into N is 30 years into
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12 months, which comes out to be 360 months,
when you put this value here. So, the present
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worth is equal to 150334.1. So, the maximum
amount he can pay for the house is, 150334.1.
21:17.540 --> 21:25.440
So, when if he sinks this P amount here, he
will get this P amount here, due to this annuity
21:25.440 --> 21:33.430
and hence he can purchase the house at t equal
to 0, whose maximum price will be rupees 150334.1.
21:33.430 --> 21:44.470
Now, take another problem the cash flow consisting
of rupees 5,000 per year is received in one
21:44.470 --> 21:50.970
discrete amount at the end of the each year
for 10 years; that means, each year end he
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is paying 5,000 rupees for 10 years. Interest
rate is 9 percent per year compounded annually
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determine the future worth of the cash flow.
22:04.500 --> 22:12.120
Now, the same problem for present worth has
been solved in example 1. So, the method 1,
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we are taking the present worth from the answer
of example 1, which comes out to be 32,088.25
22:18.470 --> 22:26.280
and this present worth is converted into future
worth, by multiplying with 1 plus i to the
22:26.280 --> 22:35.980
power N factor. So, this comes out to be 75,954.557.
Now this is a method through which we can
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find out the future value.
22:37.330 --> 22:43.560
Let us see the second method. The second method
is from first principle. So, we will find
22:43.560 --> 22:50.150
out the future worth of each payment, which
is done at the end of a year. So, the future
22:50.150 --> 22:56.420
of the payment 5,000 at the end of the first
year is equal to 5,000 into 1 plus i to the
22:56.420 --> 23:05.160
power 9 because this will earn interest for
9 years only. So, this comes out to be 10,859.4664.
23:05.160 --> 23:11.860
Similarly, the future worth for the 5,000
at the end of second year; this 5,000 into
23:11.860 --> 23:18.900
1 plus i to the power 8 is equal to 5,000,
1 plus 0.09 to the power 8 comes out to be
23:18.900 --> 23:25.710
9,962.81. So, in the similar way we find out
the future worth of all the payments up to
23:25.710 --> 23:33.890
tenth year. In the tenth year, it will not
earn any interest because it is paid at the
23:33.890 --> 23:38.710
end of tenth year and at the end of tenth
year we are finding out the future worth and
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that is why it is 5,000 only. So, when we
add up this comes out to be 75,964.65.
23:48.420 --> 23:55.130
Third method we can directly use our formula,
to find out the future worth of the annuity
23:55.130 --> 24:03.240
R. Here, we are using R symbol for annuity.
So, future R future worth of the annuity R,
24:03.240 --> 24:07.730
if you use this formula it comes out to be
75,964.65.
24:07.730 --> 24:15.430
Now, let take unequal cash flow annually.
An unequal end of the year cash flow consisting
24:15.430 --> 24:21.070
of rupees 5,000, 9,000 and 12,000 at the end
of first year, second year, third year respectively
24:21.070 --> 24:28.520
has been received. Interest rate of 10.5 per
year compounded annually. Determine the future
24:28.520 --> 24:35.050
worth of the total amount at the end of the
third year. So, any unequal cash flow has
24:35.050 --> 24:41.400
to be solved from the first principles. So,
here this is the time line at the end of the
24:41.400 --> 24:46.590
first year, 5,000 is paid, at the end of the
third year 5,000 is paid and at the rate of
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the third year 12,000 is paid. So, all these
future worth has to be calculated for these
24:53.300 --> 24:56.150
amounts.
So, the future worth of this 5,000 which is
24:56.150 --> 25:03.630
paid at the end of first year is 6,105.125.
Future worth of this 9,000, which is paid
25:03.630 --> 25:10.340
at with the end of second year is 9,945 and
the future worth of this 12,000 paid at the
25:10.340 --> 25:16.400
end of third year is 12,000. So, the few with
this can be calculated like this. Future worth
25:16.400 --> 25:23.890
of money of 5,000 invested at the end of first
year, is 5,000 into 1 plus 0.105 to the power
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2 because this will earn interest for the
first year and this second year. So, 2 years,
25:32.180 --> 25:38.190
it will earn interest. It will not earn interest
for this period of time. So, this is the earned
25:38.190 --> 25:42.440
interest for this period of time and for this
period time which is 2 years.
25:42.440 --> 25:51.570
So, 1 plus 0.105 is basically the interest
rate. So, this comes out to be 6,105.125.
