Key Formulas Reference

💰 Time Value of Money Formulas

Future Value (FV)

What will my investment be worth in the future?

FV = PV(1 + r)ᵗ

Where:

FV = Future Value
PV = Present Value (current amount)
r = interest rate per period (as decimal)
t = number of time periods

Example:

Invest $1,000 at 8% for 4 years
FV = $1,000(1 + 0.08)⁴
FV = $1,000(1.36049)
FV = $1,360.49

Present Value (PV)

How much do I need to invest today to reach a future goal?

PV = FV / (1 + r)ᵗ

Alternative notation:

PV = FV(1 + r)⁻ᵗ

Where:

PV = Present Value (amount needed today)
FV = Future Value (goal amount)
r = interest rate per period (as decimal)
t = number of time periods

Example:

Want $20,000 in 10 years at 7%
PV = $20,000 / (1 + 0.07)¹⁰
PV = $20,000 / 1.967151
PV = $10,167

Future Value of Annuity (FVA)

What will my regular savings be worth in the future?

FVA = PMT × [(1 + r)ᵗ - 1] / r

Where:

FVA = Future Value of Annuity
PMT = Payment amount per period
r = interest rate per period (as decimal)
t = number of periods

Example:

Save $1,000/year for 10 years at 10%
FVA = $1,000 × [(1 + 0.10)¹⁰ - 1] / 0.10
FVA = $1,000 × [2.59374 - 1] / 0.10
FVA = $1,000 × 15.9374
FVA = $15,937.42

Present Value of Annuity (PVA)

How much do I need today to fund regular future payments?

PVA = PMT × [1 - 1/(1 + r)ᵗ] / r

Where:

PVA = Present Value of Annuity
PMT = Payment amount per period
r = interest rate per period (as decimal)
t = number of periods

Example:

Want $30,000/year for 20 years at 7%
PVA = $30,000 × [1 - 1/(1 + 0.07)²⁰] / 0.07
PVA = $30,000 × [1 - 0.258419] / 0.07
PVA = $30,000 × 10.594
PVA = $317,820

Simple Interest

Basic interest calculation (not compounded)

i = p × r × t = prt

Where:

i = interest earned
p = principal (amount invested)
r = interest rate (as decimal)
t = time in years

Example:

$1,000 at 8% for 4 years
i = $1,000 × 0.08 × 4
i = $320

Compound Interest

Interest earned including interest on interest

Compound Interest = p[(1 + r)ᵗ - 1]

Where:

p = principal
r = interest rate (as decimal)
t = time in years

Example:

$1,000 at 8% for 4 years
i = $1,000[(1 + 0.08)⁴ - 1]
i = $1,000[1.36049 - 1]
i = $1,000 × 0.36049
i = $360.49

Rule of 72

Quick calculation for doubling time

Years to double = 72 / interest rate

Interest rate needed = 72 / years to double

Examples:

At 8%: 72 / 8 = 9 years to double
At 6%: 72 / 6 = 12 years to double
To double in 10 years: 72 / 10 = 7.2% needed

📊 Financial Ratios

Basic Liquidity Ratio

Can I pay for emergencies?

Basic Liquidity Ratio = Liquid Assets / Monthly Expenses

Benchmark: ≥ 3.0

Example:

Liquid assets: $9,000
Monthly expenses: $3,000
Ratio = $9,000 / $3,000 = 3.0 ✓

Interpretation:

Ratio ≥ 3: Good emergency fund
Ratio < 3: Need more liquid savings

Asset-to-Debt Ratio

Do I have enough assets compared to liabilities?

Asset-to-Debt Ratio = Total Assets / Total Liabilities

Benchmark: > 1.0

Example:

Total assets: $50,000
Total liabilities: $30,000
Ratio = $50,000 / $30,000 = 1.67 ✓

Interpretation:

Ratio > 1: Solvent (own more than owe)
Ratio = 1: Break even
Ratio < 1: Insolvent (owe more than own)

Debt Service-to-Income Ratio

Can I meet my debt obligations?

Debt Service-to-Income = Annual Debt Payments / Gross Annual Income

Benchmark: ≤ 0.333 (33.3%)

Example:

Annual debt payments: $18,000
Gross annual income: $60,000
Ratio = $18,000 / $60,000 = 0.30 ✓

Interpretation:

Ratio ≤ 0.333: Manageable debt
Ratio > 0.333: Too much debt

Note: Includes rent or mortgage payments

Investment Assets-to-Total Assets Ratio

Am I investing enough for my age?

