Everyday Math for Investors

Module 0.1: Everyday Math for Investors

The math of investing is not complex calculus. It is the mathematics of compounding, growth rates, and the relentless power of exponential functions. Warren Buffett has famously said that his investment success is 90% attributable to compound interest working over time. In this module, we master the essential calculations that professional investors use every day-not because they are difficult, but because their insights are profound.

Lesson 1: The Magic of Compound Interest

The Basic Formula and Real-World Magic

Compound interest is growth on growth. Unlike simple interest, which adds a fixed amount each period, compound interest reinvests earnings so they earn returns too. This is the single most important concept in investing.

The formula is straightforward: A = P(1 + r)^n

Where:

• A = final amount

• P = principal (starting amount)

• r = annual interest rate (as a decimal)

• n = number of years

Let's work through a concrete example. Suppose you invest $10,000 at age 25 in a diversified index fund earning 10% annually. How much will you have at age 65?

Calculation:

• P = $10,000

• r = 0.10 (10%)

• n = 40 years

• A = $10,000 × (1.10)^40

Breaking this down step by step:

• (1.10)^10 = 2.594

• (1.10)^20 = 6.727

• (1.10)^40 = (6.727)^2 = 45.26

So A = $10,000 × 45.26 = $452,600

This single $10,000 investment grows to nearly half a million dollars. The first $10,000 you contributed provided only 2.2% of the final amount; compound interest provided 97.8%. This is why starting early is so powerful-you are not primarily investing your money, you are investing time.

Compare this to simple interest: $10,000 at 10% simple interest for 40 years = $10,000 + (40 × $1,000) = $50,000. Compound interest delivered $452,600 versus $50,000 from simple interest. That $402,600 difference is the cost of not understanding compounding.

The Exponential Curve Compound interest creates an exponential curve, not a linear one. In early years, growth is slow. In later years, growth accelerates dramatically. This is why the final 10 years often generate more wealth than the first 30 years combined.

Compounding at Different Rates Over Different Time Horizons

Not all investments grow at the same rate. Let's compare three scenarios starting with $10,000:

5-Year Period:

• At 5% annual: $10,000 × (1.05)^5 = $12,763 | Gain: $2,763 (27.6%)

• At 8% annual: $10,000 × (1.08)^5 = $14,693 | Gain: $4,693 (46.9%)

• At 12% annual: $10,000 × (1.12)^5 = $17,623 | Gain: $7,623 (76.2%)

20-Year Period:

• At 5% annual: $10,000 × (1.05)^20 = $26,533 | Gain: $16,533 (165.3%)

• At 8% annual: $10,000 × (1.08)^20 = $46,610 | Gain: $36,610 (366.1%)

• At 12% annual: $10,000 × (1.12)^20 = $96,463 | Gain: $86,463 (864.6%)

40-Year Period:

• At 5% annual: $10,000 × (1.05)^40 = $70,400 | Gain: $60,400 (604%)

• At 8% annual: $10,000 × (1.08)^40 = $217,245 | Gain: $207,245 (2,072%)

• At 12% annual: $10,000 × (1.12)^40 = $932,319 | Gain: $922,319 (9,223%)

Notice: the difference between 5% and 8% is relatively modest in 5 years ($1,930) but becomes $190,610 over 40 years. Those 3 percentage points compound into a massive difference. This is why beating the market by even 2-3% annually creates generational wealth.

Percentage Points Compound A difference of 2% per year seems small. Over 40 years, it is the difference between $28,000 and $93,000 on the same $10,000 starting investment. Small differences in annual returns create enormous differences in final wealth.

Real Company Example: Apple's Stock Growth

From January 2004 to January 2024 (20 years), Apple stock grew from approximately $25 per share (split-adjusted) to approximately $180 per share.

Simple calculation:

• Starting price: $25

• Ending price: $180

• Ratio: $180 / $25 = 7.2x

If you invested $10,000 in Apple stock in January 2004, you would have had approximately $72,000 by January 2024. This represents a 7.2x return in 20 years.

What annual rate of return does this represent?

