Return Calculations

Accurate return measurement is essential for performance evaluation, client reporting, and manager comparison. Different return calculations serve different purposes.

Holding Period Return (HPR)

Definition

The total return earned over a specific investment period, including both capital gains and income.

Formula

HPR = (Ending Value − Beginning Value + Income) / Beginning Value

Or equivalently:

HPR = (Ending Value + Income) / Beginning Value − 1

Example Calculations

Example 1: Stock Investment

• Buy stock at $100
• Receive $3 in dividends
• Sell at $110

HPR = ($110 − $100 + $3) / $100 = $13 / $100 = 13%

Example 2: Bond Investment

• Buy bond at $1,000
• Receive $50 in interest
• Bond value at end of period: $980

HPR = ($980 − $1,000 + $50) / $1,000 = $30 / $1,000 = 3%

Annualizing Returns

To compare returns over different time periods, convert to annual basis:

Annualized Return = (1 + HPR)^(1/n) − 1

Where n = number of years

Example: 30% return over 2 years Annualized = (1.30)^(1/2) − 1 = 1.14 − 1 = 14% annually

Time-Weighted Return (TWR)

Definition

TWR measures the compound rate of return of a portfolio, eliminating the impact of external cash flows (deposits and withdrawals).

Purpose

• Measures the investment manager's performance
• Eliminates the effect of client-directed cash flows
• Required by GIPS (Global Investment Performance Standards) for fund reporting
• Allows fair comparison between managers

How TWR Works

TWR breaks the overall period into sub-periods at each cash flow, calculates the return for each sub-period, then geometrically links them.

Example:

TWR = (1.05 × 0.97 × 1.08 × 1.02) − 1 = 1.122 − 1 = 12.2%

The deposit in Q2 doesn't affect the TWR calculation.

When to Use TWR

Money-Weighted Return (MWR)

Definition

MWR is the internal rate of return (IRR) that sets the present value of all cash flows equal to zero. It reflects the actual investor experience, including the impact of cash flow timing.

Purpose

• Shows what the investor actually earned
• Accounts for timing and size of contributions/withdrawals
• Penalizes poor timing (adding money before a decline)
• Rewards good timing (adding money before a rise)

When to Use MWR

TWR vs. MWR: Key Differences

Example: Why They Differ

Scenario: Investor adds $100,000 right before market drops 20%

Scenario: Investor adds $100,000 right before market rises 20%

Other Return Measures

Arithmetic Mean Return

Simple average of periodic returns:

Arithmetic Mean = (R1 + R2 + R3 + ... + Rn) / n

Example: Returns of +10%, -5%, +15% Arithmetic Mean = (10 − 5 + 15) / 3 = 6.67%

Geometric Mean Return

Compound average return (more accurate for multi-period):

Geometric Mean = [(1+R1) × (1+R2) × ... × (1+Rn)]^(1/n) − 1

Example: Returns of +10%, -5%, +15% Geometric Mean = (1.10 × 0.95 × 1.15)^(1/3) − 1 = (1.202)^0.333 − 1 = 6.33%

Note: Geometric mean is always ≤ arithmetic mean for variable returns.

Real vs. Nominal Returns

Nominal Return

The stated return without adjusting for inflation.

Real Return

The return adjusted for inflation (purchasing power):

Real Return ≈ Nominal Return − Inflation Rate

More precise formula:

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

Example: 8% nominal return, 3% inflation Real Return = (1.08 / 1.03) − 1 = 4.85%

Benchmarking

Choosing Appropriate Benchmarks

Common Benchmarks

On the Exam

Series 65 frequently tests:

• Calculating holding period return (including dividends/interest)
• Understanding TWR eliminates cash flow impact; MWR includes it
• Knowing TWR is used for manager comparison; MWR for investor results
• GIPS requires TWR for most fund reporting

Key Takeaways

1. HPR = (Ending Value − Beginning Value + Income) / Beginning Value
2. TWR eliminates cash flow impact—used for manager evaluation and GIPS
3. MWR reflects actual investor experience—includes cash flow timing
4. TWR is better for comparing managers; MWR shows what you actually earned
5. Geometric mean is more accurate than arithmetic mean for multi-period returns
6. Real return = nominal return adjusted for inflation