On Sharpe Ratio: The Higher, The Better?

Sharpe Ratio is perhaps one of the most famous and powerful ways to evaluate the performance of a portfolio.

$Sharpe\ Ratio = \frac{Risk\ Premium}{SD\ of\ excess\ return}$

In most cases, Sharpe Ratio is used as a quick way to reflect the reward to volatility of a portfolio:

The higher Sharpe Ratio is, the more risk premium we assume that is coming from a given "volatility".

For example, some big trading companies may categorize funds with Sharpe more than 2 as spectacular, more than 1.5 as excellent, more than 1.25 as good, more than 1 as average, less than 1 as inferior...

HOWEVER, given different time horizons of a portfolio, while others remain constant, Sharpe can be completely different, and evaluation based merely on Sharpe can be misleading.

To understand this, let's first dive into the components of Sharpe Ratio:

I. Risk Premium

To begin with, assume we have all the data of monthly return of Baidu. We fear that there may be some scandals tomorrow and influence its cash flow in the future, and in the meanwhile we also hope a new version of auto-mobile car invented in the next month.

To bear this kind of risk, a risk-averse investor (we assume all the investors are risk-averse) would then expect a higher return in the future.

But how to estimate excess return? The most widely-used way is to use the arithmetic average return minus risk-free rate as the estimation of risk premium.

$\widehat{E(r)}=\frac{1}{n}\sum_{i=1}^{n} r_i$

Under the assumption of normal return distribution, the above estimation is unbiased.

Expected Return minus Risk-free Rate, and we can get our unbiased estimation of Risk Premium.

II. Volatility: Standard Deviation of Excess Return

A generally accepted valuation of expected volatility is standard deviation $\sigma $.

Some claim that we care only about the downward risk. However, Bodie argued that as long as the probability distribution is more or less symmetric about the mean, $\sigma $ is a reasonable measure of risk.

$\widehat{E(\sigma ^2)}=\frac{1}{n-1}\sum_{i=1}^{n} (r_i-\bar{r})^2$

Under normal assumption, such estimation of standard deviation is unbiased as the degrees of freedom bias is eliminated.

III. Time Horizons: a variable greatly easy to be ignored

Now, we already have the estimations of Risk Premium as well as Estimated Volatility, and simply divide them, we can derive Sharpe Ratio.

What seems to be the problem?

Given monthly average excess return r, and monthly standard deviation $\sigma$, assume that returns among each month is i.i.d., and annually estimated excess return is $12r$, while annually estimated standard deviation is $\sqrt{12}\sigma$

Therefore, now the annual Sharpe Ratio is $\sqrt{12}\frac{r}{\sigma}$, while the monthly Sharpe Ratio is $\frac{r}{\sigma}$.

This indicates that a same portfolio can be measured with drastically different Sharpe Ratio!

For example, if the previous annual Sharpe Ratio is 1.50, labeled as excellent previously, would soon become 0.43, which is inferior in a monthly basis.

However, in practice, we merely mention that "My Sharpe is 2 in an annual basis", but "The Sharpe of my strategy is 2".

We simply ignore one factor that may change the entire story.




However, misleading information (deviation from reality in return predicting) about Sharpe can also come from the part of Risk Premium itself: see the further information.

* Further Information

Geometric mean of return, is also a useful estimation, as it is more close to the reality:

$(1+g)^n=\prod_{i=1}^{n} (1+r_i)$

For example, $r_1=-0.5$, and $r_2=0.5$. Average return is 0, but if we invest 100 dollars today, and we get 50 dollars tomorrow, 75 dollars the day after tomorrow. Geometric average is -13.40%, which indicates how ridiculous the estimation of average mean could be.

However, according to Bodie, under normal assumption, geometric estimation is biased while the mean is unbiased, and the relationship between them is:

$E(geometric\ return)=E(average\ return)-\frac{1}{2}\sigma ^2$

Even though at most cases return distribution is not that normal, we can assume it to be close to be normal, and the average return is still a reasonable estimation.

* Observing frequency may also influence the accuracy of estimation, and then affect Sharpe Ratio.

* Why Sharpe Ratio should divide Standard Deviation rather than Variance? This is an interesting question, maybe we can discuss in the future.