Sharpe / Treynor / Jensen / Tỷ số thông tin

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Funds

Country focus sets the risk-free rate (4.45%) and the market index (S&P 500, expected return 10.0%, σ_m 15.0%).

Portfolio inputs

Portfolio return R_p

Portfolio σ_p

Portfolio β_p (vs market)

Benchmark / Information ratio

Benchmark return R_b

Tracking error

R_f = 4.45% (country = United States)
R_m = 10.0% (long-run mean of S&P 500)
σ_m = 15.0%

Sharpe ratio

0.419

(R_p − R_f) / σ_p

Treynor ratio

6.86%

(R_p − R_f) / β_p

Jensen's α

1.44%

vs CAPM E[r]

Information ratio

0.500

(R_p − R_b) / TE

M² (Modigliani²)

10.74%

R_f + Sharpe × σ_m

Side-by-side comparison

Portfolio Sharpe

0.42

Benchmark Sharpe

0.37

Jensen's α

1.44%

Info ratio

0.50

Reading the results: Sharpe is the canonical reward-per-σ. Treynor uses β instead of σ — useful when the portfolio is already a piece of a diversified allocation. Jensen's α is the CAPM intercept — positive means the manager beat the line. The Information Ratio is active management's quality score: excess return per unit of tracking error vs the benchmark.

Raw returns can mislead - a fund up 30% might just be running 3x the market's risk. Risk-adjusted ratios fix that by dividing return by some flavor of risk. Sharpe uses total volatility; Treynor uses beta; Jensen's alpha measures CAPM-relative outperformance; Information Ratio measures consistency vs. a benchmark.

Mỗi Lab dựa trên toán học sách giáo khoa tiêu chuẩn (Bodie / Kane / Marcus, Fabozzi, Tandelilin). Read the full methodology in the How It Works panel above.