Using the compound formula: $180 = $25 × (1 + r)^20

Solving for r: (1 + r)^20 = 7.2

Taking the 20th root: 1 + r = 7.2^(1/20) = 1.1076

Therefore: r ≈ 10.76% annually

Apple's stock compounded at roughly 10.76% per year for 20 years. This included the 2008 financial crisis, the 2020 pandemic, and numerous periods of weakness. Yet the exponential curve of compounding turned a $10,000 investment into $72,000.

Apple Stock Price History

Compound Annual Growth Rate Calculator

Lesson 2: Calculating and Interpreting CAGR

CAGR Formula and Interpretation

CAGR (Compound Annual Growth Rate) is the single uniform rate of return that converts an initial investment into a final value over a multi-year period. It smooths out volatility to show the annual "average" growth rate.

The formula is: CAGR = (Ending Value / Beginning Value)^(1/n) - 1

Where n = number of years

Let's calculate CAGR for a company's revenue growth. Suppose a software company had:

• Revenue in 2020: $50 million

• Revenue in 2024: $162 million

• Period: 4 years

Calculation:

CAGR = ($162M / $50M)^(1/4) - 1

CAGR = (3.24)^(0.25) - 1

CAGR = 1.345 - 1

CAGR ≈ 34.5%

This company's revenue grew at a compound annual rate of 34.5% from 2020 to 2024. Let's verify this makes sense:

• Year 0: $50M

• Year 1: $50M × 1.345 = $67.25M

• Year 2: $67.25M × 1.345 = $90.55M

• Year 3: $90.55M × 1.345 = $121.79M

• Year 4: $121.79M × 1.345 = $163.81M ✓ (approximately $162M)

What Makes a Good CAGR?

CAGR interpretation depends heavily on the asset class:

Stock Market Returns:

• S&P 500 long-term CAGR: approximately 10% (1950-2024)

• Small-cap stocks: 12-13%

• Individual stock beating 15%+ for 5+ years: exceptional

• Beating 20%+ for a decade: extremely rare

Business Revenue Growth:

• Mature businesses (utilities, insurance): 2-5% CAGR is respectable

• Stable growth companies (Coca-Cola): 4-7%

• Growth companies (technology, healthcare): 15-30%

• High-growth startups: 50%+, but unsustainable long-term

Real Estate Appreciation:

• Long-term residential real estate: 3-4% CAGR nationally

• Commercial property: 4-6%

CAGR Smooths Volatility CAGR shows the smooth "average" rate, but reality is volatile. A stock might return -20% one year and +50% the next, with a 10% CAGR. This smoothing is useful for comparison but can hide risk.

CAGR Example: Berkshire Hathaway vs. S&P 500

From January 1990 to December 2023 (34 years):

• Berkshire Hathaway Class A: returned approximately $15,000 per initial $1,000 (15,000x)

• S&P 500 (with dividends reinvested): returned approximately $235 per initial $1,000 (235x)

Berkshire CAGR:

(15,000)^(1/34) - 1 = 1.205 - 1 = 20.5% CAGR

S&P 500 CAGR:

(235)^(1/34) - 1 = 1.099 - 1 = 9.9% CAGR

Even accounting for higher volatility and more downside periods, Berkshire's extra 10.6 percentage points of annual return compounds into a 63x larger final value ($15,000,000 vs. $235,000 on a $1,000 investment). This demonstrates why outperformance, when sustained, becomes a moat.

Lesson 3: Time Value of Money and Present Value

Why a Dollar Today Is Worth More Than a Dollar Tomorrow

The time value of money is the principle that money available today is worth more than the same amount in the future because of its earning potential. If you have $1,000 today, you can invest it and earn returns. If someone promises you $1,000 in five years, you lose those earning opportunities.

The relationship is captured by the present value formula: PV = FV / (1 + r)^n

Where:

• PV = present value

• FV = future value

• r = discount rate (required annual return)

• n = number of years

Suppose someone offers to give you $1,000 five years from now. Your required rate of return on an alternative investment is 8%. What is that $1,000 worth in today's dollars?