Investment Assets Ratio = Investment Assets / Total Assets

Benchmarks (Age-Dependent):

20s: 10%
30s: 11-30%
40s: 30%+
50s+: 50%+

Example:

Age 35, Investment assets: $25,000
Total assets: $100,000
Ratio = $25,000 / $100,000 = 0.25 (25%) ✓

💼 Financial Statement Formulas

Net Worth

True measure of financial wealth

Net Worth = Total Assets - Total Liabilities

Example:

Assets: $222,000
Liabilities: $161,000
Net Worth = $222,000 - $161,000 = $61,000

Cash Flow Surplus/Deficit

Am I living within my means?

Surplus (Deficit) = Total Income - Total Expenses

Example:

Income: $5,000
Expenses: $4,300
Surplus = $5,000 - $4,300 = $700

Interpretation:

Positive: Surplus (save or invest)
Negative: Deficit (need to cut expenses or increase income)

Discretionary Income

Money available after necessities

Discretionary Income = Disposable Income - Necessities

Where:

Disposable Income = Take-home pay (after taxes)
Necessities = Housing, food, utilities, transportation

Example:

Disposable income: $48,000
Necessities: $30,000
Discretionary = $48,000 - $30,000 = $18,000

💵 Income & Inflation Formulas

Nominal Income Change

Percentage change in actual dollars

Nominal Change % = [(New Income - Old Income) / Old Income] × 100

Example:

Old: $37,000, New: $38,600
Nominal = [($38,600 - $37,000) / $37,000] × 100
Nominal = ($1,600 / $37,000) × 100 = 4.3%

Real Income Change

Inflation-adjusted income change

Real Change % = Nominal Change % - Inflation Rate %

Example:

Nominal change: 4.3%
Inflation: 4.0%
Real change = 4.3% - 4.0% = 0.3%

Real Income in Dollars

Actual purchasing power

Real Income = Nominal Income × (1 + Real Change as decimal)

Example:

Nominal income: $37,000
Real change: 0.3% = 0.003
Real income = $37,000 × (1 + 0.003)
Real income = $37,000 × 1.003 = $37,111

🏛️ Tax Formulas

Taxable Income

Income subject to tax

Taxable Income = Gross Income - Adjustments - Deductions - Exemptions

Step-by-step:

Start with Gross Income
Subtract Adjustments → AGI
Subtract Standard or Itemized Deductions
Subtract Exemptions → Taxable Income

Example:

Gross income: $56,000
Adjustments: -$4,000
AGI: $52,000
Itemized deductions: -$8,400
Subtotal: $43,600
Exemptions: -$3,650
Taxable income: $39,950

Progressive Tax Calculation

Tax calculated through brackets

Method: Calculate tax for each bracket separately, then sum

Example:

Taxable income: $50,650
Brackets: 10% ($0-$8,350), 15% ($8,351-$33,950), 25% ($33,951+)

First bracket:  $8,350 × 0.10 = $835.00
Second bracket: $25,600 × 0.15 = $3,840.00
Third bracket:  $16,700 × 0.25 = $4,175.00
Total Tax = $8,850.00

NOT: $50,650 × 0.25 = $12,662.50 ❌

Average Tax Rate

Overall tax burden

Average Tax Rate = Total Taxes Paid / Gross Income

Example:

Total taxes: $8,850
Gross income: $60,000
Average rate = $8,850 / $60,000 = 0.1475 = 14.75%

Note: Always less than marginal tax rate

Tax Savings from Deduction

How much a deduction saves

Tax Savings = Deduction Amount × Marginal Tax Rate

Example:

$1,000 deduction, 25% marginal rate
Savings = $1,000 × 0.25 = $250

Tax Savings from Credit

How much a credit saves

Tax Savings = Credit Amount (full dollar value)

Example:

$1,000 credit
Savings = $1,000 (dollar-for-dollar)

Comparison:

$1,000 deduction at 25% = $250 savings
$1,000 credit = $1,000 savings
Credit is 4x better!

📈 Economic Decision Making

Opportunity Cost

Value of next best alternative

Opportunity Cost = Return from Best Alternative - Return from Chosen Option

Example:

Option A: 5% return on $5,000 = $250
Option B: 8% return on $5,000 = $400
Choose A, opportunity cost = $400 - $250 = $150

Marginal Analysis

Compare incremental costs and benefits

Decision: Choose if Marginal Utility > Marginal Cost

Example:

Regular seat: $90
Premium seat: $150
Marginal cost = $150 - $90 = $60

If extra benefits worth more than $60 → Choose premium
If extra benefits worth less than $60 → Choose regular

🎓 Final Exam Formulas (Topics 9–14)

Topic pages: Final Exam Home.

Insurance: Deductible + coinsurance

Reimbursement = (1 − coinsurance %) × (Loss − Deductible)

Example: Loss $1,350, deductible $100, you pay 20% coinsurance
Loss after deductible = $1,250 → Reimbursement = 0.80 × $1,250 = $1,000 → You pay $350

Homeowners: Replacement-cost partial loss

Reimbursement = (Loss − Deductible) × (Insurance carried ÷ (Replacement value × required %))

Required % is often 80% (0.80) or 100% (1.00) of replacement value—check the policy.

