Question
which of these groups of values plugged into the TVM solver of a graphing calculator will return the same value for PV as the expression ($365)((1 + 0.002)^36 - 1) / (0.002)(1 + 0.002)^36 ?
Answers
GPT-4o mini
To find equivalent values that would give the same Present Value (PV) in the formula, we start by analyzing the expression provided:
\[
PV = \frac{365 \cdot ((1 + 0.002)^{36} - 1)}{0.002 \cdot (1 + 0.002)^{36}}
\]
Let's simplify this expression step-by-step:
1. The term \((1 + 0.002)^{36}\): This can be calculated directly.
2. The entire expression can be understood as being derived from some annuity formula for future value or present value.
Given that you want to find groups of values that yield the same PV when plugged into the TVM solver, it’s helpful to understand the TVM (Time Value of Money) inputs:
- **N**: Number of periods (in your case, 36)
- **I/Y**: Interest rate per period (0.2% or 0.002)
- **PMT**: Payment per period (which in this case seems to relate to the numerator of the PV formula)
- **FV**: Future value (though not clearly defined in the expression)
- **PV**: Present value (what we want to calculate)
- **FV**: Future Value you would get when PV is compounded.
Your expression appears to be rearranging parts of an annuity calculation into a present value result.
From the provided expression, it's apparent that the equivalent values for the PV can be found by maintaining the relationship between the payment (PMT) and the interest rate while adjusting the number of periods (N) or equivalent PMT values to keep the balance of the equation equal.
Here are possible alternative groups to consider:
1. Change **N** while adjusting **PMT** to maintain balance.
2. Adjust **I/Y** relative to **N** or **PMT** similarly, ensuring the overall relationship holds true.
To proceed, without specific value groups provided to choose from, you’d typically find that all these factors maintain the identity of the equation:
- Any combination that maintains this relationship while keeping the product related to 365 as a scalar (or monthly/annual equivalent relationships) will maintain equivalent values.
To find equivalent sets, you can use numerical iterative techniques or use a graphing calculator to directly compute PV for different N, I/Y values while adjusting PMT accordingly.
If there are specific sets of values you want to compare, please provide them, and I can verify which ones maintain that same Present Value.
\[
PV = \frac{365 \cdot ((1 + 0.002)^{36} - 1)}{0.002 \cdot (1 + 0.002)^{36}}
\]
Let's simplify this expression step-by-step:
1. The term \((1 + 0.002)^{36}\): This can be calculated directly.
2. The entire expression can be understood as being derived from some annuity formula for future value or present value.
Given that you want to find groups of values that yield the same PV when plugged into the TVM solver, it’s helpful to understand the TVM (Time Value of Money) inputs:
- **N**: Number of periods (in your case, 36)
- **I/Y**: Interest rate per period (0.2% or 0.002)
- **PMT**: Payment per period (which in this case seems to relate to the numerator of the PV formula)
- **FV**: Future value (though not clearly defined in the expression)
- **PV**: Present value (what we want to calculate)
- **FV**: Future Value you would get when PV is compounded.
Your expression appears to be rearranging parts of an annuity calculation into a present value result.
From the provided expression, it's apparent that the equivalent values for the PV can be found by maintaining the relationship between the payment (PMT) and the interest rate while adjusting the number of periods (N) or equivalent PMT values to keep the balance of the equation equal.
Here are possible alternative groups to consider:
1. Change **N** while adjusting **PMT** to maintain balance.
2. Adjust **I/Y** relative to **N** or **PMT** similarly, ensuring the overall relationship holds true.
To proceed, without specific value groups provided to choose from, you’d typically find that all these factors maintain the identity of the equation:
- Any combination that maintains this relationship while keeping the product related to 365 as a scalar (or monthly/annual equivalent relationships) will maintain equivalent values.
To find equivalent sets, you can use numerical iterative techniques or use a graphing calculator to directly compute PV for different N, I/Y values while adjusting PMT accordingly.
If there are specific sets of values you want to compare, please provide them, and I can verify which ones maintain that same Present Value.