Returns

Understanding which return measure to use (and when) is one of the most frequently tested concepts in this unit. Each measure answers a slightly different question, and the exam will test whether you can match the right measure to the right situation.

Total Return

• Total return captures all sources of investment gain: capital appreciation (or depreciation) plus income (dividends, interest)
• Formula: (Ending Value - Beginning Value + Income) / Beginning Value
• Most comprehensive single measure of investment performance
• Includes both realized and unrealized gains

Example: You buy a stock at $50, receive $2 in dividends, and sell at $55. Total Return = ($55 - $50 + $2) / $50 = 14%

Holding Period Return (HPR)

• Holding period return is the total return earned over the entire period the investment was held
• Uses the same formula as total return: (Ending Value - Beginning Value + Income) / Beginning Value
• Does not annualize; it simply measures the raw percentage gain or loss from purchase to sale
• Useful for measuring the actual result of a specific investment over a defined period

Exam Tip: Gotchas

• HPR and total return use the same formula. The key distinction is that HPR is specifically tied to the actual holding period and is not converted to an annual rate. A 30% HPR over 3 years is NOT the same as a 10% annual return.

Annualized Return

• Converts returns over any period into an equivalent annual rate
• Allows comparison of investments held for different time periods
• Uses geometric (compound) averaging, not arithmetic averaging
• Arithmetic average overstates the true compound growth rate whenever periodic returns vary (the two are equal only if every period's return is identical)

Why geometric matters: If an investment returns +50% in year one and -50% in year two, the arithmetic average is 0%. But you actually lost money: $100 becomes $150, then $75. The geometric average correctly shows the loss.

Exam Tip: Gotchas

• Annualized return uses geometric averaging, not arithmetic. Arithmetic averaging generally overstates true compound growth whenever periodic returns vary. If asked which method produces a more accurate annualized return, the answer is geometric (compound).

Time-Weighted Return (TWR)

• Measures the compound rate of growth of the portfolio
• Eliminates the effect of cash flows (deposits and withdrawals the manager cannot control)
• Preferred method for evaluating portfolio manager performance
• Generally required by the CFA Institute's Global Investment Performance Standards (GIPS); a money-weighted return is permitted instead only when the firm controls external cash flows AND the portfolio is closed-end, fixed-life, fixed-commitment, or significantly invested in illiquid assets

How it works: TWR breaks the total period into sub-periods at each cash flow point, calculates the return for each sub-period, then geometrically links them together. This isolates pure investment performance from client-driven cash flows.

When to use: Evaluating a portfolio manager's skill, comparing managers to each other, GIPS-compliant performance reporting

Dollar-Weighted Return (Money-Weighted Return)

• Commonly calculated as the internal rate of return (IRR), one method GIPS recognizes for this purpose
• Accounts for the timing and amount of all cash flows
• Reflects the actual return experienced by the investor, including the impact of when they added or withdrew money
• A large deposit before a period of strong returns will increase the dollar-weighted return relative to the time-weighted return

When to use: Evaluating an investor's personal experience, measuring the actual growth of an investor's wealth

Exam Tip: Gotchas

• Time-weighted = manager evaluation. Dollar-weighted = investor's actual experience. If the question asks about evaluating a portfolio manager, the answer is generally time-weighted. If it asks about the investor's actual return, the answer is dollar-weighted (commonly the IRR).
• GIPS generally requires time-weighted return, NOT dollar-weighted, with defined exceptions where the firm controls external cash flows AND the fund is closed-end, fixed-life, fixed-commitment, or significantly invested in illiquid assets.
• IRR is the method commonly used to calculate dollar-weighted return for a portfolio. If the exam mentions one, you can generally substitute the other.

Internal Rate of Return (IRR)

• The discount rate that makes the net present value (NPV) of all cash flows equal to zero
• Commonly used to calculate dollar-weighted return when applied to investment portfolios
• Used extensively in private equity and real estate performance measurement
• Accounts for the time value of money, unlike simple HPR

Expected Return

• The weighted average of possible returns, where weights are the probabilities of each outcome
• Formula: Sum of (Probability x Return) for each scenario
• Used in portfolio construction and risk assessment (forward-looking, not historical)

Example: If there's a 40% chance of a 12% return, a 50% chance of a 6% return, and a 10% chance of a -8% return: Expected Return = (0.40 x 12%) + (0.50 x 6%) + (0.10 x -8%) = 4.8% + 3.0% + (-0.8%) = 7.0%

Risk-Adjusted Return

Two portfolios with the same return are not equal if one took significantly more risk to achieve it. Risk-adjusted measures account for this.

