WEBVTT
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Welcome to the lecture series on Time value
of money-Concept and Calculations. In this
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lecture we will cover Future Value. The Future
value which is designated as FV is the value
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of a current asset, at a specified date in
the future based on an assumed rate of growth
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over time. The FV calculation allows investor
to predict with varying degree of accuracy,
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the amount of profit that can be generated
by different investments.
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The amount of growth, generated by holding
a given amount of cash will likely be different
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than if that same amount were invested in
stocks. So the F V equation is used to compare
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multiple options. What is Future value? Future
value is the value of an asset at a specific
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date. It measures the nominal future sum of
money that a given sum of money is worth at
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a specified time in the future assuming a
certain interest rate, or more generally rate
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of return. Future cash flows are compounded
at the rate of return, and the higher the
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rate of return, the higher the Future value
of the cash flows.
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Now, let us see this time line which shows
up to 3 years at 0 times Rupees 100 is invested.
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Now we want to find out what will be it is
value at the end of third year. Now at the
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end of third year, here the Future value of
these sum 100 Rupees will be 133.1. This value
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is dependent on the rate of return. If rate
of return will be different, this value will
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be different. This table shows if the rate
of return is 5 percent the value is 115.25,
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if the value is 7 percent this is 122.5043.
The Future value if it is 10 percent it is
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133.1, if it is 20 percent it is 172.8.
So, this Future value is a function of this
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rate of return, as well as the time where
we are calculating the Future value. For this
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the equation used is F V is equal to PV into
1 plus I to the power N. And for 10 percent
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return if I calculate this value this is 101
plus 10 by 100 to the power 3 is equal to
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133.1. Now the Future value problems can be
divided into 4 types of problems.
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The Future value of a single amount, here
in the timeline at 0 100 Rupees invested.
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This is a single amount. I want to find out
the Future value of this as the end of third
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year. So, here PV is what the second is Future
value of a multiple constant equi-time spaced
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amounts. Here we see that at 0 am I am investing
100 Rupees, at the end of first year I am
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again investing 100 Rupees, at the end of
second year again I am investing 100 Rupees
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and at the end of third year I want to know
what is the Future value of all this 3 amount.
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So, the Future value of this amount as to
be found out, Future value of this amount
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at the end of three years as to be found out,
and Future value of this amount at the end
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of third year he as to be found out and when
I add this three values the total Future value
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of this cash flows will come out.
Third type of problem is the Future value
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of multiple variable equi-time spaced amounts.
Here the amount the value of amounts are changing,
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like at t is equal to 0, the investment is
100 Rupees at the end of first year, the investment
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is 120 Rupees at the end of second year, the
investment is 130 Rupees. So, though time
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space is same that is one year, but the values
are different, one is 100 another is 120 and
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another is 130. Such type of problems as to
be solved by the original technique and not
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by a equation.
The 4th type of problem is Future value of
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multiple variable amounts variable time spaced.
Here the value of the amount is also changing
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and the value of the time is also changing,
that is here at 0 time I am investing 100
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Rupees, at the end of first year an 120 Rupees,
at the end of third year I am investing 130
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Rupees. It is not the end of second year,
but it is end of third year and I want to
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find out what is the future worth of all this
three investments at the end of 8th year.
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The last two types of problems which we have
discussed now, manual would be solved by equations
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it as to be solved by first principle.
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The Future value of a single amount; this
is FV is equal to PV into 1 plus I to the
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power N. Compounding is the process of translating
a present value into a future value. So, I
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am moving from 0th year to third year I am
moving in this direction and this is called
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Compounding.
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Now let us take an problem small problem.
What is the Future value of Rupees 500 deposited
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today after 5 years? Apart if annual interest
rate I is 15 percent and compounding is annually
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b with the same interest rate, but compounding
quarterly with same interest rate, but compounding
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continuously now the solution is that my formula
is FV is equal to PV into 1 plus I to the
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power N.
