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July 19th, 2013, 01:55 PM   #1
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Finance mathematic&engineering -annuity question

Hello, I need your help.. Thank you

Present values amounts are equal of 2 infinite series. The payments are "end of period annuites". First serie: the payments are 100 USD for the first 2 years, for the the following two years are 200 USD, for the following 2 years are 300 USD, and it continues like this to infinity. Second serie : the payments are P for the first 3 years,for the the following three years are 2P, for the following threeyears are 3P, and it continues like this to infinity. what is the value of P?
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May 13th, 2014, 06:59 PM   #2
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Quote:
Originally Posted by Kagankarakoc View Post
Hello, I need your help.. Thank you

Present values amounts are equal of 2 infinite series. The payments are "end of period annuites". First serie: the payments are 100 USD for the first 2 years, for the the following two years are 200 USD, for the following 2 years are 300 USD, and it continues like this to infinity. Second serie : the payments are P for the first 3 years,for the the following three years are 2P, for the following threeyears are 3P, and it continues like this to infinity. what is the value of P?
The first series of payments may be viewed as two different perpetuties (infinite payments) that increase by an amount

Code:
100 100 200 200 300 300 ....
Here the first payment begins at t = 1 and then increases by 100 every other year forever

Code:
100 0 200 0 300 0 ....
The present value of this first increasing perpetuity is given by

Code:
PV = [100/i + 100/{i^2}]
For the second part payment begins at t = 2 and then increases by 100 every other year forever

Code:
0 100 0 200 0 300 .....
The present value of this second perpetuity is given by

Code:
PV = [100/{i(1+i)} + 100/{i^2 (1+i)}]
Thus the persent value of the initial series is the sum of the previous two present values

Code:
PV = [100/i + 100/{i^2}] + [100/{i (1+i)} + 100/{i^2 (1+i)}]
Similarly the second series may be seen as three different perpetuties

Code:
100 100 100 200 200 200 300 300 300 ....

100 0 0 200 0 0 300 0 0 ....

0 100 0 0 200 0 0 300 0 ....

0 0 100 0 0 200 0 0 300 ....
Its present value is given as

Code:
PV = [100/i + 100/{i^2}] + [100/{i (1+i)} + 100/{i^2 (1+i)}] + [100/{i (1+i)^2} + 100/{i^2 (1+i)^2}]
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May 13th, 2014, 08:23 PM   #3
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Quote:
Originally Posted by Kagankarakoc View Post
Hello, I need your help.. Thank you

Present values amounts are equal of 2 infinite series. The payments are "end of period annuites". First serie: the payments are 100 USD for the first 2 years, for the the following two years are 200 USD, for the following 2 years are 300 USD, and it continues like this to infinity. Second serie : the payments are P for the first 3 years,for the the following three years are 2P, for the following threeyears are 3P, and it continues like this to infinity. what is the value of P?
I am sorry I overlooked your question

The present value of first series is equal to present value of second series

Code:
[100/i + 100/{i^2}] + [100/{i (1+i)} + 100/{i^2 (1+i)}]
=
[P/i + P/{i^2}] + [P/{i (1+i)} + P/{i^2 (1+i)}] + [P/{i (1+i)^2} + P/{i^2 (1+i)^2}]
Once you know the discount rate you can solve for P

Let me try by letting the discount rate = 10%

Plugging in 10% for i we solve for P

Code:
[100/0.10 + 100/{0.10^2}] + [100/{0.10 (1.1)} + 100/{0.10^2 (1.1)}] = [P/0.10 + P/{0.10^2}] + [P/{0.10 (1.1)} + P/{0.10^2 (1.1)}] + [P/{0.10 (1.1)^2} + P/{0.10^2 (1.1)^2}]

[100/0.10 + 100/0.01] + [100/0.11 + 100/0.011] = [P/0.10 + P/0.01] + [P/0.11 + P/0.011] + [P/0.121 + P/0.0121]

