Returns

Quick Answer

The Series 65 tests ten return measures: risk-adjusted (Sharpe Ratio, Alpha), time-weighted, dollar-weighted, annualized, total, holding period, internal rate of return, expected, inflation-adjusted, and after-tax. Total return and holding period return share one formula; annualize multi-year returns geometrically, never by simple division.

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 minus Rf) divided by standard deviation of the portfolio, where Rp equals portfolio return, Rf equals risk-free rate (such as a Treasury bill rate), and the standard deviation of the portfolio is 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 equals better; a negative Sharpe means the portfolio underperformed the risk-free rate.

Example: Portfolio return 12 percent, Rf 3 percent, std dev 15 percent. Sharpe equals (12 minus 3) divided by 15 equals 0.60.

What Is Alpha?

Formula: Actual Return minus Capital Asset Pricing Model (CAPM) Expected Return.

CAPM Expected Return equals Rf plus beta times (Rm minus Rf).

Where: Rf equals risk-free rate, beta equals the portfolio's beta (sensitivity to market moves), Rm equals the market's return.

Positive alpha equals outperformed what CAPM predicted for the level of systematic risk taken.

Negative alpha equals underperformed on a risk-adjusted basis.

Example: Actual 14 percent, Rf 3 percent, beta 1.0, Rm 11 percent. CAPM expected equals 3 plus 1.0(11 minus 3) equals 11 percent; Alpha equals 14 minus 11 equals plus 3 percent.

How Do Sharpe and Alpha Compare?

Measure: Sharpe Ratio. Formula: (Rp minus Rf) divided by standard deviation. Risk used: Total (std dev). Best for: Standalone portfolio. Interpretation: Excess return per unit of total risk.

Measure: Alpha. Formula: Actual minus CAPM. Risk used: Systematic (beta). Best for: Manager skill vs. CAPM expectation. Interpretation: Absolute excess return above risk-adjusted expectation.

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 equals [(1 plus R1)(1 plus R2) ... (1 plus Rn)] minus 1.

What Does a TWR Example Look Like?

Period 1 return: plus 10 percent; client deposits cash; Period 2 return: minus 5 percent.

TWR equals (1.10)(0.95) minus 1 equals 1.045 minus 1 equals 4.5 percent.

The manager's skill is evaluated on 4.5 percent, 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?

Feature: Cash flow impact. Time-weighted: Eliminated. Dollar-weighted: Fully reflected.

Feature: Measures. Time-weighted: Manager skill. Dollar-weighted: Investor experience.

Feature: Use case. Time-weighted: Manager evaluation, GIPS. Dollar-weighted: Individual investor return.

Feature: Synonym. Time-weighted: Geometric return. Dollar-weighted: Money-weighted return / IRR.

Feature: Affected by deposit timing. Time-weighted: No. Dollar-weighted: Yes.

When Do TWR and DWR Diverge?

Scenario: Investor deposited large sum before poor returns. TWR vs. DWR: TWR greater than DWR. Explanation: Investor's timing hurt returns; manager's skill was fine.

Scenario: Investor deposited large sum before strong returns. TWR vs. DWR: DWR greater than TWR. Explanation: Investor's timing helped; actual dollar gains exceeded the rate.

Scenario: No external cash flows. TWR vs. DWR: TWR equals DWR. Explanation: Identical when there are no deposits or withdrawals.

Exam Tip: Gotchas

TWR equals manager skill (eliminates cash flows). DWR equals 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 plus return) and takes the nth root: [(1 plus R1)(1 plus R2) ... (1 plus Rn)] to the power (1/n) minus 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).

Example: a portfolio that gains 50 percent one year and loses 50 percent the next has an arithmetic mean of 0 percent, but a dollar actually grows to $1.50 and then falls to $0.75. The geometric mean of about minus 13.4 percent per year reflects that real loss; the simple average hides it.

Often confused: outlier sensitivity and compounding. An extreme period does pull a simple average toward it, but that is a separate issue. The overstatement here appears even with no outlier. A portfolio that gains 10 percent and then loses 10 percent has an arithmetic mean of 0 percent, yet $100 grows to $110 and then falls to $99, a small loss.

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 equals (1 plus Holding Period Return (HPR)) to the power (1/n) minus 1, where n equals number of years.

What Do Annualized Return Examples Look Like?

3-year cumulative return of 33.1 percent: (1.331) to the power (1/3) minus 1 equals 10.0 percent per year.

6-month return of 5 percent: (1.05) to the power (1/0.5) minus 1 equals (1.05) squared minus 1 equals 10.25 percent 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 plus HPR) to the power (1/n) minus 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 equals (Ending Value minus Beginning Value plus Income) divided by Beginning Value.

Why Is Total Return Preferred?

Measure: Current yield. What it captures: Income only relative to price. Limitation: Ignores capital gains/losses.

Measure: Total return. What it captures: Income plus capital changes. Limitation: Does not risk-adjust.

Measure: Risk-adjusted return. What it captures: Return per unit of risk. Limitation: Requires risk metric selection.

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 equals (105 minus 100 plus 3) divided by 100 equals 8 percent.

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).

If a question prints this formula and total return is not among the choices, look for holding period return.

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 equals (Ending Value minus Beginning Value plus Income) divided by Beginning Value.

