Key Takeaways
- Total return includes both income (dividends/interest) AND capital appreciation.
- Time-weighted return eliminates the effect of cash flows; used to compare managers.
- Dollar-weighted return (IRR) includes cash flow timing; measures investor's actual experience.
- Sharpe ratio measures excess return per unit of TOTAL risk (standard deviation).
- Treynor ratio measures excess return per unit of SYSTEMATIC risk (beta).
- Alpha (Jensen's) measures performance ABOVE what CAPM expected; positive alpha = outperformance.
- Benchmarks must be appropriate to the portfolio's investment style and objective.
- Higher Sharpe, Treynor, or alpha indicates better risk-adjusted performance.
Measuring Portfolio Performance
Evaluating investment performance requires understanding various return measures and risk-adjusted metrics.
Return Measures
Total Return
Total return captures all sources of investment return:
Total Return = (Ending Value - Beginning Value + Income) / Beginning Value
| Component | Examples |
|---|---|
| Capital Appreciation | Price increase |
| Capital Depreciation | Price decrease |
| Income | Dividends, interest, distributions |
Time-Weighted Return (TWR)
Time-weighted return eliminates the effect of cash flows (deposits and withdrawals).
| Characteristic | Description |
|---|---|
| Purpose | Compare manager performance |
| Cash Flows | Neutralized |
| Calculation | Geometric linking of sub-period returns |
| Standard | Industry standard for comparing managers |
Why TWR? Managers don't control when clients add or withdraw money.
Dollar-Weighted Return (Money-Weighted / IRR)
Dollar-weighted return includes the effect of cash flows.
| Characteristic | Description |
|---|---|
| Purpose | Measure investor's actual experience |
| Cash Flows | Fully incorporated |
| Calculation | Internal rate of return (IRR) |
| Impact | Timing of contributions/withdrawals matters |
TWR vs. DWR Comparison
| Scenario | Better Measure |
|---|---|
| Comparing managers | Time-weighted |
| Evaluating investor experience | Dollar-weighted |
| When cash flows are frequent | Time-weighted |
| When timing decisions by investor | Dollar-weighted |
Exam Tip: Time-weighted return is used to compare MANAGERS because it eliminates cash flow timing that managers don't control. Dollar-weighted return measures the INVESTOR'S actual experience.
Risk-Adjusted Return Measures
Sharpe Ratio
The Sharpe ratio measures excess return per unit of total risk (standard deviation).
Sharpe Ratio = (Portfolio Return - Risk-Free Rate) / Standard Deviation
| Component | Meaning |
|---|---|
| Portfolio Return | Actual return earned |
| Risk-Free Rate | T-bill return |
| Standard Deviation | Total volatility (systematic + unsystematic) |
Sharpe Ratio Interpretation
| Result | Meaning |
|---|---|
| Higher | Better risk-adjusted return |
| > 1.0 | Generally considered good |
| > 2.0 | Very good |
| Negative | Underperformed risk-free rate |
Treynor Ratio
The Treynor ratio measures excess return per unit of systematic risk (beta).
Treynor Ratio = (Portfolio Return - Risk-Free Rate) / Beta
| Component | Meaning |
|---|---|
| Numerator | Excess return above risk-free |
| Denominator | Beta (systematic risk only) |
When to Use Treynor vs. Sharpe
| Use Sharpe When... | Use Treynor When... |
|---|---|
| Evaluating entire portfolio | Evaluating a component of larger portfolio |
| Portfolio is NOT fully diversified | Portfolio IS well-diversified |
| Total risk matters | Only systematic risk matters |
Alpha (Jensen's Alpha)
Alpha measures the difference between actual return and expected return based on CAPM.
Alpha = Actual Return - Expected Return (CAPM)
Or: α = Rp - [Rf + β × (Rm - Rf)]
Alpha Interpretation
| Alpha | Meaning |
|---|---|
| Positive | Outperformed expectations (manager added value) |
| Zero | Performed as expected |
| Negative | Underperformed expectations |
Alpha Example
Given: Portfolio return = 12%, Risk-free = 3%, Market = 10%, Beta = 1.0
Expected return = 3% + 1.0 × (10% - 3%) = 3% + 7% = 10%
Alpha = 12% - 10% = +2% (outperformed)
Exam Tip: Sharpe uses TOTAL risk (standard deviation). Treynor uses SYSTEMATIC risk (beta). Use Sharpe for entire portfolios, Treynor for well-diversified portfolios.
Risk-Adjusted Measures Summary
| Measure | Formula | Risk Measure | Use For |
|---|---|---|---|
| Sharpe | (Rp - Rf) / σ | Total risk | Entire portfolio |
| Treynor | (Rp - Rf) / β | Systematic risk | Diversified portfolio |
| Alpha | Rp - [Rf + β(Rm - Rf)] | Benchmark comparison | Manager evaluation |
Benchmarks
Purpose of Benchmarks
- Measure performance against a standard
- Evaluate manager skill
- Set client expectations
- Assess strategy effectiveness
Selecting Appropriate Benchmarks
| Portfolio Type | Appropriate Benchmark |
|---|---|
| Large-cap U.S. stocks | S&P 500 |
| Small-cap U.S. stocks | Russell 2000 |
| Growth stocks | Russell 1000 Growth |
| Value stocks | Russell 1000 Value |
| International stocks | MSCI EAFE |
| Emerging markets | MSCI Emerging Markets |
| U.S. bonds | Bloomberg U.S. Aggregate |
| Balanced portfolio | Blended benchmark (60/40) |
Benchmark Requirements
| Requirement | Description |
|---|---|
| Appropriate | Matches investment style and universe |
| Investable | Could actually invest in the benchmark |
| Measurable | Returns can be calculated |
| Specified in Advance | Not changed after the fact |
| Unambiguous | Clear components and weights |
Common Benchmark Issues
| Issue | Problem |
|---|---|
| Style drift | Portfolio strays from benchmark style |
| Cherry-picking | Selecting benchmark after results known |
| Inappropriate comparison | Growth fund vs. value benchmark |
| Survivorship bias | Benchmark excludes failed funds |
Standard Deviation
Standard deviation measures total volatility of returns.
| Characteristic | Description |
|---|---|
| Measures | Dispersion around average return |
| Higher value | More volatile, higher risk |
| Lower value | Less volatile, lower risk |
| Normal distribution | 68% within 1 std dev, 95% within 2 |
Standard Deviation Example
| Fund | Average Return | Standard Deviation |
|---|---|---|
| Fund A | 10% | 5% |
| Fund B | 10% | 15% |
Fund B has the same expected return but MORE risk.
Exam Tip: A portfolio with a POSITIVE alpha has outperformed what was expected based on its level of risk (beta). Positive alpha indicates the manager added value.
A portfolio with a POSITIVE alpha has:
Which return measure should be used to compare portfolio MANAGERS?
The Sharpe ratio differs from the Treynor ratio in that Sharpe uses:
An appropriate benchmark for a small-cap U.S. stock fund would be:
9.1 Qualified Retirement Plans
Chapter 9: Retirement Plans