Interest and Interest Rates

Interest Basics

Cost of borrowing money or the return on invested capital: compensating for forgoing the use of your money now

$$ F = P + I\newline F = P(1+i) \text{, where i is the interest rate for one period} $$

Compound vs Simple Interest

• Simple Interest: interest is earned only on the original principal
  • $I_s = P \cdot i \cdot N$
• Compound Interest: interest is earned on both the principal and any previously accumulated interest:
  • $F = P(1 + i)^N$

Effective vs Nominal Interest Rates

The more frequently interest is compounded, the higher the effective interest rate will be compared to the nominal rate. A payday loan charging 5% per week has a nominal rate of 260% per year but a devastating effective annual rate of over 1100%

• Nominal Interest Rate: r, the stated annual rate before accounting for compounding within the year (i..e, 12% per year)
• Compounding Period: m, how often interest is calculated and added to the principal (annually, quarterly, monthly, daily)
• Interest per Sub-period: $i_s = \frac{r}{m}$
• Effective Interest Rate: $i_e = (1 + i_s)^m - 1 = (1 + r/m)^m - 1$
  • The actual interest rate you earn or pay, after accounting for compounding

Continuous Compounding & Cash Flow Diagrams

Continuous Compounding

• Theoretical limit of compounding, where the number of periods m approaches infinity
  • Instead of using the discrete formula, we use the natural exponential constant e
• Effective interest rate formula: $i_e = e^r - 1$
  • A nominal rate of 40% compounded continuous results in an effective rate of $e^0.4 - 1 ≈ 0.492$ or 49.2%

Cash Flow Diagrams

Visual tool used to represent the timing and magnitude of all cash inflows (receipts) and outflows (disbursements) in a project or financial decision

• Time is represented on the x-axis