Calculation:

PV = $1,000 / (1.08)^5

PV = $1,000 / 1.469

PV ≈ $681

The $1,000 you'll receive in five years is worth approximately $681 today. If you invested $681 today at 8% annually, you would have $1,000 in five years:

$681 × (1.08)^5 = $681 × 1.469 = $1,001 ✓

Building a Present Value Chain: A Real Investment Example

Suppose you are considering buying a rental property. It will generate:

• Year 1: $15,000 in net rental income

• Year 2: $15,500 in net rental income

• Year 3: $16,000 in net rental income

• Year 4: $16,500 in net rental income

• Year 5: $17,000 in net rental income, plus you sell the property for $250,000

Your required rate of return is 10% (you could earn this elsewhere with similar risk).

What is this investment worth today?

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

Year 2: PV = $15,500 / (1.10)^2 = $15,500 / 1.21 = $12,810

Year 3: PV = $16,000 / (1.10)^3 = $16,000 / 1.331 = $12,016

Year 4: PV = $16,500 / (1.10)^4 = $16,500 / 1.464 = $11,271

Year 5: PV = ($17,000 + $250,000) / (1.10)^5 = $267,000 / 1.611 = $165,754

Total present value = $13,636 + $12,810 + $12,016 + $11,271 + $165,754 = $215,487

If you can buy this property for $200,000, the investment is worth $215,487 to you, so it is a good deal (assuming the projections are reasonable). If the asking price is $250,000, it is overpriced relative to your required return.

Discount Rate is Critical Changing the discount rate dramatically changes present value. At 5%, those same cash flows are worth $290,000. At 15%, they're worth $155,000. A seemingly small change in required return creates a huge valuation swing.

Introduction to Perpetuities and Annuities

A perpetuity is a stream of cash flows that continues forever. Examples include:

• Preferred stock paying a fixed dividend indefinitely

• A government bond that never matures

• Certain real estate leases

The present value of a perpetuity is surprisingly simple: PV = Annual Cash Flow / Discount Rate

If a preferred stock pays $100 per year forever and your required return is 8%, its value is:

PV = $100 / 0.08 = $1,250

If interest rates rise and your required return becomes 10%, its value falls to:

PV = $100 / 0.10 = $1,000

Notice: the same cash flow ($100) becomes less valuable when discount rates rise. This is why bond prices fall when interest rates rise.

An annuity is a finite series of equal cash flows. The formula is: PV = Payment × [1 - (1 + r)^-n] / r

Suppose you win a lottery paying $50,000 per year for 20 years. Your required return is 6%. What is the prize worth today?

PV = $50,000 × [1 - (1.06)^-20] / 0.06

PV = $50,000 × [1 - 0.3118] / 0.06

PV = $50,000 × 0.6882 / 0.06

PV = $50,000 × 11.47

PV ≈ $573,500

The $1,000,000 lottery (20 × $50,000) is worth $573,500 today. This is why lottery winners who choose the lump sum get less than the advertised total-they are receiving the present value.

Lesson 4: The Rule of 72 and Mental Math for Investors

Doubling Time with the Rule of 72

The Rule of 72 is a mental math shortcut to estimate how long it takes an investment to double at a given annual growth rate. The formula is simple: Years to Double ≈ 72 / Annual Growth Rate

At 8% growth: 72 / 8 = 9 years to double

At 10% growth: 72 / 10 = 7.2 years to double

At 12% growth: 72 / 12 = 6 years to double

Let's verify: $10,000 at 10% annually for 7.2 years:

$10,000 × (1.10)^7.2 = $10,000 × 1.998 ≈ $20,000 ✓

The Rule of 72 works so well because 72 is close to 100 × ln(2) ≈ 69.3. It is accurate for rates between 1% and 10%.

The Rule of 72 is Physical Intuition Rather than memorizing formulas, use the Rule of 72 to build intuition. At 5% (government bonds), money doubles in 14 years. At 10% (stock market), it doubles in 7 years. At 20% (exceptional businesses), it doubles in 3.6 years. See how much time advantage compounding at higher rates provides?

More Mental Math: 10% Quick Calculations

Most investors need to quickly estimate returns. A 10% annual return is a useful mental anchor. Here are shortcuts:

10 years of 10% compounding grows money 2.6x (not 2x).

