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Quantitative MethodsModule 1 of 11

Rates and Returns

8

Concepts

6

Formulas

2

Decisions

6

Quiz Questions

Key Concepts

8 concepts covered in this module.

Interest Rate Components

Nominal rate = Real risk-free rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium.

Holding Period Return (HPR)

Total return earned from holding an asset over a single period, including price change and income.

Arithmetic vs Geometric Mean

Arithmetic mean is the simple average of returns; geometric mean accounts for compounding. Geometric ≤ Arithmetic always.

Harmonic Mean

Used for averaging ratios like P/E or dollar-cost averaging. Always ≤ Geometric mean ≤ Arithmetic mean.

Money-Weighted Return (MWR)

IRR of all cash flows; sensitive to timing and size of external cash flows. Reflects investor experience.

Time-Weighted Return (TWR)

Geometric mean of sub-period returns; not affected by external cash flows. Reflects manager performance.

Continuously Compounded Return

rcc = ln(1 + HPR). Allows additive compounding across time periods.

Gross vs Net Return

Gross = before management fees; Net = after all fees. Pre-tax vs after-tax returns also critical.

Formulas

6 essential formulas for this module.

Holding Period Return

HPR = (P1 - P0 + D1) / P0

Where: P0 = beginning price, P1 = ending price, D1 = income/dividend

Geometric Mean Return

RG = [(1+R1)(1+R2)...(1+Rn)]1/n - 1

Where: Ri = return in period i, n = number of periods

Harmonic Mean

H = n / Σ(1/Xi)

Where: n = number of observations, Xi = individual values

Annualized Return

Rannual = (1 + Rperiod)c - 1

Where: c = number of periods per year

Continuously Compounded Return

rcc = ln(1 + HPR) = ln(P1/P0)

Where: ln = natural logarithm

Real Return (exact)

(1 + Rnominal) = (1 + Rreal)(1 + π)

Where: π = inflation rate

Decision Frameworks

2 decision frameworks to guide your analysis.

Which mean return to use?

  • Arithmetic mean: for forward-looking expected single-period returns
  • Geometric mean: for historical multi-period compound performance
  • Harmonic mean: for dollar-cost averaging or averaging P/E ratios

Money-weighted vs Time-weighted return?

  • MWR when evaluating the investor's actual experience
  • TWR when evaluating the portfolio manager's skill

Mind Map

Visual overview of how concepts connect in this module.

Rates and Returns
Interest Rate Components
Real risk-free rate
Inflation premium
Default risk premium
Liquidity premium
Maturity premium
Return Measures
Holding Period Return
Arithmetic Mean
Geometric Mean
Harmonic Mean
Continuously Compounded
MWR vs TWR
MWR = IRR of cash flows
TWR = geometric linking of sub-periods
MWR for investor experience
TWR for manager evaluation
Annualized Returns
Non-annual compounding
EAR vs stated rate
Continuous compounding: er
Gross/Net/Real
Gross: before fees
Net: after fees
Real: adjusted for inflation
Pre-tax vs After-tax

Study Rates and Returns

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