The Uses And Limits Of Volatility

Investors like to focus on the promise of high returns, but they should also ask how much risk they must assume in exchange for these returns. Although we often speak of risk in a general sense, there are also formal expressions of the risk-reward relationship. For example, the Sharpe ratio measures excess return per unit of risk, where risk is calculated as volatility, which is a traditional and popular risk measure. Its statistical properties are well known and it feeds into several frameworks, such as modern portfolio theory and the Black-Scholes model. In this article, we examine volatility in order to understand its uses and its limits.

Annualized Standard Deviation
Unlike implied volatility - which belongs to option pricing theory and is a forward-looking estimate based on a market consensus - regular volatility looks backward. Specifically, it is the annualized standard deviation of historical returns.

Traditional risk frameworks that rely on standard deviation generally assume that returns conform to a normal bell-shaped distribution. Normal distributions give us handy guidelines: about two-thirds of the time (68.3%), returns should fall within one standard deviation (+/-); and 95% of the time, returns should fall within two standard deviations. Two qualities of a normal distribution graph are skinny "tails" and perfect symmetry. Skinny tails imply a very low occurrence (about 0.3% of the time) of returns that are more than three standard deviations away from the average. Symmetry implies that the frequency and magnitude of upside gains is a mirror image of downside losses.

SEE: Volatility's Impact On Market Returns

Consequently, traditional models treat all uncertainty as risk, regardless of direction. As many people have shown, that's a problem if returns are not symmetrical - investors worry about their losses "to the left" of the average, but they do not worry about gains to the right of the average.

We illustrate this quirk below with two fictional stocks. The falling stock (blue line) is utterly without dispersion and therefore produces a volatility of zero, but the rising stock - because it exhibits several upside shocks but not a single drop - produces a volatility (standard deviation) of 10%.



Theoretical Properties

For example, when we calculate the volatility for the S&P 500 index as of Jan. 31, 2004, we get anywhere from 14.7% to 21.1%. Why such a range? Because we must choose both an interval and a historical period. In regard to interval, we could collect a series of monthly, weekly or daily (even intra-daily) returns. And our series of returns can extend back over a historical period of any length, such as three years, five years or 10 years. Below, we've computed the standard deviation of returns for the S&P 500 over a 10-year period, using three different intervals:


Notice that volatility increases as the interval increases, but not nearly in proportion: the weekly is not nearly five times the daily amount and monthly is not nearly four times the weekly. We've arrived at a key aspect of random walk theory: standard deviation scales (increases) in proportion to the square root of time. Therefore, if the daily standard deviation is 1.1%, and if there are 250 trading days in a year, the annualized standard deviation is the daily standard deviation of 1.1% multiplied by the square root of 250 (1.1% x 15.8 = 18.1%). Knowing this, we can annualize the interval standard deviations for the S&P 500 by multiplying by the square root of the number of intervals in a year:



Another theoretical property of volatility may or may not surprise you: it erodes returns. This is due to the key assumption of the random walk idea: that returns are expressed in percentages. Imagine you start with $100 and then gain 10% to get $110. Then you lose 10%, which nets you $99 ($110 x 90% = $99). Then you gain 10% again, to net $108.90 ($99 x 110% = $108.9). Finally, you lose 10% to net $98.01. It may be counter-intuitive, but your principal is slowly eroding even though your average gain is 0%!

If, for example, you expect an average annual gain of 10% per year (i.e., arithmetic average), it turns out that your long-run expected gain is something less than 10% per year. In fact, it will be reduced by about half the variance (where variance is the standard deviation squared). In the pure hypothetical below, we start with $100 and then imagine five years of volatility to end with $157:



The average annual returns over the five years was 10% (15% + 0% + 20% - 5% + 20% = 50% ÷ 5 = 10%), but the compound annual growth rate (CAGR, or geometric return) is a more accurate measure of the realized gain, and it was only 9.49%. Volatility eroded the result, and the difference is about half the variance of 1.1%. These results aren't from a historical example, but in terms of expectations, given a standard deviation of (variance is the square of standard deviation, ^2) and an expected average gain of , the expected annualized return is approximately - (^2 ÷ 2).

Are Returns Well-Behaved?
The theoretical framework is no doubt elegant, but it depends on well-behaved returns. Namely, a normal distribution and a random walk (i.e. independence from one period to the next). How does this compare to reality? We collected daily returns over the last 10 years for the S&P 500 and Nasdaq below (about 2,500 daily observations):



As you may expect, the volatility of Nasdaq (annualized standard deviation of 28.8%) is greater than the volatility of the S&P 500 (annualized standard deviation at 18.1%). We can observe two differences between the normal distribution and actual returns. First, the actual returns have taller peaks - meaning a greater preponderance of returns near the average. Second, actual returns have fatter tails. (Our findings align somewhat with more extensive academic studies, which also tend to find tall peaks and fat tails; the technical term for this is kurtosis). Let's say we consider minus three standard deviations to be a big loss: the S&P 500 experienced a daily loss of minus three standard deviations about -3.4% of the time. The normal curve predicts such a loss would occur about three times in 10 years, but it actually happened 14 times!

