Adjust principal, annual rate, monthly contributions and inflation to visualize wealth growth in real time. Compare compound vs simple interest, apply the Rule of 72 and track real purchasing power side-by-side.
Parameter Settings
Rule of 72: Approximate doubling time: — years (Exact value: — years)
Presets
While paused, move the sliders to update the result instantly.
Compound Growth Animation
PrincipalInterestCompound (total)SimpleContinuous
The timeline sweeps forward as interest (green) accelerates on top of the principal (blue). The gap over simple interest (orange) widens over time.
Live Results
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Elapsed Years [yr]
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Current Balance [10,000 JPY]
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Cumulative Interest [10,000 JPY]
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Effective Annual Rate EAR [%]
Theory & Key Formulas
Principal only：$A = P\left(1+\dfrac{r}{m}\right)^{mn}$
Including contributions：$A = P\left(1+\dfrac{r}{m}\right)^{mn} + C\cdot\dfrac{\left(1+\dfrac{r}{m}\right)^{mn}-1}{\dfrac{r}{m}}$
Continuous compounding：$A = Pe^{rn}$
Real return：$(1+r_{real}) = \dfrac{1+r}{1+i}$
🙋 'Interest earning interest'—how amazing is that?
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I heard Einstein said 'Compound interest is the eighth wonder of the world.' How amazing is it really? How big is the difference compared to simple interest?
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Let's look at actual numbers. If you invest 1 million yen at 5% annual interest for 30 years, with simple interest you get 100 + 100×0.05×30 = 2.5 million yen. But with compound interest, 100×1.05^30 ≈ 4.32 million yen. That's a difference of 1.82 million yen. Look at the 'Simple vs Compound' tab graph—the gap starts small but grows exponentially over time.
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What's the 'Rule of 72'? It sounds like something that might appear in an exam.
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It's a mental math rule: 'Years to double ≈ 72 ÷ annual interest rate (%)'. At 6% annual interest, 72÷6=12 years to double; at 4%, 18 years. The exact formula is ln(2)/ln(1+r), but 72 is wonderfully easy for mental math. The tool shows both the approximate and exact values in the 'Rule of 72' section—compare them. It's common knowledge in finance interviews.
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If inflation is 2%, even if assets grow, are they actually losing value? Looking at the 'Inflation Erosion' preset, the green line (real value) barely increases...
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Exactly—that's 'inflation risk'. If the annual interest rate is 1% and inflation is 3%, the real return using the Fisher equation is (1+0.01)/(1+0.03) - 1 ≈ -1.94%—your assets are effectively shrinking. So just keeping money in a bank savings account (around 0.02% annual interest) can lose to inflation and result in a real loss. The key to building wealth is whether you can keep your interest rate higher than inflation.
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I tried the iDeCo preset, and after 30 years it showed a huge amount. Is this based on contributing 15,000 yen per month at 5% annual return?
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Yes. Contributing 15,000 yen per month (180,000 yen per year) at 5% for 30 years, your total principal is 5.4 million yen, but the final asset is over 12 million yen. That's the power of 'contribution + compound interest'—it's the future value of an annuity calculation. Starting early is advantageous because the longer the time, the more the compounding effect kicks in—that's why time is called the greatest asset.
Frequently Asked Questions
Simple interest accrues only on the principal each period (A = P(1 + rn)). Compound interest adds the previous period's interest to the principal for the next period, so interest earns interest (A = P(1+r)^n). With the same interest rate and term, the larger n is, the more compound interest exceeds simple interest in final assets.
The exact doubling time is ln(2) / ln(1+r) ≈ 0.693/r (for small r), but in the practical interest rate range (r=3-15%), using 0.72 (=72/100) with a slight correction to 0.693 reduces error. Also, 72 is divisible by 2, 3, 4, 6, 8, 9, and 12, making it easy for mental math.
For the same nominal annual rate r, a higher compounding frequency m increases the effective annual rate (EAR). EAR = (1+r/m)^m - 1. For example, with a nominal annual rate of 6%, annual EAR=6.000%, monthly EAR=6.168%, daily EAR=6.183%, continuous compounding EAR = e^0.06 - 1 = 6.184%. The differences are small but accumulate over long periods.
Use the Fisher equation: (1 + r_real) = (1 + r_nominal) / (1 + i). For example, with a nominal rate of 5% and inflation of 3%, r_real = (1.05/1.03) - 1 ≈ 1.94%. The common approximation r_real ≈ r - i = 2% has little error. To see real asset value, divide the final asset amount by (1+i)^n to convert to current purchasing power.
