Risk Adjusted Returns: Returns

Posted By: Advisor Analyzer Team

In our previous article, we discussed some common measures to evaluate risk in an investment. In our second article, we will discuss methods used to assess returns after adjusting for an investment’s given risk levels.

 

Evaluating risk adjusted returns allows investors to compare apples to apples. It levels the playing field, so the investor can fairly analyze different return with varying risk levels. The more popular risk adjusted return measures are the Sharpe Ratio, Treynor Measure, The Jensen alpha and the Capital Asset Pricing Model (CAPM).     

Real World Application

We will also use a real life example to further clarify these ratios. For our example we will evaluate two stocks: CROCS Inc. (CROX) and Nike Inc. (NKE). The data used are daily closing prices from January 1, 2007-January 1, 2008. CROX’s return for the year was 67.32%, while NKE’s annual return was 33.35% (roughly half). The S & P 500 for the year 2007 returned 5.49%. The U.S. government 3-month Treasury bills yielded 4.51% in June of 2007 (we use June’s rates as oppose to another month for simplicity).

Without any additional information, CROX appears the clear winner when compared to Nike, a T-Bill or the market as a whole. However, we are comparing apples to oranges without any information on risk. Once adjusted for risk, the order of best to worst investment quickly changes.

Sharpe Ratio

Ask anyone who’s taken a finance class what the most popular risk adjusted return measure is and they would undoubtedly state the Sharpe Ratio. The Sharpe ratio measures excess returns, per unit of total risk:

 

· Sharpe Ratio = (RI – RF) / sd

 

Where:

RI = Return of the investment

RF = Risk free rate (usually T-Bills)

sd = Standard deviation (total risk) of the investment

 

So what does this mean? When we examine returns, we can look at absolute returns or excess returns. Excess returns are returns earned above the risk free rate (known as the risk premium). Since everyone can earn returns without exposing themselves to risk, the Sharpe ratio is only interested returns achieved above the risk free.

For example, suppose a government bond pays 5% annually. An investor who makes 5% in the stock market did not earn any excess returns (0=5%-5%). Therefore, the investor could have earned the same 5% returns without unnecessarily increasing his blood pressure from riding the market roller coaster.  

Once we calculate the excess returns of an investment, we examine the stock’s standard deviation. Standard deviation is used to measure the total volatility (total risk) of an investment using historical pricing data. We are dividing by the standard deviation to gauge the amount of risk taken in order to achieve excess return. A higher the ratio indicates greater excess returns were earned for a given level of total risk. Therefore, a security with a higher Sharpe ratio is considered the better investment.

 

Returning to our CROX and NKE example:

 

· Sharpe Ratio for CROX = (RCROX – RTBILL) / sdCROX = (67.32%-4.51%) / (4.17%) =  15.06

 

· Sharpe Ratio for NKE = (RNKE – RTBILL) / sdNKE = (33.35%-4.51%) / (1.39%) = 20.692  

 

· Sharpe Ratio for S & P 500 (SPY) = (RSPY – RTBILL) / sdSPY = (5.49 %-4.51%) / (1.0038%) = .9766338

 

Once we have adjusted the excess returns for CROX and NKE for the risk taken, we see that CROX was the poorer choice of the two. However, although CROX underperformed NKE, the Sharpe ratio ranks CROX as the better investment than the overall market.

Treynor Ratio

The Treynor Ratio provides a measure of excess return per unit of systematic risk (beta). The underlying assumption of the Treynor measure is that a multi-asset portfolio diversifies unsystematic risk away and the relevant risk that remains is systematic risk. Put bluntly, diversification will help reduce firm specific risks (which is why we never put all our eggs in one basket), leaving only market risks to worry about (risk that all our separated eggs simultaneously spoil).

The Treynor Ratio is very similar to the Sharpe ratio, and the relation can be proven mathematically (although we won’t go through it here). The Treynor Ratio can be calculated as:  

 

· Treynor Measure = (Ri – Rf) / bi

 

Notice the numerator of the equation is identical to the Sharpe Ratio. Therefore, both Treynor and Sharpe measure excess returns for a given level of risk. The main difference lies in the measure of risk used in the denominator: Sharpe uses standard deviation, Treynor uses Beta.

