Returns

Risk-Adjusted Returns

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

Sharpe Ratio

• Formula: (Rp - Rf) / σp
• 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

Alpha

• Formula: Actual Return - Capital Asset Pricing Model (CAPM) Expected Return
• CAPM Expected Return = Rf + β(Rm - Rf)
• 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%

Comparison Table

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.

Time-Weighted Return (TWR)

• 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

Formula

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

Example

• 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

GIPS (Global Investment Performance Standards)

• 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.

Dollar-Weighted Return (DWR) / Money-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

TWR vs. DWR Feature Comparison

When 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.

Annualized Return

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

Compound Formula (Compound Annual Growth Rate / Geometric Mean)

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

Examples

• 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

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.

Total Return

• Includes all sources of return: price appreciation (or depreciation) plus income received (dividends, interest, distributions)
• Dividends and interest are assumed reinvested for a true total-return comparison
• Most comprehensive and accurate measure of investment performance

Formula

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

Why Total Return Is Preferred

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

Example

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

Holding Period Return (HPR)

• 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

Formula

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

Example

• 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.

Internal Rate of Return (IRR)

• 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

Uses

• 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

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.

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

Probability-Weighted Formula

• E(R) = Σ [Probability x Outcome]
• Example: 30% chance of +20%; 50% chance of +8%; 20% chance of -5%
• E(R) = 0.30(20) + 0.50(8) + 0.20(-5) = 6.0 + 4.0 - 1.0 = 9.0%

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

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.

Inflation-Adjusted Return (Real Return)

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

Approximate Formula (Exam Shortcut)

• Real Return ≈ Nominal Return - Inflation Rate

Exact Formula

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

Example

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

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

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

After-Tax Return and Tax-Equivalent Yield

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

Tax Treatment by Income Type

Tax-Equivalent Yield (TEY)

• 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.