10 years of 10% = doubling in ~7 years, then another 3 years of growth.

To quickly estimate: after 7 years (doubled), you have 2x. After 10 years, you grow that 2x by another 1.10^3 = 1.33. So 2x × 1.33 = 2.6x. Correct!

For 20 years at 10%: double in 7 years (2x), then double again in another 7 years (4x), then 6 more years of growth at 1.10^6 = 1.77. So 4x × 1.77 = 7x. Verify: (1.10)^20 = 6.73. Close enough for mental math!

Fermi Estimation: Order-of-Magnitude Thinking

Fermi estimation is making rough approximations to get in the right ballpark. Investors often need to ask: "Is this business worth $1 billion or $10 billion?"

Suppose a retail company has:

• 500 stores

• $200 average order value

• 100 customers per store per day

• Operating 365 days per year

Rough revenue estimate:

500 stores × 100 customers × $200 × 365 = $3.65 billion annually

If this company has a 10% profit margin, profits are approximately $365 million. At a 25x price-to-earnings multiple (reasonable for growing retail), the enterprise value is approximately $9.1 billion.

This order-of-magnitude estimate (roughly $9 billion) can then be refined with actual data. But Fermi estimation quickly tells you if a company is in the small-cap, mid-cap, or large-cap range.

Lesson 5: How Inflation and Taxes Erode Compounding

Real Returns vs. Nominal Returns

Nominal return is the percentage gain in dollars. Real return adjusts for inflation-the percentage gain in purchasing power.

If your stock returns 8% but inflation is 3%, your real return is NOT 5%. The formula is: Real Return ≈ Nominal Return - Inflation (for rough estimates)

More precisely: Real Return = [(1 + Nominal) / (1 + Inflation)] - 1

Let's calculate precisely. You invest $10,000 earning 8% annually for 10 years with 3% inflation:

Nominal value after 10 years:

$10,000 × (1.08)^10 = $10,000 × 2.159 = $21,590

What this is worth in today's dollars:

$21,590 / (1.03)^10 = $21,590 / 1.344 = $16,070

Real return:

($16,070 / $10,000) - 1 = 60.7% over 10 years

Or approximately 4.9% annually

Your nominal gain looks strong (115.9% total, 8% annually), but your real purchasing power gain is only 60.7% total (4.9% annually). Inflation is a silent but powerful force eroding returns.

Inflation Accelerates Erosion Over Time In short periods (1-3 years), inflation seems minor. Over 20 years, inflation compounds too. Money you earn 20 years from now will buy far less. Inflation is why government bonds at 4% are not attractive-historically, inflation is 3%, leaving only 1% real return.

The Tax Drag on Compounding

Taxes are even more insidious than inflation because they are paid from investment returns, not just reducing purchasing power.

Suppose you have a $100,000 investment earning 10% annually, held in a taxable account. Your tax rate on investments is 25% (federal + state combined). Consider two scenarios:

Scenario A: Rebalancing and Selling (Paying Taxes Annually)

• Year 1: Earn $10,000, pay $2,500 in taxes. Remaining: $107,500

• Year 2: Earn $10,750, pay $2,688 in taxes. Remaining: $115,562

• ...continuing this pattern...

• Year 10: Final value ≈ $202,000

Scenario B: Buy and Hold (Paying Taxes Only at the End)

• Year 10: Pre-tax value: $100,000 × (1.10)^10 = $259,937

• Pay taxes on gains: ($259,937 - $100,000) × 0.25 = $39,984

• After-tax value: $219,953

The difference? Tax-deferred compounding delivered $219,953 versus $202,000 from annual tax drag-an extra $17,953. This is why tax-advantaged accounts (401k, IRA, HSA) are so powerful.

Moreover, if you hold long enough for capital gains to be taxed at preferential rates (15-20% instead of 25%), the advantage grows. Long-term capital gains rates are substantially lower than income tax rates in most cases.

Lesson 6: Practical Scenarios: Savings Plans and Retirement

The 30-Year Savings Plan

Many investors ask: "If I save and invest regularly, how much will I have?" This is an annuity question where you contribute a fixed amount each period.

