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July 19th, 2013, 01:55 PM  #1 
Newbie Joined: Jul 2013 Posts: 1 Thanks: 0  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? 
May 13th, 2014, 06:59 PM  #2  
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5  Quote:
Code: 100 100 200 200 300 300 .... Code: 100 0 200 0 300 0 .... Code: PV = [100/i + 100/{i^2}] Code: 0 100 0 200 0 300 ..... Code: PV = [100/{i(1+i)} + 100/{i^2 (1+i)}] Code: PV = [100/i + 100/{i^2}] + [100/{i (1+i)} + 100/{i^2 (1+i)}] 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 .... 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}]  
May 13th, 2014, 08:23 PM  #3  
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5  Quote:
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}] 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.  
May 14th, 2014, 01:32 AM  #4 
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5 
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) 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 ] Code: P (1+i)^1 + 2P(1+i)^2 + 3P(1+i)^3 + 4P(1+i)^4 + .... 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 + .... Code: P (1+i)^1 + 0 + 2P(1+i)^3 + 0 + 3P(1+i)^5 + 0 + 4P(1+i)^7 + 0 .... Code: 0 + P (1+i)^2 + 0 + 2P(1+i)^4 + 0 + 3P(1+i)^6 + 0 + 4P(1+i)^8 .... Code: P/i + P/(i^2) 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 .... 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 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 .... 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 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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