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May 29th, 2015, 08:41 AM  #1 
Newbie Joined: May 2015 From: Italy Posts: 3 Thanks: 0  Present Value Annuity Problem
Please could someone help me on this one? Consider an annuity with maturity after 8 years that is continuous with semestral density a=50 euros in the first 2 years and discrete with annual payments of 100 euros in the following period. Compute the present value of the annuity. (compound convention, annual rate i=10%). Thank you in advance! 
May 30th, 2015, 12:36 AM  #2 
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5 
i=10% p_a = 1/2 = 0.5 c_a = 0 n_a = 4 aey(i, c_a) = e^i  1 aey(10%, 0) = e^(10%)  1 aey(10%, 0) = e^(0.1)  1 aey(10%, 0) = 2.718282^(0.1)  1 aey(10%, 0) = 1.105171  1 aey(10%, 0) = 0.105171 aey(10%, 0) = 10.52% x_a = [1 + aey(i, c_a)]^(p_a) x_a = (1 + 10.52%)^(0.5) x_a = (1.1052)^(0.5) x_a = 1.051271 i=10% p_d = 1 c_d = 1 n_d = 6 aey(i, c_d) = (1+i*c)^(1/c)  1 aey(10%, 1) = (1+10%/1)^(1)  1 aey(10%, 1) = (1.1)^(1)  1 aey(10%, 1) = 1.1  1 aey(10%, 1) = 0.1 aey(10%, 1) = 10% x_d = [1 + aey(i, c_d)]^(p_d) x_d = (1 + 10%)^(1) x_d = (1.1)^(1) x_d = 1.1 PV = 50 * [1  (x_a)^(n_a)] / [x_a  1] + 100 * (x_a)^(4) * [1  (x_d)^(n_d)] / [x_d  1] PV = 50 * [1  (1.051271)^(4)] / [1.051271  1] + 100 * (1.051271)^(4) * [1  (1.1)^(6)] / [1.1  1] PV = 50 * (1  0.818731) / 0.051271 + 100 * (0.818731) * (1  0.5644739) / 0.1 PV = 50 * 0.18126925 / 0.051271 + 100 * (0.818731) * 0.4355261 / 0.1 PV = 50 * 3.53551 + 100 * (0.818731) * 4.355261 PV = 176.77 + 100 * 3.5657859 PV = 176.77 + 356.58 PV = 533.35 
May 30th, 2015, 01:10 PM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
The "continuous" is unclear...assuming it does not mean "interest compounding continuously", and that the interest is 10% EFFECTIVE annually: i = 1.1^(1/2)  1 = .04888... u = FV of the semiannual portion. u = 50[(1+i)^4  1] / i = 215.12492... i = .10 u = u(1+i)^6 = 381.10693... v = FV of the annual portion. v = 100[(1+i)^6  1] / i = 771.5600... PV = (u + v) / (1 + i)^8 = 537.7281... Have I been inspired by the Hoooooly Spook, Dexter? 
May 30th, 2015, 07:38 PM  #4 
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5  I didn't know what it meant till I looked it up in a dictionary spook (spo͞ok) n. 1. Informal A ghost; a specter. 2. Slang A secret agent; a spy. 3. Offensive Slang Used as a disparaging term for a black person. v. spooked, spook·ing, spooks v.tr. 1. To haunt. 2. To startle and cause nervous activity in; frighten: The news spooked investors, and stock prices fell. v.intr. To become frightened and nervous. 
May 30th, 2015, 07:42 PM  #5 
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5 
All of the above

May 30th, 2015, 09:23 PM  #6 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
I used holy spook = holy spirit, the 3rd member of the holy trinity. Bow your head in reverence. In the name of Big Daddy, of Sonny Boy, and of The Holy Spook. Hahmen! Be advised that it is the Holy Spook who inspires the Pope. Father and Sonny are too busy handling their real estate business. 
May 31st, 2015, 10:07 AM  #7 
Newbie Joined: May 2015 From: Italy Posts: 3 Thanks: 0 
Thank you both for your help! 

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