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Question 1185513: The present value of an annuity of R pesos payable annually for 8 years, with the first payment at the end of 10 years, is 187, 481.25. Draw the cash flow diagram and find the value of R if money is worth 5%.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**1. Cash Flow Diagram:**
The cash flow diagram will show the payments (R) occurring at the end of each year, but *shifted* by 9 years. Here's a representation:
```
Time (Years): 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Cash Flow: 0 0 0 0 0 0 0 0 0 0 R R R R R R R R
```
**2. Present Value Calculation:**
The present value of the annuity is given as 187,481.25. Since the first payment occurs at the end of year 10, we first calculate the present value of the annuity *as if it started at the end of year 9*. Then, we discount that value back to the present (time 0).
* **Present Value of Annuity (at end of year 9):** Let's call this PV₉. The formula for the present value of an ordinary annuity is:
PV = R * [1 - (1 + i)^-n] / i
where:
* PV is the present value
* R is the payment amount (what we want to find)
* i is the interest rate (5% or 0.05)
* n is the number of periods (8 years)
So, PV₉ = R * [1 - (1.05)^-8] / 0.05
* **Discounting back to the present (time 0):** We now treat PV₉ as a single future value and discount it back 9 years to time 0:
PV₀ = PV₉ / (1 + i)^9
where PV₀ is the present value at time 0, which is given as 187,481.25.
**3. Solving for R:**
Now we have the equation:
187,481.25 = {R * [1 - (1.05)^-8] / 0.05} / (1.05)^9
We need to solve for R. Here's the step-by-step process:
1. Simplify the annuity part: [1 - (1.05)^-8] / 0.05 ≈ 6.7665
2. Simplify the discounting part: (1.05)^9 ≈ 1.5513
3. Rewrite the equation: 187,481.25 = (R * 6.7665) / 1.5513
4. Multiply both sides by 1.5513: 187,481.25 * 1.5513 = R * 6.7665
5. Divide both sides by 6.7665: R = (187,481.25 * 1.5513) / 6.7665
6. Calculate R: R ≈ 42,999.99 or approximately 43,000
**Therefore, the value of R is approximately 43,000 pesos.**
Answer by ikleyn(52884)  (Show Source):
You can put this solution on YOUR website! .
I agree with @CPhill on how he sets up the equation,
but his calculations in his post to find R are not accurate.
I came to provide correct calculations.
So, I placed here only part 3 of the @CPhill solution
with my corrected/updated calculations.
**3. Solving for R:**
Now we have the equation:
187,481.25 = {R * [1 - (1.05)^-8] / 0.05} / (1.05)^9
We need to solve for R. Here's the step-by-step process:
1. Simplify the annuity part: [1 - (1.05)^-8] / 0.05 = 6.463212759 (compare with the value of 6.7665 by @CPhill)
2. Simplify the discounting part: (1.05)^9 = 1.551328216
3. R = (187481.25 * 1.551328216) / 6.463212759 = 45,000 pesos
Therefore, the value of R is approximately 45,000 pesos.
Solved/recalculated correctly.
Compare my 45000 pesos with the @CPhill's 43000 pesos.
I think, in Finance, such an error/(a discrepancy) is not allowed.
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Artificial intelligence, which can not produce correctly simple arithmetical calculations ?
It is just something unprecedented.
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