Question
suppose that the following group of values was entered into the TVM solver of a graphing calculator: N = 240; 1% = 15.6; PV = 75000; PMT = -1021.0001; FV = 0; P/Y = 12; C/Y = 12; PMT : END. also suppose that the "bal(" function was used as follows: bal(24). which of these expressions is equivalent to the value returned by the "bal(" function?
Answers
GPT-4o mini
The "bal(" function in a TVM (Time Value of Money) solver typically calculates the balance or remaining value of an investment or loan at a specific period \( N \). In this case, the solver is set with the following parameters:
- **N = 240** (total number of periods)
- **I/Y = 1%** (interest rate per period)
- **PV = 75000** (present value, or the initial amount borrowed or invested)
- **PMT = -1021.0001** (payment made each period)
- **FV = 0** (future value at the end of the number of periods)
- **P/Y = 12** (payments per year)
- **C/Y = 12** (compounding periods per year)
- **PMT: END** (annuity payments made at the end of the period)
Given that "bal(24)" is used, it is referring to the balance after 24 periods. To determine the equivalent expression returned by the "bal(24)" function, we can use the formula for the future value of annuities and account for the present value or initial investment.
The balance at period 24 can be represented by the remaining balance after considering the payments and the interest accrued. The calculation can generally be represented as follows:
\[
\text{Balance at } n = \text{PV} \times (1 + r)^n + PMT \times \left( \frac{(1 + r)^n - 1}{r} \right)
\]
Where:
- \( n \) = number of periods
- \( r \) = interest rate per period (1% or 0.01 in decimal)
- \( PMT \) = payment per period (since it is a payment to reduce debt, it is negative)
For \( n = 24 \):
- \( r = 0.01 \)
- \( PV = 75000 \)
- \( PMT = -1021.0001 \)
This becomes:
\[
\text{Balance at } 24 = 75000 \times (1 + 0.01)^{24} + (-1021.0001) \times \left( \frac{(1 + 0.01)^{24} - 1}{0.01} \right)
\]
Therefore, the expression equivalent to the value returned by the "bal(24)" function is derived from this calculation.
Make sure to correctly plug in the values into the equation to compute or further reduce the expression as necessary when performing this calculation.
- **N = 240** (total number of periods)
- **I/Y = 1%** (interest rate per period)
- **PV = 75000** (present value, or the initial amount borrowed or invested)
- **PMT = -1021.0001** (payment made each period)
- **FV = 0** (future value at the end of the number of periods)
- **P/Y = 12** (payments per year)
- **C/Y = 12** (compounding periods per year)
- **PMT: END** (annuity payments made at the end of the period)
Given that "bal(24)" is used, it is referring to the balance after 24 periods. To determine the equivalent expression returned by the "bal(24)" function, we can use the formula for the future value of annuities and account for the present value or initial investment.
The balance at period 24 can be represented by the remaining balance after considering the payments and the interest accrued. The calculation can generally be represented as follows:
\[
\text{Balance at } n = \text{PV} \times (1 + r)^n + PMT \times \left( \frac{(1 + r)^n - 1}{r} \right)
\]
Where:
- \( n \) = number of periods
- \( r \) = interest rate per period (1% or 0.01 in decimal)
- \( PMT \) = payment per period (since it is a payment to reduce debt, it is negative)
For \( n = 24 \):
- \( r = 0.01 \)
- \( PV = 75000 \)
- \( PMT = -1021.0001 \)
This becomes:
\[
\text{Balance at } 24 = 75000 \times (1 + 0.01)^{24} + (-1021.0001) \times \left( \frac{(1 + 0.01)^{24} - 1}{0.01} \right)
\]
Therefore, the expression equivalent to the value returned by the "bal(24)" function is derived from this calculation.
Make sure to correctly plug in the values into the equation to compute or further reduce the expression as necessary when performing this calculation.