25:51.570 --> 25:58.670
Similarly the money which is 9,000 it will
earn only interest for 1 year. So, this is
25:58.670 --> 26:07.350
9,000 divided by sorry 9,000 into this is
should be 9,000 into 1 plus 0.1015 to the
26:07.350 --> 26:16.930
power 1 comes out to be 9,945 and this money
12,000 will not an any interest, it is at
26:16.930 --> 26:21.710
the end of the third year and my future worth
is also calculated at the end of third year.
26:21.710 --> 26:27.480
So, it is 12,000. So, when i add them up it
comes to be 28,050.13.
26:27.480 --> 26:37.360
Now, let us take another example. A constant
end of the month cash flow with compounding
26:37.360 --> 26:42.770
monthly when nominal interest rate is 10 percent
is given below. Find the present worth and
26:42.770 --> 26:50.480
the future worth. Now if you see this cash
flow, this is being flowing from month 1 to
26:50.480 --> 27:01.100
month 24, that is 2 years and the total value
of this cash flow is 316000. Now as the payments
27:01.100 --> 27:07.620
are given at different time periods, we can
find out the present worth and future worth
27:07.620 --> 27:09.000
of this cash flow.
27:09.000 --> 27:16.270
Now, the annual interest rate is 10 percent.
So, r by m is equal to 10 divided by 12, N
27:16.270 --> 27:25.860
is 12 into 2, 12 is the months per year and
2 is the years. So, it is 24. So, my J varies
27:25.860 --> 27:33.830
from 1 to 24. Investment at the end of the
month for 2 years is given. So, the present
27:33.830 --> 27:41.400
worth factor for the first month is this,
1 divided by 1 plus r to the power m to the
27:41.400 --> 27:49.120
power J and when we put J equal to 1, this
comes up to be 0.991734. So, here for the
27:49.120 --> 27:57.250
end of the month, this is 1 the present worth
factor is this, for this and future worth
27:57.250 --> 28:03.960
factor is this. Future worth factor is 1 plus
r divided by m to the power N minus J. So,
28:03.960 --> 28:15.360
N is 24 and J is 1 for the first month. So,
it comes out to be 1.210305. So, when I multiplied
28:15.360 --> 28:25.000
this with the value that is, 15,000 then it
converts into present worth 14876.03 and 15,000
28:25.000 --> 28:33.580
into this comes out to be 18,154.58.
Similarly, for the month 2 I can calculate
28:33.580 --> 28:40.850
the values and I can calculate the PW factor
and FW factors and then find present worth
28:40.850 --> 28:47.170
and the future worth and for month 24, I can
calculate also the present worth factor and
28:47.170 --> 28:53.640
future worth. Factor present worth factor
is this future worth factor will be 1 obviously,
28:53.640 --> 28:58.180
because at the end of the second year that
is at the end of the 24th month, I am finding
28:58.180 --> 29:06.870
out the future worth this present worth becomes
12,291 and future worth becomes 15,000. Now
29:06.870 --> 29:12.420
here I have noted the future worth factor,
present worth factor, present worth and future
29:12.420 --> 29:13.420
worth.
29:13.420 --> 29:23.530
And when we add them up, so the present worth
is 325062.8 and the future worth is 396703.7
29:23.530 --> 29:32.530
as here the uniform payments have been made.
So, it can be calculated the present worth
29:32.530 --> 29:38.520
and future worth can be calculated is in formula.
So, if you see here given m equal to 12 and
29:38.520 --> 29:46.059
equal to 2, r is equal to 10 percent and r
by m we have to calculate here. So, r by m
29:46.059 --> 29:54.150
is 0.1 divided by 12. So, when you put them
into this formula, the values it becomes 325064.12.
29:54.150 --> 30:02.590
So, little bit of change in rupees about 2
rupees or so. Here because there will be errors
30:02.590 --> 30:08.690
in this calculations and rounding of errors
and that is why this 2 rupees difference has
30:08.690 --> 30:14.440
come and the future value we can calculate
from this is a present value is available.
30:14.440 --> 30:23.540
So, this comes 396705, here also 2 rupees
difference is there due to the rounding off.
30:23.540 --> 30:29.590
Now, take another problem compare two cash
flows. One constant end of the year, cash
30:29.590 --> 30:37.580
flow A and the other unequal end of the year
cash flow P, with compounding other than annually
30:37.580 --> 30:42.890
when nominal interest rate is 12 percent and
cash flow at the end of the year are given
30:42.890 --> 30:51.059
bellow using present worth factor. So, there
are two cash flows are given and one is the
30:51.059 --> 30:56.650
constant end of the year and other is unequal
end of the year. And the summation of both
30:56.650 --> 31:03.370
the cash flows is 360000. Though the summation
of both the cash flows is 360000 that present
31:03.370 --> 31:05.350
value worth will be different.
