Chapter 6- Interest Rates

📘 INTEREST RATE BASICS

Interest Rate 📘

The price paid for the use of borrowed money; expressed as a percentage of the principal per year.

Simple Interest 📘

Interest earned only on the original principal; formula 🧮 I = P × r × t.

Compound Interest 📘

Interest earned on both principal and previously earned interest; formula 🧮 FV = PV × (1 + r/m)^(m×t).

Nominal (Quoted) Interest Rate 📘

The stated annual rate before compounding adjustments.

Real Interest Rate 📘

Nominal rate adjusted for inflation; 💡 Real = Nominal – Inflation rate.

Inflation Rate 📘

The percentage increase in the general price level of goods and services.

Interest Rate Components 📈

Real rate + Inflation premium + Risk premiums (default, liquidity, maturity).

Discount Rate 📘

The rate used to calculate the present value (PV) of future cash flows.

Time Value of Money 📈

The concept that a dollar today is worth more than a dollar tomorrow due to earning potential.

🧮 APR & EAR

Annual Percentage Rate (APR) 🧮

The nominal interest rate per year without considering compounding.

Effective Annual Rate (EAR) 🧮

The true annual rate accounting for compounding; formula = (1 + r/m)^m – 1.

APR vs EAR 💡

APR understates true interest when compounding > once per year; EAR shows actual return.

Comparing Loans 📈

Always compare using EAR to reflect compounding effects accurately.

EAR Example 💡

12% APR, monthly compounding → EAR = (1 + 0.12/12)^12 – 1 = 12.68%.

Continuous Compounding 🧮

EAR = e^r – 1; uses Euler’s number for infinite compounding periods.

💡 COMPOUNDING & CONVERSIONS

Compounding Frequency 📘

Number of times interest is added per year (annual, quarterly, monthly, etc.).

More Frequent Compounding 💡

Increases EAR and total interest earned.

Periodic Rate 🧮

r_period = APR / m (rate per compounding period).

Future Value (FV) 📈

FV = PV × (1 + r/m)^(m×t); measures growth over time.

Present Value (PV) 📈

PV = FV / (1 + r/m)^(m×t); discounts future cash back to today.

Continuous Compounding Formula 🧮

FV = PV × e^(r×t).

Nominal to Periodic Conversion 🧮

r_period = Nominal Rate / m.

Periodic to Effective Conversion 🧮

EAR = (1 + r_period)^m – 1.

Annualizing Short-Term Rates 📈

Annual Rate ≈ Periodic Rate × (Periods per Year).

Daily Compounding 💡

Use m = 365 for typical bank savings calculations.

📈 TIME VALUE OF MONEY APPLICATIONS

Future Value of an Annuity 📘

FV = PMT × [(1 + r/m)^(m×t) – 1] / (r/m).

Present Value of an Annuity 📘

PV = PMT × [1 – (1 + r/m)^(-m×t)] / (r/m).

Perpetuity 📘

Stream of equal payments forever; PV = PMT / r.

Growing Perpetuity 📘

Payments grow by g each period; PV = PMT / (r – g).

Loan Amortization 📈

Process of repaying a loan in fixed payments including interest + principal.

Interest Portion 📘

Interest = Beginning Balance × Rate per period.

Principal Portion 📘

Payment – Interest = Principal Reduction.

Discounting 💡

Finding the PV of future sums using a discount rate.

Compounding 💡

Finding the FV of a present sum using a growth rate.

Rule of 72 💡

Approximation for doubling time = 72 / Annual Rate (%).

💰 DETERMINANTS OF INTEREST RATES

Nominal Interest Rate Components 📈

Nominal = Real + Inflation + Risk premiums.

Real Risk-Free Rate (r*) 📘

Rate with no inflation or risk; baseline return on pure time value of money.

Inflation Premium (IP) 📘

Compensation for expected inflation over the term.

Default Risk Premium (DRP) 📘

Added rate for potential borrower default.

Liquidity Premium (LP) 📘

Added rate for less easily tradable securities.

Maturity Risk Premium (MRP) 📘

Compensation for longer maturities and greater interest rate risk.

Treasury Security Rate Formula 🧮

r_T = r* + IP + MRP.

Corporate Bond Rate Formula 🧮

r_C = r* + IP + DRP + LP + MRP.

Term Structure 📘

Relationship between interest rates and maturities for securities of equal risk.

Real vs Nominal Rate Example 💡

If nominal = 9% and inflation = 3%, real = 6%.

📊 YIELD CURVE & THEORIES

Yield Curve 📈

Graph showing relationship between bond yields and maturities.

Normal Yield Curve 📘

Upward sloping; long-term rates > short-term rates due to inflation & risk.

Inverted Yield Curve 📘

Downward sloping; indicates possible economic slowdown or recession.

Flat Yield Curve 📘

Minimal difference between short- and long-term yields.

Term Structure Theories 📘

Explain shape of yield curve (expectations, liquidity, segmentation).

Pure Expectations Theory 📘

Long-term rates = average of current + expected future short-term rates.

Liquidity Premium Theory 📘

Investors demand extra yield for longer maturities → curve slopes upward.

Market Segmentation Theory 📘

Curve shape depends on supply/demand in separate maturity markets.

Expectations Example 💡

If investors expect higher future rates, yield curve slopes upward.

Yield Curve Signals 📈

Upward = growth/inflation ahead; Inverted = recession warning.

💡 MISC. RELATIONSHIPS

Inflation & Interest 📈

Higher expected inflation → higher nominal rates.

Risk & Return 📈

Higher risk → higher required interest rate (compensation).

Loan Types 💰

Discount loan (interest deducted upfront), interest-only loan, amortized loan.

APR on Credit Cards 💳

Reflects nominal cost of borrowing; compounding makes true cost higher (EAR).

Continuous Compounding Insight 💡

Useful for advanced finance models and exponential growth.

Nominal vs Real Insight 💡

Nominal measures face return; Real measures purchasing power.

Time Value Summary 📈

PV ↓ as r or t ↑; FV ↑ as r or t ↑.