Returns

This page works through each measure's formula, what it captures, and the manager-versus-investor distinction (time-weighted versus dollar-weighted) the exam returns to repeatedly.

What Are the Risk-Adjusted Return Measures?

Two measures tested on the Series 65: the Sharpe Ratio (uses total risk) and Alpha (uses systematic risk via CAPM).

What Is the Sharpe Ratio?

• Formula: (Rp - Rf) / σp
• Where: Rp = portfolio return, Rf = risk-free rate (such as a Treasury bill rate), σp = standard deviation of the portfolio (total risk)
• Risk measure: standard deviation (total risk, both systematic and unsystematic)
• Best for: evaluating a standalone portfolio or any investment held in isolation
• Higher = better; a negative Sharpe means the portfolio underperformed the risk-free rate
• Example: Portfolio return 12%, Rf 3%, std dev 15% → Sharpe = (12 - 3) / 15 = 0.60

What Is Alpha?

• Formula: Actual Return - Capital Asset Pricing Model (CAPM) Expected Return
• CAPM Expected Return = Rf + β(Rm - Rf)
• Where: Rf = risk-free rate, β = the portfolio's beta (sensitivity to market moves), Rm = the market's return
• Positive alpha = outperformed what CAPM predicted for the level of systematic risk taken
• Negative alpha = underperformed on a risk-adjusted basis
• Example: Actual 14%, Rf 3%, β 1.0, Rm 11% → CAPM expected = 3 + 1.0(11 - 3) = 11%; Alpha = 14 - 11 = +3%

How Do Sharpe and Alpha Compare?

When to use which:

• Undiversified portfolio (investor's only holding) - use the Sharpe ratio (total risk matters because unsystematic risk has not been diversified away)
• Manager skill evaluation on a CAPM basis - use alpha (isolates outperformance against the systematic-risk-adjusted expected return)

Exam Tip: Gotchas

• Sharpe uses standard deviation (total risk). Alpha uses beta via CAPM (systematic risk only). The exam will test which measure is appropriate based on whether the portfolio is the investor's ENTIRE holding (Sharpe) vs. evaluating manager skill against a CAPM expectation (alpha).
• Alpha is NOT simply portfolio return minus market return. You must first calculate the CAPM-expected return using the portfolio's beta, then subtract.

What Is Time-Weighted Return?

• Measures manager performance by eliminating the distorting effect of client cash flows (deposits and withdrawals outside the manager's control)
• Divide the measurement period at each cash flow date, calculate a sub-period return for each segment, then chain them together

What Is the TWR Formula?

• TWR = [(1 + R1)(1 + R2) ... (1 + Rn)] - 1

What Does a TWR Example Look Like?

• Period 1 return: +10%; client deposits cash; Period 2 return: -5%
• TWR = (1.10)(0.95) - 1 = 1.045 - 1 = 4.5%
• The manager's skill is evaluated on 4.5%, regardless of how much money was deposited or when

What Does GIPS Require?

• Requires TWR for reporting manager returns to clients
• TWR is the industry standard for manager evaluation

Exam Tip: Gotchas

• If a question asks how to evaluate a portfolio manager's performance, the answer is time-weighted return. GIPS requires it specifically because it removes the effect of client-driven cash flows.

What Is Dollar-Weighted Return?

• Measures the investor's actual experience, including the effect of the timing and size of cash flows
• Mathematically equivalent to the Internal Rate of Return (IRR): the discount rate that makes the present value of all cash inflows equal the present value of all cash outflows

How Do TWR and DWR Compare?

When Do TWR and DWR Diverge?

Exam Tip: Gotchas

• TWR = manager skill (eliminates cash flows). DWR = investor experience (includes cash flows).
• GIPS requires TWR for performance reporting precisely because DWR is influenced by factors outside the manager's control.

What Is Annualized Return?

• Converts a return for any holding period into an equivalent annual rate for comparison across time periods

What Is a Geometric Mean, and Why Do Returns Use It?

