How do you calculate compound interest?

A = P(1 + r/n)^(nt) where P is principal, r is annual rate (decimal), n is compounding frequency per year, and t is years. For example, $5,000 at 7% monthly for 10 years = $10,048.

How do you calculate a mortgage payment?

M = P × [r(1+r)^n] / [(1+r)^n - 1] where P is loan amount, r is monthly rate (annual/12), n is total payments. A $300K loan at 6.5% for 30 years = $1,896/month.

What is the Rule of 72?

Divide 72 by the interest rate to estimate doubling time. At 8% interest, money doubles in approximately 72/8 = 9 years.

How much do I need to retire using the 4% rule?

Multiply your annual retirement expenses by 25. If you need $60,000/year, you need $1,500,000 saved. Adjust expenses for inflation before calculating.

Comprehensive Finance Calculators - Master Your Money

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Our finance calculators cover everything from simple interest to complex investment portfolios, tax planning, and business valuation. Whether you're planning for retirement, analyzing investments, comparing loans, budgeting for a home, or managing business finances, CalcREX provides professional-grade tools with clear explanations and actionable insights. All calculators are free to use on any device with no registration required.

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Common Financial Terms

APR Annual Percentage Rate

ROI Return on Investment

APY Annual Percentage Yield

NPV Net Present Value

IRR Internal Rate of Return

CAGR Compound Annual Growth

Financial Planning Tips

💡 Pay yourself first - Save before spending

📊 Diversify investments to reduce risk

⏰ Start early - Compound interest works best over time

💳 Pay off high-interest debt first

Finance & Money - In-Depth Guides with Worked Examples

Master the core financial formulas every student, investor, and professional needs - from compound interest to loan amortization, ROI, and retirement planning.

Compound Interest - The Most Powerful Force in Finance

How money grows when interest earns interest - and why starting early matters more than investing more.

1 Simple vs. Compound Interest

Simple Interest

I = P × r × t

Interest only on original principal. Linear growth. Used in short-term loans.

Compound Interest

A = P × (1 + r/n)^(nt)

Interest on principal + accumulated interest. Exponential growth. Used in savings, investments.

2 The Compound Interest Formula

A = P(1 + r/n)^(nt)

A = final amountP = principal (initial investment)r = annual interest rate (decimal)n = compounding frequency per yeart = time in years

n = 1

Annually

n = 4

Quarterly

n = 12

Monthly

n = 365

Daily

n → ∞

Continuous: A = Pe^(rt)

Worked Example 1

Compound Interest - Monthly Compounding

Problem: You invest $5,000 at 7% annual interest, compounded monthly, for 10 years. What is the final amount? How much is interest earned?

Step 1: Identify variables

P = $5,000, r = 0.07, n = 12, t = 10

Step 2: Apply formula

A = 5000 × (1 + 0.07/12)^(12 × 10)

= 5000 × (1.005833)^120

= 5000 × 2.00966

A = $10,048.31

Step 3: Interest earned

Interest = A − P = $10,048.31 − $5,000

Interest = $5,048.31

Answer: $10,048.31 total, with $5,048.31 in interest - you've more than doubled your money. Simple interest would have yielded only $3,500 in interest ($8,500 total). That extra $1,548 is compound interest working on accumulated interest.

3 The Rule of 72

A quick mental math shortcut to estimate how long it takes to double your money.

Years to double ≈ 72 / interest rate (%)

At 6%: 72/6 = 12 yearsAt 8%: 72/8 = 9 yearsAt 12%: 72/12 = 6 years

Works in reverse too: If prices double in 10 years, inflation ≈ 72/10 = 7.2%. If your debt doubles in 6 years, you're paying ≈ 72/6 = 12% interest.

Worked Example 2

The Power of Starting Early

Problem: Alice invests $200/month from age 25 to 35 (10 years), then stops. Bob invests $200/month from age 35 to 65 (30 years). Both earn 8% annually. Who has more at 65?

