SUBSTANDARD
DEVIATIONS
David C. Shimko
PRINCIPAL, RISK MANAGEMENT ADVISORY
GROUP, BANKERS TRUST
JeanPaul St. Germain
VICE PRESIDENT, RISK MANAGEMENT ADVISORY
GROUP, BANKERS TRUST
(originally published by PMA OnLine Magazine: 05/98)
Your school professor didn't tell you
everything you need to know about
calculating volatility. David Shimko and JP StGermain explain the implications
The colossus of VaR (do we need to spell out VaR any more?)
depends on a humble volatility calculation, but few seem to question how volatility is
computed. We all learned at some point how to calculate return volatility in a financial
instrument:
How to calculate annual percent volatility, 101
1. Choose a frequency (daily, weekly, monthly)
2. List the sequence of prices for a financial instrument
3. Calculate the percentage price changes from one period to the next
(The erudite insist on log changes instead of percentage changes)
4. Calculate the standard deviation of the percentage price changes
5. Multiply by the square root of the number of periods per year
For example, the annual percent volatility of the tenweek price sequence below:
Table
1: Power Prices In The US
PJM
OnPeak 
6/6/95 
25.37 
6/13/95 
25.37 
6/20/95 
25.5 
6/27/95 
21.5 
7/4/95 
22.5 
7/11/95 
32.5 
7/18/95 
27.5 
7/25/95 
27.5 
8/1/95 
36.5 
8/8/95 
24.5 
Ann % vol 
173% 
using percentage changes is
173%. The price sequence shows PennsylvaniaNew JerseyMaryland (PJM) onpeak weekly power
prices in the US. Based on these data, you would say that electricity volatility is high.
In fact, if you did this calculation using all the data from June 6, 1995 to December 30,
1997, you would have computed a volatility of 273% per year. Depending on the period of
time you choose, you might find volatility estimates of over 1000% per year!
Before you faint from electric shock, or try to plug these numbers
into your BlackScholes calculator, take a look at the historical prices:
Your eye tells you that volatility cannot be
really 273% over this period, at least in the conventional way we think about volatility.
Prices seem to bounce up readily, only to bounce back the next week. So whatÕs wrong with
the calculation?
What you might have forgotten
When you calculate volatility the usual (101) way, you are
implicitly making a number of assumptions:
A1. The percentage change has a constant mean and volatility.
A2. The percentage change is independent of previous changes.
That is, the price follows a random walk. [The log change does
nothing other than convert a periodic return into a continuously compounded return, and
avoids the possibility of negative prices.] These assumptions are approximately true when
dealing with stocks, although certainly there are many exceptions. They are less reliable
for bonds, where volatility declines as a bond approaches maturity. And certainly,
assumption A2 is the most often violated, especially when referring to interest rates,
some currency rates and commodity prices.
Why might prices not follow random walks? If the price of power
jumps to $100, it will certainly not stay there; everyone knows that the price will soon
revert to a more reasonable level. In statistics, this is known as mean reversion. In the
context of this note, the mean price change, or expected price change is negative when the
price is very high, and positive when the price is very low.
In equation form, this can be represented as follows
P_{t+1}  P_{t} = k (m  P_{t}) + s e _{t}
which might look complicated at first. It says merely that the
projected price change (P_{t+1}  P_{t}) is proportional on average to the
distance the price lies from its true longrun mean m . The term k is often called the
speed of mean reversion, since the higher it is, the faster the price moves toward its
longrun mean.
This equation is not difficult to estimate or implement. Here's the
recipe:
 Regress each price change on the previous price.
 The negative of the slope estimate is the speed of mean reversion.
 The intercept estimate, when divided by the mean reversion speed, is
the longrun mean.
 The residual standard deviation is the volatility of dollar price
changes after adjusting for a variable mean.
Table 2: Calculating
the impact of mean reversion

PJM 

PJM 




OnPeak 
Change 
OnPeak 

Excel^{TM}
functions 
6/6/95 
25.37 



SLOPE 
1.02 
6/13/95 
25.37 
0 
25.37 

INTERCEPT 
27.45 
6/20/95 
25.5 
0.13 
25.37 

STEYX 
5.11 
6/27/95 
21.5 
4 
25.5 



7/4/95 
22.5 
1 
21.5 

Speed 
1.02 
7/11/95 
32.5 
10 
22.5 

Mean (Slope/speed) 
27.04 
7/18/95 
27.5 
5 
32.5 

Standard deviation 
5.11 
7/25/95 
27.5 
0 
27.5 



8/1/95 
36.5 
9 
27.5 

Forecasts 

8/8/95 
24.5 
12 
36.5 

Mean 
27.08 


"Y" 
"X" 

Volatility of 1yr forecast 
18% 
The regression data are tabulated easily in Excel^{TM},
using the SLOPE, INTERCEPT and STEYX (standard error of a prediction) functions. If the
slope estimate is insignificant, you needn't worry about conventional mean reversion.
However, this regression results show a very high degree of mean reversion  in fact,
the mean prediction for any week (based on these very limited data) would be around $27 regardless
of what it was the previous week.
This is anathema to the random walk concept, where next week's
expected price is equal to this week's price. The residual standard deviation of
$5.11 can be divided by the mean forecast to get a percentage volatility, but any attempt
to annualize that volatility is completely inappropriate  since the price changes are
not independent of each other over time. The volatility of a oneyear forecasted price
using these data is not much more than $5.11, or about 18% of the forecasted price level.
This is a full order of magnitude below the 173% volatility estimated from percentage
changes.
So why is this important to VaR?
Lest you think this problem is limited to the esoteric electricity
market, let us caution you. Mean reversion is extremely important for all industrial
commodities, for interest rates, and even for some currencies. The ways we calculate
volatility are vulnerable to ignoring this important phenomenon. As a result, many of the
models used to make volatility estimates may be grossly overestimating the actual
voilatility one should expect as a result of holding the instrument. High VaR estimates
may seem conservative and therefore appropriate at first, but overestimating risk leads to
penalizing performance measures, earning subpar returns on capital, and discouraging
profitable and appropriate trading activity.
If you take away nothing else from this column, think twice before
you calculate price volatility.
David Shimko is a principal in Bankers Trust's awardwinning
Risk Management Advisory group, specializing in risk management consulting to the power
and energy industry. Prior to joining Bankers, he held trading floor and risk management
positions at JPMorgan.
A former asst. professor of finance at the University of Southern California,
Shimko has authored over 50 publications in the areas of strategic risk management and
asset valuation, including a PhD level textbook, Finance in Continuous Time: A Primer.
Shimko is a monthly columnist appearing in Risk magazine.
David Shimko
Principal
Bankers Trust Company
130 Liberty Street, MS 2344
New York, NY 10006
(212) 2504715 Tel  (212)
2506969 Fax
david.shimko@bankerstrust.com
Reprinted with permission, David Shimko, Risk
Magazine Advisory group. All rights reserved. Copyright 1998. 