How do you quantify investment risk?
"The question is: how do you quantify investment risk? As Michael S says, one approach is to treat investment returns as a random variable. Bill Goetzmann (Yale finance professor) told me that if you accept that markets are efficient or that the price of an asset reflects it's underlying value, then changes in price represent changes in value, so standard deviation naturally becomes the appropriate measure for riskiness of an asset. Essentially, the more volatile an asset, the riskier it is. There is another school of thought that comes from Ben Graham and Warren Buffett, which says that volatility is not inherently risky. Rather, risk should be defined as the permanent loss of capital, so the riskiness of an asset is the probability of a permanent loss of capital invested.  This is easy to do in casino games, based on basic probability such as   roulette or slots. But what has been done with the various kinds of   investment risks? My point is saying that certain bonds are ""low risk""   isn't good enough; I'd like some numbers--or at least a range of   numbers--and therefore one could calculate expected payoff (in the   statistics sense). Or can it not be done--and if not, why not? Investing is more art than science. In theory, a Triple-A bond rating means the asset is riskless or nearly riskless, but we saw that this was obviously wrong since several of the AAA mortgage backed securities (MBS) went under prior to the recent US recession. More recently, the current threat of default suggests that bond ratings are not entirely accurate, since US Treasuries are considered riskless assets.  Investors often use bond ratings to evaluate investments - a bond is considered investment grade if it's BBB- or higher. To adequately price bonds and evaluate risk, there are too many factors to simply refer to a chart because things like the issuer, credit quality, liquidity risk, systematic risk, and unsystematic risk all play a factor.  Another factor you have to consider is the overall portfolio. Markowitz showed that adding a riskier asset can actually lower the overall risk of a portfolio because of diversification. This is all under the assumption that risk = variance, which I think is bunk.  I'm aware that Wall Street is nothing like roulette, but then again   there must be some math and heavy economics behind calculating risk   for individual investors. This is, after all, what ""quants"" are paid   to do, in part. Is it all voodoo? I suspect some of it is, but not all   of it. Quants are often involved in high frequency trading as well, but that's another note. There are complicated risk management products, such as the Aladdin system by BlackRock, which incorporate modern portfolio theory (Markowitz, Fama, Sharpe, Samuelson, etc) and financial formulas to manage risk.  Crouhy's Risk Management covers some of the concepts applied.  I also tend to think that when people point to the last x number of   years of stock market performance, that is of less value than they   expect. Even going back to 1900 provides ""only"" 110 years of data, and   in my view, complex systems need more data than those 40,500 data   points. 10,000 years' worth of data, ok, but not 110. Any books or articles that address these issues, or your own informed   views, would be helfpul. I fully agree with you here. A lot of work is done in the Santa Fe Institute to study ""complex adaptive systems,"" and we don't have any big, clear theory as of yet. Conventional risk management is based on the ideas of modern portfolio theory, but a lot of that is seen to be wrong. Behavioral finance is introducing new ideas on how investors behave and why the old models are wrong, which is why I cannot suggest you study risk management and risk models because I and many skilled investors consider them to be largely wrong. There are many good books on investing, the best of which is Benjamin Graham's The Intelligent Investor. Although not a book on risk solely, it provides a different viewpoint on how to invest and covers how to protect investments via a ""Margin of Safety."" Lastly, I'd recommend Against the Gods by Peter Bernstein, which covers the history of risk and risk analysis. It's not solely a finance book but rather a fascinating historical view of risk, and it helps but many things in context. Hope it helps!" "Another approach would be more personalized, which is to measure the risk of missing your goals, rather than measuring the risk of an investment in some abstract sense. Financial planners do this for example with Monte Carlo simulation software (see http://en.wikipedia.org/wiki/Monte_Carlo_method). They would put in a goal such as not running out of money before you die, with assumptions such as the longest you might live and how much you'll withdraw every year. You'd also assume an asset allocation. The Monte Carlo simulation then generates random market movements over the time period, considering historical behavior of your asset allocation, and each run of the simulation would either succeed (you are able to support yourself until death) or fail (you run out of money). The risk measure is the percentage of simulation runs that fail. You can do this to plan saving for retirement in addition to planning withdrawals; then your goal would be to have X amount of money in real after-inflation dollars, perhaps, and success is if you end up with it, and failure is if you don't. The great thing about this risk measure is that it's relevant and personal; ""10% chance of being impoverished at age 85,"" ""20% chance of having to work an extra decade because you don't have enough at 65,"" these kinds of answers. Which is a lot easier to act on than ""the variance is 10"" or ""the beta is 1.5"" - would you rather know your plan has a 90% chance of success, or know that you have a variance of 10? Both numbers are probably just guesses, but at least the ""chance of success"" measure is actionable and relevant. Some tangential thoughts FWIW:" "The standard measure of risk is the variance of the asset. The return on investment of the asset is understood as a random variable with a particular distribution. One can make inferences about the underlying distribution using historical data. As you say, this is what the quants do. There are other, more sophisticated measures of risk that allow for such things as skewed distributions and Markov switching. If you are interested in learning more, I suggest starting with the foundations of Modern Portfolio Theory: ""Portfolio Selection"" by Harry Markowitz and ""Capital Asset Prices"" by William Sharpe." "For a retail investor who isn't a Physics or Math major, the ""Beta"" of the stock is probably the best way to quantify risk.  