How Are Your Alternative Investments Performing?

Measuring Performance Using The Information Ratio And T-statistic … — ᴊᴇғғʀᴇʏ sᴇᴀᴛᴏɴ & ᴀʟᴇxᴀɴᴅᴇʀ ᴢʜᴀɴɢ

Dealing in Alternative Investments requires a bit of statistical knowledge (the more the better). So I decided to pick out one component that would benefit our members who handle their investments personally and, at the same time, others who hire an advisor — because it never hurts to ask the right questions.

High Frequency Trading & Unconventional Return Periods

Here we will be using a second form of the Information Ratio (IR) that measures residual return to residual risk; it measures the components of performance that come from active management unrelated to the benchmark. This is essentially an ‘active return/active risk’ ratio. Active risk is also known as ‘tracking error’.

In Forex, when returns are realized at higher frequencies (many, many times per year), Information ratios and the corresponding t-statistics can be calculated in a straightforward manner.

Keep in mind that forex is a zero-sum game, so the benchmark is zero to slightly negative (due to transaction costs), so our numerator is mean minus 0, which is just the mean (μ).

Assuming there are N return occurrences per year, and the mean (μ) and standard deviation (σ) of the returns are μ and σ, the annualized IR can be calculated as (μ×N)/(σ×√N) …or (μ/σ)×√N.

• The corresponding t-statistic is (μ/σ)×√(N×Number of years).

For monthly returns, the annualized IR and the corresponding t-statistic are (μ/σ)×√12 and (μ/σ)×√(12×Number of years), respectively. Here, μ and σ are the monthly mean and standard deviation of on the monthly returns.

Similarly, assuming μ and σ are the daily mean and standard deviation for daily returns (you traded every day the market was open) and there are 252 trading days in the year, the annualized IR is (μ/σ)×√252 …the corresponding t-stat is (μ/σ)×√(252×Number of years).

Remember, an information ratio tells us whether the manager of the investment outperformed his or her benchmark (on a risk-adjusted basis), but it cannot specifically tell us if the manager was just lucky or has skillz.

That is why we calculate the test-statistic for each investment.

The Test-Statistic

Test statistics (t-stat,t-statistic) are tricky creatures. Essentially when evaluating performance, I require a t-stat of 4 or more (the higher the better) before increasing the stake. In the future, I will explain a simple model I use to allocate cash among accounts and strategies according to their t-stats.

Now, here is a simplified way to estimate a t-statistic for the unusual return periods that most alternative investments have:

• Test statistic= [ (μ minus benchmark)/σ) ]×√(N return occurrences×number of years).

Note that “N return occurrences×number of years” is just the total number of return occurrences resulting from that specific investment or strategy (either positive or negative). So, if you closed out 3 trades (at 1%, -2.3% and 3%), that counts as N=3.

Or, if your minority equity investment reconciled every 6 weeks, for the past 1.5 years then N=13, (78 weeks / 6).

Remember, it is important to convert your daily/weekly/monthly returns to an annual (yearly) number. This makes it easier to compare performance between and among your alternative investments.

And since the volatility adjustment is built-in, it is an apples-to-apples comparison.

“One of the biggest pitfalls for performance measurement is to measure the ‘part’ with ignorance of the ‘whole.’” — Pearl Zhu

Thank you to Deborah Kidd, CFA for her excellent article on performance ratios at cfapubs.org

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