If you contribute C at the end of each year, the total assets after n years are A = P(1+r)^n + C × [(1+r)^n - 1] / r. The second term is the 'Future Value of Annuity', representing the contribution effect. If you contribute at the beginning of each year, A = P(1+r)^n + C(1+r) × [(1+r)^n - 1] / r, which is (1+r) times more than the end-of-year method.
What is Compound Interest Simulator?
Compound Interest Simulator is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Physical Model & Key Equations
The simulator is based on the governing equations behind Compound Interest Simulator. Understanding these equations is key to interpreting the results correctly.
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Real-World Applications
Engineering Design: The concepts behind Compound Interest Simulator are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.
Enter principal amount in v_p (e.g., $50,000 initial investment)
Set annual interest rate using sl_r (e.g., 7% for equity index funds, 4% for bonds)
Specify time period in v_n (years 1–50) to observe exponential growth curves
Optional: add annual contributions via sl_p to model regular deposits like 401(k) additions
Compare compound vs. simple interest output and note effective doubling time using Rule of 72
Worked Example
Real estate investment scenario:
How to Use
Enter principal amount in v_p (e.g., $50,000 initial investment)
Set annual interest rate using sl_r (e.g., 7% for equity index funds, 4% for bonds)
Specify time period in v_n (years 1–50) to observe exponential growth curves
Optional: add annual contributions via sl_p to model regular deposits like 401(k) additions
Compare compound vs. simple interest output and note effective doubling time using Rule of 72
Worked Example
Real estate investment scenario: $100,000 principal at 6.5% annual return over 25 years with $5,000 annual contributions. Compound interest yields $766,428 final value; simple interest produces only $412,500. Rule of 72 predicts doubling in 11.1 years (72÷6.5), confirmed by simulator output at year 11 showing ~$198,000. Inflation adjustment at 2.8% annually reduces real purchasing power to $382,153 in today's dollars—critical for retirement planning accuracy.
Practical Notes
Adjust sl_r sensitivity: a 1% rate increase on $200k principal compounds to ~$67k additional wealth over 20 years—test dividend reinvestment scenarios
Use contributions slider to model monthly/annual savings discipline; $250/month ($3,000/year) outpaces inflation at 3.2% on modest starting balances within 8–10 years
Rule of 72 breaks down below 2% rates; verify manually for treasury bonds yielding 1.5%–2.0%
Compare municipal bond yields (often 4%–5% tax-exempt) against taxable alternatives requiring higher nominal rates for equivalent after-tax returns
00,000 principal at 6.5% annual return over 25 years with $5,000 annual contributions. Compound interest yields $766,428 final value; simple interest produces only $412,500. Rule of 72 predicts doubling in 11.1 years (72÷6.5), confirmed by simulator output at year 11 showing ~
How to Use
Enter principal amount in v_p (e.g., $50,000 initial investment)
Set annual interest rate using sl_r (e.g., 7% for equity index funds, 4% for bonds)
Specify time period in v_n (years 1–50) to observe exponential growth curves
Optional: add annual contributions via sl_p to model regular deposits like 401(k) additions
Compare compound vs. simple interest output and note effective doubling time using Rule of 72
Worked Example
Real estate investment scenario: $100,000 principal at 6.5% annual return over 25 years with $5,000 annual contributions. Compound interest yields $766,428 final value; simple interest produces only $412,500. Rule of 72 predicts doubling in 11.1 years (72÷6.5), confirmed by simulator output at year 11 showing ~$198,000. Inflation adjustment at 2.8% annually reduces real purchasing power to $382,153 in today's dollars—critical for retirement planning accuracy.
Practical Notes
Adjust sl_r sensitivity: a 1% rate increase on $200k principal compounds to ~$67k additional wealth over 20 years—test dividend reinvestment scenarios
Use contributions slider to model monthly/annual savings discipline; $250/month ($3,000/year) outpaces inflation at 3.2% on modest starting balances within 8–10 years
Rule of 72 breaks down below 2% rates; verify manually for treasury bonds yielding 1.5%–2.0%
Compare municipal bond yields (often 4%–5% tax-exempt) against taxable alternatives requiring higher nominal rates for equivalent after-tax returns
98,000. Inflation adjustment at 2.8% annually reduces real purchasing power to $382,153 in today's dollars—critical for retirement planning accuracy.
Practical Notes
Adjust sl_r sensitivity: a 1% rate increase on $200k principal compounds to ~$67k additional wealth over 20 years—test dividend reinvestment scenarios
Use contributions slider to model monthly/annual savings discipline; $250/month ($3,000/year) outpaces inflation at 3.2% on modest starting balances within 8–10 years
Rule of 72 breaks down below 2% rates; verify manually for treasury bonds yielding 1.5%–2.0%
Compare municipal bond yields (often 4%–5% tax-exempt) against taxable alternatives requiring higher nominal rates for equivalent after-tax returns