These are distributions of separate interval returns, but what does theory say about returns over time? As a test, let's take a look at the actual daily distributions of the S&P 500 above. In this case, the average annual return (over the last 10 years) was about 10.6% and, as discussed, the annualized volatility was 18.1%. Here we perform a hypothetical trial by starting with $100 and holding it over 10 years, but we expose the investment each year to a random outcome that averaged 10.6% with a standard deviation of 18.1%. This trial was done 500 times, making it a so-called Monte Carlo simulation. The final price outcomes of 500 trials are shown below:


A normal distribution is shown as backdrop solely to highlight the very non-normal price outcomes. Technically, the final price outcomes are lognormal (meaning that if the x-axis were converted to natural log of x, the distribution would look more normal). The point is that several price outcomes are way over to the right: out of 500 trials, six outcomes produced a $700 end-of-period result! These precious few outcomes managed to earn over 20% on average, each year, over 10 years. On the left hand side, because a declining balance reduces the cumulative effects of percentage losses, we only got a handful of final outcomes that were less than $50. To summarize a difficult idea, we can say that interval returns - expressed in percentage terms - are normally distributed, but final price outcomes are log-normally distributed.

SEE: Multivariate Models: The Monte Carlo Analysis

Finally, another finding of our trials is consistent with the "erosion effects" of volatility: if your investment earned exactly the average each year, you would hold about $273 at the end (10.6% compounded over 10 years). But in this experiment, our overall expected gain was closer to $250. In other words, the average (arithmetic) annual gain was 10.6%, but the cumulative (geometric) gain was less.

It is critical to keep in mind that our simulation assumes a random walk: it assumes that returns from one period to the next are totally independent. We have not proved that by any means, and it is not a trivial assumption. If you believe returns follow trends, you are technically saying they show positive serial correlation. If you think they revert to the mean, then technically you are saying they show negative serial correlation. Neither stance is consistent with independence.

The Bottom Line
Volatility is annualized standard deviation of returns. In the traditional theoretical framework, it not only measures risk, but affects the expectation of long-term (multi-period) returns. As such, it asks us to accept the dubious assumptions that interval returns are normally distributed and independent. If these assumptions are true, high volatility is a double-edged sword: it erodes your expected long-term return (it reduces the arithmetic average to the geometric average), but it also provides you with more chances to make a few big gains.

SEE: Implied Volatility: Buy Low And Sell High

More From Investopedia

--

Get stories like this on the Yahoo app and discover more every day.
Download it now.
Loading...
  • IMF urges BOJ to clarify policy guidance, strengthen communication Reuters - 9 minutes ago

    The Bank of Japan must continue to clarify its monetary policy guidance and strengthen communication after adopting a new policy framework, an official of the International Monetary Fund said on Monday. IMF Japan mission chief Luc Everaert told a … More »

  • Oil prices rebound after Algeria says all options open at OPEC meeting
    Oil prices rebound after Algeria says all options open at OPEC meeting Reuters - 9 minutes ago

    Crude prices rebounded on Monday after Algeria's energy minister said the day before that all options were possible for an oil output cut or freeze at this week's informal meeting of OPEC producers. Members of the Organization of the Petroleum … More »

  • Exclusive: Thailand considering tougher tax collection rules for internet, tech firms
    Exclusive: Thailand considering tougher tax collection rules for internet, tech firms Reuters - 10 minutes ago

    BANGKOK/JAKARTA (Reuters) - Thailand is studying plans to toughen tax collection rules for internet and technology firms like Alphabet Inc's Google, the head of the Revenue Department told Reuters, as the tax affairs of these firms comes under … More »

  • BOJ's Kuroda says ready to use every possible tool to hit price goal
    BOJ's Kuroda says ready to use every possible tool to hit price goal Reuters - 22 minutes ago

    Bank of Japan Governor Haruhiko Kuroda, in his first speech since the BOJ's decision last week to overhaul its radical stimulus programme, said the central bank stood ready to use every available tool to achieve its 2 percent inflation target. … More »

  • Closely watching yen rise impact on economy - BOJ Kuroda Reuters - 23 minutes ago

    OSAKA (Reuters) - Bank of Japan Governor Haruhiko Kuroda said on Monday the central bank would closely watch how recent yen rises could affect the country's economy and prices. "It's basically desirable ... … More »

