Financial
Treynor-Black Model formulas: Appraisal Ratio & Optimal Active Sleeve Weight
Explore the model-implied active weight, appraisal ratio and theoretical Sharpe ratio under a single-index portfolio model.
A model calculation for an active sleeve and market portfolio
Alpha is the expected intercept in a single-index excess-return model, not simply return minus the market. Residual volatility is the standard deviation of the unexplained component. Express alpha, market excess return and both volatilities for the same period.
The formulas assume residual risk is uncorrelated with market returns and use known model parameters with unconstrained positions. Short positions or leverage may be needed. Estimated alpha, fees, borrowing costs and investment constraints can materially change the result; a nonzero historical alpha does not promise improved future performance.
The active weight is a fraction of the combined risky portfolio, not an allocation of total personal wealth. Values below zero or above one are retained, not clipped to a long-only allocation. The square-root result is the theoretical maximum positive Sharpe magnitude under the model, not the realized Sharpe of an arbitrary mix.
| Core-Satellite Parameter | Cell / Formula | Institutional Inputs | Treynor-Black Output |
|---|---|---|---|
| Active Alpha & Residual Vol | B2 = 2.4%, B3 = 12.0% | AR = B2 / B3 | Appraisal Ratio = 0.20 |
| Market Risk Prem & Vol | B4 = 6.0%, B5 = 15.0% | Passive Sharpe = 0.40 | Market ratio = 2.6667 |
| Optimal Active Weight w_A* | beta_A B6 = 1.0 | (0.024/0.0144) / (0.06/0.0225) | 62.5% Active / 37.5% Index |
| Combined Portfolio Sharpe | SQRT((B4/B5)^2 + (B2/B3)^2) | SQRT(0.16 + 0.04) | 0.4472 (up from 0.4000!) |
Treynor-Black Optimal Active Sleeve Weight w_A*
Compute the unadjusted weight and apply the beta adjustment. The denominator must be nonzero. A tolerance of 1e-10 flags near-singular normalization as Undefined weight; this numerical policy does not make nearby large allocations economically stable. The example requires positive market premium and positive volatilities.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B3<=0,B4<=0,B5<=0),"Invalid inputs",LET(rawWeight,(B2/(B3^2))/(B4/(B5^2)),normalizer,1+(1-B6)*rawWeight,IF(ABS(normalizer)<1E-10,"Undefined weight",ROUND(rawWeight/normalizer,4)))))Appraisal Ratio (alpha / sigma_e) and Combined Sharpe Ratio
Calculate the theoretical square-root combination under the model assumptions. This is neither a forecast guarantee nor evidence of manager skill.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B3<=0,B4<=0,B5<=0),"Invalid inputs",ROUND(SQRT(((B4 / B5) ^ 2) + ((B2 / B3) ^ 2)), 4)))Appraisal ratio from alpha and residual volatility
Divide alpha by residual standard deviation, not residual variance. A negative alpha produces a negative appraisal ratio.
=IF(COUNT(B2:B3)<>2,"Invalid inputs",IF(B3<=0,"Invalid inputs",ROUND(B2 / B3, 4)))Try the example
Use the original values or change them to test your understanding.