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Maths

Financial Mathematics

Master the mathematics of money: interest calculations, loans, investments, present and future value. This is the foundation for building wealth and making sound financial decisions.

Time Value of Money

Core Concept: Money today is worth more than the same amount in the future.

Why?

  • Can invest today's money and earn returns
  • Inflation reduces future purchasing power
  • Opportunity cost of waiting

Example: Would you rather have $1,000 today or $1,000 in 5 years?

  • Take it today! Invest at 7% → worth $1,403 in 5 years

Simple Interest

Interest calculated only on the principal amount.

Formula: I = Prt

  • I = Interest earned
  • P = Principal (starting amount)
  • r = Annual interest rate (as decimal)
  • t = Time (in years)

Total Amount: A = P + I = P(1 + rt)

Examples

Savings: Deposit $5,000 at 3% simple interest for 4 years

  • Interest: I = 5000 × 0.03 × 4 = $600
  • Total: $5,000 + $600 = $5,600

Loan: Borrow $10,000 at 8% simple interest for 2 years

  • Interest: I = 10,000 × 0.08 × 2 = $1,600
  • Total owed: $10,000 + $1,600 = $11,600

Finding Other Variables

Find Principal: P = I / (rt)

Example: Want to earn $500 interest in 2 years at 5%

  • P = 500 / (0.05 × 2) = 500 / 0.1 = $5,000 needed

Find Rate: r = I / (Pt)

Example: $2,000 earns $240 in 3 years

  • r = 240 / (2000 × 3) = 240 / 6000 = 0.04 or 4%

Find Time: t = I / (Pr)

Example: $8,000 at 6% to earn $1,200

  • t = 1200 / (8000 × 0.06) = 1200 / 480 = 2.5 years

Compound Interest

Interest calculated on principal plus accumulated interest: the most powerful force in wealth building.

Formula: A = P(1 + r)^t

  • A = Final amount
  • P = Principal
  • r = Annual interest rate (as decimal)
  • t = Time (in years)

Interest Earned: I = A − P

Examples

Investment: $10,000 at 6% compounded annually for 5 years

  • A = 10,000(1.06)^5
  • A = 10,000 × 1.3382 = $13,382.26
  • Interest: $13,382.26 − $10,000 = $3,382.26

Compare to Simple Interest:

  • Simple: I = 10,000 × 0.06 × 5 = $3,000
  • Compound earned $382.26 more!

The Rule of 72

Quick estimation: How long to double your money?

Formula: Years to double ≈ 72 / Interest Rate

Examples:

  • 6% rate: 72 / 6 = 12 years to double
  • 8% rate: 72 / 8 = 9 years to double
  • 4% rate: 72 / 4 = 18 years to double

Reverse Use: What rate needed to double in 10 years?

  • Rate ≈ 72 / 10 = 7.2%

Real Application:

  • $50,000 at 7% doubles every ~10 years
  • Age 30: $50,000
  • Age 40: $100,000
  • Age 50: $200,000
  • Age 60: $400,000

Compound Frequency

Interest can compound at different intervals.

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

  • n = compounding periods per year

Compounding Frequencies:

  • Annually: n = 1
  • Semi-annually: n = 2
  • Quarterly: n = 4
  • Monthly: n = 12
  • Daily: n = 365

Example: $5,000 at 6% for 3 years

Annually: A = 5000(1.06)^3 = $5,955.08

Quarterly: A = 5000(1 + 0.06/4)^(4×3) = 5000(1.015)^12 = $5,978.09

Monthly: A = 5000(1 + 0.06/12)^(12×3) = 5000(1.005)^36 = $5,983.40

More frequent compounding = slightly more interest

Loans and Mortgages

Loan Payment Formula

Formula: M = P × [r(1+r)^n] / [(1+r)^n − 1]

  • M = Monthly payment
  • P = Loan principal
  • r = Monthly interest rate (annual rate / 12)
  • n = Total number of payments (years × 12)

Example: $200,000 mortgage at 5% for 30 years

  • r = 0.05 / 12 = 0.00417
  • n = 30 × 12 = 360
  • M = 200,000 × [0.00417(1.00417)^360] / [(1.00417)^360 − 1]
  • M ≈ $1,073.64/month

Total Interest Paid

Total paid: Total = Monthly Payment × Number of Payments
Interest paid: Interest = Total − Principal

Example (continued):

  • Total: $1,073.64 × 360 = $386,510.40
  • Interest: $386,510.40 − $200,000 = $186,510.40
  • Pay almost as much in interest as the principal!

