Key Formulas & Frameworks
Future Value: FV = PV × (1 + r)n
Present Value: PV = FV ÷ (1 + r)n
Where r = interest rate per period, n = number of periods
Accept if NPV > 0; reject if NPV < 0
Accept if IRR > cost of capital; reject if IRR < cost of capital
Rf = risk-free rate, β = systematic risk, Rm = market return
WACC = (E/V × Re) + (D/V × Rd × (1 − Tc))
E = equity value, D = debt value, V = E+D, Tc = corporate tax rate
D₁ = next year's dividend, r = required return, g = constant growth rate
Portfolio Variance: σp² = Σ Σ wiwjσij
w = weight of asset i, σij = covariance between assets i and j
Measures risk-adjusted return; higher is better
C = S₀N(d₁) − Xe−rTN(d₂)
d₁ = [ln(S₀/X) + (r + σ²/2)T] ÷ (σ√T)
d₂ = d₁ − σ√T
C = coupon payment, F = face value, y = yield to maturity, n = periods
ROE = Profit Margin × Asset Turnover × Equity Multiplier
Enterprise Value: EV = Market Cap + Total Debt − Cash
Core Concepts
Time Value of Money (TVM)
A dollar today is worth more than a dollar in the future due to its potential earning capacity. Core principles: compounding (growing present value forward), discounting (bringing future value back to present), and the opportunity cost of capital.
Risk-Return Tradeoff
The principle that potential return rises with an increase in risk. Low-risk investments (government bonds) offer modest returns; high-risk investments (stocks, venture capital) offer higher expected returns to compensate for greater uncertainty. Beta (β) measures systematic risk relative to the market.
Capital Structure
The mix of debt and equity used to finance a company's operations. The Modigliani-Miller Theorem (without taxes) states that firm value is independent of capital structure. With taxes, debt provides a tax shield (interest is tax-deductible), creating an optimal capital structure that balances tax benefits against bankruptcy costs.
Dividend Policy
The decision of how much earnings to distribute to shareholders vs. retain for reinvestment. Theories include: Irrelevance Theory (MM), Bird-in-Hand Theory (Gordon/Lintner — investors prefer certain dividends), and Tax Preference Theory (capital gains taxed lower than dividends).
Market Efficiency
The Efficient Market Hypothesis (EMH) states that asset prices fully reflect all available information. Three forms: Weak (prices reflect past prices), Semi-Strong (prices reflect all public information), Strong (prices reflect all public and private information). Implications: active management cannot consistently beat the market.
Derivatives & Hedging
Financial instruments whose value derives from an underlying asset. Types: Futures (standardized, exchange-traded), Options (right but not obligation), Swaps (exchange cash flows). Used for hedging (reducing risk), speculation (betting on price movements), and arbitrage (exploiting price discrepancies).
Behavioral Finance
The study of psychological influences on investor behavior. Key biases: Overconfidence, Loss Aversion (losses hurt ~2x more than gains please), Herding, Anchoring, Confirmation Bias, and Disposition Effect (selling winners too early, holding losers too long).
Financial Intermediation
The role of financial institutions in connecting savers and borrowers. Banks transform short-term deposits into long-term loans (maturity transformation), diversify risk (pooling), and reduce transaction costs. Capital markets (stock and bond markets) provide direct financing.
Worked Examples & Case Studies
Example 1: Net Present Value (NPV)
Problem: A project requires $100,000 initial investment and generates $30,000/year for 5 years. Cost of capital is 10%. Should the project be accepted?
Solution: NPV = 30,000 × [1 − (1.10)−5] ÷ 0.10 − 100,000 = 30,000 × 3.7908 − 100,000 = 113,724 − 100,000 = $13,724. Since NPV > 0, accept the project.
Example 2: CAPM Calculation
Problem: Risk-free rate = 3%, market return = 10%, stock beta = 1.5. Calculate expected return.
Solution: E(Ri) = 3% + 1.5 × (10% − 3%) = 3% + 1.5 × 7% = 3% + 10.5% = 13.5%. The stock should return 13.5% to compensate for its systematic risk.