31:05.350 --> 31:12.790
So, will see that; so here again we are calculating
because this is a discretely compounding problem.
31:12.790 --> 31:20.510
Annual interest rate is 12 percent r by m
is 0.12 divided by 12, N is 12 into 2, 12
31:20.510 --> 31:28.920
is for 12 months per year and J varies from
1 to 24. So, present worth factor for the
31:28.920 --> 31:38.500
first month is 0.9909901 and similarly we
have calculated the present worth at the end
31:38.500 --> 31:45.540
of the month, equal amount. So, cash flow
this is for equal cash flow. So, this comes
31:45.540 --> 31:54.010
out to 14,851 and present worth for the unequal
cash flow, comes out to be 7920.79207.
31:54.010 --> 32:03.220
So, here the factor present worth factor is
this. So, the present worth of the first end
32:03.220 --> 32:10.920
of the month investment cash flow is 15,000
into this factor comes out to be 14,851 where
32:10.920 --> 32:20.980
this 14,851 is let in here. Now present worth
of the first end of the month investment for
32:20.980 --> 32:30.591
cash flow B; this is 8,000 invested into the
same factor comes out to be 7,920.79208 this
32:30.591 --> 32:43.670
is written here. Now for twenty 24 months,
we can see here the factor is 0.787566127
32:43.670 --> 32:50.400
this and the present worth of the cash flow
is this.
32:50.400 --> 32:56.290
And present worth of cash flow B is this.
Similarly, we have calculated all the cash
32:56.290 --> 33:06.650
flows present worth and then when we add them
up for cash flow A, the present worth is 318650
33:06.650 --> 33:15.950
and for cash flow B, this is 317121.1192.
So, what conclusion we make, though the sum
33:15.950 --> 33:24.490
up of the cash flow A and B are same; that
is 360000 their present worth are different.
33:24.490 --> 33:30.250
Indicating that more amount has been paid
through cash flow A than cash flow B (Refer
33:30.250 --> 33:35.350
Time: 30:31) I am paying more in cash flow
A than cash flow B as the present worth of
33:35.350 --> 33:47.300
the cash flow A, is 3,18,650.8089. Whereas,
that of the cash flow B it is 317121.1192.
33:47.300 --> 33:52.290
So, this is the conclusion withdraw out of
it.
33:52.290 --> 33:59.179
Let us take an example, example 9. Compare
two cash flows, one constant end of the year
33:59.179 --> 34:06.880
cash flow A and the other unequal end of the
year cash flow B, with compounding other than
34:06.880 --> 34:11.349
annually when annual interest rate is 12 percent.
34:11.349 --> 34:18.429
And cash flow at the end of the year is given
below is using future worth method. Now here,
34:18.429 --> 34:24.679
we find the cash flow is for about 2 years
and every month there is a cash flow. Now
34:24.679 --> 34:31.800
for constant monthly cash flow it is 15,000
and for unequal it is varying
34:31.800 --> 34:38.919
Let us see the solution. Now the annual interest
rate is 12 percent r by m is 0.12 divided
34:38.919 --> 34:44.450
by 12, N is 12 into to 2; that is for 2 years
and 12 month's per year which comes out to
34:44.450 --> 34:52.379
24. Investment at the end of the month for
2 years is given. Future, worth factor for
34:52.379 --> 35:00.170
the first month 1 plus r divided by 8 into
the power N minus J; so N is 24 and for the
35:00.170 --> 35:08.260
first month it is J is 1. So, it is 24 minus
1 and the future worth factor comes out to
35:08.260 --> 35:19.190
be 1.25163018 and this factor is this.
So, the future worth for the cash flow A,
35:19.190 --> 35:28.069
will be 15,000 into this factor comes out
to be 18,857.4452 this value and for the cash
35:28.069 --> 35:37.550
flow B this will be, 8,000 into this factor
which comes out to be 10,057.30415. Similarly
35:37.550 --> 35:44.390
for the second month can be calculated and
for the 24th month the future worth factor
35:44.390 --> 35:51.559
will be 1. So, this will be 15,000 and this
will be 14,000 and if you calculate like this.
35:51.559 --> 35:59.259
So, you can fill up in this table and we find
that, this future worth of this is 404601.9728,
35:59.259 --> 36:11.599
this is 402659.6728. So, what conclusion we
derive out of it is that, though the sum of
36:11.599 --> 36:19.579
the cash flows A and B are the same that is
63,60,000 their future worth's are different,
36:19.579 --> 36:25.489
indicating that more money has been paid through
cash flow A than the cash flow B, as the future
36:25.489 --> 36:34.599
of worth of the cash flow A is 4,04,601.97.
