WEBVTT
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Welcome to the lecture series on Time value
of money-Concepts and Calculations. The topic
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of the present lecture is Present Value. What
is a present value? Present value it is given
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by PV is the current worth of a future sum
of money or stream of cash flows under a specified
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rate of return.
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Future cash flows are discounted at the discounted
rate and the higher the discounted rate the
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lower is the present value of the future cash
flows. Now this shows a Time line where after
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3 years Rupees 100 some is available and the
discounter rate is 10 percent.
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Now, what will the present value of these
100 Rupees at year 0? At year 0, when we transfer
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this money to year 0 we get the present value.
The present value is 75.13; that means if
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we invest 73.13 at t equal to 0, at 10 percent
interest rate, I will be able to get 100 Rupees
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after 3 years. Now this is given by PV equal
to FV divided 1 plus i to the power N and
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in this case FV is 100 i is 10 percent. So,
1 plus 10 divided by 100 whole cubes and this
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get 75.13 Rupees.
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Now, present value problems can be divided
into 4 types of problems. The first problem
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is present value of the single amount here
if we see in the time line at the end of third
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year, we have 100 Rupees and we want to find
out what is the present value of it here.
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The second type of problem is present value
of multiple constant equi-time spaced amounts.
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Here we see that after the end of first year
100 Rupees is invented and at the end of second
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year again 100 Rupees is invested, at the
end of third year again 100 Rupees is invested.
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So, we want to find out the present value
for all these investments at PV equal to 0.
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The third type of problem is present value
of multiple variable equi-time spaced amounts.
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Here we are investing money at equi space
time intervals, but the amount of money are
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different. Like at the end of third year I
am investing 130 Rupees at the end of second
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year I am investing 120 Rupees and at the
end of first year I am investing 100 Rupees.
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All this 3 amounts are different, but they
are invested at equi-time spaced. The fourth
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type of problem is present value of multiple
variable amounts at variable time spaced.
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Now here we see that at the end of first year
we are investing 100 Rupees, at the end of
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third year we are investing 120 Rupees and
at the end of 8 year we are investing 130
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Rupees. So, 130,120 and 100 are different
amounts and they are invested at different
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time line. The first one is after 1 year,
second one after 3rd year and third one after
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8 year.
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Present value of a single amount; present
value of a single amount can be found out
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from this equation FV is equal to PV into
1 plus i to the power N. From where you can
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calculate PV is equal to FV divided by 1 plus
i to the power N; discounting this is called
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discounting. Discounting is the process of
translating a future value or a set of future
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cash flows into a present value. So, we are
moving in this direction of time and this
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is called Discounting.
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Example 1: what is the present value of Rupees
500 due in 5 years; part a, if annual interest
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is 15 percent and compounding is annually;
part b, with same interest rate but compounding
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quarterly; c with same interest rate but compounding
continuously.
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So, my formula is PV is equal to FV divided
by 1 plus i to the power N. Where, FV is equal
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to Rupees 500, i is equal to 15 percent and
N is equal to 5. Finding the PV of a single
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cash flow when compound interest is applied
is called discounting, that is reverse of
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compounding, the PV source, and the value
of amount in terms of today’s purchasing
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power. So, you can calculate. PV from the
above equation PV is equal to FV divided by
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1 plus i to the power n is equal to 500 divided
by 1 plus 15 this is the i value divided by
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100 to the power 5 comes out to be 248.59.
That means, if I am investing 500 Rupees at
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the end of 5 years. So it is value at t equal
to 0, the present value will be 248.59, when
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mine discounting rate is 15 percent. Note;
PV is less than money available at the end
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of 5th year. This happens when we move that.
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Now part b, given FV is equal to 500, N is
equal to 5 m is equal to 4 because it is restrictly
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compounding and it is quarterly compounding
that is why m is equal to 4 i or r is equal
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to 15 percent. So, my formula for this cases
PV is equal to FV divided by 1 plus r divided
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by m to the power mN, m is 4 and N is 5. So,
the multiplication is 20. So, 500 divided
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by 1 plus 15 divided by 100 divided by 4 to
the power 20 that comes out to be Rupees 249.45.
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Part c; given FV is equal to 500, N is equal
to 5, r is equal to 15 percent. Now it is
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a continuous interest forming. So, PV is equal
to FV divided by e to the power rN is equal
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to 500 divided by e to the power 0.15 into
5 comes out with 236.183. Please note down
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that present value of Rupees 500 in part a
is greater than part b is greater than part
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c.
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Now, let us see the derivation of present
worth of an annuity A for annually compounding.
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Now P will be equal to A, If here we see this
is the time line, at t equal to 0 we want
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the present value at t equal to 1 we are investing
Rupees A, t equal to 2, again A amount is
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invested and t equal to N and another A amount
is invested.
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So, continuously we are investing A amount
each year up to Nth year and our interest
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rate is half set. So, the present value I
have to transport all this a values to this
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time line A equal to 0 to this and this to
this. So, I can write down this equation P
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is equal to a 1 plus i to the power minus
1 plus A1 plus i to the power minus 2 like
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this up to A into 1 plus i to the power minus
N.