The future value of an annuity formula is: FV = Payment × [((1 + r)^n - 1) / r]

Suppose you save $500 per month ($6,000 per year) for 30 years, earning an average of 8% annually:

FV = $500 × [((1.08)^30 - 1) / 0.08] × 12 months (adjusting for monthly contributions and compounding)

For monthly contributions at 8% annual (0.667% monthly):

FV = $500 × {[((1.00667)^360 - 1) / 0.00667]}

FV = $500 × 1,339.6

FV ≈ $669,800

By contributing $500 per month for 30 years, you invest $180,000 of your own money ($500 × 12 × 30). The remaining $489,800 is investment returns. Compounding on both your contributions and the returns generates nearly twice what you invested.

This is why starting young is not optional-it is the single most important factor in building wealth. Starting at age 25 versus 35 adds a full decade of compounding. That decade often creates more wealth than all other years combined.

Tax-Advantaged Retirement Accounts

The previous example assumed a 8% after-tax return. In a 401(k) or IRA:

• You contribute pre-tax dollars (tax-deferred growth)

• You earn 8% gross returns with no annual tax drag

• Your $180,000 contribution in pre-tax dollars might actually represent only $140,000 in your pre-tax income (saving 25% immediately)

With tax deferral and higher gross returns, the final value exceeds $750,000 easily. Combined with employer matching (if available), the wealth-building accelerates further.

This is why contributions to tax-advantaged accounts should be maxed out before taxable investing:

• 2024 401(k) limit: $23,500

• 2024 IRA limit: $7,000

• 2024 HSA limit: $4,150 (also triple tax-advantaged)

A worker contributing $23,500 annually to a 401(k) earning 8% for 35 years accumulates approximately $4.2 million pre-tax (or roughly $3 million after-tax at 28% tax rate in retirement). This is achievable through disciplined, compound-based wealth-building.

Lesson 7: Practice Problems and Self-Assessment

The following problems test your mastery of the concepts in this module. Work through each one carefully, checking your arithmetic.

Problem 1: Apple Stock Compounding

You bought 100 shares of Apple in January 2004 at $25 per share (split-adjusted). In January 2024, it trades at $180. What is your total return? What is your CAGR? If Apple continues growing at the same CAGR for another 20 years, what will the share price be in 2044?

Answer: Total return = ($180 - $25) / $25 = 620%. CAGR = 10.76% (approximately). At 10.76% for 20 more years: $180 × (1.1076)^20 = $1,296. If this seems high, verify: does a 10.76% annual return for 40 years turn $25 into $1,296? Yes: (1.1076)^40 = 51.8, and $25 × 51.8 = $1,295.

Problem 2: Rental Property Valuation

A property rents for $24,000 per year and will rent for the next 30 years. After 30 years, you can sell it for $400,000. Your required return is 7%. What is this property worth today? (Hint: use an annuity formula for the rentals, then add the present value of the sale price.)

Answer: PV of annuity: $24,000 × [1 - (1.07)^-30] / 0.07 = $24,000 × 12.41 = $297,840. PV of sale: $400,000 / (1.07)^30 = $400,000 / 7.612 = $52,546. Total = $350,386.

Problem 3: Rule of 72 Application

The stock market has historically returned 10% annually. At this rate, how long does it take for an investment to triple? (Hint: doubling takes 7.2 years; you need to think about what happens next.)

Answer: Doubling takes 7.2 years. To triple, you need to grow by 1.5x after the doubling. This is an additional (1.5)^(1/7.2) = 1.054, or 5.4% of another year. So triple takes approximately 11.4 years. Verify: (1.10)^11.4 = 3.0. Correct!

Self-Practice Prompt 1: How much would you need to invest today at 8% annual returns to have $1 million in 30 years? Use the present value formula.

Self-Practice Prompt 2: Coca-Cola has paid dividends for 62 consecutive years. Suppose it pays $2.00 per share annually forever. Your required return is 8%. What is the perpetual dividend stream worth today?

Self-Practice Prompt 3: Create a personal savings scenario: assume you save $X per month for 40 years at Y% returns. What do you end up with? How much is your own money versus investment returns?