The two measures of risk are mathematically linked. In fact, beta is a portion of standard deviation. Very simply:

 

· Total Risk (=Standard Dev) = Market Risk (=Beta) + Firm Specific Risk

 

When applying the Treynor Measure to our CROX example:

 

· Treynor Ratio for CROX = (RCROX – RTBILL) / sCROX = (67.32%-4.51%) / (2.29) = .2742

 

· Treynor Ratio for NKE = (RNKE – RTBILL) / sNKE = (33.35%-4.51%) / (.45) = .6408  

 

· Treynor Ratio for S & P 500 (SPY) = (RSPY – RTBILL) / sSPY = (5.49 %-4.51%) / (1) = .0098  

 

Note the market’s beta equals 1. This is always the case since beta measures an investment’s percentage change for a 1% change in the market. The market’s change to itself is therefore always a 1-to-1 ratio. Therefore, the beta of the market to itself is always 1. Evaluating the results for CROX, NKE and the overall market again ranks NKE as the best investment. The results are actually identical to the Sharpe ratio. Therefore, it can be inferred that NKE is the better performing investment after adjusting for total risk as well as adjusting for market risk.

Jensen Alpha and Capital Asset Pricing Model

The word alpha has been a widely used term in the financial industry. Alpha is a performance measure relative to an overall market index, and is gauged against an appropriate benchmark. A small cap stock should be assessed relative to a small cap index, as oppose to the S & P 500 (which is an index that represents 500 of the largest publically traded companies). The actual calculations are quite simple:

 

· Alpha = (R Investment – R Benchmark)

 

For example, if the market return was 7% and an investment’s return was 10%, alpha is 3%.

The capital asset pricing model (CAPM) is a performance measure, when coupled with alpha, creates the Jensen ratio. While we will not go into detail of the CAPM in this article, there is a mathematical relationship between CAPM, Beta, Sharpe Ratio and Treynor. The CAPM is a method used to estimate expected excess returns of an investment relative to beta risk (market risk). In essence, we are comparing the expected return to the actual returns earned from an investment:

 

· R Investment using CAPM = [R Risk Free + bp (R Market – R Risk Free)]

 

The right hand side of the CAPM equation determines what the investment should have earned. Using this equation, alpha can be calculated by establishing the expected returns generated from the CAPM as the benchmark return, and taking the difference from the realized returns.

 

Therefore, to calculate alpha, we combine the two equations:

 

o ap = (R Investment –R Benchmark)

o ap = (R Investment –R Investment Using CAPM)

o ap = R Investment – [R Risk Free + bp (R Market – R Risk Free)]  

When we apply this relationship to evaluate our real world investment applications, we find that:

 

· R CROX using CAPM = [R Risk Free + bCROX (R SPY – R Risk Free)] = [4.51% + 2.29(5.49 – 4.51%)] = 6.7542

 

· a CROX = (R CROX ACTUAL –R CROX USING CAPM) = 67.32% - 6.7542% = 60.5658 %

 

· R NKE using CAPM = [R Risk Free + bNKE (R SPY – R Risk Free)] = [4.51% + .45(5.49 – 4.51%)] = 4.951

 

· a NKE = (R CROX ACTUAL ��R NKE USING CAPM) = 33.35% - 4.91% =28.44%

 

Notice when using alpha, we get different results than Sharpe or Treynor method. Using the Jensen alpha method, CROX was the better investment between the two.

The difference in rankings most likely stems from improper benchmark selection (although not a great benchmark, the Wilshire returns were marginally better than the S&P 500). However, it should be noted that inconsistent recommendations are not uncommon: two methods may yield diametrically differing results.

Real World Results

For 2008 year to date (1/1/08 – 9/19/08), CROX had a return of (-89.13%) while NKE, for the same period, experienced a return of (+1.77%). The S&P 500 tracker (SPY) fell 13% during this time. Therefore, although the profits earned by CROX doubled that of NKE for 2007, we find that Treynor and Sharpe ratios correctly determined NKE as a better risk adjusted investment for 2008.