Personal Finance Math

Why Financial Math Matters

Personal finance: Using math to make smart money decisions

Applications:

• Saving and investing
• Taking loans
• Budgeting
• Retirement planning

Key skill: Understanding how money grows or shrinks over time

Simple Interest

Simple interest: Interest calculated only on principal

Formula: I = Prt

Where:

• I = interest earned
• P = principal (initial amount)
• r = annual interest rate (as decimal)
• t = time (in years)

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

Example 1: Simple Interest

Invest $1000 at 5% simple interest for 3 years

Calculate interest:

I = Prt
I = 1000(0.05)(3)
I = 150

Total amount: $1000 + $150 = $1150

Example 2: Find Rate

Borrow $2000, pay back $2300 after 2 years

Interest paid: $2300 - $2000 = $300

Find rate:

300 = 2000(r)(2)
300 = 4000r
r = 0.075 = 7.5%

Interest rate: 7.5% per year

Compound Interest

Compound interest: Interest calculated on principal PLUS accumulated interest

More realistic for savings accounts and investments

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

Where:

• A = final amount
• P = principal
• r = annual interest rate (as decimal)
• n = number of times interest compounds per year
• t = time in years

Example 1: Annual Compounding

Invest $1000 at 5% compounded annually for 3 years

n = 1 (compounds once per year)

Calculate:

A = 1000(1 + 0.05/1)^(1·3)
A = 1000(1.05)³
A = 1000(1.157625)
A ≈ $1157.63

Compare to simple interest: $1150 Earn extra: $7.63 from compounding

Example 2: Monthly Compounding

Invest $5000 at 6% compounded monthly for 2 years

n = 12 (compounds 12 times per year)

Calculate:

A = 5000(1 + 0.06/12)^(12·2)
A = 5000(1 + 0.005)^24
A = 5000(1.005)^24
A = 5000(1.127159)
A ≈ $5635.80

Interest earned: $5635.80 - $5000 = $635.80

Continuous Compounding

Maximum compounding: Interest compounds every instant

Formula: A = Pe^(rt)

Where e ≈ 2.71828 (Euler's number)

Example: Continuous Compounding

Invest $1000 at 5% compounded continuously for 3 years

Calculate:

A = 1000·e^(0.05·3)
A = 1000·e^0.15
A = 1000·1.16183
A ≈ $1161.83

Slightly more than monthly compounding

Loan Payments (Amortization)

Amortization: Paying off loan with regular payments

Monthly payment formula:

M = P[r(1+r)^n] / [(1+r)^n - 1]

Where:

• M = monthly payment
• P = loan amount (principal)
• r = monthly interest rate (annual rate / 12)
• n = number of payments

Example: Car Loan

Loan: $20,000 at 6% annual interest for 5 years

Monthly rate: r = 0.06/12 = 0.005 Number of payments: n = 5 × 12 = 60

Calculate monthly payment:

M = 20000[0.005(1.005)^60] / [(1.005)^60 - 1]
M = 20000[0.005(1.34885)] / [1.34885 - 1]
M = 20000[0.00674425] / [0.34885]
M = 134.885 / 0.34885
M ≈ $386.66

Total paid: $386.66 × 60 = $23,199.60 Total interest: $23,199.60 - $20,000 = $3,199.60

Credit Cards

Credit cards: Revolving credit with monthly interest

APR (Annual Percentage Rate): Yearly interest rate

Daily rate: APR / 365

Most cards compound daily

Example: Credit Card Balance

Balance: $1000 APR: 18% Pay minimum $25/month

Monthly interest rate: 0.18/12 = 0.015

After 1 month (if no payment):

Balance = 1000(1.015) = $1015

After 1 month with $25 payment:

New balance = 1000(1.015) - 25 = $990

Important: Paying only minimum takes YEARS to pay off

Savings Goals

Future value planning: How much to save regularly

Annuity formula: FV = PMT × [(1 + r)^n - 1] / r

Where:

• FV = future value (goal)
• PMT = regular payment
• r = interest rate per period
• n = number of periods

Example: Save for College

Goal: $50,000 in 10 years Interest: 5% annual, compounded monthly r = 0.05/12, n = 120 months

Find monthly payment:

50000 = PMT × [(1 + 0.05/12)^120 - 1] / (0.05/12)
50000 = PMT × [1.64701 - 1] / 0.004167
50000 = PMT × 155.28
PMT ≈ $322

Save $322 per month to reach goal

Present Value

Present value: How much future money is worth today

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

Use: Compare money received at different times

Example: Lottery Payout

Option A: $1,000,000 today Option B: $1,200,000 in 5 years

Assume 6% annual return

Present value of Option B:

PV = 1,200,000 / (1.06)^5
PV = 1,200,000 / 1.33823
PV ≈ $896,663

Option A is better! ($1,000,000 today > $896,663 equivalent)

Rule of 72

Quick estimate: How long to double money

Formula: Years to double ≈ 72 / (interest rate)

Example: Doubling Time

6% interest: 72 / 6 = 12 years

8% interest: 72 / 8 = 9 years

3% interest: 72 / 3 = 24 years

Not exact, but close approximation

Budgeting Math

Budget: Plan for income and expenses

Key equation: Income - Expenses = Savings

50/30/20 rule:

• 50% needs (housing, food, utilities)
• 30% wants (entertainment, dining out)
• 20% savings and debt repayment

Example: Monthly Budget

Income: $3000/month

50/30/20 allocation:

• Needs: 0.50 × $3000 = $1500
• Wants: 0.30 × $3000 = $900
• Savings: 0.20 × $3000 = $600

Investment Returns

Return on investment (ROI): Profit as percentage

Formula: ROI = (Final Value - Initial Value) / Initial Value × 100%

Example: Stock Investment

Buy stock: $2000 Sell stock: $2500

Calculate ROI:

ROI = (2500 - 2000) / 2000 × 100%
ROI = 500 / 2000 × 100%
ROI = 25%

25% return on investment

Inflation

Inflation: Decrease in purchasing power over time

Real interest rate: Nominal rate - Inflation rate

Example: Real Return

Investment earns 7% per year Inflation is 3% per year

Real return: 7% - 3% = 4%

Your purchasing power grows 4% per year

Tax Considerations

Tax brackets: Different rates for different income levels

Marginal rate: Rate on last dollar earned

Effective rate: Total tax / Total income

Example: Progressive Tax

Income: $60,000

Brackets:

• First $10,000: 10% = $1,000
• Next $30,000: 15% = $4,500
• Next $20,000: 20% = $4,000

Total tax: $9,500 Effective rate: $9,500 / $60,000 = 15.8% Marginal rate: 20%

Retirement Planning

Compound growth over decades

Earlier you start, more time for growth

Example: Start Early vs Start Late

Scenario A: Save $200/month from age 25 to 35 (10 years) Scenario B: Save $200/month from age 35 to 65 (30 years)

Both earn 7% annual return

Scenario A:

• Total invested: $200 × 120 = $24,000
• Value at 65: Grows from age 35 to 65 (30 more years)
• Approximate value at 65: $200,000

Scenario B:

• Total invested: $200 × 360 = $72,000
• Approximate value at 65: $240,000

Starting early matters!

Practice