Volatility and Standard Deviation

Volatility: the language of risk
Volatility ($\sigma$, sigma) is a statistical measure of the dispersion of asset returns relative to the mean value. This is the “standard language” of the financial industry for describing risk. The higher the volatility, the less predictable the outcome.

Mathematics of volatility
Standard deviation formula:
$ \sigma = \sqrt{\frac{\sum(R_i - \bar{R})^2}{n-1}} $
Where $R_i$ is the return for period $i$, $\bar{R}$ is the average return, $n$ is the number of observations.

The rule of the normal distribution
If returns are normally distributed (the Gaussian bell curve):

Range	Probability	Interpretation
$\mu \pm 1\sigma$	68.3%	“Usual” year
$\mu \pm 2\sigma$	95.4%	Almost always falls here
$\mu \pm 3\sigma$	99.7%	Extreme events
gt;3\sigma$	0.3%	“Black swans” (in theory, once in 370 years)

Practical example: stocks vs bonds

Asset	Average return	Volatility	95% Range
S&P 500	10%	16%	-22% to +42%
US Treasuries 10Y	5%	7%	-9% to +19%
Bitcoin	50%	80%	-110% to +210%
Gold	7%	15%	-23% to +37%

Note: Bitcoin technically can show -110%, but in practice the maximum loss = 100%

Annualization of volatility
Daily volatility is recalculated to annual:
$ \sigma_{annual} = \sigma_{daily} \times \sqrt{252} $
252 is the number of trading days in a year. For monthly data: $\times \sqrt{12}$

Limitations of volatility as a measure of risk

Symmetry — “penalizes” both rises and falls equally. Investor doesn’t mind positive surprises!
Normality — actual returns have “fat tails,” extreme events happen more often than normal distribution predicts
Retrospectiveness — past volatility does not guarantee future volatility
Clustering — periods of high volatility group together (GARCH effect)

Volatility across asset classes

Class	Typical $\sigma$	Crisis $\sigma$
Money Market	0.5%	1%
IG Bonds	5-7%	10-15%
HY Bonds	8-12%	20-30%
DM Equities	15-18%	30-50%
EM Equities	20-25%	40-60%
Commodities	20-30%	50%+
Crypto	60-100%	150%+

Practical application for CIO

Risk budget — IPS sets the maximum portfolio volatility (for example, 10%)
Position assessment — highly volatile assets require smaller weight
Stress tests — what if volatility doubles?
VaR calculations — Value at Risk is based on volatility