Investment Performance Measures

Performance measurement is heavily tested on the CFP exam. Understanding when to use each measure--and how to calculate them--is essential for evaluating investments and managers.

Holding Period Return (HPR)

Holding period return measures the total return earned on an investment during the period it was held. HPR does NOT consider the length of the holding period--a 10% return is a 10% HPR whether earned in one month or five years.

Formula

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

Or equivalently:

HPR = (Selling Price - Purchase Price +/- Cash Flows) / Initial Investment

Simple Example

An investor buys a stock for $50 and sells it for $55 after receiving $2 in dividends.

HPR = ($55 - $50 + $2) / $50 = $7 / $50 = 14%

Margin Transaction Example

This is commonly tested. An investor must calculate the return on their actual equity (out-of-pocket investment), not the total position value.

Example: Joe buys 100 shares at $20 with 60% initial margin. After one year, he pays 10% margin interest and sells at $30. What is the HPR?

Step 1: Calculate out-of-pocket investment

• Total purchase: $20 x 100 = $2,000
• Equity invested (60%): $2,000 x 0.60 = $1,200
• Amount borrowed (40%): $2,000 x 0.40 = $800

Step 2: Calculate total gain and cash flows

• Sale proceeds: $30 x 100 = $3,000
• Total purchase price: $2,000
• Gain on position: $3,000 - $2,000 = $1,000
• Interest paid: $800 x 10% = $80

Step 3: Calculate HPR

• HPR = ($1,000 - $80) / $1,200 = $920 / $1,200 = 76.67%

Multi-Period HPR

When given periodic returns, compound them:

HPR = [(1 + r1) x (1 + r2) x ... x (1 + rn)] - 1

Example: Monthly returns of 2%, 3.5%, and -1.5%

HPR = (1.02 x 1.035 x 0.985) - 1 = 1.0399 - 1 = 3.99%

Arithmetic vs. Geometric Mean

Arithmetic Mean

Arithmetic mean is a simple average of returns. Add the returns and divide by the number of periods.

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

Example: Returns of 10%, 15%, and -5% Arithmetic Mean = (10% + 15% + (-5%)) / 3 = 20% / 3 = 6.67%

Geometric Mean

Geometric mean is a time-weighted, compounded return. It accounts for the compounding effect.

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

Example: Returns of 10%, 15%, and -5%

On calculator:

• PV = -1 (initial investment of $1)
• FV = (1.10)(1.15)(0.95) = 1.2017
• N = 3
• I/Y = ? = 6.30%

Key Differences

Important: When there is ANY volatility in returns, the geometric mean will ALWAYS be lower than the arithmetic mean. The more volatile the returns, the greater the difference.

Example of Volatility Impact:

• Year 1: +50%, Year 2: -50%
• Arithmetic Mean: (50% + (-50%)) / 2 = 0%
• Geometric Mean: [(1.50)(0.50)]^0.5 - 1 = [0.75]^0.5 - 1 = -13.4%

Despite an arithmetic average of 0%, the investor actually lost 25% ($1 to $1.50 to $0.75).

Time-Weighted Return (TWR)

Time-weighted return measures the compound rate of growth of $1 invested, eliminating the effect of cash flow timing. It treats each sub-period equally, regardless of how much money was invested.

When to Use TWR

• Evaluating MANAGER performance - The manager does not control when clients add/withdraw money
• Comparing fund managers - Fair comparison regardless of fund flows
• Mutual fund reporting - Funds report on a time-weighted basis

How TWR Works

TWR breaks the investment period into sub-periods at each cash flow, calculates the return for each sub-period, then compounds them.

Example Calculation

An investor buys 1 share of stock for $50. After one year, the stock is worth $65 and pays a $4 dividend. The investor buys another share at $65. After another year, each share is worth $75.

Period 1 (before second purchase): HPR1 = ($65 + $4 - $50) / $50 = $19 / $50 = 38%

Period 2 (after second purchase): HPR2 = ($75 - $65) / $65 = $10 / $65 = 15.38%

Time-Weighted Return (compounded): TWR = [(1 + 0.38)(1 + 0.1538)]^(1/2) - 1 = [1.593]^0.5 - 1 = 26.2% annually

Notice: TWR ignores that the investor had more money invested in Period 2.

Dollar-Weighted Return (Money-Weighted Return/IRR)

Dollar-weighted return is the internal rate of return (IRR) of all cash flows. It considers when money was added or withdrawn, reflecting the investor's actual experience.