• A geometric mean averages a set of values by multiplying them and taking the root, instead of adding them and dividing (that is the arithmetic mean, the simple average from descriptive statistics).
• For returns, it multiplies each period's growth factor (1 + return) and takes the nth root: [(1 + R1)(1 + R2) ... (1 + Rn)]^(1/n) - 1.
• It is the correct average for multi-period returns because it captures compounding: each period builds on the result of the last, not on the original starting value.
• The arithmetic mean overstates multi-period performance whenever returns are volatile (the two are equal only when every period's return is the same). A portfolio that gains 50% one year and loses 50% the next has an arithmetic mean of 0%, but a dollar actually grows to $1.50 and then falls to $0.75. The geometric mean of about -13.4% per year reflects that real loss; the simple average hides it.

Where a geometric mean is tested:

• Annualized return / Compound Annual Growth Rate (CAGR): the geometric mean of the annual growth factors (the formula below).
• Time-weighted return (TWR): chain-links each period's return the same way, which is why it is also called the geometric return (see the Time-Weighted Return section above).

What Is the Compound Formula (CAGR / Geometric Mean)?

• Annualized Return = (1 + Holding Period Return (HPR))^(1/n) - 1, where n = number of years

What Do Annualized Return Examples Look Like?

• 3-year cumulative return of 33.1% → (1.331)^(1/3) - 1 = 10.0% per year
• 6-month return of 5% → (1.05)^(1/0.5) - 1 = (1.05)² - 1 = 10.25% per year

What Is the Simple Approximation Tested on Exam?

• Holding period less than 1 year: multiply HPR by the number of periods in a year
• Holding period greater than 1 year: use the geometric formula above

Exam Tip: Gotchas

• Do not simply divide a multi-year return by the number of years. That ignores compounding. Use the geometric formula: (1 + HPR)^(1/n) - 1.

What Is Total Return?

• Includes all sources of return: price appreciation (or depreciation) plus income received (dividends, interest, distributions)
• Most comprehensive measure of investment performance

What Is the Total Return Formula?

• Total Return = (Ending Value - Beginning Value + Income) / Beginning Value

Why Is Total Return Preferred?

• Preferred over yield alone because yield ignores capital changes
• Used for comparing performance across different asset classes

What Does a Total Return Example Look Like?

• Buy stock at $100; receive $3 dividend; sell at $105
• Total Return = (105 - 100 + 3) / 100 = 8%

What Is Holding Period Return?

• Return for the specific period an investment was held, regardless of how long that period was
• Identical formula to total return; the distinction is that HPR can cover any time span (1 day, 6 months, 3 years)
• Does not annualize; does not account for the length of time
• Must annualize separately before comparing HPRs of different lengths

What Is the HPR Formula?

• HPR = (Ending Value - Beginning Value + Income) / Beginning Value

What Does an HPR Example Look Like?

• Bought at $50, sold at $60, received $2 in dividends
• HPR = (60 - 50 + 2) / 50 = 24% (for whatever period was held)

Exam Tip: Gotchas

• A 24% HPR over 3 years is very different from a 24% HPR over 6 months. Always annualize before comparing returns across different holding periods.

What Is Internal Rate of Return?

• The discount rate that makes Net Present Value (NPV) = 0 (present value of all cash inflows equals present value of all cash outflows)
• Mathematically equivalent to the dollar-weighted return

What Is IRR Used For?

• Capital budgeting: accept a project if IRR > hurdle rate (required return)
• Real estate and private equity: evaluates returns on irregular cash-flow streams
• Annuities: evaluates investments with irregular cash flows

What Is the IRR Decision Rule?

• IRR > required return (hurdle rate) → accept the investment
• IRR < hurdle rate → reject

Exam Tip: Gotchas

• IRR and dollar-weighted return are the same concept applied in different contexts. The exam may use either term.
• IRR assumes reinvestment of interim cash flows at the IRR itself, which can overstate returns for projects with very high IRRs.

What Is Expected Return?

• Expected return is the probability-weighted average of all possible outcomes
• Based on forward-looking estimates, not historical data
• Used in portfolio construction and risk assessment

What Is the Probability-Weighted Formula?

