Analytical Methods

The whole quantitative unit on one sheet: the time value formulas, the risk statistics, and the ratios the exam loves.

The One-Liners That Win Points

• NPV is dollars; IRR is a percentage. When the two conflict on mutually exclusive projects, NPV wins because it assumes reinvestment at the discount rate (realistic), while IRR assumes reinvestment at the IRR itself (often unrealistic).
• IRR is the discount rate that makes NPV equal zero. For a bond, IRR equals yield to maturity (YTM).
• Standard deviation measures total risk (systematic plus unsystematic); beta measures only systematic (market) risk. Diversification cuts unsystematic risk but does NOT reduce beta.
• Sharpe ratio uses standard deviation (total risk) in the denominator; the numerator is excess return above the risk-free rate.
• Alpha is EXTRA return above the risk-adjusted expectation, not raw return. Positive alpha means the manager added value.
• Correlation of +1.0 gives NO diversification benefit; maximum benefit comes from negative correlation. Zero correlation still helps.
• Mean > median = right-skewed (high outliers pulled it up); mean < median = left-skewed (low outliers dragged it down).
• Quick ratio is the stricter liquidity test because it removes inventory; it is always equal to or less than the current ratio.

Numbers to Lock In

• Future value: FV = PV × (1 + r)^n
• Present value: PV = FV / (1 + r)^n (as periods rise, PV falls)
• Rule of 72: Years to double = 72 / annual rate of return (8% doubles in ~9 years; 6% in ~12)
• NPV = Present Value of Cash Flows − Cost of Investment. Positive = accept, negative = reject, zero = earns exactly the required return
• IRR decision rule: accept if IRR > required (hurdle) rate; reject if below
• Normal distribution: ±1 standard deviation = 68% of returns, ±2 = 95%, ±3 = 99%
• Correlation coefficient ranges −1.0 to +1.0; R-squared = correlation coefficient squared (percent of movement explained by the benchmark)
• Beta: market rise 10% × beta 1.2 = stock rises 12%; S&P 500 beta = 1.0
• Sharpe Ratio: $\text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p}$ (risk-free rate = 91-day/3-month Treasury bill)
• Current Ratio = Current Assets / Current Liabilities
• Quick Ratio = (Current Assets − Inventory) / Current Liabilities
• Debt-to-Equity = Total Liabilities / Shareholders' Equity (above 1.0 = more debt than equity)
• Price-to-Earnings (P/E) = Market Price Per Share / Earnings Per Share (EPS)
• Price-to-Book (P/B) = Market Price Per Share / Book Value Per Share

Top Gotchas

• More time is not always more value: longer periods increase future value (more compounding) but decrease present value (more discounting).
• A zero NPV is not zero profit; it means the investment earns exactly the discount rate used.
• Beta 1.0 still carries unsystematic risk; it just moves with the market on average.
• The median ignores actual values (only position matters); the mean is skewed by outliers. If a distribution has no repeating value, there is no mode.
• P/E is meaningless with negative earnings; a high P/E can reflect justified growth, a low P/E legitimate concern. Always compare to peers.
• Book value is an accounting (historical-cost) measure; P/B below 1.0 can signal a bargain OR distress.

One-Breath Recap

Time value of money is the backbone: future value compounds (FV = PV × (1 + r)^n), present value discounts, and the Rule of 72 approximates doubling time. Accept a positive NPV in dollars or an IRR above the hurdle rate in percent, and prefer NPV when they conflict. Statistics split into central tendency (mean, median, mode) and risk: standard deviation for total risk, beta for systematic, Sharpe for return per unit of total risk, alpha for excess return, and correlation for diversification. Round it out with liquidity ratios (current, quick), the debt-to-equity leverage ratio, and the P/E and P/B valuation ratios.

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.