What is the Capital Market Line (CML)?
– Definition: The capital market line (CML) is a theoretical straight line that shows the best possible combinations of expected return and total risk (standard deviation of returns) that can be achieved by mixing a risk-free asset with a fully diversified market portfolio of risky assets. Portfolios that lie on the CML are, in theory, the most efficient: they give the highest expected return for a given level of total risk.
Key concepts and definitions
– Risk-free rate (rf): The return available on a security assumed to have zero risk of default (e.g., a short-term government bill).
– Market portfolio (T): A fully diversified portfolio of risky assets representing the market; its expected return is RT and its standard deviation is σT.
– Portfolio standard deviation (σp): The total volatility (risk) of a specific portfolio.
– Sharpe ratio: (Expected return − risk-free rate) / standard deviation. It measures excess return per unit of total risk.
– Tangency (market) portfolio: The point on the efficient frontier that is tangent to the line drawn from the risk-free rate; it is the risky-asset mix that maximizes the Sharpe ratio.
– Capital Allocation Line (CAL): Any line showing combinations of a risk-free asset and some risky portfolio. CML is the specific CAL where the risky portfolio is the market portfolio.
– Security Market Line (SML): A related CAPM construct that uses beta (systematic risk) rather than total risk; it describes expected return of individual assets as a function of beta.
Formula and how to use it
– CML equation:
Rp = rf + [(RT − rf) / σT] × σp
where:
– Rp = expected return of the portfolio on the CML
– rf = risk-free rate
– RT = expected return of the market portfolio
– σT = standard deviation of market returns
– σp = standard deviation of the portfolio
– Interpretation: The slope [(RT − rf) / σT] is the Sharpe ratio of the market portfolio. For any chosen total risk level σp, the CML gives the highest feasible expected return Rp by combining rf and the market portfolio.
Numeric example (worked)
Assumptions:
– Risk-free rate rf = 2.0% (0.02)
– Market expected return RT = 8.0% (0.08)
– Market standard deviation σT = 12.0% (0.12)
– You target a portfolio with total volatility σp = 6.0% (0.06)
Step 1 — compute market Sharpe (slope):
slope = (RT − rf) / σT = (0.08 − 0.02) / 0.12 = 0.06 / 0.12 = 0.5
Step 2 — plug into CML:
Rp = rf + slope × σp = 0.02 + 0.5 × 0.06 = 0.02 + 0.03 = 0.05 → 5.0%
Interpretation: Under the CML assumptions, a portfolio with
Interpretation: Under the CML assumptions, a portfolio with total volatility σp = 6.0% would be formed by holding 50% of wealth in the market portfolio and 50% in the risk-free asset, and it would have the highest possible expected return (5.0%) for that level of total risk.
Why the 50% weight? Two equivalent ways to see it:
– By volatility proportions: weight in market wT = σp / σT = 0.06 / 0.12 = 0.5.
– By expected-return decomposition: Rp = wT · RT + (1 − wT) · rf. Plugging wT = 0.5 gives Rp = 0.5·0.08 + 0.5·0.02 = 0.05 (5.0%).
Quick extension: leverage (borrowing) example
– Suppose you want σp = 18% (greater than σT = 12%). Then wT = 0.18 / 0.12 = 1.5. That implies 150% long the market and 50% borrowed at the risk-free rate (i.e., −50% in the risk-free asset).
– Using the CML: Rp = rf + slope × σp = 0.02 + 0.5 × 0.18 = 0.11 → 11.0%.
– Check with weights: Rp = 1.5·0.08 + (−0.5)·0.02 = 0.12 − 0.01 = 0.11.
Key formulas (summary)
– CML: Rp = rf + [(RT − rf) / σT] × σp
where Rp = portfolio expected return, rf = risk-free rate, RT = market expected return, σT = market standard deviation, σp = portfolio standard deviation.
– Market weight needed to achieve σp: wT = σp / σT.
– Expected return from weights: Rp = wT·RT + (1 − wT)·rf.
Practical checklist when you try to apply the CML
1. Confirm the model assumptions (or at least be aware of them): single-period horizon, frictionless markets, same expectations across investors, ability to lend/borrow at rf, normally distributed returns or mean–variance preferences.
2. Estimate inputs:
– rf: current short-term yield (e.g., Treasury bill).
– RT: expected market return (use historical average, forward dividend/yield models, or analyst consensus). Be explicit about the horizon and method.
