Risk and Return in Practice: Worked Example and Portfolio Context
Direct answer: A worked example grounds risk-return theory: two portfolios with the same expected return of 7% per year have very different Sharpe ratios depending on their composition. Portfolio A (100% equities) might have a standard deviation of 15% and a Sharpe ratio of 0.47. Portfolio B (70% equities, 30% bonds) might have a standard deviation of 11% and a Sharpe ratio of 0.64. The second portfolio achieves the same expected return with less risk, which is the practical meaning of efficient asset class combination.
Why Worked Examples Matter for Risk-Return Understanding
Risk-return theory is easy to state and difficult to internalize. The Sharpe ratio is a formula; what it means for a portfolio decision requires working through numbers. This article does that work concretely, using two hypothetical portfolios with the same expected return but different compositions, and showing the calculation step by step.
The goal is not to replicate the exact numbers for your own portfolio, which depends on your specific holdings, correlation assumptions, and time horizon. The goal is to build the intuition: adding a less-volatile asset class can maintain expected returns while reducing total portfolio risk, and the Sharpe ratio quantifies how much better or worse the trade is.
The Setup: Two Portfolios, Same Expected Return
Assume a simplified two-asset world: U.S. equities and U.S. investment-grade bonds. We set the following assumptions, which are broadly consistent with long-run historical evidence and academic estimates:
- Equity expected return: 9% per year (gross, before costs)
- Bond expected return: 4% per year
- Equity standard deviation: 15% per year
- Bond standard deviation: 5% per year
- Correlation between equities and bonds: -0.10 (slightly negative, consistent with recent historical data; this varies over time)
- Risk-free rate: 2% per year
We construct two portfolios:
- Portfolio A: 100% equities. Expected return = 9%. Standard deviation = 15%.
- Portfolio B: 70% equities, 30% bonds. We calculate the expected return and standard deviation below.
Worked Sharpe Ratio Calculation
Step 1: Portfolio B Expected Return
The expected return of a two-asset portfolio is the weighted average of the two asset expected returns:
E(RB) = wequity × E(Requity) + wbond × E(Rbond)
E(RB) = 0.70 × 9% + 0.30 × 4% = 6.3% + 1.2% = 7.5%
Step 2: Portfolio B Standard Deviation
The standard deviation of a two-asset portfolio is not simply the weighted average of the two standard deviations. Correlation between the assets reduces it. The formula is:
σB = √(we2σe2 + wb2σb2 + 2 × we × wb × ρ × σe × σb)
Plugging in:
- we = 0.70, wb = 0.30
- σe = 15%, σb = 5%
- ρ = -0.10
Term 1: (0.70)2 × (0.15)2 = 0.49 × 0.0225 = 0.011025
Term 2: (0.30)2 × (0.05)2 = 0.09 × 0.0025 = 0.000225
Term 3: 2 × 0.70 × 0.30 × (-0.10) × 0.15 × 0.05 = 2 × 0.21 × (-0.10) × 0.0075 = -0.000315
Sum = 0.011025 + 0.000225 + (-0.000315) = 0.010935
σB = √0.010935 ≈ 10.46% (rounded to approximately 10.5%)
Step 3: Sharpe Ratio for Both Portfolios
The Sharpe ratio is: S = (E(R) - Rf) / σ
Portfolio A: S = (9% - 2%) / 15% = 7% / 15% = 0.47
Portfolio B: S = (7.5% - 2%) / 10.46% = 5.5% / 10.46% = 0.53
What These Numbers Mean
Portfolio A delivers 0.47 units of excess return per unit of risk. Portfolio B delivers 0.53 units. Portfolio B has a higher Sharpe ratio despite having slightly lower expected return, because the reduction in volatility is proportionally larger than the reduction in return.
In plain terms: for every percentage point of return above the risk-free rate, Portfolio A requires you to bear 2.14 percentage points of risk (σ/excess return = 15/7). Portfolio B requires only 1.90 percentage points (10.46/5.5). Portfolio B is more efficient.
| Metric | Portfolio A (100% equity) | Portfolio B (70/30) |
|---|---|---|
| Expected return | 9.0% | 7.5% |
| Standard deviation | 15.0% | 10.5% |
| Risk-free rate | 2.0% | 2.0% |
| Sharpe ratio | 0.47 | 0.53 |
| Return per unit of risk | Lower | Higher |
Applying This to a Real Portfolio Review
The two-portfolio math above is simplified. A real portfolio review applies the same thinking but with more assets and messier correlation data. The process has four steps.
Step 1: Identify Risk Concentration
List every holding and assign it to a risk factor: equity market (domestic), equity market (international), credit, interest rate, real estate, commodity, currency. A portfolio that looks diversified across 20 funds may have 95% of its risk in equity market beta if most of those funds are correlated equity funds. Concentration in one factor is not inherently wrong, but it should be a deliberate choice, not an accident.
Step 2: Calculate Implied Expected Return
For each major holding, estimate the gross expected return (using long-run historical data or factor-based models), subtract the expense ratio and estimated tax drag, and weight by position size. This gives you a portfolio-level net expected return to compare against the risk you are taking.