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Now, I both the side I multiply it with 1
plus I, then this A1 plus I to the power minus
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becomes A and this becomes minus N minus 1
in brackets. So, subtracting equation 2 from
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equation 1 we have P minus P, in brackets
1 plus i is equal to A, in bracket 1 plus
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i minus N minus A. So if we solve this we
can write down minus Pi is equal to A in brackets
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1 minus 1 plus i to the power N divided by
1 plus i to the power N. Now if I take this
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negative into consideration, I can write down
this equation this reverse is A equal to 1
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plus i to the power N minus 1 divided by i
into 1 plus i to the power minus 1. This is
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called Present worth of Annuity of annually
compounding.
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Now you can find out based on this value,
we can find out the present worth of annuity
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A for discretely compounding. The starting
point is present worth of annuity A for annually
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compounding. This equation we have seen P
is equal to A into 1 plus i to the power minus
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1 whole divided by i into 1 plus i to the
power N. To get the formula for present worth
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of annuity a for discretely compounding replace
i in PV of an annuity for annual compounding
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by i by m. So, if i replace this i by i by
m i can get the formula for present worth
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of an annuity for discretely compounding.
So you will see that here I have changed i
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with i by m and N with m into N. So, if it
is done, So I get this equations. The above
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formula is valid for the case when number
of payments is equal to number of compounding
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periods, this has to be remembered. This is
only valid for the case when number of payments
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is equal to the number of compounding periods.
Now let see the derivation of present worth
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of annuity A for continuous compounding. His
is our time line at the end of first year
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I am investing A money, end of second year
again A money and at the end of Nth year again
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A money. So, if I take this money to time
line t equal to 0, I can find out the present
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value, but here the compounding is continuous.
So when this A is converted into present value.
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So, this is A into e to the power minus r.
Similarly, if I write for all n values, then
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the last value is A into e to the power minus
rN. Now if I multiply the left hand side and
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right hand side with e to the power r, then
this becomes A, this becomes A into e to the
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power minus r and the last one becomes Ae
to the power minus r in brackets N minus 1.
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Now if you subtract equation 8 from equation
7, then this is P minus Pe to the power r
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is equal to Ae to the power minus rN minus
A. So you can write down P is equal to A in
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brackets 1 minus e to the power rN divided
by e to the power rN in brackets 1 minus e
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to the power r. So, this equation links the
annuity with the present value. This is the
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present value and this is the value of the
annuity.
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So present worth of annuity for continuously
compounding, we will be using this equation.
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Now let us see the different problems which
can be worked out based on this information.
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Now this example to a cash flow consisting
of 1 10000 per year is received in one discrete
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amount at the end of each years for 10 years.
So, for 10 years we are getting 10000 Rupees
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each year. Interest will be 10 percent per
year compounded annually determine the present
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worth that is at the 0 time. The solution
is cash flow for each year up to 10 years
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is Rupees 10000 interest at 10 percent per
annum, N is equal to 10 years and what is
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required is the PV value that is present value.
So, my equation is PV equal to A in brackets
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1 plus i to the power N minus 1 and divided
by i into 1 plus 1 plus i to the power N.
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So, when we put values into this equation
the value of P comes out to be 61446. So the
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cash flow over the 10 years is 1000000. Now
let us please note that the total cash flow
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in 10 years is more than the present worth
and this is obvious because we are discounting
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the values; now present value of multiple
constant equi-time spaced amounts.
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So my amount is not changing. Let us see the
problem, example 3 a cash flow consisting
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of rupee 1000 per year is received in one
discrete amount at the end of each year for
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3 years; that means, at the end of first year
I am getting 1000 at the end of second year
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again 1000, at the end of third year again
1000. Interest will be 10 percent per year
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compounded annually, determine the present
worth at time equal to 10. So, this I am solving
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from the first principle as well as using
the equation. So, cash flow per year up to
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3 years is equal to 1000, i is equal to 10
percent, N is equal to 3. Now if I take this
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value that is 1000, which is given at the
end of third year. Transport it to 0 time
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line and this can be done by finding out the
present value of this 1000 Rupees.
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So this will be 1000 divided by 1 plus 10
divided by 10 to the power 3, which comes
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out to be 751.31, that means, if this money
1000 present value will be calculated it comes
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out to be 751.31 as written here. Now similarly
the 1000 Rupees which we have received at
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the end of second year, its present value
is calculated, it comes out to be 826.45,
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this is equal to 1000 divided by 1 plus 10
divided by 100 to the power 2. Similarly 1000
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which is available at the end of first year
it is present value will be calculated it
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will be 909.09.
Now, let see here, the present value of this
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1000 which will available at the end of third
year is 751 the present value of this is 826,
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the present value of this 1000 is 909.09.
Have you observed a trend this value is less,
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this value is bit more and this value is more
than this value this will happen always. Now
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when I add these three values here I get 2486.85.
So what I have done, I have added the present
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worth of this value, present worth of this
value and present worth of this value and
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this 3 values comes out to be 2486.85. So,
this is our answer because you are interested
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in finding out the worth of these 3 values.
