Understanding how money's value changes over time through Napoleon's rose promise and the power of compound interest
On March 19, 1797, during a speech at the Luxembourg National Primary School, Napoleon Bonaparte made a seemingly simple promise to the people of Luxembourg. He pledged to send a bouquet of roses of equal value each year as a symbol of the friendship between France and Luxembourg. However, due to wars and political events, this promise was forgotten.
In 1984, Luxembourg formally requested that France fulfill this promise or provide compensation. The calculation method proposed was striking: using 3 Louis as the principal amount, compounded at 5% annual interest from 1797 onward.
The result was astonishing - after 187 years of compound interest, the amount reached 11,003,550 francs (equivalent to approximately €1.7 million in today's value). Each Louis was worth 20 francs, making the final calculation particularly impressive.
Classic Case Revelation: The Power of Compound Interest
This story perfectly demonstrates the magical power of compound interest - a small amount of money, after 187 years of compound growth, produced an astonishing time value effect, embodying the classic financial theory that "time is money."
The Time Value of Money (also known as the time value of funds) refers to the value appreciation that occurs as money moves through time, meaning that a certain amount of money held now is worth more than the same amount received in the future.
Essential Principle
Money has time value because it can be invested to earn returns, making current money more valuable than future money of the same nominal amount.
Investment Opportunity Cost
Current funds can be used for investment, business operations, or deposits, generating interest, profits, or returns through these economic activities.
Inflation Factor
Inflation reduces the real purchasing power of money, making future money typically worth less than current money.
Risk Factor
Future returns carry uncertainty, requiring risk compensation from investors.
Only when funds participate in economic activities can they generate value appreciation. Idle funds do not create time value.
Value creation requires a time process. Time is a necessary factor for value appreciation.
Time Value Rate (Relative)
The social average fund profit rate without risk and inflation, usually represented by short-term treasury bill rates.
Time Value Amount (Absolute)
The actual appreciation amount that funds bring during production and operation, commonly known as interest or investment returns.
Important Distinction
The time value of money ≠ market interest rate. Market interest rates include risk premiums and inflation compensation.
The current worth of future cash flows, discounted at an appropriate rate
The value of current money at a future date, considering compound interest
Interest calculated on both principal and previously earned interest
The rate used to calculate present value of future cash flows
Future Value (FV): The value of current money at a future point in time, representing the concept of principal plus interest.
Present Value (PV): The current value of future money discounted to the present, representing the principal amount.
Simple Interest: Interest calculated only on the principal amount, with interest not participating in subsequent interest calculations.
Compound Interest: Interest calculated on both principal and previously accumulated interest (interest on interest).
The Eighth Wonder of the World
Albert Einstein famously called compound interest "the most powerful force in the universe" and "the eighth wonder of the world."
This reflects the incredible wealth accumulation effect created by the combination of time and compound returns.
Simple Interest Future Value
FV = PV + PV × i × n = PV × (1 + i × n)
Where: FV = Future Value, PV = Present Value, i = Interest Rate, n = Number of Periods
Simple Interest Present Value
PV = FV / (1 + i × n)
The reciprocal relationship between present value and future value coefficients
Compound Interest Future Value
FV = PV × (1 + i)ⁿ
(1 + i)ⁿ is called the compound interest future value coefficient
Compound Interest Present Value
PV = FV / (1 + i)ⁿ = FV × (1 + i)⁻ⁿ
(1 + i)⁻ⁿ is called the discount coefficient
Nominal Interest Rate (r)
The annual interest rate announced by banks or financial institutions, not considering compounding frequency.
Effective Interest Rate (i)
The actual annual return rate after considering the compounding effect.
Effective Interest Rate Calculation
i = (1 + r/M)ᴹ - 1
or equivalently: 1 + i = (1 + r/M)ᴹ
Where: i = effective rate, r = nominal rate, M = compounding frequency per year
Conditions: Deposit $1,000 now at 6% annual interest, compounded annually. Calculate the value after 5 years.