Alice: 10 years of contributions, then 30 years of growth

FV after 10 yrs = 200 × [((1+0.08/12)^120 − 1)/(0.08/12)]

= $36,589

Then grows 30 more years: $36,589 × (1.00667)^360

Alice at 65: $398,807

Total contributed: $200 × 120 = $24,000

Bob: 30 years of contributions

FV = 200 × [((1+0.08/12)^360 − 1)/(0.08/12)]

Bob at 65: $298,072

Total contributed: $200 × 360 = $72,000

Answer: Alice: $398,807 (invested $24K). Bob: $298,072 (invested $72K). Alice invested 3× less money but ends up with $100K more - because her money had 30 extra years of compounding. Start early.

Worked Example 3

APR vs. APY - What You Actually Earn

Problem: A savings account advertises 5% APR compounded daily. What is the actual APY (Annual Percentage Yield)?

Step 1: APY formula

APY = (1 + r/n)^n − 1

= (1 + 0.05/365)^365 − 1

= (1.000137)^365 − 1

APY = 5.127%

Answer: 5.127% APY. The more frequently interest compounds, the higher the effective yield. APR is the nominal rate; APY is what you actually earn. Always compare APY when choosing savings accounts.

Loans & Mortgages - Monthly Payments, Amortization & Total Cost

How to calculate what you'll actually pay - and strategies to save thousands in interest.

1 The Loan Payment Formula

The standard formula for fixed-rate loan payments (mortgages, car loans, personal loans).

M = P × [r(1+r)^n] / [(1+r)^n − 1]

M = monthly paymentP = principal (loan amount)r = monthly interest rate (annual rate / 12)n = total number of payments (years × 12)

Worked Example 1

Mortgage Payment Calculation

Problem: You take a $300,000 mortgage at 6.5% for 30 years. What is the monthly payment? How much total interest will you pay?

Step 1: Convert to monthly values

P = $300,000

r = 0.065/12 = 0.005417

n = 30 × 12 = 360 payments

Step 2: Apply formula

M = 300000 × [0.005417(1.005417)^360] / [(1.005417)^360 − 1]

= 300000 × [0.005417 × 6.9916] / [6.9916 − 1]

= 300000 × 0.03788 / 5.9916

M = $1,896.20/month

Step 3: Total cost & interest

Total paid = $1,896.20 × 360 = $682,632

Total interest = $682,632 − $300,000 = $382,632

Answer: $1,896.20/month, with $382,632 in total interest - more than the original loan! This is why even small rate differences matter enormously over 30 years.

2 Amortization - Where Your Payment Goes

Each payment is split between interest and principal. Early payments are mostly interest; later payments are mostly principal.

First payment breakdown on our $300K mortgage: Interest = $300,000 × 0.005417 = $1,625. Principal = $1,896.20 − $1,625 = $271.20. Only $271 reduces your balance! By payment #300, it flips: ~$1,700 goes to principal.

Worked Example 2

Extra Payments - How Much Do You Save?

Problem: On the same $300,000 mortgage at 6.5%, you pay an extra $200/month. How many years do you save? How much interest?

Step 1: New payment amount

New M = $1,896.20 + $200 = $2,096.20

Step 2: Solve for new n

n = −ln(1 − P×r/M) / ln(1+r)

= −ln(1 − 300000×0.005417/2096.20) / ln(1.005417)

= −ln(0.2248) / 0.005402

n ≈ 277 months = 23.1 years (saves ~7 years!)

Step 3: Interest saved

New total = $2,096.20 × 277 = $580,647

Old total = $682,632

Interest saved = $101,985

Answer: 7 years off the mortgage and ~$102,000 saved in interest - by adding just $200/month. Extra payments go 100% to principal, which reduces future interest calculations.

Worked Example 3

Car Loan - Total Cost Comparison

Problem: A $25,000 car loan at 4.9% for 5 years vs. 7.2% for 6 years. Compare monthly payments and total cost.

Option A: 4.9%, 5 years

r = 0.049/12 = 0.004083, n = 60

M = 25000 × 0.004083(1.004083)^60 / [(1.004083)^60 − 1]

M = $471.78/month | Total = $28,307

Option B: 7.2%, 6 years

r = 0.072/12 = 0.006, n = 72

M = 25000 × 0.006(1.006)^72 / [(1.006)^72 − 1]

M = $429.65/month | Total = $30,935

Answer: Option B has lower payments ($42 less/month) but costs $2,628 more overall. The "affordable" payment costs more. Always compare total cost, not just monthly payments.