This can be done by a formula which is PV
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is equal to A within brackets 1 plus I to
the power N minus 1 divided by i 1 plus i
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to the power, if I use this formula and put
my numbers, it comes out to be 2486.85 the
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same value.
Now, let us take a problem which is little
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bit of miss miscellaneous in nature. Mr. and
Mrs. Sharma wish to have an annuity for their
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daughter, when she goes to university. They
wish to invest into an annuity that will pay
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their daughter Rupees 2000 per month for 4
years. What is the present value of that annuity
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given that current interest rate is 9 percent
per annum?
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Here you will note that the payments are per
month and not per year, whereas my interest
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rate is per year. So, this is a case where
the discounting is done per period which is
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month. So, for this case A is equal to 2000
m is equal to 12 because there are 12 months
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in a year interest per month is r by m which
is 0.09 divided by 12, which comes out to
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be 0.0075 and number of periods is equal to
12 into 4 which is 48. So, I use this equation
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P is equal to A in brackets 1 plus 0.007 to
the power 48 minus 1 divided by 0.0075 into
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1 plus 0.0075 to the power 48. This comes
out to be 80369.85.
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So, this is a different problem and uses a
compounding method which is not annually.
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So, this means that if I invest 80369.85 invested
now at 9 percent will provide Rupees 2000
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per month for 4 years.
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Now, let us take another example, which is
based on continuous compounding. A cash flow
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consisting of Rupees 10000 per year is received
in one discrete amount at the end of each
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year for 10 years; interest will be 10 percent
per year, compounded continuously determine
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the present worth at zero time. The solution
is given that cash flow each year up to 10
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years is 10000; that means, each year will
have 10000 Rupees that means, in first year
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I will have 10000 Rupees, in the second year
I will have 10000 Rupees, at the end of third
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year I will have 10000 Rupees in so and so
for, up to the 10 year interest rate r is
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10 percent per annum N is equal to 10 years
and compounding method is continuous.
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So, we use the equation present worth PV is
equal to A in brackets e to the power rN minus
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1 in divided by e to the power rN into e to
the power r minus 1, when we put our values
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into this equation then the value of the PV
which comes out to be 60104.07. The cash flow
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over 10 year is 100000. Please note that the
total cash flow in 10 years is more than the
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present worth of it.
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Then another problem present value of multiple
variable equi-time spaced amounts. When the
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values are different; that means, end of the
first year value of 1000 end of second year
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1500, end of the third year it is 2000. In
such problems I cannot use an equation to
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find out the values and hence has to be solved
from the first principle.
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So, a cash flow consisting of Rupees 1000,
1500 and 2000 per year is received has discrete
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amounts, 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 present worth at 0 times. So,
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we have to find out the present worth of this
values which were invested at different period
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of time. So, from the first principle we see
that the cash flows are available 1000, 1500
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and 2000 and i is equal to 10 percent. So,
let us see this, these value is 2000 invested
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at the end of third year, when I find out
the present value of this, this will be 2000
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divided by 1 plus 10 divided by 100 to the
power 3, which comes out to be 1502.63.
22:40.580 --> 22:46.950
Similarly for this 1500 which is at the end
of second year if I want to find out the present
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value of it here and the 0th here and it is
1500 divided by 1 plus 10 divided by 100 to
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the power 2 which comes out to Rupees 1239.67.
Now, the value which is 1000 which is available
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at the end of first year, it is present value
is found out at time equal to 0 and this is
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10000, 1000 divided by 1 plus 10 divided by
100 to the power 1 is 909.09. So, when I add
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these 3 values here I get a value Rupees 3651.39.
So, the present value of all this 3 amounts
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1000, 1500 and two 2000 invested at different
interval of time is equal to 3651.39.
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Now, let us take another example, here cash
flow consisting of 1000, 2500 and 5000 per
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year is received as discrete amount at the
end of 1st year, 6th year and 8th year respectively.
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Interest rate is 10 percent per year compounded
annually determine the present worth at zero
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time. This is a problem, where present value
of a multiple variable amounts with variable
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time spaced. So time is different and amount
is different here. So, it will be solved by
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using the first principle.
24:19.440 --> 24:25.570
Now, for this what we will do at the end of
8th year the amount available is 5000 to will
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find out the present value of this 5000 Rupees
that means, we will transport this amount
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from here to this time line and this will
be done by 5000 divided by 1 plus 10 divided
24:38.990 --> 24:46.890
by 100 to the power 8, which comes out to
be 2332.54. Now there is another amount that
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the end of 6th is which 2500 it is is present
value has to be find out by taking it to the
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0th here line.
So, this will be 2500 divided by 1 plus 10
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divided by 100 to the power 6 is comes out
to be Rupees 1411.18 and at the end of 1st
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year we have 1000 amount and this has to be
brought down to the present value. This is
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1000 divided by 1 plus 10 divided by 100 to
the power 1, is comes out to be Rupees 909.09.
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Now if we add these 3 values, which is the
present worth of these investments 5000, 2500
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and 1000 they is added hero, to find out the
present value of the cash flow which is 4652.81.
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Thank you.