Systematic Risk & Beta Analytics Lab

Treynor Ratio Calculator

Model the Treynor Ratio, systematic beta exposure, Jensen's Alpha, and CAPM benchmark performance in an interactive institutional risk simulator.

Treynor Ratio (Systematic Efficiency)
10.19%
Superior Systematic Alpha (≥ 10.0%)

1. Portfolio Return & Systematic Risk Inputs

%
Annualized realized or projected portfolio return.
%
Current yield on benchmark short-term US Treasuries.
β
Sensitivity to systematic market swings (Market = 1.0).
%
Expected annual return of broad market index (e.g. S&P 500).
%
Annualized standard deviation (used for Sharpe comparison).
%
Target excess return per unit of beta hurdle.

2. CAPM Benchmark & Diversification Analysis

Excess Return (Rp - Rf)
+13.75%
Spread over Risk-Free Rate
Jensen's Alpha (α)
+4.64%
Rp - CAPM Expected
CAPM Expected Return
13.36%
Rf + β × (Rm - Rf)
Sharpe Ratio (Total Risk)
0.63x
(Rp - Rf) / σtotal
Required Return at Target Treynor: 16.40%
Performance Spread vs. Target: +1.60% (Surplus over Target)

3. Executive Risk & Performance Evaluation

Analyzing portfolio systematic beta risk and market risk-adjusted return...

Matrix 1: Portfolio Return (%) vs. Beta on Treynor Ratio (%)

Displays projected Treynor Ratio across variations in portfolio annual return and systematic market beta exposure.

Matrix 2: Market Return (%) vs. Beta on Jensen's Alpha (%)

Displays Jensen's Alpha sensitivity as the broad market return and portfolio beta shift.

4. Core Mathematical Formulas & Systematic Risk Principles

A. Primary Treynor Ratio Equation

Treynor Ratio = (Rp - Rf) / βp
Rp = Portfolio Annualized Return, Rf = Risk-Free Rate, βp = Portfolio Beta

Quantifies excess return generated per unit of systematic, non-diversifiable market risk.

B. Capital Asset Pricing Model (CAPM)

E(Rp) = Rf + βp × [E(Rm) - Rf]

Represents the theoretical minimum required return demanded by rational investors for assuming beta exposure βp.

C. Jensen's Alpha (α) Equation

α = Rp - E(Rp) = Rp - [ Rf + βp × (Rm - Rf) ]

Measures true abnormal managerial performance and excess value creation above the CAPM Security Market Line (SML).

D. Sharpe vs. Treynor Diversification Appraisal

Sharpe Ratio = (Rp - Rf) / σtotal
Total Variance = β2 × σm2 + σε2 (Unsystematic Risk)

When a portfolio holds substantial firm-specific unhedged risk, its Sharpe Ratio suffers while Treynor remains unaffected.

5. Institutional Risk Benchmarks Across Asset Strategies

Strategy / Portfolio Profile Typical Beta (β) Excess Return Typical Treynor Risk & Diversification Characteristics
Quality Large-Cap Compounders 0.80 – 0.95 +9.0% – 12.0% 10.0% – 13.0% Low leverage, high ROIC, and defensive balance sheets generate superior alpha per unit of market beta.
Aggressive Tech & Disruption 1.25 – 1.60 +12.0% – 18.0% 9.0% – 11.5% High market beta amplifies bull market gains; requires significant excess return to maintain high Treynor ratios.
Broad Market Index (S&P 500) 1.00 +6.0% – 8.0% 6.0% – 8.0% Baseline benchmark. Zero unsystematic risk; Treynor ratio equals exact market risk premium (R_m - R_f).
Defensive Utilities & Staples 0.45 – 0.65 +4.0% – 6.0% 7.5% – 10.0% Low systematic volatility; stable cash flows provide respectable Treynor ratios even during modest equity rallies.
Distressed / Cyclical Turnarounds 1.50 – 2.00 +3.0% – 6.0% 2.0% – 3.5% High debt and operational leverage elevate beta, creating poor Treynor efficiency unless a massive turnaround occurs.

6. Frequently Asked Questions

The Treynor Ratio, introduced by Jack Treynor, measures the risk-adjusted excess return per unit of systematic (undiversifiable market) risk. The formula is: Treynor Ratio = (Portfolio Return - Risk-Free Rate) / Portfolio Beta.

The Sharpe Ratio divides excess return by total risk (standard deviation), which includes both systematic market risk and unsystematic (firm-specific) risk. The Treynor Ratio divides excess return only by systematic risk (beta). If a portfolio is fully diversified, its unsystematic risk approaches zero and Treynor and Sharpe rankings converge; if poorly diversified, Sharpe will penalize the unsystematic risk while Treynor will ignore it.

Both metrics evaluate performance relative to the Capital Asset Pricing Model (CAPM). While Jensen's Alpha measures the absolute percentage excess return above the security market line (Alpha = R_p - [R_f + Beta*(R_m - R_f)]), the Treynor Ratio measures the slope of the line connecting the risk-free rate to the portfolio on the Security Market Line (SML).

A good Treynor Ratio is one that exceeds the benchmark market Treynor Ratio (Market Return - Risk-Free Rate). For example, if the equity market premium is 6.75%, any portfolio Treynor Ratio above 6.75% indicates positive risk-adjusted excess return per unit of market volatility.

The Treynor Ratio is optimal when evaluating an investment or sub-fund that will be added to an already well-diversified broader portfolio. Because the investor's overall portfolio eliminates unsystematic risk, only the systematic risk (beta) of the added asset matters.

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