**Lecture Notes: Future Value (FV)**

**1. Understanding Future Value (FV)**

**Definition:** Future Value (FV) represents the amount of money an investment or sum of money will grow to over a specified period at a given interest rate. It allows us to determine how much a present sum of money will be worth in the future, considering the effects of interest or growth.

**Formula:** FV = PV × (1 + r)^n

**FV**: Future Value**PV**: Present Value (the initial amount of money)**r**: Interest Rate per period**n**: Number of periods

**Purpose:** The FV formula helps in understanding how investments or savings will grow over time due to interest or investment returns. It is crucial for financial planning, investment decisions, and assessing the future worth of financial assets.

**2. Components of the FV Formula**

**Present Value (PV):**The current value of the sum of money or investment.**Interest Rate (r):**The rate at which the money will grow per period. It could be expressed as a decimal (e.g., 5% as 0.05).**Number of Periods (n):**The number of time periods (such as years, months) the money is invested or borrowed.

**3. Example Calculation**

Suppose you have $5,000 today, and you want to invest it in a savings account that offers an annual interest rate of 4% for 10 years. To find out how much this investment will be worth in 10 years, use the FV formula.

**Given:**

**PV**= $5,000**r**= 0.04 (4% annual interest rate)**n**= 10 years

**Calculation:**

FV = 5000 × (1 + 0.04)^10

FV = 5000 × (1.04)^10

FV = 5000 × 1.48024

FV = 7401.20

After 10 years, the investment will grow to $7,401.20.

**Interactive Scenario Example**

**Scenario: Retirement Savings**

**Scenario Description:**

Imagine you are planning for your retirement and want to determine how much your current savings will grow over time with a certain interest rate. You plan to invest $20,000 in a retirement fund that earns an annual interest rate of 6%, compounded yearly, for 25 years.

**Steps to Analyze Using Modern Techniques:**

**Input Variables:****Present Value (PV)**: $20,000**Interest Rate (r)**: 6% or 0.06**Number of Periods (n)**: 25 years

**Use Interactive Tools:****Future Value Calculator:**Online calculators or financial software (like Excel or Google Sheets) can be used to input these values and compute FV.

**Calculation:**Using an online FV calculator or Excel:**Formula in Excel/Google Sheets:**`=FV(0.06, 25, 0, -20000)`

**Result:**The future value will be approximately $85,771.97.

**Impact of Modern Techniques:** Modern tools such as financial calculators and spreadsheet software make it easy to perform FV calculations without manual computation, allowing for quick adjustments and scenario analysis.

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