This formula is used for when compounding
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is annually and not compounding quarterly
which is discrete compounding or compounding
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continuously, where PV is equal to Rupees
500 I is equal to 15 percent N is equal to
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5 and I have to calculate what is the FV,
that is final worth or final value. Finding
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the final value of a single cash flow, when
compounding interest is applied is called
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compounding and this is the reverse of the
discounting.
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The FV shows the value of amount at a future
date in commensuration with the purchasing
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power of the date. Now FV is equal to PV into
1 plus I to the power N is equal to 500 into
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1 plus 15 divided by 100 to the power 5 that
comes out to be 1005.68, that means, if I
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am spending 500 Rupees today 15 percent and
interest rate than at the end of 5th year
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I will have a sum which will be 1005.68.
Note FV is more than money available at the
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start of first year or the start or of 0th
year basically this happens when we move to
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future.
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The part two is PV is equal to 500, N is equal
to five m is equal to four and I equal to
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r equal to 15 percent. Now this is a discrete
compounding problem. So my equation changes
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FV is equal to PV 1 plus r divided by m to
the power mN, this is the equation from discrete
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compounding. So FV is equal to 500 into 1
plus 15 by 100 into 4 in brackets and then
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overall I have 4 into 5 the power is 4 into
5 that is 20 which comes out be one 1044.076.
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Now, in the c part the compounding is continuous.
So again the equation changes PV is equal
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to Rupees 500, N is equal to 5, r is equal
to 15 percent. So, FV is equal to PV into
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e to the power rN, which is 500 into e to
the power 0.15 into 5, which comes out to
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be 1058.5. Now from here it is very clear
that annual compounding if say FV value which
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is lower than the discrete compounding and
discrete compounding if say FV value which
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is lower than continuous compounding. So,
please note that the Future value of Rupees
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500 in part c is greater than part, b is greater
than part a.
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Now derivation of future worth of an annuity
for annually compounding; now this derivation
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I am not gone for detailed derivation because
this is derivation is a small one if I use
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the Future value is equal to present value
1 plus I to the power N.
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And if I know the present value equation then
I can put this and find out the future value.
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So, FV is equal to a into 1 plus I to the
power N minus 1 divided by I in the brackets
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1 plus I to the power n, this when you to
be multiplied by 1 plus I to the power N this
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gives you Future value. Because this equation
is for present value and Future value is equal
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to present value into this. So, this is for
present value, when I multiply this with this
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part this gives future value.
So, Future value is equal to a in brackets
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in small bracket 1 plus I to the power N minus
1 divided by i. So this is future worth of
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annuity for annually compounding. Now here
the investments are like this in the first
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year I have A at the end of first year at
the end. So, the second year I have again
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A and N at the end of the nth year I have
again A. So, you will remember this one for
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this type of investment this is valid, I am
not investing at t equal to 0 year, so this
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equation as been derived for such type of
investment.
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Future worth of annuity A for annually compounding
is this
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and future worth of annuity A for discretely
compounding. Than the calculated based on
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this to get the formula for future worth of
an annuity A for discretely compounding replace
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i in FV equation this of an annuity for annual
compounding by i by m and replace N by m into
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N.
So, if I do that the future worth FV is equal
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to this into in brackets 1 plus i by m to
the power mN minus 1 bracket close divided
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by i by m. The above formula is valid for
the case when number payments is equal to
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number of compounding periods this as to be
remembered. This formula is derived based
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on this assumption is valid for the case when
number of payments is equal to the number
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of compounding periods. Now let us see the
derivation of the Future value of an annuity
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A for continuous compounding.
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Obliviously we are not deriving it from first
principles we are using the equation this
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FV is equal to PV e to the power rN, now this
is for PV part of it. So, PV can be written
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a into e to the power rN minus 1 divided by
e to the power rN in brackets e to the power
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r minus 1 and we only multiplied this part
with this and this cancels out. So, FV is
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equal to A e to the power rN minus 1 divided
by e to the power r minus 1, here also you
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see that the investment is done like this,
at the end of first year the first investment,
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end of the second year the second d investment
and eight of the Nth year that is another
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investment A.