– σT: market portfolio standard deviation (use historical returns of a broad market index).
3. Compute slope (market Sharpe): (RT − rf) / σT.
4. For a target σp, compute wT = σp / σT and Rp from the CML formula or by weights.
5. Perform sensitivity checks: vary RT and σT to see how estimates change.
6. Consider practical frictions: transaction costs, taxes, borrowing constraints, and estimation error.
Limitations and common pitfalls
– CML applies to efficient portfolios (combinations of the market portfolio and risk-free asset). It does not apply to individual securities; for single assets, use the Security Market Line (SML) and CAPM beta.
– Input uncertainty: small changes in RT or σT can materially change the slope and hence Rp.
– Borrowing and lending at a single rf rate is often unrealistic; differences between borrowing and lending rates break the simple leverage implication.
– Real markets have frictions, heterogeneous expectations, and may not have a single well-defined “market portfolio” (the theoretical market portfolio should include all assets, not just a stock index).
Short worked recap (your original numbers)
– rf = 2.0%, RT = 8.0%, σT = 12.0%, target σp = 6.0%.
– slope = (0.08 − 0.02) / 0.12 = 0.5.
– wT = 0.06 / 0.12 = 0.5 (50% market, 50% risk-free).
– Rp = 0.02 + 0.5 × 0.06 = 0.05 → 5.0%.
Further reading (select reputable sources)
– Investopedia — Capital Market Line (C
— the Capital Market Line (CML). https://www.investopedia.com/terms/c/cml.asp
– Wikipedia — Capital market line. https://en.wikipedia.org/wiki/Capital_market_line
– Aswath Damodaran (NYU Stern) — Lecture notes on portfolio theory and CAPM (useful for intuition and worked examples). https://pages.stern.nyu.edu/~adamodar/
– Sharpe, W. F. (1964) — “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk” (original CAPM paper). https://www.jstor.org/stable/1924119
Practical checklist: applying the CML to build a target-risk portfolio
1. Choose a risk-free rate (rf). Use a short-term government yield consistent with your horizon (e.g., 3‑month T-bill yield for short horizon). Note assumptions: single rf for borrowing and lending.
2. Select a proxy for the market portfolio (Rm) and estimate its expected return and standard deviation (σm). Common proxies: broad market index (e.g., MSCI World, S&P 500). Recognize the proxy is an approximation.
3. Decide your target portfolio volatility (σp), reflecting your risk tolerance and horizon.
4. Compute the market-price-of-risk (slope): slope = (Rm − rf) / σm.
5. Compute the portfolio expected return using the CML: Rp = rf + slope × σp.
6. Compute the weight in the market portfolio (w) and in the risk-free asset: w = σp / σm; weight in rf = 1 − w. If w > 1 you are levering; if w rf, leveraged positions are costed at rb (see step 8).
– Margin/leverage limits: verify broker margin requirements and maximum allowable leverage.
– Short-sale or position limits: ensure you can hold negative weights if w 1 (you must borrow to lever), and borrowing rate is rb > rf, expected return becomes:
Rp = rb + [(Rm − rb) / σm] × σp
Equivalently, using weights:
Rp = w × Rm + (1 − w) × rb,
with w = σp / σm.
– This produces a kinked (piecewise) capital allocation line: one slope up to w = 1 and a shallower slope beyond (higher borrowing cost reduces the reward per unit risk when levering).
9. Address estimation error and robustness.
– Expected-return and volatility inputs are noisy. Small errors can change w materially.
– Practical mitigations:
– Use long-horizon or bootstrapped estimates for Rm and σm.
– Apply shrinkage toward long-run means.
– Stress-test results across plausible Rm, σm, and rf scenarios.
– Use a range of σp values to see sensitivity of w and Rp.
– Track realized tracking error and compare ex-post portfolio σ to target σp.
10. Implementation checklist and monitoring rules.
– Pre-trade: confirm target σp, compute w and rf/borrow rate to use, verify margin and costs.
– Execute: allocate w to the market proxy and 1−w to the risk-free instrument (or borrow rb and invest extra in the market).
– Rebalancing: set rules (calendar-based, threshold-based, or volatility-targeting) and account for transaction costs.
– Reporting: monitor realized return, realized volatility, leverage, and deviations from target.
– Review inputs quarterly (or when macro conditions change materially).
Worked numeric example (step-by-step)
Assumptions:
– rf = 2.0% (0.02)
– Rm = 8.0