Amortization

Early payments mostly interest, later payments mostly principal.

First Payment Breakdown ($200k at 5%):

  • Interest: $200,000 × 0.00417 = $833.33
  • Principal: $1,073.64 − $833.33 = $240.31
  • Remaining balance: $200,000 − $240.31 = $199,759.69

Why Extra Payments Matter:

  • Extra $100/month saves ~$30,000 in interest over life of loan
  • Pays off loan ~5 years early

APR vs Interest Rate

APR (Annual Percentage Rate): Includes fees and closing costs
Interest Rate: Just the interest

Example:

  • Interest rate: 4.5%
  • Closing costs: $5,000
  • APR: 4.7% (higher, reflects true cost)

Always compare APRs when shopping for loans

Investment Returns

Rate of Return

Formula: Return = [(Final − Initial) / Initial] × 100

Example: Bought stock at $50, sold at $65

  • Return: [($65 − $50) / $50] × 100 = 30%

Annualized Return

For investments held multiple years.

Formula: Annualized Return = [(Final / Initial)^(1/years) − 1] × 100

Example: Invested $10,000, worth $15,000 after 3 years

  • [(15,000 / 10,000)^(1/3) − 1] × 100
  • [1.5^0.333 − 1] × 100
  • [1.1447 − 1] × 100 = 14.47%/year

Average Return vs Compound Annual Growth Rate (CAGR)

Example: Investment over 3 years

  • Year 1: +20%
  • Year 2: −10%
  • Year 3: +15%

Simple Average: (20 − 10 + 15) / 3 = 8.33% ❌ Misleading!

Actual Growth:

  • Start: $1,000
  • After Y1: $1,000 × 1.20 = $1,200
  • After Y2: $1,200 × 0.90 = $1,080
  • After Y3: $1,080 × 1.15 = $1,242

CAGR: [(1,242/1,000)^(1/3) − 1] × 100 = 7.49% ✓ More accurate

Key: Use CAGR for multi-year performance, not simple average

Present Value and Future Value

Future Value (FV)

What money today will be worth in the future.

Formula: FV = PV(1 + r)^t

Example: $15,000 today at 7% for 10 years

  • FV = 15,000(1.07)^10 = $29,519.05

Present Value (PV)

What future money is worth today.

Formula: PV = FV / (1 + r)^t

Example: Need $50,000 in 8 years, 6% return available

  • PV = 50,000 / (1.06)^8 = $31,327.95
  • Invest $31,328 today to have $50,000 in 8 years

Application: Compare job offers

  • Job A: $80,000 now
  • Job B: $95,000 in 2 years
  • Using 5% discount rate: PV = 95,000 / (1.05)^2 = $86,167
  • Job B is worth more (in today's dollars)

Retirement and Annuities

Future Value of Regular Deposits

Formula: FV = PMT × [((1+r)^n − 1) / r]

  • PMT = Regular payment amount
  • r = Interest rate per period
  • n = Number of periods

Example: Save $500/month for 30 years at 7% annual (0.583% monthly)

  • r = 0.07/12 = 0.00583
  • n = 30 × 12 = 360
  • FV = 500 × [((1.00583)^360 − 1) / 0.00583]
  • FV ≈ $612,439

Key Insight: Regular investing builds substantial wealth

  • Total deposited: $500 × 360 = $180,000
  • Growth from compounding: $432,439

Required Monthly Savings

Rearrange to find needed monthly amount.