Component: Capital gains/losses. Included in HPR? Yes.

Component: Dividends received. Included in HPR? Yes.

Component: Interest received. Included in HPR? Yes.

Component: Transaction costs. Included in HPR? Depends on context (typically excluded on exam).

What Does an HPR Example Look Like?

Bought at $50, sold at $60, received $2 in dividends.

HPR equals (60 minus 50 plus 2) divided by 50 equals 24 percent (for whatever period was held).

Exam Tip: Gotchas

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

HPR counts all income received during the holding period, not an annual rate. Dividing only that income by the purchase price gives an income-only ratio, not HPR. It is also not current yield, which uses annual income and the current market price. HPR adds the price change.

What Is Internal Rate of Return?

The discount rate that makes Net Present Value (NPV) equals 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 is greater than hurdle rate (required return).

Real estate and private equity: evaluates returns on irregular cash-flow streams.

Annuities: evaluates investments with irregular cash flows.

Bonds: yield to maturity (YTM) is the IRR of a bond's cash flows, the discount rate that equates the present value of the remaining coupon payments and par value to the bond's current price.

What Is the IRR Decision Rule?

IRR greater than required return (hurdle rate): accept the investment.

IRR less than 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) equals sum of [Probability x Outcome].

The 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 percent (or 1.0) across all scenarios; that is how you know you have accounted for every outcome.

Example: 30 percent chance of plus 20 percent; 50 percent chance of plus 8 percent; 20 percent chance of minus 5 percent (30 percent plus 50 percent plus 20 percent equals 100 percent, so all scenarios are covered).

E(R) equals 0.30(20) plus 0.50(8) plus 0.20(minus 5) equals 6.0 plus 4.0 minus 1.0 equals 9.0 percent.

What Is CAPM Expected Return?

Formula: Rf plus beta times (Rm minus Rf).

This is the required return for the level of systematic risk taken.

If the actual return exceeds this, alpha is positive.

How Do Expected Return and Required Return Compare?

Expected return is a forecast: a probability-weighted estimate of what an investment may earn, built from scenario analysis or a historical average.

Required return is the minimum return demanded for the risk taken. On the Series 65, the required return is the CAPM figure, Rf plus beta times (Rm minus Rf).

The two are distinct concepts and are not interchangeable, even though a question may hand you both numbers for the same investment.

Decision rule: compare the forecast expected return with the CAPM required return for the same risk level.

Expected return above required return: the investment is attractive.

Expected return below required return: the investment is not attractive.

Expected return equal to required return: the investment is fairly valued.

Example: an analyst forecasts a 12 percent expected return; CAPM gives a 10 percent required return for that risk level: attractive, because the expected return exceeds the required return.

What Are the Expected Return Methods?

Method: Probability-weighted. Inputs: Scenario probabilities and returns. Use case: Forecasting with discrete outcomes.

Method: CAPM. Inputs: Risk-free rate, beta, market premium. Use case: Determining required return for a given risk level.

Method: Historical average. Inputs: Past returns. Use case: Estimating future based on historical patterns.

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 approximately equals Nominal Return minus Inflation Rate.

What Is the Exact Formula?

Real Return equals [(1 plus Nominal) divided by (1 plus Inflation)] minus 1.

What Does a Real Return Example Look Like?

Nominal 8 percent, inflation 3 percent.

Approximate: 8 percent minus 3 percent equals 5.0 percent.

Exact: (1.08 divided by 1.03) minus 1 equals 4.85 percent.

What Is the Planning Implication?

A 6 percent nominal return with 5 percent inflation produces only about 1 percent 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 equals Pre-Tax Return x (1 minus Tax Rate).

Example: 10 percent pre-tax return, investor in 25 percent bracket: 10 percent x (1 minus 0.25) equals 7.5 percent after-tax.

How Is Investment Income Taxed by Type?

Income type: Ordinary income (taxable interest, short-term gains). Tax treatment: Taxed at marginal income tax rate.

Income type: Qualified dividends. Tax treatment: Preferential rate (0 percent, 15 percent, or 20 percent).

Income type: Long-term capital gains (held greater than 1 year). Tax treatment: Preferential rate (0 percent, 15 percent, or 20 percent).

Income type: Municipal bond interest. Tax treatment: Generally exempt from federal tax.

Income type: Tax-deferred accounts (Individual Retirement Account (IRA), 401(k)). Tax treatment: No current tax; taxed on withdrawal.

What Is Tax-Equivalent Yield?

Used to compare tax-exempt bonds (municipals) with taxable bonds on an equal basis.

Formula: TEY equals Tax-Exempt Yield divided by (1 minus Tax Rate).

Example: Municipal bond yields 3 percent, investor in 32 percent bracket.

TEY equals 3 percent divided by (1 minus 0.32) equals 3 percent divided by 0.68 equals 4.41 percent.

If a comparable taxable bond yields less than 4.41 percent, 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 minus 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 minus beginning plus income) divided by beginning; with no total return option, look for HPR.

Annualize multi-year returns geometrically, (1 plus HPR) to the power (1/n) minus 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 plus Nominal) by (1 plus Inflation).

After-tax return multiplies by (1 minus tax rate); tax-equivalent yield divides by (1 minus tax rate).