Quantitative Methods

An interest rate can be viewed as (1) a required rate of return investors demand to delay consumption, (2) a discount rate used to find present value of future cash flows, and (3) an opportunity cost of consuming or investing funds now. All three describe the same number from different perspectives.

Nominal risk-free rate ≈ real risk-free rate + expected inflation premium. For example, a 2% real rate plus 3% expected inflation gives an approximate 5% nominal risk-free rate (exact form: (1.02)(1.03) − 1 = 5.06%).

Required return = real risk-free rate + inflation premium + default-risk premium + liquidity premium + maturity premium. Each premium compensates the investor for an additional source of risk beyond the real time value of money.

HPR = (P1 − P0 + income) / P0 = (ending price − beginning price + cash flow) / beginning price. Example: buy at 100, sell at 105, receive 2 dividend: HPR = (105 − 100 + 2)/100 = 7%.

The geometric mean measures the compound annual growth rate actually earned over multiple periods and is correct for past performance; the arithmetic mean is the best estimate of a single future period's expected return. Geometric mean ≤ arithmetic mean, with equality only when all returns are identical.

Geometric mean = [(1 + R1)(1 + R2)...(1 + Rn)]^(1/n) − 1. Example: returns of +50% and −50%: [(1.5)(0.5)]^(1/2) − 1 = 0.75^0.5 − 1 = −13.4%, reflecting the actual compound loss.