# Sharpe Ratio, Explained in Plain English

In today’s near-zero interest rate economy, the reward versus risk of an investment portfolio can be measured using the Sharpe ratio.  Like a batting average, higher numbers are better, and 0.400 is very good.

If portfolio Z has a forward-looking Sharpe ratio of 0.400, and an expected return of 8%, there is a 68% chance its 1-year return will be between -12% and +28%.

The math is surprisingly easy.  Because the Sharpe ratio is a return/risk ratio it can be transformed into a risk/return ratio by finding its inverse (using the “1/x” button on a calculator).  The inverse of 0.400 is 2.5.  The return is 8%, so the “risk” is 2.5 times 8% which is 20%.

For the Sharpe ratio, the downside risk and the upside “risk” are the same.  So the downside is 8% -20%, or -12%.   The upside risk is 8%+20%, or 28%.  Easy!

### Sharpe Ratios and Risk (more detail)

Where did the “68% chance” come from?  The answer is a bit more complicated, but still fairly easy to understand.

It comes from the 3-sigma1 rule of statistics.  The range of -12% to +28% comes from 1 standard deviations of the mean (or plus or minus one sigma).  The 3-sigma rule also says that 95% of outcomes will fall within two standard deviations.  Double the deviation means  two times the upside and downside risk, so the 95% confidence range becomes -32% to 48%.  Finally the 3-sigma rule means triple the upside and downside risk, meaning outcomes from -52% to +68% will occur 99.7 percent of the time.

Almost every investor will be be pleased with a positive sigma event, where the return is above 8%.   For example a +1 sigma (+1σ) occurrence has a +28% return — quite nice.

A downside event is potentially quite troublesome.  Even a -1σ event means a 12% loss.  A -2σ is a much worse 32% loss.

### Ex Ante and Ex Post Sharpe Ratios

Forward-looking (ex ante) Sharpe ratios are predictions “prior to the event(s)”.  They are always positive, because no rational investor would invest in a negative expected return.  The assumptions baked into an ex ante Sharpe ratio predictions are 1) expected standard deviation of total return, σ,  2) expected future return.

Backward-looking, or after the fact, (ex post) Sharpe ratios can be negative or positive.  In fact, assuming “normal distributions of return”, there is a reasonable (but less than 50%) chance of a negative ex post Sharpe ratio.

Sigma1 HAL0 software optimizes for Sharpe ratios by optimizing for return and standard deviation.  It also optimizes for semivariance.  More “plain English” on that advantage later.

# Benchmarking Financial Algorithms

In my last post I showed that there are far more that a googol permutations of portfolio of 100 assets with (positive, non-zero) weights in increments of 10 basis points, or 0.1%.    That number can be expressed as C(999,99), or C(999,900) or 999!/(99!*900!), or ~6.385*10138.  Out of sheer audacity, I will call this number Balhiser’s first constant (Kβ1).  [Wouldn’t it be ironic and embarrassing if my math was incorrect?]

In the spirit of Alan Turing’s 100th birthday today and David Hilbert’s 23 unsolved problems of 1900, I propose the creation of an initial set of financial problems to rate the general effectiveness of various portfolio-optimization algorithms.  These problems would be of a similar form:  each having a search space of Kβ1. There would be 23 initial problems P1…P23.  Each would have a series of 37 monthly absolute returns.  Each security will have an expected annualized 3-year return (some based on the historic 37-month returns, others independent).  The challenge for any algorithm A to score the best average score on these problems.

I propose the following scoring measures:  1) S”(A) (S double prime) which simply computes the least average semi-variance portfolio independent of expected return.  2) S'(A) which computes the best average semi-variance and expected return efficient frontier versus a baseline frontier.  3) S(A) which computes the best average semi-variance, variance, and expected return efficient frontier surface versus a baseline surface.  Any algorithm would be disqualified if any single test took longer than 10 minutes.  Similarly any algorithm would be disqualified if it failed to produce a “sufficient solution density and breadth” for S’ and S” on any test.  Obviously, a standard benchmark computer would be required.  Any OS, supporting software, etc could be used for purposes of benchmarking.

The benchmark computer would likely be a well-equipped multi-core system such as a 32 GB Intel  i7-3770 system.  There could be separate benchmarks for parallel computing, where the algorithm + hardware was tested as holistic system.

I propose these initial portfolio benchmarks for a variety of reasons.  1)  Similar standardized benchmarks have been very helpful in evaluating and improving algorithms in other fields such as electrical engineering.  2)  Providing a standard that helps separate statistically significant from anecdotal inference. 3)  Illustrate both the challenge and the opportunity for financial algorithms to solve important investing problems. 4)  Lowering barriers to entry for financial algorithm developers (and thus lowering the cost of high-quality algorithms to financial businesses).  5)  I believe HAL0 can provide superior results.