Where as that of the cash flow B is 402659.67.
36:34.599 --> 36:46.430
Now, let us take another example. Example
10 compare two cash flows, one constant end
36:46.430 --> 36:51.690
of the year and which is called cash flow
A and the other unequal end of the year which
36:51.690 --> 36:56.920
was called cash flow B, with compounding other
than annually, when nominal interest rate
36:56.920 --> 37:02.299
is 12 percent and cash flows at the end of
the year are given below. End of the months
37:02.299 --> 37:08.410
are given below, basically using future worth
method. So, here we have two cash flows. The
37:08.410 --> 37:18.019
summation of these cash flows is different.
One is 360000 rupees another is 361 rupees
37:18.019 --> 37:19.019
556.
37:19.019 --> 37:28.299
Now, if we calculate it, the annual rate of
interest r is 12 percent, r by m is 0.12 divided
37:28.299 --> 37:37.089
by 12, N is 12 into 2 that is 24; investment
at the end of the month for 2 years is given
37:37.089 --> 37:47.010
future worth factor of the month is 1 plus
r divided by m to the power N minus J. So,
37:47.010 --> 37:55.859
for the first month J is equal to 1. So, this
comes out to be 1.257163018, this is the factor,
37:55.859 --> 38:00.869
future worth factor. So, this future worth
factor is written here.
38:00.869 --> 38:10.700
Now, the amount is future worth amount is
15,000 for cash flow A 15,000 into this factor
38:10.700 --> 38:19.309
which comes out to be 18,857.4453. So, the
amount is written here and future worth for
38:19.309 --> 38:30.170
the cash flow B is, this is cash flow B is
8,000 into this factor comes out to be 10,057.3042.
38:30.170 --> 38:38.019
Similarly, we filled up all the future worth's
at 24 month, the future worth factor is 1
38:38.019 --> 38:43.570
that is, why the values remain same. It is
15,000 here and this is 14,000 here. The one
38:43.570 --> 38:52.450
way fill up this table, the future worth of
both the cash flows are same. Whereas the
38:52.450 --> 38:57.230
sum of both the cash flows was not same, but
the future worth of the cash flows is the
38:57.230 --> 38:58.380
same conclusion.
38:58.380 --> 39:05.020
Though the some of the cash flows A and B
are different their future worth are almost
39:05.020 --> 39:10.559
same only difference in paisa, indicating
that same amount has been paid through the
39:10.559 --> 39:19.500
cash flow A and cash flow B as their future
worth of the cash flow A and B are 4,04,601.92.
39:19.500 --> 39:30.010
Whereas the cash flow B is, it is only 404601.92.
So, this is the conclusion you make. Though
39:30.010 --> 39:35.069
that primary summation of the cash flows are
different, but the future worth is same; that
39:35.069 --> 39:39.519
means, so we are investing same amount of
money in both the cash flows.
39:39.519 --> 39:46.960
Now, this is the last question. Example number
11; a bonus package pays an employee rupees
39:46.960 --> 39:54.200
2,000 at the end of the first year 2,600 at
the end of the second year and so on. For
39:54.200 --> 40:01.869
the first 9 years of employment. What is the
present worth of the bonus package at 8 percent
40:01.869 --> 40:02.869
interest?
40:02.869 --> 40:14.410
So, this is can be converted into to two series,
one series which is put at t equal to 1, 2,000
40:14.410 --> 40:24.770
t equal to 2,000, t equal to 3,200 so on so
forth up to t equal to N 2,000. And then 2,600
40:24.770 --> 40:33.180
minus 2,000 its start from t equal to 2,600
and then it grows up to N minus 1G. So, this
40:33.180 --> 40:38.549
can be converted into so solution of the problem
can be divided into two parts and annuity
40:38.549 --> 40:44.479
of 2,000 up to 10 years and then arithmetic
gradient series annuity having G is equal
40:44.479 --> 40:53.140
to rupees 600. So, the present worth of the
package is P equal to rupees 2,000 into this
40:53.140 --> 41:00.329
factor P by A 8 percent 10 and then plus rupees
600 to the factor P by G 8 percent 10.
41:00.329 --> 41:09.089
So, here we have written the values of these
factors. So, P by A 8 percent comma 10 factor
41:09.089 --> 41:13.820
is 1 plus i to the power N minus 1 divided
by i into 1 plus i to the power of N. This
41:13.820 --> 41:25.180
is the summation of annuity of 2,000 rupees
per year up to N periods and this is a arithmetic
41:25.180 --> 41:33.730
gradient series present worth. So, when we
solve this problem we get P is equal to rupees
41:33.730 --> 41:47.339
29,006.26.
Thank you.