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February 2nd, 2013, 08:13 AM   #1
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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.
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February 2nd, 2013, 10:30 AM   #2
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Re: Find the present value

Quote:
Originally Posted by Keroro
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.
P = 1000, n=120 (months)

Step 1: calculate the present value as at 10th year of the monthly annuity of $1000 payable
over the next 10 years @ 2.5 semi-annually.
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
You're lucky the Ottawa Senators are not playing hockey this afternoon!
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February 3rd, 2013, 04:15 AM   #3
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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?
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February 3rd, 2013, 10:19 AM   #4
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Re: Find the present value

Quote:
Originally Posted by Keroro
Thanks a lot of helping out! I've managed to solve this using complex annuities
and actuarial science symbols.
Yikes...why? All you need is 2 basic financial formulas as I've shown on my post...
If you didn't "follow", then what is your intent? Simply get an answer you can "copy"?

Quote:
Do you have a program that prints out bank statements?
"Bank statement printing" is more a routine than a program:
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
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February 24th, 2013, 09:05 AM   #5
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Re: Find the present value

Ah thanks a lot! That's some hectic programming!
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