What is the difference between a Sharpe relationship and a Treynor relationship?


The Sharpe index and the Treynor index are two indices used to measure risk-adjusted rate of return. Both are named after their creators, the Nobel laureate William Sharpe and the American economist Jack Treynor, respectively. While they can help investors understand investments and risks, they offer different approaches to evaluating investment performance. The Sharpe index helps investors understand an investment’s performance compared to its risk, while the Treynor index explores the excess return generated by each unit of risk in a portfolio.

This short article explores how each ratio works and how they differ.

How Sharpe’s relationship works

First developed in 1966 and revised in 1994, the Sharpe index aims to reveal how well an asset performs compared to a risk-free investment. The common benchmark used to represent risk-free investment is US Treasury bills, especially the 90-day Treasury bill. The Sharpe index calculates the expected or actual investment return for an investment portfolio (or even an individual equity investment), subtracts the risk-free investment return, and then divides that number by the standard deviation for the portfolio of investments. Generally, the higher the value of the Sharpe index, the more attractive the risk-adjusted return.

MR

=

(

r

X


R

F

)

S

D

where:

r

X

=

Expected or actual return on investment

investment

R

F

=

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Return on investment without risk

S

D

=

Standard deviation of

r

X

begin {aligned} & textnormal {SR} = frac {(r_ {x} – R_ {F})} {SD} \ & textbf {where:} \ & begin {aligned} r_ { x} = & text {Expected or actual return on investment} \ & text {on investment} end {aligned} \ & R_ {F} = text {Return on investment without risk} \ & SD = text {Standard deviation of} r_ {x} \ end {aligned}

MR=SD(rXRF)where:rX= Expected or actual return on investment investmentRF=Return on investment without riskSD=Standard deviation of rX

The expected or actual rate of return can be measured at any frequency, as long as the measurement is consistent. Once the expected or actual rate of return is subtracted from the risk-free investment return, it can be divided by the standard deviation. The larger the deviation, the better the performance.

The primary purpose of the Sharpe index is to determine whether you are getting a significantly higher return on your investment in exchange for accepting the additional risk inherent in investing in stocks compared to investing in risk-free instruments.

How Treynor’s relationship works

Developed around the same time as the Sharpe index, the Treynor index also seeks to assess the risk-adjusted performance of an investment portfolio, but measures the portfolio’s performance against a different benchmark. Rather than measuring the performance of a portfolio only against the rate of return of a risk-free investment, the Treynor index seeks to examine how well a portfolio outperforms the stock market as a whole. It does this by substituting beta for standard deviation in the Sharpe relationship equation, with beta defined as the rate of return due to overall market performance.

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For example, if a standard stock index shows a 10% rate of return, that constitutes beta. An investment portfolio that shows a 13% rate of return, according to the Treynor index, only receives credit for the additional 3% return it generated above the overall market return. The Treynor index can be considered to determine whether your investment portfolio is significantly outperforming average market returns.

Limitations of each proportion

There are certain drawbacks to each of these proportions. Where Sharpe’s relationship fails is that it is accentuated by investments that do not have a normal distribution of returns like hedge funds. Many of them use dynamic trading strategies and options that can skew your returns.

The main disadvantage of the Treynor relationship is that it is retrospective and relies on the use of a specific benchmark to measure beta. However, most investments do not necessarily behave in the same way in the future as they did in the past.

The bottom line

The difference between the two metrics is that the Treynor index uses beta, or market risk, to measure volatility rather than using total risk (standard deviation) like the Sharpe index.

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Mark Holland

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