Investments - ROI, CAGR & Portfolio Returns

How to measure investment performance - the formulas that separate real gains from misleading numbers.

1 Return on Investment (ROI)

ROI = (Gain − Cost) / Cost × 100%

Simple percentage returnDoes NOT account for time

2 CAGR - Compound Annual Growth Rate

CAGR is the smoothed annual rate of return, accounting for compounding. It's the best way to compare investments of different durations.

CAGR = (Ending/Beginning)^(1/years) − 1

Annualized return - allows apples-to-apples comparisonSmooths out volatility over the period

Worked Example 1

ROI vs. CAGR - Which Is Better?

Problem: Investment A: $10,000 → $18,000 in 5 years. Investment B: $10,000 → $15,000 in 3 years. Which performed better?

Step 1: Simple ROI

ROI(A) = (18000 − 10000)/10000 = 80%

ROI(B) = (15000 − 10000)/10000 = 50%

ROI says A is better - but that's misleading!

Step 2: CAGR (annualized)

CAGR(A) = (18000/10000)^(1/5) − 1 = 1.8^0.2 − 1

CAGR(A) = 12.47% per year

CAGR(B) = (15000/10000)^(1/3) − 1 = 1.5^0.333 − 1

CAGR(B) = 14.47% per year

Answer: B is actually better (14.47% vs 12.47% annually). If you reinvested B's returns for 5 years at the same rate, you'd have $19,687 - more than A's $18,000. Always use CAGR to compare investments of different lengths.

3 Dollar-Cost Averaging (DCA)

Investing a fixed dollar amount at regular intervals, regardless of price. You buy more shares when prices are low and fewer when prices are high.

Why DCA works: Your average cost per share is always lower than the average price per share (harmonic mean < arithmetic mean). It also removes the emotional temptation to time the market.

Worked Example 2

Dollar-Cost Averaging in Action

Problem: You invest $500/month for 4 months. Share prices are: $50, $40, $25, $50. What is your average cost vs. average price?

Step 1: Shares purchased each month

Month 1: $500/$50 = 10 shares

Month 2: $500/$40 = 12.5 shares

Month 3: $500/$25 = 20 shares

Month 4: $500/$50 = 10 shares

Step 2: Compare averages

Total shares = 52.5

Total invested = $2,000

Avg cost/share = $2000/52.5 = $38.10

Avg price = ($50+$40+$25+$50)/4 = $41.25

Current value = 52.5 × $50 = $2,625

Gain = $625 (31.25% return)

Answer: Average cost $38.10 vs average price $41.25 - DCA saved $3.15/share. Even though the price ended where it started ($50), you're up 31.25% because you bought heavily during the dip. DCA turns volatility into an advantage.

Worked Example 3

Real Return After Inflation

Problem: Your portfolio returned 9% this year. Inflation was 3.5%. What is your real (inflation-adjusted) return?

Step 1: Fisher equation (precise method)

Real return = [(1 + nominal)/(1 + inflation)] − 1

= (1.09/1.035) − 1

Real return = 5.31%

Quick approximation

Real ≈ Nominal − Inflation = 9% − 3.5%

≈ 5.5% (close but slightly overestimates)

Answer: 5.31% real return. Always adjust for inflation when evaluating long-term performance. A "12% return" during 8% inflation is only 3.7% real growth in purchasing power.

Time Value of Money - Present Value, Future Value & Annuities

The central principle of all finance - a dollar today is worth more than a dollar tomorrow.

1 Present Value & Future Value

Money has a time dimension. Present value (PV) answers "what is a future sum worth today?" Future value (FV) answers "what will today's money be worth later?"

Future Value

FV = PV × (1 + r)^n

Growing money forward. What will $X become?

Present Value

PV = FV / (1 + r)^n

Discounting money back. What is future $X worth now?

Worked Example 1

Present Value - Is This Deal Worth It?

Problem: Someone offers you $50,000 five years from now, or $38,000 cash today. If you can earn 6% annually on investments, which is better?

Step 1: Find PV of the $50,000 offer

PV = FV / (1 + r)^n = 50000 / (1.06)^5

= 50000 / 1.3382

PV = $37,363

Step 2: Compare

$38,000 today > $37,363 (PV of future $50K)

Take the $38,000 cash today

Answer: Take $38,000 today. Invested at 6%, it grows to $50,853 in 5 years - $853 more than the future offer. The discount rate (your alternative return) is what makes TVM decisions possible.

