Time Value of Money: Compounding and Future Value

Recap: Previously discussed Time Value of Money (TVM), basic definitions, and began Future Value (FV) calculations (also known as compounding).

Simple Interest Definition: Interest earned only on the original principal amount.
- Characteristic: The amount of interest remains constant over time because the principal amount does not change.
- Example: A 1,000 deposit at a simple interest rate of 10\% means an annual interest of 10\% \times \$1,000 = \$100 every year, with no change in the interest amount.
Compound Interest Definition: Interest earned on both the original principal and on any accumulated interest received.
- Characteristic: The principal amount effectively changes each year to a new balance, as previously earned interest is reinvested and also earns interest (interest on interest).
Comparative Example: Calculating Future Value of 100 invested at 10\% for 2 years.
- Simple Interest:
  - Annual interest: 10\% \times \$100 = \$10.
  - Year 1 interest: 10
  - Year 2 interest: 10
  - Total FV: Principal + Total Interest = \$100 + \$10 + \$10 = \$120.
- Compound Interest (using formula):
  - Formula: FV = PV \times (1 + r)^t
  - Calculation: \$100 \times (1 + 0.10)^2 = \$100 \times (1.1)^2 = \$100 \times 1.21 = \$121.
- Difference: There is a \$1 difference (\$121 - \$120 = \$1). This extra \$1 comes from the interest earned on the first year's interest (\$10 \times 10\% = \$1). The new beginning balance for the second year using compounding is \$110. This demonstrates the power of compounding.

Effectiveness: Compounding significantly increases wealth accumulation, especially over longer investment horizons.
Early Investment: Many investment experts advocate for starting investments early (in younger years) to maximize the benefits of compounding. This allows more time for the balance to grow and potentially recover from losses.
Risk and Age:
- Younger investors can afford to take more risk because they have sufficient time to recover from potential losses.
- Older investors (e.g., 50s, 60s) generally cannot afford significant losses, as there is less time to recover, making risk-taking less advisable.
Historical Market Trends:
- Historically, the stock market in the United States has shown an upward trend over time (e.g., S&P 500 index).
- While there are fluctuations, the overall trend suggests consistent growth.
- This trend provides a rationale for long-term stock investment and suggests investing in broader market index funds for a safer approach.

General Formula:
- FV = PV \times (1 + r)^t
- Where:
  - FV = Future Value
  - PV = Present Value
  - r = Interest Rate (in decimal form)
  - t = Number of Periods
Excel Spreadsheet: Utilize the FV function.
- Syntax: =FV(rate, nper, pmt, pv, type).
- For lump sum calculations (Chapter 4), the pmt (payment) argument is set to 0 or left blank; pv is often entered as a negative number to represent a cash outflow/investment.
Time Value of Money (TVM) Table: Provides approximate future value interest factors (FVIF).
- To use: Multiply the present value by the corresponding FVIF from the table.
- Limitation: Not always suitable for interest rates with decimals (e.g., 5.5\%) as tables typically provide factors only for whole percentages.
Online Financial Calculator: Enter present value (often as a negative cash outflow), annual rate, number of periods, and ensure compounding is set to 'annual' (default 'end mode' for cash flows).
- Note on Present Value Input: For investments (cash outflows), the present value should often be entered as a negative number in financial calculators to ensure the future value is a positive cash inflow.

Investment of \$100 for 5 years at 10\% annual interest:
- Using formula: FV = \$100 \times (1 + 0.10)^5 = \$100 \times 1.61051 = \$161.05.
- Using TVM table (for 10\% and 5 years): The multiplier is 1.61051. \$100 \times 1.61051 = \$161.05.
- Using online calculator: PV = -100, Rate = 10, Periods = 5, FV = \$161.05.
Relative's deposit of \$10 at 5.5\% interest, 200 years ago:
- This is still a future value question: How much is that investment worth today?
- PV = \$10, r = 5.5\% = 0.055, t = 200 years.
- Using formula: FV = \$10 \times (1 + 0.055)^{200} = \$447,108.84.
- Cannot use TVM table due to the 5.5\% interest rate not being in typical tables.
- Rule of 72: Used to estimate the number of years it takes for an investment to double.
  - Formula: \text{Years to double} = 72 / \text{interest rate (as a whole number)}.
  - Example: If interest rate is 3\%, years to double = 72/3 = 24 years. If 6\%, years to double = 72/6 = 12 years.
Widget Sales Growth: Company expects unit sales to increase by 15\% per year for 5 years; currently sells 3,000,000 widgets.
- Treat sales growth rate as the interest rate (r = 0.15).
- Current sales are the present value (PV = 3,000,000 widgets).
- Number of periods (t = 5 years).
- Using formula: FV = 3,000,000 \times (1 + 0.15)^5 = 3,000,000 \times 2.011357 = 6,034,071.50 \text{ widgets}.
- The key is to correctly identify the components (PV, r, t) from the problem description.

Relationship 1: Time Period (t)
- For a given interest rate (r), the longer the time period (t), the higher the future value (FV).
- As t increases, FV increases.
Relationship 2: Interest Rate (r)
- For a given time period (t), the higher the interest rate (r), the higher the future value (FV).
- As r increases, FV increases.
Synergy: The power of compounding is most evident with longer time periods and higher interest rates. These relationships are directly observable from the future value formula.

FV Function: Calculates Future Value (=FV(rate, nper, pmt, pv, type)).
PV Function: Calculates Present Value (=PV(rate, nper, pmt, fv, type)).
RATE Function: Calculates the interest rate (=RATE(nper, pmt, pv, fv, type, guess)).
NPER Function: Calculates the number of periods (=NPER(rate, pmt, pv, fv, type)).
Formula Icon: Use the 'formula icon' in Excel if you cannot recall the exact function syntax; it provides a list of functions and their arguments.

Definition: Present value is the current value (value at time 0 on a timeline) of a future cash flow discounted at an appropriate discount rate.
Core Question: PV answers questions like:
- "How much do I have to invest today to have a certain amount in the future?"
- "What is the current worth of an investment to be received at a later date?"
Application: PV calculations are highly useful for setting financial goals (e.g., determining the lump sum amount to set aside today to reach a future financial target).
Distinction (Chapter 4 vs. 5):
- Chapter 4: Focuses exclusively on single cash flows (lump sums) for both future and present value calculations.
- Chapter 5: Will introduce the concept of annuities, which deal with regular, periodic cash flows (e.g., monthly or quarterly savings).
Conceptualization: Think of present value as the "price tag today" for money you will receive in the future.