Step 3: Stress-Test Against Historical Drawdowns
Apply historical stress scenarios to your allocation. A 60/40 stock-bond portfolio fell approximately 35% from peak to trough during the 2008 financial crisis. A 100% equity portfolio fell roughly 55%. An all-bond portfolio fell 15 to 20% in the 2022 rate-rise environment. Ask: if my portfolio fell by this amount tomorrow, would I hold without selling? If the honest answer is no, you are taking more risk than your temperament can support.
Step 4: Check Liquidity Against Near-Term Needs
Separate your assets into buckets by time horizon. Cash and short-term T-bills for expenses in the next 1 to 2 years. Intermediate bonds or balanced funds for expenses in years 3 to 7. Long-horizon equities for everything beyond. This structure ensures you are never forced to sell equities during a drawdown to fund living expenses, which is the primary mechanism by which sequence-of-returns risk actually hurts investors.
Sequence-of-Returns Risk: Why Timing Matters
Two investors have identical average annual returns of 6% over 20 years. One retires in 2000 and experiences early large losses. The other retires in 2010 and experiences early gains. Their final wealth differs substantially if they are making withdrawals throughout.
Why the sequence matters in withdrawal phase:
- An investor withdrawing 4% of initial portfolio value per year who suffers a 40% loss in year one must now withdraw from a much smaller base. The same nominal withdrawal is a much higher percentage of the depleted portfolio.
- The depleted portfolio has fewer assets to participate in the subsequent recovery. Even a full market recovery leaves the investor worse off than if they had experienced that same loss at a later stage.
- The loss compounds forward: fewer shares remaining, less participation in recovery, higher percentage withdrawals from remaining capital.
An illustration: Portfolio starts at $1,000,000. Annual withdrawal: $40,000 (4%). If the first year produces a -30% return: portfolio falls to $700,000 before withdrawal, then to $660,000 after. The $40,000 withdrawal now represents 6.1% of the remaining portfolio, not 4%. In year two, with a 30% rebound, the portfolio grows to $858,000 before withdrawal, then falls to $818,000 after. The investor has effectively recovered the market loss but their portfolio is at $818,000 vs. an investor who had no early loss ($1,060,000 at the same point). The gap persists for the rest of retirement.
Mitigating sequence risk: Hold enough in short-term stable assets (cash, T-bills, short-term bonds) to cover 2 to 3 years of withdrawals. This means you are never forced to sell equities in a down market. The equity portfolio can recover before it must fund withdrawals.
Portfolio Risk Review Checklist
- Is your equity allocation appropriate for your time horizon? Investors within 5 years of significant withdrawals should reduce equity exposure below what might be appropriate for a 30-year horizon.
- Are you genuinely diversified across risk factors, or are you holding equity market beta expressed as many different funds?
- Could you hold your portfolio through a 40% drawdown without selling? If no, reduce equity allocation until you can honestly answer yes.
- Do you have 1 to 3 years of expenses in stable, liquid assets (cash, T-bills, money market funds)?
- Have you calculated net expected return after fees and estimated tax drag? Compare against a simple index fund alternative.
- Are your most volatile holdings truly earning a higher Sharpe ratio than a simpler, cheaper alternative, or are they adding complexity without improving efficiency?
Frequently Asked Questions
What does the Sharpe ratio actually measure and what is a good Sharpe ratio?
What does the Sharpe ratio actually measure and what is a good Sharpe ratio? The Sharpe ratio measures the excess return per unit of risk. It answers: for every percentage point of return above the risk-free rate, how many percentage points of standard deviation did you accept? A higher Sharpe ratio means you received more return per unit of risk taken. As a rough guide: a Sharpe ratio below 0.5 is below-average for a diversified portfolio, 0.5 to 1.0 is reasonable, and above 1.0 is strong over a full market cycle. U.S. equities have historically produced Sharpe ratios of around 0.4 to 0.6 over long periods.
What is sequence-of-returns risk and who is most affected by it?
What is sequence-of-returns risk and who is most affected by it? Sequence-of-returns risk is the risk that the timing of investment returns, not just their average, determines your final wealth. Two investors with identical average returns but different sequences end up with very different outcomes if they are withdrawing money. An investor who retires into a severe bear market and continues withdrawing to fund living expenses is forced to sell assets at depressed prices, depleting the portfolio before it can recover. The same average return experienced in a different order, with strong early returns and poor later returns, produces a much better outcome.
How do I know if my portfolio's risk level is appropriate for my objectives?
How do I know if my portfolio's risk level is appropriate for my objectives? Start by stress-testing against historical drawdowns. If your portfolio has the same allocation as the 60/40 stock-bond blend, it declined roughly 30 to 35% in 2008 and 2009. Ask yourself: could you hold without selling if your portfolio fell by that amount? Then assess your time horizon. A 35-year-old saving for retirement who can hold through multiple cycles can afford more equity risk than a 60-year-old who will begin withdrawing in 5 years. Finally, match liquid assets to near-term spending needs.