I was in SF this week talking with friends in tech and startups, and kept thinking about one idea from finance.
In finance, Sharpe ratio measures expected excess return relative to volatility.
I started playing with a very simple question:
Even if the underlying strategy is good, how often should it still look bad?
First, the assumptions, because the numbers below depend on them.
I am using a toy model where monthly excess returns are independent and identically distributed, normally distributed, and generated from the same mean and volatility every month. “Underperform” means the monthly excess return is below zero. I assume the true Sharpe is known and constant, and I look at 60 months, or five years.
Obviously markets do not behave exactly like this. Returns can have fat tails, autocorrelation, changing volatility, and regime shifts.
This is a clean baseline, not a model of reality.
Start with Sharpe = 2
Suppose annual Sharpe is 2.
Under the usual square root of time annualization,
\[S_{month}=\frac{2}{\sqrt{12}}\approx0.577\]Why does this imply a 28.2% chance of underperforming in one month?
Let monthly excess return be
\[r\sim N(\mu,\sigma^2)\]Monthly Sharpe is simply
\[\frac{\mu}{\sigma}=0.577\]I call a month bad when \(r<0\).
Standardizing:
\[P(r<0) = P\left( \frac{r-\mu}{\sigma} < -\frac{\mu}{\sigma} \right)\]So:
\[P(r<0)=\Phi(-0.577)\]where \(\Phi\) is the standard normal CDF.
And
\[\Phi(-0.577)\approx0.282\]So even in this very clean Sharpe 2 toy world, about 28% of individual months underperform.
That was already more than I expected.
Four bad months in a row
The next calculation is slightly different.
Because I assumed monthly returns are i.i.d., every month has the same bad month probability
\[p=0.282\]and one month’s outcome does not change the next month’s probability.
For one specific four month window, the probability that all four months are bad is just
\[p^4 = 0.282^4 \approx0.0063\]or about 0.63%.
But over five years, there are many opportunities for a four month streak to occur.
There are 57 possible four month windows inside 60 months.
We cannot simply multiply \(57\times0.63\%\), because those windows overlap. January through April and February through May, for example, share three months.
So I calculate the exact probability with a tiny Markov chain.
At any point, the strategy can be in one of four states:
\[0,\;1,\;2,\;3\]meaning the current run contains zero, one, two, or three consecutive bad months.
Each new month does one of two things.
With probability \(1-p\), the month is good and the state returns to 0.
With probability \(p\), the month is bad and the streak increases by one.
Once state 3 gets another bad month, we have hit four consecutive underperforming months.
Run that process for 60 months and calculate the probability that we ever hit four.
The result is:
\[P(\text{at least one 4 month streak in 60 months}) \approx23.3\%\]So under these assumptions, even a Sharpe 2 strategy has roughly a one in four chance of experiencing at least one four month underperformance streak over five years.
That is the part I actually find interesting.
A few months of bad outcomes can contain information.
They can also just be variance.
The difficult part is knowing which one you are looking at.
A more ordinary Sharpe
Now take annual Sharpe = 1.2.
Monthly Sharpe becomes
\[\frac{1.2}{\sqrt{12}}\approx0.346\]So the probability of a bad month is
\[p=\Phi(-0.346)\approx36.5\%\]Using exactly the same 60 month Markov calculation:
\[P(\text{at least one 4 month streak}) \approx49.3\%\]Basically a coin flip.
Again, that is not a claim that real Sharpe 1.2 strategies have exactly a 49.3% probability of doing this.
It is what falls out of this particular i.i.d. normal model.
That distinction matters.
The diversification part
I also played with combining two strategies.
This calculation needs another set of assumptions.
Assume the two strategies have the same expected excess return \(\mu\), the same volatility \(\sigma\), the same Sharpe \(S=\mu/\sigma\), constant correlation \(\rho\), and we put 50% into each.
Then
\[r_p=\frac{r_1+r_2}{2}\]Expected return stays
\[E[r_p]=\mu\]while portfolio variance is
\[\operatorname{Var}(r_p) = \frac14 \left( \sigma^2+\sigma^2+2\rho\sigma^2 \right)\]so
\[\sigma_p = \sigma\sqrt{\frac{1+\rho}{2}}\]Therefore:
\[S_p = \frac{\mu} {\sigma\sqrt{(1+\rho)/2}} = S\sqrt{\frac{2}{1+\rho}}\]For two Sharpe 1.2 strategies:
\[\rho=0 \quad\Rightarrow\quad S_p = 1.2\sqrt2 \approx1.70\]At correlation 0.2:
\[S_p = 1.2\sqrt{\frac{2}{1.2}} \approx1.55\]At correlation −0.3:
\[S_p = 1.2\sqrt{\frac{2}{0.7}} \approx2.03\]That is just standard diversification math under equal means, equal volatilities, fixed correlation, and no transaction costs.
Obviously, people are not portfolios :)
But I like the analogy.
Some of the people I value most are useful precisely because their strengths and mistakes are different from mine.
One is better at execution. Someone else is calmer when things go wrong. Another catches assumptions I tend to miss.
The value is not that everyone has a higher individual “Sharpe.”
It is that we are not perfectly correlated.
And the first calculation gives me a related reminder:
a bad stretch is evidence, but it is not automatically a verdict.
Sometimes the process is broken.
Sometimes you are seeing variance.
I think knowing when to tell those two apart is probably more useful than trying to avoid bad months entirely.