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February 2nd, 2013, 08:13 AM  #1 
Member Joined: Jun 2012 Posts: 64 Thanks: 0  Find the present value
Find the present value of an annuity certain of $12000 p.a. payable monthly in arrear for 20 years when the rate of interest is 5% p.a. for the first 10 years and thereafter 2.5% per half year. [Ans: $152 783.23] I am unsure with what to do with the "2.5% per half year" part. I am approaching this problem with complex annuities but I keep getting an amount few thousand off. I think my problem lies in the 2.5%. Please help, thanks. 
February 2nd, 2013, 10:30 AM  #2  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: Find the present value Quote:
Step 1: calculate the present value as at 10th year of the monthly annuity of $1000 payable over the next 10 years @ 2.5 semiannually. Rate needs to be converted to monthly equivalent (to match payment frequency): (1 + i)^12 = (1 + .025)^2 ..... i = .0041239.... PV1 = P[1  1/(1 + i)^n] / i : that'll give you 94504.618... (required value after 10 years) Step 2: calculate present value of above as of today. Rate must be converted again, to the monthly equivalent of 5% p.a.: (1 + i)^12 = 1.05 ..... i = .0040741.... PV2 = 94504.618 / (1 + i)^n = 58017.638... Step 3:calculate the present value as of today of the monthly annuity of $1000 payable over the next 10 years @ 5% p.a. We use the rate above: i = .0040741... PV3 = P[1  1/(1 + i)^n] / i : that'll give you 94765.591... So net PV = PV2 + PV3 = 58017.638... + 94765.591... = 152783.229.... It'll all look like this (bank statement basis): Code: MTH PAYMENT INTEREST BALANCE 0 152,783.23 : PV2 + PV3 1 1,000.00 622.46 152,405.69 ... 119 1,000.00 390.01 95,117.09 120 1,000.00 387.53 94,504.62 : see PV1 121 1,000.00 390.01 93,894.35 : new rate 2.5% s.a. starts ... 239 1,000.00 8.20 995.89 240 1,000.00 4.11 .00  
February 3rd, 2013, 04:15 AM  #3 
Member Joined: Jun 2012 Posts: 64 Thanks: 0  Re: Find the present value
Thanks a lot of helping out! I've managed to solve this using complex annuities and actuarial science symbols. Do you have a program that prints out bank statements? 
February 3rd, 2013, 10:19 AM  #4  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: Find the present value Quote:
If you didn't "follow", then what is your intent? Simply get an answer you can "copy"? Quote:
all it is is print the transactions in some fancy manner... Anyhow, here's my program that fully handles your problem: [square brackets show results or information] p = 1000 : n = 120 i = 1.05^(1/12)  1 [converted annual rate = .0040741...] j = 1.025^(1/6)  1 [converted s.a. rate = .0041239...] pv1 = p*(1  1/(1 + j)^n / j [94504.618...] pv2 = pv1 / (1 + i)^n [58017.638...] pv3 = p*(1  1/(1 + i)^n / i [94765.591...] b = pv2 + pv3 [balance in account at start = 152783.228...] Following does the bank statement printing (m = month): loop m from 1 to 240 if m = 121 then i = j [use s.a. rate from month 121] k = b * i [interest for current month] b = b  p + k [resulting account balance for current month] print m, p, k, b [current month's transaction printed] endloop That's it...if you TRY to understand it, you'll see there's no mystery nowhere  
February 24th, 2013, 09:05 AM  #5 
Member Joined: Jun 2012 Posts: 64 Thanks: 0  Re: Find the present value
Ah thanks a lot! That's some hectic programming! 

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