2 Annuities - Streams of Equal Payments

An annuity is a series of equal payments at regular intervals - like loan payments, pension income, or systematic investments.

FV(annuity) = PMT × [(1+r)^n − 1] / r

PMT = payment per periodr = interest rate per periodn = number of periods

PV(annuity) = PMT × [1 − (1+r)^(−n)] / r

Used to value loan streams, pensions, leases

Worked Example 2

Annuity - Saving for a Goal

Problem: You want $100,000 in 15 years. You can earn 7% annually, compounded monthly. How much must you save each month?

Step 1: Rearrange FV annuity for PMT

PMT = FV × r / [(1+r)^n − 1]

r = 0.07/12 = 0.005833, n = 180

Step 2: Calculate

PMT = 100000 × 0.005833 / [(1.005833)^180 − 1]

= 583.33 / [2.8484 − 1]

= 583.33 / 1.8484

PMT = $315.58/month

Answer: $315.58/month. Total contributions: $56,804. Interest earned: $43,196. Almost half the goal comes from compound returns - the longer your timeline, the more compounding does the heavy lifting.

Worked Example 3

PV of Annuity - What's a Pension Worth?

Problem: A pension pays $2,000/month for 25 years. If the discount rate is 5%, what is the lump-sum present value?

Step 1: Monthly rate and periods

r = 0.05/12 = 0.004167, n = 300

Step 2: PV annuity formula

PV = 2000 × [1 − (1.004167)^(−300)] / 0.004167

= 2000 × [1 − 0.2873] / 0.004167

= 2000 × 170.95

PV = $341,902

Answer: $341,902. The pension will pay $600,000 total, but discounted to today it's worth $342K. This is how insurance companies price annuity buyouts and how courts value future income in lawsuits.

Business Finance - Break-Even, Profit Margin & Cash Flow

The numbers every business owner and finance student needs - from pricing decisions to profitability analysis.

1 Break-Even Analysis

The break-even point is where total revenue equals total costs - no profit, no loss. Every unit sold beyond this point generates profit.

BEP (units) = Fixed Costs / (Price − Variable Cost)

Price − Variable Cost = Contribution Margin per unitBEP (revenue) = Fixed Costs / CM RatioCM Ratio = Contribution Margin / Price

Worked Example 1

Break-Even Point

Problem: A bakery has fixed costs of $4,500/month. Each cake costs $8 to make and sells for $25. How many cakes must they sell to break even? What revenue is needed?

Step 1: Contribution margin

CM = Price − Variable Cost = $25 − $8

CM = $17 per cake

Step 2: Break-even units

BEP = $4,500 / $17

BEP = 265 cakes/month

Step 3: Break-even revenue

Revenue = 265 × $25

= $6,625/month

Answer: 265 cakes ($6,625 revenue). Every cake after #265 generates $17 profit. If they sell 350 cakes: profit = (350 − 265) × $17 = $1,445/month.

2 Profit Margins

Gross Margin

(Revenue − COGS) / Revenue × 100

Operating Margin

(Revenue − COGS − OpEx) / Revenue × 100

Net Margin

Net Income / Revenue × 100

Markup

(Price − Cost) / Cost × 100

Margin ≠ Markup! A product bought for $60 and sold for $100 has 40% margin but 66.7% markup. Margin is based on selling price; markup is based on cost. Confusing them is a costly mistake in pricing.

Worked Example 2

Margin vs. Markup

Problem: A retailer wants a 40% profit margin on a product that costs $45 wholesale. What should the selling price be? What is the markup percentage?

Step 1: Selling price from margin

Margin = (Price − Cost)/Price

0.40 = (Price − 45)/Price

Price − 0.40 × Price = 45

0.60 × Price = 45

Price = $75.00

Step 2: Markup percentage

Markup = (75 − 45)/45 × 100

Markup = 66.7%

Answer: Sell at $75 for 40% margin (66.7% markup). Quick conversion: Margin = Markup/(1+Markup). Markup = Margin/(1−Margin). A 50% margin = 100% markup (doubling the cost).

Worked Example 3

Net Present Value (NPV) - Should You Take the Project?

