You have decided to buy a car, the price of the car is $18,000. The car dealer presents you with two choices: Purchase the car for cash and receive $2000 instant cash rebate – your out of pocket expense is $16,000 today. Purchase the car for $18,000 with zero percent interest 36-month loan with monthly payments. The market interest rate is 4%. Which of the option above is cheaper? How much do you save? Formula attached
You have decided to buy a car, the price of the car is $18,000. The car dealer presents you with two choices: Purchase the car for cash and receive $2000 instant cash rebate – your out of pocket expense is $16,000 today. Purchase the car for $18,000 with zero percent interest 36-month loan with monthly payments. The market interest rate is 4%. Which of the option above is cheaper? How much do you save? Formula attached
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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- You have decided to buy a car, the price of the car is $18,000. The car dealer presents you with two choices:
- Purchase the car for cash and receive $2000 instant cash rebate – your out of pocket expense is $16,000 today.
- Purchase the car for $18,000 with zero percent interest 36-month loan with monthly payments.
The market interest rate is 4%. Which of the option above is cheaper? How much do you save?
Formula attached
![### Future and Present Value
1. **\[ FV = C(1 + r)^T \]**
- Future Value (FV) is calculated using the initial investment (C) and the rate of return (r) over time (T).
2. **\[ PV = \frac{D}{(1+r)^T} \]**
- Present Value (PV) is derived from future payment (D) discounted back by rate (r) over time (T).
3. **\[ FV = PV(1 + r)^T \]**
- An alternative representation of Future Value derived from Present Value increased by rate (r) over time (T).
4. **\[ r = \left(\frac{FV}{PV}\right)^{\frac{1}{T}} - 1 \]**
- Calculation of interest rate (r) based on Future Value, Present Value over time.
5. **\[ T = \frac{\ln\left(\frac{FV}{PV}\right)}{\ln(1+r)} \]**
- Duration (T) can be extracted using logarithms of the ratio of Future to Present Value and rate.
### Annuity
1. **\[ PV = \frac{pmt}{r} \left[ 1 - \frac{1}{(1+r)^T} \right] \]**
- Present Value of an annuity is based on regular payment (pmt) adjusted for interest rate and period.
2. **\[ pmt = \frac{PV \cdot r}{1 - \frac{1}{(1+r)^T}} \]**
- The annuity payment calculation from Present Value.
3. **\[ T = \frac{\ln(pmt) - \ln(pmt-PV \cdot r)}{\ln(1+r)} \]**
- Determines the period (T) by evaluating payment dynamics and rate.
4. **\[ FV = \frac{pmt}{r} \left[(1 + r)^T - 1\right] \]**
- Future Value of annuity from regular payments at interest rate over time.
### Annuity Due
5. **\[ PV = \frac{pmt}{r} \left[ 1 - \frac{1}{(1+r)^T} \right](1 + r) \]**
- Present Value for annuity due](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F10c7525c-c416-4320-97b0-471febd6b171%2F285d95a7-aa3a-4423-9d9f-560a4f7ef622%2F920owc8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Future and Present Value
1. **\[ FV = C(1 + r)^T \]**
- Future Value (FV) is calculated using the initial investment (C) and the rate of return (r) over time (T).
2. **\[ PV = \frac{D}{(1+r)^T} \]**
- Present Value (PV) is derived from future payment (D) discounted back by rate (r) over time (T).
3. **\[ FV = PV(1 + r)^T \]**
- An alternative representation of Future Value derived from Present Value increased by rate (r) over time (T).
4. **\[ r = \left(\frac{FV}{PV}\right)^{\frac{1}{T}} - 1 \]**
- Calculation of interest rate (r) based on Future Value, Present Value over time.
5. **\[ T = \frac{\ln\left(\frac{FV}{PV}\right)}{\ln(1+r)} \]**
- Duration (T) can be extracted using logarithms of the ratio of Future to Present Value and rate.
### Annuity
1. **\[ PV = \frac{pmt}{r} \left[ 1 - \frac{1}{(1+r)^T} \right] \]**
- Present Value of an annuity is based on regular payment (pmt) adjusted for interest rate and period.
2. **\[ pmt = \frac{PV \cdot r}{1 - \frac{1}{(1+r)^T}} \]**
- The annuity payment calculation from Present Value.
3. **\[ T = \frac{\ln(pmt) - \ln(pmt-PV \cdot r)}{\ln(1+r)} \]**
- Determines the period (T) by evaluating payment dynamics and rate.
4. **\[ FV = \frac{pmt}{r} \left[(1 + r)^T - 1\right] \]**
- Future Value of annuity from regular payments at interest rate over time.
### Annuity Due
5. **\[ PV = \frac{pmt}{r} \left[ 1 - \frac{1}{(1+r)^T} \right](1 + r) \]**
- Present Value for annuity due
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