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This is a problem which is given in example
2. It is related to Future value of multiple
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constant equi-time spaced amounts. Here basically
the Future value of a annuity is been calculated
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the example two is a cash flow consisting
of Rupees 10000 per year is received in one
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discrete amount at the end of each year for
10 years, interest will be 10 percent per
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year compounded annually.
Determine the Future value at the 10 years.
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So, What is demanding that the at the end
of first year 1000 is invested, at the end
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of second year 10000, is invested at the end
of third year 10000 is invested so on so for
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up to the end of for 10th year and at the
end of 10th year we want the future value.
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So, the Future value of this Future value
of, this Future value of, this Future value
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of, this and Future value of this amounts
are to be added up to find out the total Future
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value for. This can be done using equation
also. So here the cash flow each year up to
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10 years is Rupees 10000 interest rate is
10 percent per annum, N is ten years. So,
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Future value is equal to A into in brackets
1 plus i to the power N minus 1 divided by
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i ,when I put values into this equation this
is a is ten thousand one plus i is 10 percent
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to the power N. N is 10 minus 1 divided by
0.1 this is the value of i. Then it becomes
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equal to 159374.25. The cash flow over 10
years is about 100000, please note that the
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total cash flow in 10 years is less than the
future worth. Future value of multiple constant
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equi-time spaced amounts.
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Question 3, a cash flow consisting of 1000
per year is received in one discrete amount
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at the end of each year for 3 years; interest
will be 10 percent per year compounded annually.
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Determine the future worth at the end of the
third year.
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Now solution based on first principle will
do both solution based on the first principle
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and then is in the equation. Solution based
on the first principle gives you inside of
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the problem you solved. Now what is with ask
is that cash flow per year up to 3 years is
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1000, that means, at the end of first year
1000 is paid, end of second year another 1000
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is paid and the third year another 1000 is
paid. So, if I find out the Future value of
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this 1000 which is paid at the end of second
year it is there for 2 years. So, the Future
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value at the end of first year is 1000, 1
plus 100 by 100 to the power 2.
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This 2 is because it is invested for 2 years
not for 3 years. If I invest here then it
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will remain for 3 years, but investment at
the end of first year it remain for 2 years.
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So, this is 1210 and which has been invested
at the end of second year it remains for only
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1 year. So, this is 1001 plus 10 divided by
100 to the power 1 it is 1100 and what is
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been invested in the third year it remains
for 0 year and hence this is 1000 only.
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When we add this 3 half then it becomes Rupees
3310. So, what we have done, we have found
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out the Future value of this amount, we have
found out the Future value of this amount,
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we have found out the Future value of this
amount and we have added them together to
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find out the total Future value the same can
be done through this formula. The Future value
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is equal to A, in brackets 1 plus i to the
power N minus 1 divided by i, when I put my
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values here, then it comes out to be 3310.
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Let us taken mix problem, which is example
number 4. Mr. and Mrs. Sharma wish to create
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an annuity for their daughter. So, that she
gets a sum when she goes to university. They
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wish to invest into an annuity of Rupees 2000
per month for 4 years. So, that she gets the
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sum after four years when she will be requiring
it. What is the Future value of the annuity,
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given that the current interest rates are
nine percent per annum?
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Now, here basically he is investing per month
and interest rates are in annum. So, this
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is a problem of discrete compounding and hence
the formula which will be used for this case
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is this. This is a formula for converting
annuities to Future value when there is a
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discrete compounding. So, here A that is annuity,
is 2000 per month, m is equal to 12 because
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there are 12 months per year, interest rate
per month is r by m is equal to 0.09, which
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is the value of r divided by m which is 12
it comes out to be 0075. Number of periods
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is equal to 12 into 4.