FV = 1000 × (1 + 6%)⁵ = 1000 × 1.3382 = $1,338.20
Appreciation: $338.20, appreciation rate: 33.82%
Conditions: Same as Example 1, but compounded semi-annually (twice per year).
FV = 1000 × (1 + 3%)¹⁰ = 1000 × 1.3439 = $1,343.90
Comparison: $5.70 more than annual compounding, demonstrating the advantage of higher frequency compounding.
Scenario: College student Xiao Ming plans to receive $1,000 in 3 years at 6% annual interest rate.
PV = 1000 × (1 + 6%)⁻³ = 1000 × 0.8396 = $839.60
Meaning: Xiao Ming needs to deposit $839.60 now to achieve his $1,000 goal in 3 years.
Time Effect
As time increases, the compound interest effect shows exponential amplification.
Interest Rate Effect
Higher interest rates accelerate value growth.
Accelerated Growth
The combination creates astonishing wealth accumulation effects.
Quick Estimation Formula
Doubling Years ≈ 72 ÷ Annual Interest Rate (%)
Example: At 6% annual interest rate, 72 ÷ 6 = 12 years to double the principal.
Application: Rough estimation tool with reasonable error margins for practical use.
Evaluating investment opportunities using NPV and IRR calculations
Understanding loan structures and optimal repayment strategies
Calculating required savings for future financial security
Making long-term investment decisions for businesses
Project Evaluation
Using NPV and IRR calculations to assess investment opportunities.
Stock Valuation
Dividend discount models for determining stock intrinsic value.
Bond Pricing
Present value calculations for fixed income securities.
Retirement Planning
Calculating required savings for future financial security.
Education Funding
Planning savings for children's education expenses.
Home Purchase Decisions
Comparing different mortgage financing options.
Characteristics
Each month pays a fixed principal amount plus decreasing interest.
Monthly Payment = Loan Principal ÷ Repayment Months + Remaining Principal × Monthly Interest Rate
Advantages: Lower total interest expense
Disadvantages: Higher early payment pressure
Characteristics
Fixed monthly payment amount throughout the loan term.
Monthly Payment = Loan Principal × [Monthly Rate × (1 + Monthly Rate)ⁿ] ÷ [(1 + Monthly Rate)ⁿ - 1]
Advantages: Stable monthly payment pressure
Disadvantages: Higher total interest expense
Decision Recommendation
Choose the appropriate repayment method based on personal cash flow situation and risk preference. The equal principal payment method is suitable for those with strong early repayment ability, while the equal monthly payment method is suitable for those who prefer stable payments.
Present Value Formula
PV = Σ[cₜ / (1 + r)ᵗ]
For a continuous n-year period, where each year-end cash flow is c₁, c₂, ..., cₙ, and the discount rate is r:
PV = c₁/(1+r) + c₂/(1+r)² + ... + cₙ/(1+r)ⁿ
NPV Function
Calculates net present value of cash flows
PV Function
Calculates present value of investments
FV Function
Calculates future value of investments
PMT Function
Calculates equal periodic payments
Essential questions and answers about the time value of money for better understanding and SEO optimization.
A: The fundamental principle is that money has time value because current funds can be invested to earn returns, making them more valuable than future money of the same nominal amount.
Napoleon's Example: Using 3 Louis as principal with 5% compound interest over 187 years resulted in 11,003,550 francs. This demonstrates how compound interest transforms small amounts into substantial sums over time, proving that "time is money" in financial mathematics.
Mathematical Foundation: FV = PV × (1 + i)ⁿ shows exponential growth where each period's interest earns interest in subsequent periods, creating the compound interest effect.
A: Present value represents how much future money is worth today, while future value shows how much current money will grow over time with compound interest.
Key Relationship: PV and FV are reciprocals in compound interest calculations. If you know one, you can calculate the other using the compound interest formula.
