SOLUTION: A sequence of semi-annual payments of P2,500 will start at the end of 3 years and the last at the end of 9 years. If money is worth 12.75% compounded semi-annually, find the sum of
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Question 1193987: A sequence of semi-annual payments of P2,500 will start at the end of 3 years and the last at the end of 9 years. If money is worth 12.75% compounded semi-annually, find the sum of these values
a. At the end of 2 years
b. At the end of 5 years
Found 2 solutions by parmen, ikleyn:
Answer by parmen(42) (Show Source): You can put this solution on YOUR website!
**1. Determine the key parameters:**
* **Payment amount (PMT):** P2,500
* **Interest rate per period:** 12.75% per year compounded semi-annually, so the periodic interest rate is 12.75% / 2 = 6.375% or 0.06375
* **Number of periods:**
* **Total periods:** 9 years * 2 periods/year = 18 periods
* **Periods to the start of the annuity:** 3 years * 2 periods/year = 6 periods
* **Periods within the annuity:** 18 periods - 6 periods = 12 periods
**2. Calculate the present value of the annuity at the start of the annuity:**
We can use the formula for the present value of an ordinary annuity:
* **Present Value (PV) = PMT * [(1 - (1 + r)^-n) / r]**
Where:
* PMT = payment amount
* r = periodic interest rate
* n = number of periods
* PV = 2500 * [(1 - (1 + 0.06375)^-12) / 0.06375]
* PV ≈ 2500 * 7.0462
* PV ≈ 17,615.50
**3. Calculate the present value of the annuity at the end of 2 years:**
* **Periods to the end of 2 years:** 2 years * 2 periods/year = 4 periods
* **Present Value at the end of 2 years = PV at the start of the annuity / (1 + r)^number of periods**
* Present Value at the end of 2 years = 17,615.50 / (1 + 0.06375)^4
* Present Value at the end of 2 years ≈ 17,615.50 / 1.2838
* Present Value at the end of 2 years ≈ 13,708.68
**4. Calculate the present value of the annuity at the end of 5 years:**
* **Periods to the end of 5 years:** 5 years * 2 periods/year = 10 periods
* **Periods between the end of 5 years and the start of the annuity:** 10 periods - 6 periods = 4 periods
* **Present Value at the end of 5 years = PV at the start of the annuity / (1 + r)^number of periods**
* Present Value at the end of 5 years = 17,615.50 / (1 + 0.06375)^4
* Present Value at the end of 5 years ≈ 17,615.50 / 1.2838
* Present Value at the end of 5 years ≈ 13,708.68
**Therefore:**
* **a. At the end of 2 years, the sum of these values is approximately P13,708.68.**
* **b. At the end of 5 years, the sum of these values is also approximately P13,708.68.**
**Note:** These calculations assume that the interest rate remains constant throughout the entire period.
Answer by ikleyn(52908) (Show Source): You can put this solution on YOUR website!
.
As I read this problem formulation, I see FATAL DEFICIENCIES in its formulation.
From the problem's description, it is unclear, if it is an Ordinary Annuity, which accumulates
deposits making regular compounding semi-annually, or it is a Sinking Fund, which provides
outpayments semiannually. From the text, it is impossible to determine/(to distinct),
which version is considered.
Second note regarding the formulation: it asks "find the sum of these values".
- Find the sum of WHICH values ? In problems of this kind, a question is NEVER posed this way.
If an Ordinary Annuity is considered, then normally the problem should ask to find the future value.
So, my conclusion regarding the problem formulation is that its formulation is FATALLY ILLITERATE.
It is not appropriate form for a Math problem, and is not appropriate form to post to the forum.
Now, about the solution.
The "tutor" @parmen considers it as an Ordinary Annuity, but for some reason, unknown to me,
calculates the Present Value instead of the Future Value, which makes few sense (if any).
In this situation, the solution normally should calculate a Future Value.
Besides of this, in the solution by @parmen, there is a systematic error in all calculations.
From the end of the 3-rd year to the end of the 9-th year, there are
2*(9-3)+1 = 2*6+1 = 13 semi-annual payments - not 12.
From the end of the 3-rd year to the end of the 5-th year, there are
2*(5-3)+1 = 2*2+1 = 5 semi-annual payments - not 4.
Regarding the solution, I think that is not appropriate for a human-tutor or for an AI-tutor
to treat such problem without adequate explicit comments/corrections of its formulation,
as well as it is not appropriate to make such errors.
Again, my conclusion after writing these my notes is that the problem formulation is illiterate
and the solution treats the problem incorrectly from pedagogical point of vies and is also INCORRECT in its essence.
////////////////////////
I have a mathematical education (MS in Math) from renowned university,
PhD degree in Physics and Mathematics, and I worked internationally
and in the US for many years as a computer engineer in Civil Engineering.
Our major everyday task/duty was performing/supporting numerical simulation for complex phenomena
in Civil Engineering (from conductive heat transfer, phase changes, natural and forced convection, radiation heat exchange
to stability of engineering constructions, including hydrodynamics, liquid sloshing, elastic stress and strain in solids,
vibrations, fluid-structure interactions etc.)
Our other everyday task/duty was to make mutual cross-checking to calculations of other colleagues.
I can say, based on my experience, that with such errors, that AI makes every day at this forum, such a colleague
would be fired in the next day - - - because other colleagues and consumers would ask the management to do it.
I say these words not to offend you personally.
In opposite, working on your posts and checking your solutions, I am on your side and work for your benefits.
I simply want to correctly and adequately describe you the current state/level/status of your artificial intelligence.
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