Treynor quotient

The Treynor quotient , and the Treynor measure or Treynor ratio called ( English Treynor ratio ), is a financial ratio that the ratio of the excess return to the beta factor and thus the risk premium per unit of the received systematic risk is measured. The key figure was presented by Jack Treynor in 1965 as part of his work on the capital goods price model (CAPM).

Formal representation

Investments above or below the security line are overvalued or undervalued.

The key figure is derived from the central equation of the CAPM, the securities line :

{\ displaystyle E [R_ {i}] = r_ {f} + \ beta _ {i} (E [R_ {M}] - r_ {f})}

.

In the financial market equilibrium, the current price of a security adapts in such a way that the expected return exceeds the risk-free interest rate by a risk premium that increases proportionally with the security beta factor . This equation is transformed and it results:

{\ displaystyle T_ {i} \ equiv E [R_ {M}] - r_ {f} = {\ frac {E [R_ {i}] - r_ {f}} {\ beta _ {i}}}}

where represents the return on the portfolio, the return on the risk free investment and the beta of the portfolio. The Treynor quotient is therefore a measure of the excess return achieved per unit of non-diversifiable risk taken over. The market portfolio has by definition a beta of 1 and thus the Treynor quotient obtained directly as excess return. ${\ displaystyle R_ {i}}$ ${\ displaystyle r_ {f}}$ ${\ displaystyle \ beta _ {i}}$

In the adjacent figure, two securities are drawn outside the securities line. The respective Treynor quotient corresponds to the slope of a line through these points (see blue dashed lines). Such a straight line shows all - combinations that investors can realize by investing in the security at risk-free interest rates. If there are two portfolios to choose from under the same general conditions, the portfolio with the larger Treynor quotient achieves its return with a lower systematic risk . In other words, one would choose between two portfolios with the same beta factor that one that generates the greater excess return. ${\ displaystyle {\ frac {E [R_ {i}] - r_ {f}} {\ beta _ {i}}}}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ beta}$

rating

In contrast to the Treynor quotient, the Sharpe quotient uses the standard deviation (volatility) instead of the beta factor and thus measures the overall risk, i.e. in addition to the systematic risk also the unsystematic risk that arises from insufficient diversification of the portfolio. ${\ displaystyle \ sigma _ {i}}$

If you compare two portfolios that do not consist of stocks from the same market, the Sharpe quotient is more suitable, since the beta factor of the Treynor quotient expresses the sensitivity of a portfolio to fluctuations in the respective market. The Sharpe quotient can be used across all markets, as the calculation is based on the standard deviation.

Since the Treynor quotient and Sharpe quotient are relative risk measures, both can be used to rank portfolios with a different systematic risk. The Jensen Alpha is an absolute benchmark for performance .

Obtaining the beta values required for the Treynor quotient is often problematic. Due to the poor quality of the data, such a calculation is very complicated for hedge funds . The Treynor measure as a whole is subject to the CAPM's criticism and is dependent on its premises.

literature

Jack L. Treynor: How to Rate Management of Investment Funds . In: Harvard Business Review . tape 43 , no. 1 , 1965, p. 63-75 .
Klaus Spremann: Portfolio Management . De Oldenbourg Gruyter, 2008, ISBN 978-3-486-58779-1 . In particular Chapter 11.2.3 on p. 356ff.
Marco Wilkens, Hendrik Scholz: From the Treynor Ratio to Market Risk-Adjusted Performance . In: Finance Operations . 1999, p. 308–315 ( at uni-augsburg.de [PDF; 170 kB ]).

Individual evidence

↑ Trautmann, Siegfried. Investments: assessment, selection and risk management. Springer-Verlag, 2007. p. 178.
↑ Bernd R. Fischer: Performance Analysis in Practice: Performance Measures, Attribution Analysis, Global Investment Performance Standards . Walter de Gruyter, 2010, p. 454-455 .
↑ Dieter G. Kaiser: Hedge Funds: Demystification of an Asset Class; Structures, opportunities, risks . Springer-Verlag, 2004, p. 184 .
↑ André Rutkis: hedge funds as alternative investments. Styles, risks, performance . Frankfurt School Verlag, Frankfurt am Main 2002, p. 72 .

[1] Trautmann, Siegfried. Investments: assessment, selection and risk management. Springer-Verlag, 2007. p. 178.

[2] Bernd R. Fischer: Performance Analysis in Practice: Performance Measures, Attribution Analysis, Global Investment Performance Standards . Walter de Gruyter, 2010, p. 454-455 .

[3] Dieter G. Kaiser: Hedge Funds: Demystification of an Asset Class; Structures, opportunities, risks . Springer-Verlag, 2004, p. 184 .

[4] André Rutkis: hedge funds as alternative investments. Styles, risks, performance . Frankfurt School Verlag, Frankfurt am Main 2002, p. 72 .