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2012-10-26 11:22:25
"When it comes to money, everybody is of the same religion." - Voltaire
Most people would agree that they want to make and have money, but very few
people would agree to the level of risk they are willing to take on to make
that money. Therefore, risk must be the first issue you address when you are
looking at choosing your investments. (For more insight, see Determining Risk
And The Risk Pyramid.)
Tutorial: Top Stock-Picking Strategies
In this article, we'll show you why the Sharpe ratio can help you determine
which asset classes will deliver the highest returns while considering its
risk.
The Sharpe ratio is designed to measure a unit of reward for each unit of risk
taken. Let's take a look at this simple ratio in more detail.
Sharpe Ratio Dynamics
The Sharpe ratio, developed by Nobel Laureate William Sharpe, is designed to
measure how many excess units of returns an investor can achieve over the
risk-free rate for each unit of risk taken.
Thus, the Shape Ratio measures the risk/reward value of investors' assets class
choices beyond the U.S. Treasury.
Let's take a look at the efficient frontier chart below to better illustrate
the concept of risk, return and the Sharpe ratio.
Figure 1: Efficient Frontier - if you plot all the investment choices that you
have at your disposal - stocks, bonds and portfolios of stocks and bonds, etc.
- on the chart above, the resulting chart will be bounded by an upward sloping
curve known as the efficient frontier.
Return Dynamics
Without taking on risk, you can achieve a level of return as indicated on the
chart by the risk-free portfolio, the U.S. Treasury.
To achieve an additional X percent of return, you will need to take Z level of
risk. Portfolio A represents your risk and return payoff. The Sharpe ratio of
Portfolio A can simply be defined as X divided by Z. Portfolios B and C will
deliver a higher level of returns should you choose to take additional risk
beyond Z.
Unlike portfolio B and C, portfolios A' and A'' will deliver a higher level of
returns for the same level of risk Z. Thus, A'' is preferable to A' and A' is
preferable to A. The Sharpe ratio of A' is defined as X+Y divided by Z.
Therefore, the Sharpe ratio of A' is higher than that of A. Given the same
level of risk Z, it can be concluded that any portfolio providing X plus
additional returns should be considered superior. The additional achievable
returns will be limited by the efficient frontier. Applying this same
methodology, we can also presume that Portfolios B and C are superior if their
Sharpe ratios are shown to be higher to that of A. (To learn more, check out
Understanding The Sharpe Ratio and The Sharpe Ratio Can Oversimplify Risk.)
Breaking Down the Sharpe Ratio
A common mathematical definition of the Sharpe ratio for a portfolio is the
excess returns of the portfolio over the risk-free rate divided by the
portfolio's standard deviation.
Here is an illustration of the Sharpe ratio in the same efficient frontier
chart:
Figure 2
It can be concluded that for a given level of risk (sp), Portfolio A can
achieve a higher Sharpe ratio by following the blue arrow toward the efficient
frontier or, for a given level of return (Rp), Portfolio A can also achieve
higher Sharpe ratio by following the red arrow toward the efficient frontier.
Sharpe Ratio and Risk
The charts and the formula demonstrate that the Sharpe ratio penalizes the
excess returns by adding of risk as defined by standard deviation. The standard
deviation is also commonly referred to as the total risk. Mathematically, the
square of standard deviation is the variance, Markowitz's definition of risk.
(For further reading, see Understanding Volatility Measurements.)
So why did Sharpe choose the standard deviation to adjust excess returns for
risk and why should we care? We know that Markowitz defined variance as
something not to be desired by investors. Variance is defined as a measure of
statistical dispersion or an indication of how far away it is from the expected
value. The square root of variance, or standard deviation, has the same unit
form as the data series being analyzed and is such more commonly used to
measure risk.
The following example illustrates why investors should care about variance:
An investor has a choice of three portfolios, all with expected returns of 10%
for the next 10 years. The average returns in the table below indicates the
stated expectation. The returns achieved for the investment horizon is
indicated by annualized returns, which takes compounding into account. As the
data table and the chart clearly illustrates below, the standard deviation
takes returns away from the expected return. If there is no risk, zero standard
deviation, your returns will equal your expected returns.
Expected Average Returns
Year Portfolio A Portfolio B Portfolio C
Year 1 10.00% 9.00% 2.00%
Year 2 10.00% 15.00% -2.00%
Year 3 10.00% 23.00% 18.00%
Year 4 10.00% 10.00% 12.00%
Year 5 10.00% 11.00% 15.00%
Year 6 10.00% 8.00% 2.00%
Year 7 10.00% 7.00% 7.00%
Year 8 10.00% 6.00% 21.00%
Year 9 10.00% 6.00% 8.00%
Year 10 10.00% 5.00% 17.00%
Average Returns 10.00% 10.00% 10.00%
Annualized Returns 10.00% 9.88% 9.75%
Standard Deviation 0.00% 5.44% 7.80%
Figure 3
Figure 4
Conclusion
Risk and reward must be evaluated together when considering investment choices;
this is focal point presented in modern portfolio theory. In a common
definition of risk, the standard deviation or variance takes rewards away from
the investor. As such, the risk must always be addressed along with the reward
when you are looking to choose your investments. The Sharpe ratio can help you
determine the investment choice that will deliver the highest returns while
considering its risk.
by Edward Aw
Edward Aw currently heads the quantitative research efforts at a privately held
wealth management company based in New York. His main responsibilities include
developing analyst tools, enhancing security selection and risk management. He
served as a quantitative analyst for five years at Deutsche Investment
Management Americas. Aw also held analytical roles at the Dreyfus Corporation,
Goldman Sachs and Morgan Stanley. He holds a Chartered Financial Analyst
designation, a Bachelor of Arts from the State University of New York and an
Master of Business Administration from Hofstra University. He is a member of
the New York Society of Security Analysts.