New on FinoLib · Volume II: Capital Markets
My latest article is now live on FinoLib, the free finance library I write: Sharpe Ratio Formula: Risk-Adjusted Return and the CML, from Volume II: Capital Markets. Read the summary below for the substance of the topic, then continue to FinoLib for the complete treatment.
5.3 The Capital Market Line & the Market Portfolio
The idea in plain words
Add a safe asset like cash to the menu, and the curved frontier becomes a straight line, the Capital Market Line. It runs from the risk-free rate to one special portfolio, the market portfolio. Every investor then picks that one risky portfolio and mixes it with cash or borrowing to set their level of risk, which is the theory behind index funds. Tracking error and active share measure how far a fund strays from its index, and the evidence that most active funds trail their index after costs is strong.
Key points
■CML: E(Rp) = Rf + [(E(RT) − Rf) ÷ σT] × σp; in the stock–bond example the slope is 0.467.
■A 5%-volatility investor earns 5.75% on the CML, a target the risky frontier alone cannot reach.
■Levering the tangency portfolio beats 100% stocks only if borrowing costs less than about 4.18%.
Decision rule
Choose the risky portfolio once (broad, diversified, low cost), then set your risk by the split between it and cash, not by changing what the risky part holds. Before paying for active management, divide the fee premium over an index fund by the fund’s active share: that is the annual cost per dollar actually managed actively. If it exceeds about 3% a year, the active positions must beat the benchmark by more than that just to break even, so choose the index fund unless you have specific evidence of skill.
The costliest mistake
Paying active fees for a closet index fund. Take a fund with 15% active share charging 0.85% a year against a 0.05% index fund (illustrative fees). On $500,000 the fee gap is 0.80% × $500,000 = $4,000 a year, but only 15% × $500,000 = $75,000 is invested differently from the index, so you pay $4,000 ÷ $75,000 = 5.33% a year on the active slice. Check active share before the fee.
Recap
With a risk-free asset, the best portfolios lie on the Capital Market Line through the tangency portfolio, and investors set risk by the cash or borrowing mix rather than by changing the risky holdings. Tracking error and active share measure how far a fund departs from its index, and Sharpe’s arithmetic and SPIVA data show most active funds trail their index after costs.
5.4 The Sharpe Ratio & Risk-Adjusted Return
The idea in plain words
The Sharpe ratio asks how much extra return you got for each unit of risk. It takes a portfolio’s return above the risk-free rate and divides it by its volatility, so funds with different risk can be compared fairly. A fund with a higher Sharpe ratio could be levered up to match a riskier fund’s volatility and would earn more. Other measures, called Treynor, Sortino and information ratios, answer different questions, and they can rank the same two funds in opposite order.
Key points
■Sharpe = (R − Rf) ÷ σ; at a 4% risk-free rate Fund B (0.80) beats Fund A (0.60) despite a lower return.
■Held at A’s 20% volatility, B earns 4% + 0.80 × 20% = 20%, against A’s 16%.
■Treynor uses beta, Sortino uses downside deviation, the information ratio uses tracking error against a benchmark.
A closer look
Comparing two investments purely by their raw returns is misleading if they carry very different levels of risk. The Sharpe ratio, developed by William Sharpe, standardizes this comparison by measuring return earned per unit of risk taken.
Decision rule
Choose the ratio by the decision. Use Sharpe when the fund will be most of your risky money; Treynor when it is one sleeve among many; the information ratio when you are hiring a manager against a benchmark; Sortino when returns are visibly skewed. Compute every ratio from the same frequency and period, annualizing a monthly Sharpe ratio by multiplying by √12. Require at least 36 monthly observations before ranking.
The costliest mistake
Ranking funds on raw return. In the example, choosing Fund A for its 16% gives up the 20% that Fund B delivers at the same 20% volatility when held with 2× leverage at the risk-free rate: four points a year. Compounded on $1 million for ten years, $1 million × (1.20^10 − 1.16^10) = $6.19 million − $4.41 million = $1.78 million. Real borrowing costs more than the risk-free rate, which shrinks the gap, but the ranking error remains. Compare risk-matched returns, not headline returns.
What the full article covers
In the full article I take you through these sections:
■5.3 The Capital Market Line & the Market Portfolio
■5.4 The Sharpe Ratio & Risk-Adjusted Return
It includes worked examples, a look under the hood at the mechanics, a clear decision rule, the one mistake that costs people the most, a recap, a reveal-on-click quiz and the official sources behind every figure.
Volume II: Capital Markets
Read the full article on FinoLib
The complete article, Sharpe Ratio Formula: Risk-Adjusted Return and the CML, with every worked example, diagram, quiz and source, is free to read in Volume II, Capital Markets on finolib.com.
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