• E(R) = Σ [Probability x Outcome]
• The Σ (sum) runs across every possible scenario the question gives you, not an open-ended range. If a question lists three economic scenarios (recession, normal, expansion), the sum has exactly three terms. Add one Probability x Outcome term per scenario until every scenario listed has been included.
• The probabilities themselves must add up to 100% (or 1.0) across all scenarios; that is how you know you have accounted for every outcome.
• Example: 30% chance of +20%; 50% chance of +8%; 20% chance of -5% (30% + 50% + 20% = 100%, so all scenarios are covered)
• E(R) = 0.30(20) + 0.50(8) + 0.20(-5) = 6.0 + 4.0 - 1.0 = 9.0%

What Is CAPM Expected Return?

• Formula: Rf + β(Rm - Rf)
• This is the required return for the level of systematic risk taken
• If the actual return exceeds this, alpha is positive

What Are the Expected Return Methods?

Exam Tip: Gotchas

• Expected return is NOT a guaranteed return. It is a weighted average of possible outcomes. The exam may present a scenario with three possible economic conditions (recession, normal, expansion), each with a probability and return, and ask you to calculate the expected return.

What Is Inflation-Adjusted (Real) Return?

• Measures increase in purchasing power by removing the distorting effect of inflation

What Is the Approximate Formula (Exam Shortcut)?

• Real Return ≈ Nominal Return - Inflation Rate

What Is the Exact Formula?

• Real Return = [(1 + Nominal) / (1 + Inflation)] - 1

What Does a Real Return Example Look Like?

• Nominal 8%, inflation 3%
• Approximate: 8% - 3% = 5.0%
• Exact: (1.08 / 1.03) - 1 = 4.85%

What Is the Planning Implication?

• A 6% nominal return with 5% inflation produces only ~1% real return, barely growing purchasing power
• Long-term investors must account for inflation when evaluating whether portfolio growth is sufficient

What Are TIPS (Treasury Inflation-Protected Securities)?

• Principal adjusts with the Consumer Price Index (CPI); coupon is paid on the adjusted principal
• Real yield is known at purchase; nominal yield varies with inflation

What Is After-Tax Return and Tax-Equivalent Yield?

What Is After-Tax Return?

• Formula: After-Tax Return = Pre-Tax Return x (1 - Tax Rate)
• Example: 10% pre-tax return, investor in 25% bracket → 10% x (1 - 0.25) = 7.5% after-tax

How Is Investment Income Taxed by Type?

What Is Tax-Equivalent Yield?

• Used to compare tax-exempt bonds (municipals) with taxable bonds on an equal basis
• Formula: TEY = Tax-Exempt Yield / (1 - Tax Rate)
• Example: Municipal bond yields 3%, investor in 32% bracket
  • TEY = 3% / (1 - 0.32) = 3% / 0.68 = 4.41%
  • If a comparable taxable bond yields less than 4.41%, the muni is more attractive on an after-tax basis

Exam Tip: Gotchas

• The TEY formula is frequently tested. You divide the tax-exempt yield by (1 - tax rate), not multiply.
• A higher tax bracket makes municipal bonds more attractive because the tax-equivalent yield increases.

What Should You Check on Exam Day?

• Total return and holding period return share one formula: (ending - beginning + income) / beginning
• Annualize multi-year returns geometrically, (1 + HPR)^(1/n) - 1, never by simple division
• Sharpe ratio uses total risk (standard deviation); alpha uses systematic risk (beta via CAPM)
• Time-weighted return measures the manager and is required by GIPS; dollar-weighted return measures the investor and equals IRR
• IRR is the discount rate that sets NPV to zero and is mathematically equivalent to dollar-weighted return
• Expected return is a probability-weighted forecast, not a guarantee
• Real return subtracts inflation from nominal return; the exact formula divides (1 + Nominal) by (1 + Inflation)
• After-tax return multiplies by (1 - tax rate); tax-equivalent yield divides by (1 - tax rate)