Sharpe Ratio

• Formula: (Portfolio Return - Risk-Free Rate) / Standard Deviation of Portfolio
• Measures excess return per unit of total risk (standard deviation)
• Uses standard deviation as the risk measure (total risk = systematic + unsystematic)
• Best for evaluating a portfolio that represents the investor's entire investment
• Higher Sharpe ratio = better risk-adjusted performance

Treynor Ratio

• Formula: (Portfolio Return - Risk-Free Rate) / Beta of Portfolio
• Measures excess return per unit of systematic risk (beta)
• Uses beta as the risk measure (systematic risk only)
• Best for evaluating a portfolio that is one of many diversified portfolios held by the investor
• Higher Treynor ratio = better risk-adjusted performance

Jensen's Alpha (Alpha)

• Formula: Portfolio Return - [Risk-Free Rate + Beta x (Market Return - Risk-Free Rate)]
• Measures the abnormal return above what the Capital Asset Pricing Model (CAPM) predicts
• Positive alpha = portfolio outperformed its expected risk-adjusted return (manager added value)
• Negative alpha = portfolio underperformed expectations
• Zero alpha = portfolio performed exactly as CAPM predicted

Exam Tip: Gotchas

• Sharpe uses standard deviation (total risk). Treynor uses beta (systematic risk). If the portfolio is well-diversified, unsystematic risk has been eliminated, so Sharpe and Treynor should give similar rankings. If the portfolio is NOT well-diversified, the rankings may differ.
• Positive alpha means the manager beat CAPM expectations, not just that the portfolio went up. A portfolio can have a positive return but negative alpha if it underperformed what CAPM predicted for its level of risk.

Inflation-Adjusted (Real) Return

• Nominal return adjusted for the erosion of purchasing power from inflation
• Approximate formula: Real Return = Nominal Return - Inflation Rate
• More accurate formula: Real Return = (1 + Nominal) / (1 + Inflation) - 1
• Shows the actual increase in purchasing power, not just dollar value

Example: A portfolio earns 8% nominal in a year when inflation is 3%.

• Approximate real return: 8% - 3% = 5%
• Precise real return: (1.08 / 1.03) - 1 = 4.85%

After-Tax Return

• Return after accounting for taxes on dividends, interest, and capital gains
• Varies by the investor's tax bracket and the type of income received
• Tax-deferred accounts (Traditional IRA, 401(k)): No annual tax drag; the taxable portion is generally taxed on withdrawal (after-tax contributions and qualifying rollovers are not taxed again)
• Tax-free accounts (Roth IRA): No tax on growth or qualified withdrawals
• Municipal bond interest: Generally exempt from federal income tax; state and local exemption typically applies only to in-state bonds. Qualified private-activity bond interest stays exempt from regular federal tax but can be an alternative minimum tax (AMT) preference item

Tax-Equivalent Yield

• Allows comparison of tax-free and taxable investments on an equal basis
• Formula: Tax-Equivalent Yield = Municipal Bond Yield / (1 - Tax Rate)
• Include both federal and state tax rates when the muni is exempt from both

Example: A municipal bond yields 3.5% and the investor is in the 32% federal bracket. Tax-Equivalent Yield = 3.5% / (1 - 0.32) = 3.5% / 0.68 = 5.15%

This means a taxable bond would need to yield at least 5.15% to match the muni's after-tax return.

What Should You Check on Exam Day?

• Total return and HPR share the same formula; HPR is just not annualized.
• Annualized return uses geometric (compound) averaging, not arithmetic, because arithmetic generally overstates true compound growth whenever periodic returns vary.
• Time-weighted return evaluates the manager and strips out client cash flows (generally required by GIPS); dollar-weighted return (commonly the IRR) reflects the investor's actual experience and is affected by cash-flow timing.
• Sharpe ratio uses standard deviation (total risk) and fits an investor's entire portfolio; Treynor ratio uses beta (systematic risk) and fits one of several diversified portfolios; positive alpha means the manager beat the CAPM-predicted return, not just that the portfolio rose.
• Real return approximates as Nominal - Inflation, or precisely as (1 + Nominal) / (1 + Inflation) - 1; tax-equivalent yield grosses up a muni's yield for comparison to a taxable bond.