Practical Example: $1,000 invested today at 6% annual compound interest becomes $1,338.20 in 5 years (FV). Conversely, to receive $1,000 in 5 years at 6% interest, you need $839.60 today (PV).
A: The Rule of 72 is a quick estimation tool to calculate how long it takes for an investment to double at a given annual interest rate.
Formula: Doubling Years ≈ 72 ÷ Annual Interest Rate (%)
Examples: At 6% annual interest: 72 ÷ 6 = 12 years to double. At 8% annual interest: 72 ÷ 8 = 9 years to double. At 12% annual interest: 72 ÷ 12 = 6 years to double.
Financial Planning Use: Helps estimate investment growth for retirement planning, education funding, or any long-term financial goal setting.
A: Higher compounding frequency leads to faster growth because interest is calculated and added to the principal more frequently, allowing interest to earn interest sooner.
Comparison Example: $1,000 at 6% annual interest:
• Annual compounding (once per year): FV = 1000 × (1 + 6%)⁵ = $1,338.20
• Semi-annual compounding (twice per year): FV = 1000 × (1 + 3%)¹⁰ = $1,343.90
Difference: Semi-annual compounding yields $5.70 more, demonstrating the advantage of more frequent compounding in long-term investments.
A: Simple interest is calculated only on the principal amount, while compound interest includes interest on previously earned interest.
Simple Interest: Best for short-term loans and simple borrowing where interest doesn't compound. Formula: I = P × r × t
Compound Interest: Essential for long-term investments, savings accounts, and retirement planning where exponential growth matters. Formula: A = P × (1 + r/n)ⁿᵗ
Key Insight: Compound interest creates exponential growth while simple interest grows linearly, making compound interest crucial for long-term wealth building.
A: Inflation reduces the purchasing power of money over time, making future money worth less than current money in real terms.
Real vs Nominal Returns: If an investment earns 8% nominal return but inflation is 3%, the real return is only 5%. This means your money buys 5% more goods and services, not 8%.
Investment Impact: High inflation periods require higher nominal returns to maintain purchasing power. For example, during high inflation, you need investments returning more than the inflation rate to preserve wealth.
A: The discount rate is the rate used to calculate present value of future cash flows. Higher discount rates result in lower present values.
Formula: PV = FV / (1 + r)ⁿ where r is the discount rate. The discount rate reflects opportunity cost, inflation, and risk.
Choosing Discount Rate: Use risk-free rate (Treasury bills) for low-risk projects, or risk-adjusted rates for higher-risk investments. Consider opportunity cost and inflation expectations.
A: Time value of money helps compare different loan structures by calculating total interest costs and cash flow implications.
Equal Principal Payment: Fixed principal payment + decreasing interest. Lower total interest but higher early payments. Good for those with strong early cash flow.
Equal Monthly Payment: Fixed monthly payment throughout the term. Higher total interest but stable payments. Better for consistent cash flow situations.
A: Time value of money is crucial for calculating how much to save today to achieve future retirement income goals.
Retirement Calculation: To accumulate $1,000,000 in 30 years at 7% annual return, you need to save approximately $12,077 annually, calculated using the future value of annuity formula.
Key Factors: Starting age, retirement age, expected return rate, and desired retirement income all affect the required savings amount through time value calculations.
A: Bond valuation uses present value calculations to determine fair price based on future cash flows (coupon payments and principal).
Bond Pricing Formula: P = Σ(C/(1+r)ᵗ) + F/(1+r)ⁿ where C is coupon payment, F is face value, r is yield to maturity.
Yield to Maturity: The discount rate that equates the bond's present value to its market price, representing the total return if held to maturity.
Mastering the time value of money is the cornerstone of making scientific financial decisions. It helps us establish comparable value foundations between cash flows at different time points, enabling more rational and optimized financial choices across various domains including investment analysis, loan planning, retirement preparation, and business decision-making.