Formula: PMT = FV × [r / ((1+r)^n − 1)]

Example: Want $1,000,000 in 35 years at 8% annual

  • r = 0.08/12 = 0.00667
  • n = 35 × 12 = 420
  • PMT = 1,000,000 × [0.00667 / ((1.00667)^420 − 1)]
  • PMT ≈ $671/month

Starting early makes a HUGE difference:

  • Start at 25: $671/month
  • Start at 35: $1,386/month (10 years later)
  • Start at 45: $3,204/month (20 years later)

Inflation Impact

Real Return = Nominal Return − Inflation Rate

Example: Investment earns 7%, inflation is 3%

  • Real return: 7% − 3% = 4%
  • This is your actual purchasing power growth

Future Value Adjusted for Inflation:

Formula: Real FV = FV / (1 + inflation)^t

Example: $100,000 in 20 years with 3% inflation

  • Real value = 100,000 / (1.03)^20 = $55,368
  • $100,000 future dollars only buys what $55,368 does today

Planning Tip: Factor inflation into retirement calculations

  • Need $50,000/year today
  • In 30 years with 3% inflation: $50,000 × (1.03)^30 = $121,363/year needed

Investment Comparison

Comparing Investment Options

Option A: $10,000 at 6% simple interest for 5 years
Option B: $10,000 at 5% compounded annually for 5 years

Option A: A = 10,000(1 + 0.06×5) = $13,000
Option B: A = 10,000(1.05)^5 = $12,762.82

Option A wins (simple interest can win for short terms with higher rates)

But at 10 years: Option A: A = 10,000(1 + 0.06×10) = $16,000
Option B: A = 10,000(1.05)^10 = $16,288.95

Option B wins (compound always wins long-term)

Practice Problems

Simple Interest

  1. $8,000 at 4.5% for 3 years. How much interest?
  2. Want $900 interest from $6,000 in 3 years. What rate needed?

Compound Interest

  1. $12,000 at 7% compounded annually for 6 years. Final amount?
  2. Use Rule of 72: How long to double money at 9%?

Loans

  1. What's the monthly payment on a $150,000, 20-year loan at 4.5%?
  2. From #5, how much total interest is paid?

Investment Returns

  1. Stock bought at $40, sold at $52. What's the return percentage?
  2. Investment went from $20,000 to $35,000 in 5 years. What's the annualized return?

Present/Future Value

  1. Need $75,000 in 12 years. How much to invest today at 6%?
  2. Saving $400/month for 25 years at 8% annual. How much at end?

Solutions

  1. $1,080 (8000 × 0.045 × 3)
  2. 5% (900 / (6000 × 3) = 0.05)
  3. $17,950.83 (12000 × 1.07^6)
  4. 8 years (72 / 9)
  5. ~$948.10 (using loan payment formula)
  6. ~$77,544 (948.10 × 240 − 150,000)
  7. 30% ((52 − 40) / 40 × 100)
  8. 11.84% ((35000/20000)^(1/5) − 1 × 100)
  9. $37,264.63 (75000 / 1.06^12)
  10. ~$376,000 (using annuity formula)

Key Takeaways

✓ Compound interest is powerful: start investing early
✓ Time is more valuable than rate: even small differences compound hugely
✓ Frequency matters: more frequent compounding means more growth
✓ Extra loan payments save thousands: especially early in loan life
✓ Factor in inflation: don't plan with nominal dollars
✓ APR reflects true cost: compare APRs, not just interest rates

Real-World Applications

  • Retirement Planning: Calculate required monthly savings
  • Home Buying: Understand mortgage costs and payment structures
  • Debt Management: Compare loan options, plan payoff strategies
  • Investment Decisions: Compare returns, project portfolio growth
  • Business Finance: Evaluate project returns, financing options
  • Education: Calculate student loan costs, 529 plan growth

Next Steps

Move to Chapter 07: Business Metrics to learn how to calculate and interpret key business numbers like ROI, profit margins, break-even points, and growth rates.