Examples: A Beta of 1 means that a stock moves in line with the market. Over 1 means that you would expect the stock to move up or down faster than the market as a whole. Under 1 means that you would expect the stock to move slower than the market as a whole." I use two measures to define investment risk: What's the longest period of time over which this investment has had negative returns? What's the worst-case fall in the value of this investment (peak to trough)? I find that the former works best for long-term investments, like retirement. As a concrete example, I have most of my retirement money in equity, since the Sensex has had zero returns over as long as a decade. Since my investment time-frame is longer, equity is risk-free, by this measure. For short-term investments, like money put aside to buy a car next year, the second measure works better. For this purpose, I might choose a debt fund that isn't the safest, and has had a worst-case 8% loss over the past decade. I can afford that loss, putting in more money from my pocket to buy the car, if needed. So, I might choose this fund for this purpose, taking a slight risk to earn higher return. In any case, how much money I need for a car can only be a rough guess, so having 8% less than originally planned may turn out to be enough. Or it may turn out that the entire amount originally planned for is insufficient, in which case a further 8% shortfall may not be a big deal. These two measures I've defined are simple to explain and understand, unlike academic stuff like beta, standard deviation, information ratio or other mumbo-jumbo. And they are simple to apply to a practical problem, as I've illustrated with the two examples above. On the other hand, if someone tells me that the standard deviation of a mutual fund is 15%, I'll have no idea what that means, or how to apply that to my financial situation. All this suffers from the problem of being limited to historical data, and the future may not be like the past. But that affects any risk statistic, and you can't do better unless you have a time machine.
Fund performance at NAV (%) for latest quarter, YTD, and average annual total returns for 1, 3, 5, 10 years. P/E ratio (1 yr. forecast), P/B ratio, Beta, Sharpe ratio, Wtd. avg. market cap, fund assets.  I guess I would want to calculate all these things based off of the data that I would be working with. I will assume I am working with daily fund values per share over 10+ years. "In addition to individual stocks, your entire portfolio will also have a beta.  It would be equal to the weighted sum of the individual asset betas  So a beta portfolio of 1 would have approximate risk equal to a market index. You would use this to construct a risk level that you were comfortable with, given the expected return of the individual assets.  You are also interested in obtaining a high level of ""alpha"" which means that your portfolio is earning more than what would be expected, given it's level of risk." Ah, pardon me, so it's not *either* 10x return or zero, but also includes points in between there?  Is it path dependent? Do you have any history on the asset?  The usual crutches for dealing with unknown probabilities are using risk neutrality or arbitrage pricing, but if the market is inefficient then that will be an estimation at best. "Every investment comes with a risk. There is also a bit of speculation involved. In there is an anticipation that one expects the value to go up in normal course of events. By your definition ""If I buy this equipment, I could produce more widgets, or sell more widgets,"" as an investment. Here again there is an anticipation that the widgets you sell will give you more return. If you are investing in stock/share, you are essentially holding a small portion of value in company and to that extent you are owining some equipment that is producing some widget .... Hence when you are purchasing Stocks, it would be looked as investment if you have done your home work and have a good plan of how you want to invest along with weiging the risk involved. However if you are investing only for the purpose of making quick bucks following so called hot tips, then you are not investing but speculating." "First of all, setting some basics: What is a sound way to measure the risk of each investment in order to compare them with each other ? There is no single way that can be used across all asset classes / risks. Generally speaking, you want to perform both a quantitative and qualitative assessment of risks that you identify.  