When to Use Dollar-Weighted Return

• Evaluating INVESTOR results - What did the investor actually earn?
• When timing matters - Cash flow timing significantly impacts return
• Personal performance assessment - How did MY investment decisions work out?

How Dollar-Weighted Return Works

It is the discount rate that makes the present value of all cash inflows equal to the present value of all cash outflows--the IRR.

Example Calculation

Same scenario as above: Buy 1 share at $50, receive $4 dividend, buy another share at $65, sell both shares at $75.

Cash Flow Timeline:

• Time 0: -$50 (initial purchase)
• Time 1: $4 - $65 = -$61 (dividend received, second share purchased)
• Time 2: $150 (sell 2 shares at $75 each)

Using IRR function on calculator: CF0 = -50 CF1 = -61 CF2 = 150 IRR = 22.63%

TWR vs. Dollar-Weighted Return Comparison

When They Differ Significantly

• Large cash flows relative to portfolio size
• High performance volatility during the period
• If investor adds money before good performance: DWR > TWR
• If investor adds money before poor performance: DWR < TWR

CFP Exam Tip: Mutual funds report on a TIME-WEIGHTED basis. This allows fair comparison between funds, regardless of investor cash flow timing.

Risk-Adjusted Performance Measures

These measures evaluate return relative to the risk taken. Higher values indicate better risk-adjusted performance.

Sharpe Ratio

The Sharpe ratio measures excess return per unit of total risk (standard deviation).

Formula: Sharpe Ratio = (Rp - Rf) / Standard Deviation p

Where:

• Rp = Portfolio return
• Rf = Risk-free rate
• Standard Deviation p = Portfolio standard deviation

Key Points:

• Uses standard deviation (total risk)
• Appropriate for non-diversified portfolios
• Higher Sharpe ratio = better performance
• Relative measure - must compare to other funds

Example: Portfolio return = 12%, Risk-free rate = 3%, Standard deviation = 18% Sharpe = (12% - 3%) / 18% = 9% / 18% = 0.50

For every 1% of total risk, the portfolio earned 0.50% of excess return.

Treynor Ratio

The Treynor ratio measures excess return per unit of systematic risk (beta).

Formula: Treynor Ratio = (Rp - Rf) / Beta p

Where:

• Rp = Portfolio return
• Rf = Risk-free rate
• Beta p = Portfolio beta

Key Points:

• Uses beta (systematic risk only)
• Appropriate for well-diversified portfolios
• Higher Treynor ratio = better performance
• Relative measure - must compare to other funds

Example: Portfolio return = 12%, Risk-free rate = 3%, Beta = 1.5 Treynor = (12% - 3%) / 1.5 = 9% / 1.5 = 6.0

For every unit of systematic risk, the portfolio earned 6% excess return.

Jensen's Alpha

Jensen's alpha measures the portfolio's excess return above what CAPM predicts. It shows absolute outperformance.

Formula: Alpha = Rp - [Rf + Beta p(Rm - Rf)]

Where:

• Rp = Portfolio return
• Rf = Risk-free rate
• Beta p = Portfolio beta
• Rm = Market return

Key Points:

• Absolute measure - positive alpha is good by itself
• Measures manager skill in generating excess returns
• Positive alpha = outperformed risk-adjusted expectations
• Negative alpha = underperformed risk-adjusted expectations

Example: Portfolio return = 15%, Risk-free rate = 3%, Beta = 2.0, Market return = 8% Expected return = 3% + 2.0(8% - 3%) = 3% + 10% = 13% Alpha = 15% - 13% = +2%

The manager outperformed by 2% on a risk-adjusted basis.

Information Ratio

The information ratio measures excess return relative to a benchmark, per unit of tracking error.

Formula: Information Ratio = (Rp - RB) / Tracking Error

Where:

• Rp = Portfolio return
• RB = Benchmark return
• Tracking Error = Standard deviation of excess return

Key Points:

• Measures consistency of outperformance relative to benchmark
• Higher information ratio = more consistent outperformance
• Relative measure - compare to other actively managed funds

Choosing the Right Performance Measure

Summary Table of Performance Measures

CFP Exam Tip: If the exam does not give you R-squared to determine if beta is reliable, use Sharpe ratio as the default. When in doubt, total risk (standard deviation) is the safer choice.