Example 3: WACC Calculation
Problem: A company has $60 million equity (cost = 12%) and $40 million debt (cost = 6%, tax rate = 30%). Calculate WACC.
Solution: V = 100. E/V = 0.60, D/V = 0.40. WACC = (0.60 × 12%) + (0.40 × 6% × 0.70) = 7.2% + 1.68% = 8.88%. This is the minimum return the company must earn on its investments.
Case Study: 2008 Financial Crisis & Subprime Mortgages
Background: Banks issued mortgages to borrowers with poor credit (subprime), packaged them into Mortgage-Backed Securities (MBS) and Collateralized Debt Obligations (CDOs), and sold them globally. Credit rating agencies gave AAA ratings to risky tranches.
Financial Failures: (1) Moral hazard — originators didn't bear default risk, (2) Information asymmetry — investors couldn't assess underlying quality, (3) Excessive leverage — banks operated at 30:1 debt-to-equity, (4) Systemic risk — interconnectedness meant one failure cascaded. Result: Global recession, $20 trillion in wealth destroyed, Dodd-Frank Act regulation.
Case Study: Apple Share Buybacks
Background: Apple initiated a $100 billion share buyback program in 2018, returning capital to shareholders through repurchases and dividends.
Financial Analysis: (1) Buybacks reduce shares outstanding, increasing EPS, (2) Signal effect — management believes stock is undervalued, (3) Tax efficiency vs. dividends, (4) Capital structure optimization — Apple had massive cash reserves exceeding $200B. Effect: EPS grew faster than net income, supporting stock price appreciation from $40 (split-adjusted, 2018) to $190+ (2024).
Practice Problems
Problem 1: How much will $10,000 grow to in 10 years at 7% annual interest, compounded annually?
Answer: FV = 10,000 × (1.07)10 = 10,000 × 1.9672 = $19,672
Problem 2: What is the present value of $50,000 received 5 years from now, discounted at 8%?
Answer: PV = 50,000 ÷ (1.08)5 = 50,000 ÷ 1.4693 = $34,029
Problem 3: A stock has beta = 1.2, risk-free rate = 4%, market risk premium = 6%. Calculate expected return using CAPM.
Answer: E(R) = 4% + 1.2 × 6% = 11.2%
Problem 4: A bond with face value $1,000 pays 5% coupon annually and matures in 3 years. If YTM is 6%, what is the bond price?
Answer: P = 50/1.06 + 50/1.06² + 1,050/1.06³ = 47.17 + 44.50 + 881.60 = $973.27
Problem 5: Company X has net income of $50M, sales of $200M, assets of $400M, equity of $200M. Use DuPont analysis to find ROE and its components.
Answer: Profit Margin = 50/200 = 25%. Asset Turnover = 200/400 = 0.5. Equity Multiplier = 400/200 = 2.0. ROE = 25% × 0.5 × 2.0 = 25%
Problem 6: Calculate the Sharpe Ratio for a portfolio with return 12%, risk-free rate 3%, and standard deviation 15%.
Answer: S = (12% − 3%) ÷ 15% = 9% ÷ 15% = 0.60
Problem 7: A company expects to pay a $2 dividend next year. Dividends are expected to grow at 5% indefinitely. Required return is 10%. Calculate the stock price using Gordon Growth Model.
Answer: P₀ = 2 ÷ (0.10 − 0.05) = 2 ÷ 0.05 = $40
Problem 8: A portfolio has $60,000 in Stock A (E(R)=10%, σ=15%) and $40,000 in Stock B (E(R)=8%, σ=10%). Correlation is 0.3. Calculate portfolio expected return and standard deviation.
Answer: E(Rp) = 0.6 × 10% + 0.4 × 8% = 9.2%. Cov = 0.3 × 0.15 × 0.10 = 0.0045. σp² = 0.6²×0.15² + 0.4²×0.10² + 2×0.6×0.4×0.0045 = 0.0081 + 0.0016 + 0.00216 = 0.01186. σp = 10.9%