Quantitative risk assessment may involve historical data and/or parametric or non-parametric models. Using historical data is often simple but may be hard in cases where the amount of data you have on a given event is low (e.g. risk of bust by investing in a cryptocurrency). Parametric and non-parametric risk quantification models exist (e.g. Value at Risk (VaR), Expected Shortfall (ES), etc) and abound but a lot of them are more complicated than necessary for an individual's requirements. Qualitative risk assessment is ""simply"" assessing the likelihood and severity of risks by using intuition, expert judgment (where that applies), etc. One may consult with outside parties (e.g. lawyers, accountants, bankers, etc) where their advisory may help highlighting some risks or understanding them better. To ease comparing investment opportunities, you may want to perform a risk assessment on categories of risks (e.g. investing in the stock market vs bond market). To compare between those categories, one should look at the whole picture (quantitative and qualitative) with their risk appetite in mind. Of course, after taking those macro decisions, you would need to further assess risks on more micro decisions (e.g. Microsoft or Google ?). You would then most likely end up with better comparatives as you would be comparing items similar in nature. Should I always consider the worst case scenario ? Because when I do that, I always can lose everything. Generally speaking, you want to consider everything so that you can perform a risk assessment and decide on your risk mitigating strategy (see Q4). By assessing the likelihood and severity of risks you may find that even in cases where you are comparatively as worse-off (e.g. in case of complete bust), the likelihood may differ. For example, keeping gold in a personal stash at home vs your employer going bankrupt if you are working for a large firm. Do note that you want to compare risks (both likelihood and severity) after any risk mitigation strategy you may want to put in place (e.g. maybe putting your gold in a safety box in a secure bank would make the likelihood of losing your gold essentially null). Is there a way to estimate the probability of such events, better than intuition ? Estimating probability or likelihood is largely dependent on data on hand and your capacity to model events. For most practical purposes of an individual, modelling would be way off in terms of reward-benefits. You may therefore want to simply research on past events and assign them a 1-5 (1 being very low, 5 being very high) risk rating based on your assessment of the likelihood. For example, you may assign a 1 on your employer going bankrupt and a 2 or 3 on being burglarized. This is only slightly better than intuition but has the merit of being based on data (e.g. frequency of burglary in your neighborhood). Should I only consider more probable outcomes and have a plan for them if they occur? This depends largely on your risk appetite. The more risk averse you are, the more thorough you will want to be in identifying, tracking and mitigating risks. For the risks that you have identified as relevant, or of concern, you may opt to establish a risk mitigating strategy, which is conventionally one of accepting, sharing (by taking insurance, for example), avoiding and reducing. It may not be possible to share or reduce some risks, especially for individuals, and so often the response will be either to accept or avoid the given risks by opting in or out on an opportunity." There are many ways to calculate the return, and every way will give you a different results in terms of a percentage-value. One way to always get something meaningful - count the cash. You had 977 (+ 31) and in the end you have 1.370, which means you have earned 363 dollars. But what is your return in terms of percentage? One way to look at it, is by pretending that it is a fund in which you invest 1 dollar. What is the fund worth in the beginning and in the end? The tricky part in your example is, you injected new capital into the equation. Initially you invested 977 dollars which later, in the second period became worth 1.473. You then sold off 200 shares for 950 dollars. Remember your portfolio is still worth 1.473, split between 950 in cash and 523 in Shares. So far so good - still easy to calculate return (1.473 / 977 -1 = 50.8% return). Now you buy share for 981 dollars, but you only had 950 in cash? We now need to consider 2 scenarios. Either you (or someone else) injected 31 dollars into the fund - or you actually had the 31 dollars in the fund to begin with. If you already had the cash in the fund to begin with, your initial investment is 1.008 and not 977 (977 in shares and 31 in cash). In the end the value of the fund is 1.370, which means your return is 1.370 / 1.007 = 36%. Consider if the 31 dollars was paid in to the fund by someone other than you. You will then need to recalculate how much you each own of the fund. Just before the injection, the fund was worth 950 in cash and 387 in stock (310 - 200 = 110 x 3.54) = 1.339 dollars - then 31 dollars are injected, bringing the value of the fund up to 1.370. The ownership of the fund is split with 1.339 / 1.370 = 97.8% of the value for the old capital and 2.2% for the new capital. If the value of the fund was to change from here, you could calculate the return for each investor individually by applying their share of the funds value respective to their investment. Because the value of the fund has not changed since the last period (bullet 3), the return on the original investment is (977 / 1.339 - 1 = 37.2%) and the return on the new capital is (31 / 31 = 0%). If you (and not someone else) injected the 31 dollar into the fund, you will need to calculate the weight of each share of capital in each period and get the average return for each period to get to a total return. In this specific case you will still get 37.2% return - but it gets even more comlex for each time you inject new capital. The investment return for a given strategy is directly proportional to the amount invested. Invest twice as much, profit (or lose) twice as much. It's a straight multiplier. However, there are some strategies which are less risky with a larger investment, and some investments which have