INTEREST RATES AND RETURN MEASUREMENT :
- Interest Rates represent the time value of money and are interpreted as required rates of return, discount rates, or opportunity costs.
- An interest rate is composed of a real risk-free rate and premiums for default risk, liquidity risk, and maturity risk.
- The nominal risk-free rate (e.g., T-bills) includes an inflation premium; the real risk-free rate excludes inflation and default risk.
- Holding Period Return (HPR) is the percentage gain over a period, including price changes and income (like dividends).
- Returns over multiple periods are compounded; annualized returns standardize these over a year.
- The arithmetic mean is the average of returns and is an unbiased estimator of the true mean.
- The geometric mean reflects compounded growth and is always ≤ arithmetic mean; it’s used when measuring multi-period performance.
- The harmonic mean is suitable for averaging share prices from equal monetary investments over time.
- Order of means (for unequal values): Harmonic < Geometric < Arithmetic.
- Outliers can distort averages; trimmed and winsorized means are used to reduce their impact.
TIME-WEIGHTED AND MONEY-WEIGHTED RETURNS:
- Money-weighted return (MWR) is equivalent to the internal rate of return (IRR) for a portfolio, accounting for all cash inflows and outflows.
- MWR considers deposits as inflows and withdrawals (including ending value) as outflows, requiring IRR calculation via financial calculators.
- Time-weighted return (TWR) measures the compound rate of growth of $1 over the evaluation period, independent of cash flow timing.
- TWR is calculated by dividing the period into subperiods at each cash flow, computing holding period returns (HPRs), and taking their geometric mean.
- In the given example, TWR (15.84%) exceeded MWR (13.86%) due to higher weighting of poorer performance in Year 2 by MWR.
- TWR is preferred in investment management, as it reflects the manager’s skill without being distorted by client-driven cash flows.
- MWR is appropriate when the portfolio manager controls the timing of cash inflows and outflows.
COMMON MEASURES OF RETURN:
- Annualized Returns: Returns are often expressed annually for comparability; short- or long-period returns can be annualized using appropriate formulas.
- Impact of Compounding: Increasing compounding frequency raises the effective interest rate, which increases future value and decreases present value of cash flows.
- Continuous Compounding: Achieved via natural logarithms; continuously compounded returns are additive across time periods and based on the price relative (end value / start value).
- Gross vs. Net Returns:
- Gross return: Before management fees.
- Net return: After deducting management fees; both exclude trading costs.
- Tax Considerations:
- Pretax nominal return: Before taxes.
- After-tax nominal return: Post-tax.
- Different income types (dividends, interest) are taxed differently.
- Real Return: Adjusted for inflation; reflects change in purchasing power, calculated approximately as nominal return minus inflation.
- Leveraged Returns: Amplify gains/losses using borrowed funds; calculated as return on the investor’s actual cash investment, factoring in borrowing costs.
KEY CONCEPTS:
- Interest Rate Concepts
- Interest rates reflect return requirements, discount rates, or opportunity costs.
- Nominal rate ≈ real risk-free rate + expected inflation.
- Risk premiums (default, liquidity, maturity) are added to reflect security risks.
- Return Measures
- Holding period return assesses return over time.
- Arithmetic mean is a simple average; geometric mean reflects compounded growth.
- Harmonic mean is useful for averaging prices; trimmed/winsorized means reduce outlier effects.
- Money-Weighted vs. Time-Weighted Return
- Money-weighted return is IRR based on actual cash flows.
- Time-weighted return measures compound growth, ideal for performance evaluation.
- Choice depends on who controls cash flows.
- Annualization
- Returns are often expressed annually; continuous compounding can be used.
- Types of Returns
- Gross return excludes fees; net return includes management/administration fees.
- Pretax and after-tax returns differ by tax effect.
- Real return adjusts for inflation; leveraged return reflects gains/losses relative to investor’s equity.