Problem: A project costs $50,000 upfront and generates $15,000/year for 5 years. Required return is 10%. Should you invest?

Step 1: PV of each year's cash flow

Year 1: $15,000/1.10 = $13,636

Year 2: $15,000/1.21 = $12,397

Year 3: $15,000/1.331 = $11,270

Year 4: $15,000/1.4641 = $10,245

Year 5: $15,000/1.6105 = $9,314

Step 2: Calculate NPV

Sum of PVs = $56,862

NPV = $56,862 − $50,000

NPV = +$6,862 → ACCEPT the project

Answer: NPV = +$6,862. Positive NPV means the project returns more than the 10% required rate. The total undiscounted cash is $75,000, but adjusted for time and risk, the net gain is $6,862 in today's dollars.

Retirement Planning - Savings Goals, Withdrawal Rates & Inflation

How much do you actually need to retire? The math behind the 4% rule, required savings, and making your money last.

1 The 4% Rule (Safe Withdrawal Rate)

The 4% rule suggests you can withdraw 4% of your portfolio in year 1, then adjust for inflation each year, and your money should last 30+ years.

Retirement Nest Egg = Annual Expenses × 25

Based on 4% withdrawal rate (1/0.04 = 25)Need $40K/year → save $1,000,000Need $60K/year → save $1,500,000

Why 25×? The Trinity Study showed that a 4% initial withdrawal rate, adjusted for inflation, survived 30-year periods 95% of the time with a 50/50 stock/bond portfolio. It's a starting guideline, not a guarantee.

Worked Example 1

How Much Do You Need to Retire?

Problem: You spend $4,500/month now and expect to need 80% of that in retirement. Inflation averages 3%. You retire in 20 years. What nest egg do you need (in future dollars)?

Step 1: Current retirement spending need

Annual need = $4,500 × 12 × 0.80 = $43,200

Step 2: Adjust for inflation (20 years)

Future annual need = $43,200 × (1.03)^20

= $43,200 × 1.8061

= $78,024/year in future dollars

Step 3: Apply 25× rule

Nest egg = $78,024 × 25 = $1,950,600

Answer: ~$1.95 million. Without adjusting for inflation, you'd target only $1.08M - and run out of money because $43,200 won't buy the same goods in 20 years. Always plan in future dollars.

2 Required Monthly Savings

Once you know your target nest egg, work backwards to find the monthly savings needed using the FV annuity formula.

PMT = FV × r / [(1+r)^n − 1]

FV = retirement targetr = monthly return raten = months until retirement

Worked Example 2

Monthly Savings to Reach Your Goal

Problem: You need $1,950,600 in 20 years. You already have $50,000 saved. Expected return: 8% annually. How much must you save monthly?

Step 1: Future value of existing savings

FV(existing) = $50,000 × (1 + 0.08/12)^240

= $50,000 × 4.9268 = $246,341

Step 2: Remaining gap

Gap = $1,950,600 − $246,341 = $1,704,259

Step 3: Monthly savings needed

r = 0.08/12 = 0.00667, n = 240

PMT = 1,704,259 × 0.00667 / [(1.00667)^240 − 1]

= 11,364 / [4.9268 − 1]

PMT = $2,893/month

Answer: $2,893/month. Total contributions: $50,000 + ($2,893 × 240) = $744,320. The remaining $1.2M comes from investment returns - compounding does most of the work over 20 years.

Worked Example 3

How Long Will Your Savings Last?

Problem: You retire with $800,000 and withdraw $3,500/month. Your portfolio earns 5% annually. How many years will it last?

Step 1: Monthly rate

r = 0.05/12 = 0.004167

Step 2: Solve for n (months)

n = −ln(1 − PV×r/PMT) / ln(1+r)

= −ln(1 − 800000×0.004167/3500) / ln(1.004167)

= −ln(1 − 0.9524) / 0.004158

= −ln(0.0476) / 0.004158

= 3.045 / 0.004158

n ≈ 732 months ≈ 61 years

Answer: ~61 years - well beyond a typical retirement. The 5% return almost covers the withdrawals ($3,333/month in interest vs $3,500 withdrawn), so the principal depletes very slowly. If returns drop to 3%, it lasts only 28 years - highlighting why return assumptions matter enormously.