a minimum unit of purchase that puts them out of reach of smaller investors. "A kid can lose everything he owns in a crap shoot and live. But a senior citizen might not afford medical treatment if interest rates turn and their bonds underperform. In modern portfolio theory, risk/""aggression"" is measured by beta and you get more return by increasing risk. Risk-adjusted return is measured by the Sharpe ratio and the efficient frontier shows how much return you get for each level of risk. For simplicity, we will assume that choosing beta is the only investment choice you make. You are buying a house tomorrow all cash, you should set aside that much in liquid assets today. (Return = who cares, Beta = 0) Your kids go to college in 5 years, so you invest funds now with a 5 year investment horizon to produce, with a reasonable level of certainty, the needed cash then. (Beta = low) You wish to leave money in your estate. Invest for the highest return with a horizon of your lifetime. (Return = maximum, Beta = who cares) In other words, you set risk based on how important your expenses are now or later. And your portfolio is a weighted average. On paper, let's say you have sold yourself into indentured servitude. In return you have received a paid-up-front annuity which pays dividends and increases annually. For someone in their twenties: This adds up to a present value of $1 million. When young, the value of lifetime remaining wages is high. It is also low risk, you will probably find a job eventually in any market condition. If your portfolio is significantly smaller than $1 million this means that the low risk of future wages pulls down your beta, and therefore: Youth invest aggressively with available funds because they compensate large, low-risk future earnings to meet their desired risk appetite." "Defining risk tolerance is often aided with a series of questions. Such as - You are 30 and have saved 3 years salary in your 401(k). The market drops 33% and since you are 100% S&P, you are down the same. How do you respond?  (a) move to cash - I don't want to lose more money. (b) ride it out. Keep my deposits to the maximum each year. Sleep like a baby.  A pro will have a series of this type of question. In the end, the question resolves to ""what keeps you up at night?"" I recall a conversation with a coworker who was so risk averse, that CDs were the only right investment for her. I had to explain in painstaking detail, that our company short term bond fund (sub 1 year government paper) was a safe place to invest while getting our deposits matched dollar for dollar. In our conversations, I realized that long term expectations (of 8% or more) came with too high a risk for her, at any level of her allocation. Zero it was." "Specifically, what does my broker mean when they say an asset or investment strategy is high risk?  In this context, it is a statement based on past events and probability. It is based on how confident s/he is that the investment will perform to certain benchmarks. This is a math question, primarily (with some opinion mixed in, granted). This is where the Sharpe ratio and others fit well. How am I supposed to answer a question like ""rate your risk tolerance from low to high""? This is the hard question, as you have seen. In this context, risk tolerance is derived from your current position and future plans (goals). This is a planning, goal setting, and strategy question, primarily (with some math mixed in, granted). How vulnerable is your current position and future plans to an under-performing investment? If you answer ""very"", then you choose investments that have a lower probability of under-performing.  The Sharpe ratio has little to do with answering this question. It is a tool to find investments that better match your answer to this question." Basically, diversifying narrows the spread of possible results, raising  the center of the returns bell-curve by reducing the likelihood of extreme results at either the high or low end.  It's largely a matter of basic statistics. Bet double-or-nothing on a single coin flip, and those are the only possible results, and your odds of a disaster (losing most or all of the money) are 50%. Bet half of it on each of two coin flips, and your odds of losing are reduced to 25% at the cost of reducing your odds of winning to 25%, with 50% odds that you retain your money and can try the game again. Three coins divides the space further; the extremes are reduced to 12.5% each, with the middle being most likely. If that was all there was, this would be a zero-sum game and pure gambling. But the stock market is actually positive-sum, since companies are delivering part of their profits to their stockholder owners. This moves the center of the bell curve up a bit from break-even, historically to about +8%. This is why index funds produce a profit with very little active decision; they treat the variation as mostly random (which seems to work statistically) and just try to capture average results of a (hopefully) slightly above-average bucket of stocks and/or bonds. This approach is boring. It will never double your money overnight. On the other hand, it will never wipe you out overnight. If you have patience and are willing to let compound interest work for you, and trust that most market swings regress to the mean in the long run, it quietly builds your savings while not driving you crazy worrying about it. If all you are looking for is better return than the banks, and you have a reasonable amount of time before you need to pull the funds out, it's one of the more reliably predictable risk/reward trade-off points. You may want to refine this by biasing the mix of what you're holding. The simplest adjustment is how much you keep in each of several major investment categories. Large cap stocks, small cap stocks, bonds, and real estate (in the