Investment: Total return and yield

Total return = Current income + Capital gain (or loss)

Yield = Total return ÷ Price paid

Example: Paid $4,500; dividends $300; capital gain $500 → Total return $800 → Yield = 800 ÷ 4,500 ≈ 17.78%

Stock ratios

P/E ratio = Market price per share ÷ EPS

EPS = Annual profit (after preferred) ÷ Common shares outstanding

Dividend yield = Annual dividend per share ÷ Market price per share

Payout ratio = Dividends ÷ Earnings  (or DPS ÷ EPS)

P/S ratio = Market capitalization ÷ Annual sales

Lecture rule of thumb: avoid P/S > 1.5; < 0.75 often favorable.

Book value per share = Shareholders' equity ÷ Shares outstanding

Mutual fund: Net asset value (NAV)

NAV = (Market value of assets − Market value of liabilities) ÷ Shares outstanding

Example: Assets $100M, liabilities $5M, 10M shares → NAV = 95M ÷ 10M = $9.50

Mutual fund: Load (amount invested)

Amount invested = Amount paid × (1 − load %)

Example: $10,000 with 8.5% load → 10,000 × 0.915 = $9,150 invested

Real estate

Price-to-rent = Property price ÷ (12 × monthly rent)

Rental yield = (Annual rent ÷ 2) ÷ Purchase price

(Lecture screen: half of rent to non-debt expenses.)

LTV (loan-to-value) = Debt ÷ Property value

Capital gain = Sale price − Purchase price − Capital improvements

(Repairs may be treated separately in examples.)

Annual depreciation (building only) = Building value ÷ Depreciation years

Discounted cash flow (property value):

Price = CF1/(1+r)¹ + CF2/(1+r)² + CF3/(1+r)³ + CF4/(1+r)⁴ + CF5/(1+r)⁵

(Last period often includes sale proceeds.)

Retirement

Income shortfall = Desired retirement income − Social Security − Pension

Annual withdrawal (given PVA factor) = Nest egg ÷ PVA factor(r, years)

Nest egg planning: Find PV of the annual shortfall over retirement years, subtract future value of current savings, then use annuity formula to find required PMT (annual save) to close the gap.

Income replacement (rule of thumb): often 80–100% of pre-retirement gross income (including SS), depending on lifestyle.

Final exam quick table

What you need	Formula
Insurer pays (deductible + coinsurance)	(1 − CP) × (L − D)
Partial loss reimbursement (home)	(L − D) × (I ÷ (RV × 0.8 or 1.0))
Fund share price	NAV
Stock valuation ratios	P/E, EPS, yield, P/S, book value/sh
Rental deal screen	Price-to-rent; rental yield
Leverage	LTV
Max price for property (DCF)	Sum of PV of cash flows
How much can I withdraw?	Nest egg ÷ PVA factor

🎯 Quick Reference Table

What You Want to Find	Formula to Use
Future value of lump sum	FV = PV(1 + r)ᵗ
Present value of lump sum	PV = FV / (1 + r)ᵗ
Future value of payments	FVA = PMT[(1+r)ᵗ-1]/r
Present value of payments	PVA = PMT[1-1/(1+r)ᵗ]/r
Years to double money	72 / interest rate
Emergency fund adequacy	Liquid Assets / Monthly Expenses
Solvency	Total Assets / Total Liabilities
Debt burden	Annual Debt / Gross Income
Investment progress	Investment Assets / Total Assets
Financial wealth	Assets - Liabilities
Living within means	Income - Expenses
Inflation-adjusted income	Nominal % - Inflation %
Income subject to tax	Gross - Adjustments - Deductions - Exemptions
Overall tax burden	Total Tax / Gross Income

⚠️ Common Formula Mistakes

Time Value of Money

❌ Using 8 instead of 0.08 for 8%
❌ Using monthly rate with annual periods
❌ Forgetting to subtract 1 in annuity formulas
❌ Confusing which formula to use (PV vs FV)

Financial Ratios

❌ Using monthly debt instead of annual
❌ Using net income instead of gross
❌ Including non-liquid assets in liquidity ratio

Taxes

❌ Multiplying entire income by marginal rate
❌ Confusing deductions with credits
❌ Forgetting progressive tax calculation

💡 Formula Tips

Always write the formula first before plugging in numbers
Convert percentages to decimals (8% = 0.08)
Match time periods (annual rate needs annual periods)
Use parentheses for order of operations
Show your work for partial credit
Label your answer with correct units
Double-check your calculation
Verify reasonableness of answer

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