Analytical Methods

The whole unit on one sheet: time value of money, the risk-and-return statistics, and the ratios the exam loves.

Time Value of Money

• A dollar received today is worth more than a dollar received in the future, because today's dollar can be invested and earn a return. The return you give up by waiting is opportunity cost.
• Future value (FV): what a current investment grows to at a given rate over time. Driven by compounding (earning returns on your returns). Higher rate AND longer time both increase it.
• Net present value (NPV): present value of an investment's future cash inflows minus the initial cost.
  • Formula: NPV = Present Value of Cash Inflows - Initial Investment
  • NPV > 0 = adds value = Accept; NPV < 0 = destroys value = Reject; NPV = 0 = earns exactly the required rate = Indifferent.
• Internal rate of return (IRR): the discount rate that makes NPV of all cash flows equal to zero.
  • Accept if IRR > required rate of return (hurdle rate); reject if IRR < required rate of return.
  • For bonds, IRR equals the yield to maturity (YTM).
• When NPV and IRR conflict on mutually exclusive projects, follow NPV (it measures actual dollar value added). For independent projects they always agree.

Descriptive Statistics: Central Tendency

• Mean (arithmetic average): sum of all values divided by the number of values. Most common average return, but sensitive to outliers.
• Median: the middle value in ordered data. Resistant to outliers; the better "typical" measure when data is skewed.
• Mode: the most frequent value; a set can have multiple modes or none.
• Range: highest value minus lowest; simplest dispersion measure, ignores everything in between.

Descriptive Statistics: Risk and Return

• Standard deviation: how far individual returns typically deviate from the mean. Measures total risk (systematic + unsystematic). Higher = more volatility = more risk.
• Beta: sensitivity to overall market moves; measures systematic (market) risk only.
  • Beta = 1.0 moves with the market; > 1.0 more volatile (aggressive); < 1.0 less volatile (defensive); 0 no correlation (Treasury bills); < 0 moves opposite the market.
  • A beta of 1.5 is expected to move 1.5% for every 1% market move.
• Alpha: excess return relative to what beta predicted; the primary measure of active management skill. Positive = beat the risk-adjusted benchmark.
  • Jensen's Alpha formula: Alpha = Actual Return - [Risk-Free Rate + Beta x (Market Return - Risk-Free Rate)]
• Sharpe ratio: excess return per unit of total risk.
  • Formula: Sharpe Ratio = (Portfolio Return - Risk-Free Rate) / Standard Deviation
  • Higher = better risk-adjusted performance. Uses standard deviation, NOT beta.
• Correlation: how two investments move relative to each other, ranging from -1.0 to +1.0.
  • +1.0 perfect positive (no diversification benefit); 0 no relationship (moderate); -1.0 perfect negative (maximum benefit).
  • Combining low- or negative-correlation assets reduces risk; anything below +1.0 helps.
  • Diversification eliminates unsystematic (company-specific) risk only; systematic (market) risk always remains. Roughly 30 or more securities diversifies away most unsystematic risk.

Financial Ratios: Liquidity and Leverage

• Current ratio = Current Assets / Current Liabilities. Current assets include cash, receivables, marketable securities, and inventory. Above 1.0 = adequate liquidity; below 1.0 = may struggle to pay short-term debts.
• Quick ratio (acid-test) = (Current Assets - Inventory) / Current Liabilities. Excludes inventory (the least liquid current asset); more conservative; better gauge of immediate payment ability. 1.0 or higher is generally healthy.
• Debt-to-equity ratio = Total Debt / Shareholders' Equity. Measures financial leverage. Higher = more debt reliance = higher financial risk; leverage amplifies both gains and losses.
• Ratios are only meaningful in context: compare to industry peers, competitors, and the company's own history, and read multiple ratios together.

Valuation Ratios

• Price-to-earnings (P/E) ratio = Market Price Per Share / Earnings Per Share (EPS). Shows how much investors pay per dollar of earnings; a P/E of 20 = $20 paid per $1 of earnings.
  • High P/E: strong expected growth OR overvalued. Low P/E: possibly undervalued OR declining prospects.
  • Trailing P/E uses past 12 months of actual earnings; forward P/E uses estimated future earnings.
• Price-to-book (P/B) ratio = Market Price Per Share / Book Value Per Share. Book value = total assets minus total liabilities (shareholders' equity), divided by shares outstanding.
  • P/B < 1.0 trades below book value (maybe undervalued, maybe impaired assets); P/B > 1.0 market values it above net asset value; P/B = 1.0 price equals book value.
  • Most useful for capital-intensive, asset-heavy industries (banking, insurance, manufacturing, real estate); less useful for asset-light tech and service firms.

The One-Liners That Win Points

• NPV is a dollar amount; IRR is a percentage. The exam tests whether you can distinguish the two.
• IRR and YTM are the same concept applied differently (YTM is IRR applied to a bond's cash flows).
• Standard deviation = total risk; beta = systematic risk only. "Total risk" or "volatility" points to standard deviation; "market risk" points to beta.
• The Sharpe ratio uses standard deviation, not beta. Per unit of systematic risk would be the Treynor ratio.
• Positive alpha means beating expectations after adjusting for risk, not merely that the fund went up.
• The quick ratio is always equal to or less than the current ratio for the same company.
• P/E uses earnings per share; P/B uses book value per share. These are often confused.
• All ratios are meaningless in isolation and require comparison to peers, competitors, and history.

Top Gotchas

• Doubling the time period does NOT double the future value. Compounding grows it faster than a simple doubling.
• NPV > 0 does not mean risk-free. It means expected returns exceed the required rate after accounting for the discount rate.
• IRR assumes reinvestment at the IRR itself, which may be unrealistic for very high-IRR projects.
• When a question mentions "skewed data" or "outliers," pick the median, not the mean.
• Diversification does NOT reduce all risk; systematic (market) risk remains no matter how many securities you hold.
• Correlation of +1.0 means zero diversification benefit, not maximum.
• Leverage is not liquidity. A company can be highly leveraged yet still liquid.
• A current ratio below 1.0 is a warning sign, not a guarantee of bankruptcy (credit lines and other resources may exist).
• A low P/E does not automatically mean "buy," and a high P/E does not automatically mean overvalued. Context is everything.
• P/B < 1.0 does not automatically mean undervalued; assets may be overstated or impaired.

Memory Aid: Risk-Adjusted Metrics

• Standard deviation = Sum of all risk (total)
• Beta = market risk only (Broad market sensitivity)
• Alpha = Added value by the manager
• Sharpe ratio = Standard deviation in the denominator (total risk)

One-Breath Recap

Analytical methods are the math behind every recommendation: time value of money says a dollar today beats a dollar tomorrow, so you value cash flows with future value, net present value (accept when above zero), and internal rate of return (which equals a bond's yield to maturity, and when it conflicts with NPV on mutually exclusive projects you follow NPV). Descriptive statistics measure the ride: standard deviation is total risk, beta is systematic risk only, alpha is manager skill, the Sharpe ratio is return per unit of total risk, and low or negative correlation drives diversification that kills unsystematic but never systematic risk. Financial ratios size up the company (current and quick for liquidity, debt-to-equity for leverage), and valuation ratios ask whether the price is right (price-to-earnings per dollar of earnings, price-to-book against net asset value). The thread through all of it: every ratio and metric is meaningless in isolation, so always compare to peers, competitors, and history.

Need more than the recap? This is a condensed summary. If it is not enough, read the full Analytical Methods unit for the complete lesson.