This is 48 because there are 4 years and m
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is 12. So, m into N is equal to 48 and what
is the value of F is demanded. So, if I use
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this equation and put my values then it comes
out to be 115041.41. So, this means if 2000
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per month of annuity is paid for 4 years at
an interest rate of 9 percent it will grow
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to 115041.41 at the end of 4 years.
Example number 5, this is related to continuous
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compounding. A cash flow consisting of 10000
per year is received in one discrete amount
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at the end of each year for 10 years; that
means, at the end of first year I am getting
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10000 at the end of second year I am again
getting 10000 at the end of third year again
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getting 10000 likewise I am getting up to
10 years interest will be 10 percent per year
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compounded continuously, determine the future
worth at the end of 10th year.
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Here the cash flow each year up to ten years
is 10000, interest rate r is 10 percent per
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annum, N is 10 years compounding is continuous.
So, future worth for continuous compounding
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is this value, that is A in brackets e to
the power rN minus 1 divided by e to the power
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r minus 1. So, when we put value on this equation
this is 10000 the value of A is 10000 and
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this is e to the power 0.1 into N is equal
to 10 minus 1 and divided by e to the power
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0.1 minus 1 it comes out to be 163379.86.
The cash flow over 10 year is only 100000,
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Please note that the total cash flow in 10
years is less than the future worth.
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Now this is a problem for Future value of
multiple variable equi-time spaced amounts,
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here the amount is changing. So, it is example
number 6 and such type examples are to be
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solved from first principle. A cash flow consisting
of 1000, 1500 and 2000 per year is received
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as discrete amount at the end of first year,
second year and third year respectively. Interest
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rate is 10 percent per year compounded annually
determine the future worth at the end of the
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third year.
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So, what we have to do this is the timeline
in which investment is own at the end of first
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year this is 10000, at the end of second year
this is 1500 and the end of third year it
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is 2000 and what I am suppose to find what
are the Future value of this 3 investments
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at the end of third year.
So, the Future value of the amount received
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at the end of third year is 2000. So, it remains
2000 because there is it will not earn any
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interest because at the end of third year
this amount is there. Now at the end of second
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year this will earn interest for only one
year. So, this is 1501 plus 10 divided by
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100 to the power 1 is equal to 1650 and the
investment which is done at the end of first
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year it will earn interest for 2 years. So,
this is 1000 into 1 plus 10 divided by 100
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to the power 2. This is 1210. So if I add
of this 3, 2000, 1650 and 1210 here it comes
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out to be Rupees 4860. So, the combined Future
value of all this 3 investments is 4860.
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Now, let us take Future value of multiple
variable amounts in variable time spaced.
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So, here the investments are different here
the investment is 1000, 2500 and 5000 and
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that to also invested at different time, it
is at the end of first year, this is at the
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end of 6 year and this is the end of 8 year.
So, they are not uniform. So such types of
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problems are done using first principle.
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So given cash flow in first 6th and 8th year
are 1000, 2500 and 5000, i is equal to 10
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percent. So, the Future value of the amount
received at the end of 8th year is 5000, because
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I am finding out the Future value at the end
of 8th year.
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So, whatever investment which is done at the
end of 8th year will not draw any interest
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and that is why that value 5000 remains 5000.
At the end of 6th year, my investment is 2500.
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So it will draw interest for 2 years 627 and
728 and that is why the Future value will
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be 2500 into 1 plus 10 by 100 to the power
2 which comes out to be 3025 and the value
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which has been invested at the end of first
year will draw interest rate up to 7 years.
25:34.489 --> 25:41.110
So, this is 1000 into 1 plus 10 divided by
100 to the power 7 which comes out to be 1948.72.
25:41.110 --> 25:48.309
So, when I add up the Future value of all
these 3 investments that is 1000, 2005 and
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5000. This is the value I get is Rupees 9973.72.
Thank you.