  • China investigating North Korean bank, report says Associated Press - 30 minutes ago

    Chinese authorities are investigating a North Korean bank suspected of financing its government's imports of goods that might be used by the North's nuclear weapons program, a South Korean newspaper reported ... … More »

  • Dollar edges lower ahead of US presidential debate AFP - 45 minutes ago

    The dollar slipped Monday as traders shift focus from central bank policy to the US presidential election, while the Philippine peso tumbled to a seven-year low on worries over President Rodrigo Duterte's ... … More »

  • How Russia's central bank chief held the line Reuters - 1 hour 0 minutes ago

    One Thursday evening in March, Elvira Nabiullina, governor of Russia's central bank, faced down a rival in the struggle for influence inside President Vladimir Putin’s entourage. At a meeting chaired by Putin, Economy Minister Alexei Ulyukayev … More »

  • Barclays to Reduce Office Space in Tokyo After Cutting Jobs (1) Bloomberg - 1 hour 9 minutes ago

    The U.K. bank, which rents two floors of the Mori Tower in Tokyo’s Roppongi Hills office complex, will vacate about half of the 31st floor as early as this year while keeping its space on the 32nd, people with knowledge of the matter said, asking … More »

  • The 10 most important things in the world right now
    The 10 most important things in the world right now Business Insider - 1 hour 25 minutes ago

    REUTERS/Mohamed Abd El Ghany Students walk to school … Continued The post The 10 most important things in the world right now appeared first on Business Insider. … More »

  • Budget 2017: Khazanah sees continuation of pro-growth, pro-society focus Business Times - 1 hour 27 minutes ago

    KUALA LUMPUR: Khazanah Nasional Bhd sees a continuation of a pro-growth and pro-society focus in the upcoming Budget 2017. Managing Director Tan Sri Azman Mokhtar said the budget would also like to be fiscally responsible on the back of a … More »

  • Hong Kong Stocks Drop Most in Two Weeks as Financial Shares Fall Bloomberg - 1 hour 29 minutes ago

    AAC Technologies Holdings Inc. and Bank of Communications Co. were among the biggest losers on the gauge. Wharf Holdings Ltd. retreated for the first time in seven days, while Sands China Ltd. extended a slump from this month’s high to almost 10 … More »

  • Dallas police and fire pension in crisis, retirees concerned
    Dallas police and fire pension in crisis, retirees concerned Associated Press - 1 hour 37 minutes ago

    Still recovering from the July sniper attack that left five law enforcement officers dead, the Dallas Police Department is facing a new crisis as its pension fund approaches insolvency and scores of officers, ... … More »

  • BRIEF-Scientex says qtrly revenue 561.1 mln rgt Reuters - 1 hour 42 minutes ago

    Sept 26 (Reuters) - Scientex Bhd * Qtrly revenue 561.1 million rgt versus 452.5 million rgt; qtrly net profit 54.1 million rgt versus 48.9 million rgt Source text:( http://bit.ly/2d2pxXH ) Further company ... … More »

  • African elephants 'suffer worst decline in 25 years'
    African elephants 'suffer worst decline in 25 years' AFP News - 1 hour 43 minutes ago

    The number of African elephants has dropped by around 111,000 in the past decade, a new report released Sunday at the Johannesburg conference on the wildlife trade said, blaming the plummeting figures on poaching. The revelation, the worst drop in … More »

  • Sakakibara Says Yen May Gain to 90 per Dollar by End-2017 (1) Bloomberg - 1 hour 45 minutes ago

    Given the monetary policy divergence between the U.S. and Japan, some appreciation of the Japanese currency “is only natural,” said Sakakibara, dubbed “Mr. Yen” for his ability to influence the exchange rate while a senior Ministry of Finance … More »

  • Russia under fire at UN as air strikes pound Syria's Aleppo
    Russia under fire at UN as air strikes pound Syria's Aleppo AFP News - 1 hour 46 minutes ago

    The US accused Moscow of "barbarism" over the worsening carnage in Aleppo, as Syrian and Russian warplanes pounded the city in one of the heaviest bombing raids of the five-year war. At an emergency meeting of the UN Security Council to demand … More »

  • Asian Stocks Decline as Investors Await OPEC Meeting Outcome Bloomberg - 2 hours 0 minutes ago

    The MSCI Asia Pacific Index fell 0.8 percent to 140.87 as of 1:39 p.m. in Tokyo, headed for its steepest drop since Sept. 12. The Topix index lost 0.8 percent before a speech by Bank of Japan Governor Haruhiko Kuroda. Saudi Arabia, the world’s … More »

  • China flies army planes over strait near Japan
    China flies army planes over strait near Japan AFP News - 2 hours 11 minutes ago