[1000 + 10000] + [909.09 + 9090.91] = [10P + 100P] + [9.09P + 90.91P] + [8.26P + 82.64P]

21000 = 110P + 100P + 90.9P

300.9 P = 21000

P = 69.79

Last edited by AbrahamA; May 13th, 2014 at 08:25 PM.
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May 14th, 2014, 01:32 AM   #4
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I want to discuss this with other members of the forum

The present value of a perpetuity where each an initial payment P increases by a amount P per period forever is given by

Code:
P/i + P/(i^2)
where i is the discount rate

This is found by taking the limits of N as it tends to infinity in the formula for present value of an annuity with linear gradient

Code:
P [ 1 - 1/(1+i)^n } / n ] + P/i [ {1 - 1/(1+i)^n } / n} - n/(1+i)^n ]
This is the short form formula for present value of an ordinary annuity that is the sum of discounted payments as follows

Code:
P (1+i)^-1 + 2P(1+i)^-2 + 3P(1+i)^-3 + 4P(1+i)^-4 + ....
Where as our first investment is of a summation form such as

Code:
P (1+i)^-1 + P(1+i)^-2 + 2P(1+i)^-3 + 2P(1+i)^-4 + 3P(1+i)^-5 + 3P(1+i)^-6 + 4P(1+i)^-7 + 4P(1+i)^-8 +  ....
But since we have divided the investment into two sets of perpetuties

Code:
P (1+i)^-1 + 0 + 2P(1+i)^-3 + 0 + 3P(1+i)^-5 + 0 + 4P(1+i)^-7 + 0 ....
and

Code:
0 + P (1+i)^-2 + 0 + 2P(1+i)^-4 + 0 + 3P(1+i)^-6 + 0 + 4P(1+i)^-8 ....
Thus applying the formula for present value of a prepetuity with linear gradient was incorrectly used

Code:
P/i + P/(i^2)
We can only apply this formula if the payments are discounted at time periods 1,2,3,4,5, ....

Whereas the payments for our first perpetuity are coming in at time periods 1,3,5,7,9, ..... and payments for second perpetuity are coming in at time periods 2,4,6,8,10 .....

To apply the present value formula, we have to discount each of the perpetuity payments at time periods 1,2,3,4,5, ....

To accomplish this for the first set, we divide each discounted payment by discount factor where t goes from 0,1,2,3,4....

This would have the same affect had we multiplied the present value of perpetuity with a capital recovery factor of an annuity due

These were the cash flows of the first investment

Code:
100 100 200 200 300 300 400 400 500 500 ....

t  =     1       3       5       7      9  
CF = 100 0 200 0 300 0 400 0 500 0 .... 

actual t for PV = -1 -3 -5 -7 -9 ....
t used for PV   = -1 -2 -3 -4 -5 ....
difference in t  =  0  1  2  3  4 ....
difference in time periods used in discounting means that the present value of a growing perpetuity that we have calculated has to be multiplied by capital recovery factor

The capital recovery factor of an annuity due for an amount of 1 dollar at 10% discount rate for infinite periods is

Code:
i/(1+i)

11,000 i/(1+i)
11,000 0.1/(1.1)
11,000 

1,000
And for the second part of the perpetuity, we divide each discounted payment by a discount factor where t goes from 1,2,3,4, ....

Code:
t  =        2       4      6       8     10
CF = 0 100 0 200 0 300 0 400 0 500 ....

actual t for PV = -2 -4 -6 -8 -10 ....
t used for PV   = -1 -2 -3 -4 -5 ....
difference in t  =  1  2  3  4  5 ....
We can do the same for the present value calculations of the second term of the perpetuity this time multiplying it by capital recovery factor of an ordinary annuity

The capital recovery factor of an ordinary annuity for an amount of 1 dollar at 10% discount rate for infinite periods is

Code:
i

= 10,000 (0.1)
= 1,000
Thus the present value of first investment at a discount rate of 10% is 2,000

The original calculation I presented earlier produced a present value of 21,000 at a discount rate of 10%

So what do think is going on here.
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