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### Compound Interest Calculation Scenarios #### Scenario 1: Annual Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded annually? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** #### Scenario 2: Semi-Annual Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded semi-annually? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** #### Scenario 3: Quarterly Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded quarterly? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** ### Analysis **2. Reasonableness Test:** In each of the three scenarios, the present value (PV) should continue to decrease as compounding frequency increases. This occurs because more frequent compounding periods lead to accrued interest being compounded more often, thereby requiring a lower initial investment to reach the same future value. By analyzing the scenarios under the assumption of constant interest rates but varying compounding frequencies, the exercise illustrates the time value of money concept, emphasizing how different compounding intervals impact the amount initially invested to achieve

### Compound Interest Calculation Scenarios #### Scenario 1: Annual Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded annually? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** #### Scenario 2: Semi-Annual Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded semi-annually? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** #### Scenario 3: Quarterly Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded quarterly? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** ### Analysis **2. Reasonableness Test:** In each of the three scenarios, the present value (PV) should continue to decrease as compounding frequency increases. This occurs because more frequent compounding periods lead to accrued interest being compounded more often, thereby requiring a lower initial investment to reach the same future value. By analyzing the scenarios under the assumption of constant interest rates but varying compounding frequencies, the exercise illustrates the time value of money concept, emphasizing how different compounding intervals impact the amount initially invested to achieve

Essentials Of Investments
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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### Compound Interest Calculation Scenarios #### Scenario 1: Annual Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded annually? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** #### Scenario 2: Semi-Annual Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded semi-annually? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** #### Scenario 3: Quarterly Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded quarterly? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** ### Analysis **2. Reasonableness Test:** In each of the three scenarios, the present value (PV) should continue to decrease as compounding frequency increases. This occurs because more frequent compounding periods lead to accrued interest being compounded more often, thereby requiring a lower initial investment to reach the same future value. By analyzing the scenarios under the assumption of constant interest rates but varying compounding frequencies, the exercise illustrates the time value of money concept, emphasizing how different compounding intervals impact the amount initially invested to achieve
Transcribed Image Text:### Compound Interest Calculation Scenarios #### Scenario 1: Annual Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded annually? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** #### Scenario 2: Semi-Annual Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded semi-annually? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** #### Scenario 3: Quarterly Compounding **Problem Statement:** Assume that a deposit made on January 1, 2020, earns 6% interest. The deposit plus interest accumulated to $50,000 on January 1, 2026. How much was invested on January 1, 2020, if interest was compounded quarterly? **Parameters:** - **N (period of time):** - **I (Interest):** - **PV (Present Value):** - **FV (Future Value):** $50,000 - **PMT (Annuity):** ### Analysis **2. Reasonableness Test:** In each of the three scenarios, the present value (PV) should continue to decrease as compounding frequency increases. This occurs because more frequent compounding periods lead to accrued interest being compounded more often, thereby requiring a lower initial investment to reach the same future value. By analyzing the scenarios under the assumption of constant interest rates but varying compounding frequencies, the exercise illustrates the time value of money concept, emphasizing how different compounding intervals impact the amount initially invested to achieve
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