    More than 40 Chinese military aircraft have traversed a strait near Japan, China's defence ministry said Monday, after Tokyo announced it may patrol with the US in the contested South China Sea region. China's Air Force on Sunday sent 40 planes … More »

  • SE Asia Stocks-Fall in line with Asian peers; U.S. presidential debate awaited Reuters - 2 hours 30 minutes ago

    By Suhail Hassan Bhat Sept 26 (Reuters) - Southeast Asian stock markets fell on Monday, in line with Asian peers, after last week's gains while investors shifted their attention from central banks to American ... … More »

  • Oil edges higher ahead of OPEC meeting AFP - 3 hours ago

    Oil prices rose modestly in Asia on Monday ahead of a producers' meeting this week that might agree to cap supplies. But analysts warned the wellspring of optimism should be tempered by experience of two ... … More »

  • Malaysia's closing tin prices for Sept 26 Reuters - 3 hours ago

    KUALA LUMPUR, Sept 26 (Reuters) - Following is a table of Kuala Lumpur tin market prices provided by a Malaysian trader. (Prices in $/tonne, and volumes in tonnes unless stated) Date KL Premium/ Spot Bid ... … More »

  • Singapore Shipbuilder Says Australian Units Face Court Action Bloomberg - 3 hours ago

    Otto Marine Ltd. said three creditors filed three such applications with the Supreme Court of Western Australia to wind up Go Inshore Pty and Go Marine Group Pty. The total claimed against the units is A$787,204 ($599,220), according to a filing to … More »

  • Asian markets slip as Fed rally vanishes
    Asian markets slip as Fed rally vanishes AFP News - 3 hours ago

    Asian markets slipped at the start of the new trading week, picking up where they left off after last week's Fed-inspired rally ran out of steam. The stay of execution for easy money served also to bolster the Japanese yen, knocking some of the … More »

  • China to extend anti-dumping measures on U.S. broiler chicken product imports Reuters - 3 hours ago

    China's commerce ministry said it will extend anti-dumping measures on imports of U.S. broiler chicken products for a further five years, effective from Sept 27. A suspension of the measures would potentially hurt Chinese firms, according to a … More »

 
Recent Quotes
Symbol Price Change % Chg 
Your most recently viewed tickers will automatically show up here if you type a ticker in the "Enter symbol/company" at the bottom of this module.
You need to enable your browser cookies to view your most recent quotes.
 
Sign-in to view quotes in your portfolios.

MARKET MOVERS

  • Most Actives
    Most Actives
    NamePriceChange% Chg
    0.139+0.003+2.21%
    N21.SI
    0.735+0.015+2.08%
    BS6.SI
    0.144+0.001+0.70%
    5DM.SI
    0.270.000.00%
    5ME.SI
    0.445+0.025+5.95%
    5SO.SI
  • % Gainers
    % Gainers
    NamePriceChange% Chg
    0.197+0.015+8.24%
    43B.SI
    0.60+0.04+7.14%
    BDR.SI
    0.62+0.04+6.90%
    BMT.SI
    0.56+0.04+6.67%
    BJE.SI
    0.24+0.01+4.35%
    5RE.SI
  • % Losers
    % Losers
    NamePriceChange% Chg
    0.31-0.06-17.33%
    AZT.SI
    0.345-0.060-14.81%
    BLR.SI
    0.15-0.02-10.71%
    A04.SI
    0.123-0.013-9.56%
    5NF.SI
    0.171-0.018-9.52%
    AXH.SI

Market Data

  • Currencies
    Currencies
    NamePriceChange% Chg
    1.3614+0.0033+0.24%
    USDSGD=X
    1.5285+0.0049+0.32%
    EURSGD=X
    1.7625+0.0053+0.30%
    GBPSGD=X
    74.0500-0.072-0.10%
    SGDJPY=X
    5.6945-0.011-0.19%
    SGDHKD=X
    3.0371+0.0103+0.34%
    SGDMYR=X
    9,606.9658-48.5674-0.50%
    SGDIDR=X
    4.9013-0.0053-0.11%
    SGDCNY=X
    1.0374+0.0024+0.23%
    AUDSGD=X
  • Commodities
    Commodities
    NamePriceChange% Chg
    1,338.80-2.90-0.22%
    GCZ16.CMX
    19.62-0.18-0.93%
    SIZ16.CMX
    95.76-3.02-3.05%
    ^XAU
    2.19-0.01-0.39%
    HGZ16.CMX
    43.85+0.41+0.94%
    CLV16.NYM
  • Bonds
    Bonds
    TreasuryYield (%)Yield Change
    1.15-0.02
    ^FVX
    1.62-0.02
    ^TNX
    2.34-0.01
    ^TYX