form of REITs) each have different baseline risk/return curves, and move in different ways in response to news, so maintaining a selected ratio between these buckets and adding the resulting curves together is one simple way to make fairly predictable adjustments to the width (and centerline) of the total bell curve. If you think you can do better than this, go for it. But index funds have been outperforming professionally managed funds (after the management fees are accounted for), and unless you are interested in spending a lot of time researching and playing with your money the odds of your doing much better aren't great unless you're willing to risk doing much worse. For me, boring is good. I want my savings to work for me rather than the other way around, and I don't consider the market at all interesting as a game. Others will feel differently. There's probably a risk committee and an investment committee where several high level executives analyze and discuss the investments they will do and which risk level to take.  Is all based on numbers and evidence, but in the end people decide how much risk to take.  Then there Risk Management I suppose supervises the risk and that they don't go over the threshold (measured by VAR or whatever) "You're talking about modern portfolio theory. The wiki article goes into the math. Here's the gist: Modern portfolio theory (MPT) is a theory of finance that attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets. At the most basic level, you either a) pick a level of risk (standard deviation of your whole portfolio) that you're ok with and find the maximum return you can achieve while not exceeding your risk level, or b) pick a level of expected return that you want and minimize risk (again, the standard deviation of your portfolio). You don't maximize both moments at once. The techniques behind actually solving them in all but the most trivial cases (portfolios of two or three assets are trivial cases) are basically quadratic programming because to be realistic, you might have a portfolio that a) doesn't allow short sales for all instruments, and/or b) has some securities that can't be held in fractional amounts (like ETF's or bonds). Then there isn't a closed form solution and you need computational techniques like mixed integer quadratic programming Plenty of firms and people use these techniques, even in their most basic form. Also your terms are a bit strange: It has correlation table p11, p12, ... pij, pnn for i and j running from 1 to n This is usually called the covariance matrix. I want to maximize 2 variables. Namely the expected return and the additive inverse of the standard deviation of the mixed investments. Like I said above you don't maximize two moments (return and inverse of risk). I realize that you're trying to minimize risk by maximizing ""negative risk"" so to speak but since risk and return are inherently a tradeoff you can't achieve the best of both worlds. Maybe I should point out that although the above sounds nice, and, theoretically, it's sound, as one of the comments points out, it's harder to apply in practice. For example it's easy to calculate a covariance matrix between the returns of two or more assets, but in the simplest case of modern portfolio theory, the assumption is that those covariances don't change over your time horizon. Also coming up with a realistic measure of your level of risk can be tricky. For example you may be ok with a standard deviation of 20% in the positive direction but only be ok with a standard deviation of 5% in the negative direction. Basically in your head, the distribution of returns you want probably has negative skewness:  because on the whole you want more positive returns than negative returns. Like I said this can get complicated because then you start minimizing other forms of risk like value at risk, for example, and then modern portfolio theory doesn't necessarily give you closed form solutions anymore. Any actively managed fund that applies this in practice (since obviously a completely passive fund will just replicate the index and not try to minimize risk or anything like that) will probably be using something like the above, or at least something that's more complicated than the basic undergrad portfolio optimization that I talked about above. We'll quickly get beyond what I know at this rate, so maybe I should stop there." Your asset mix should reflect your own risk tolerance.  Whatever the ideal answer to your question, it requires you to have good timing, not once, but twice. Let me offer a personal example. In 2007, the S&P hit its short term peak at 1550 or so. As it tanked in the crisis, a coworker shared with me that he went to cash, on the way down, selling out at about 1100. At the bottom, 670 or so, I congratulated his brilliance (sarcasm here) and as it passed 1300 just 2 years later, again mentions how he must be thrilled he doubled his money. He admitted he was still in cash. Done with stocks. So he was worse off than had he held on to his pre-crash assets. For sake of disclosure, my own mix at the time was 100% stock. That's not a recommendation, just a reflection of how my wife and I were invested.  We retired early, and after the 2013 excellent year, moved to a mix closer to 75/25. At any time, a crisis hits, and we have 5-6 years spending money to let the market recover. If a Japanesque long term decline occurs, Social Security kicks in for us in 8 years.  If my intent wasn't 100% clear, I'm suggesting your long term investing should always reflect your own risk tolerance, not some short term gut feel that disaster is around the corner. Many of the above comments are correct about illiquidity.  If someone needs to trade at a time of low liquidity, for instance when the markets are closed, the bid/ask spread can often be large to induce someone to trade at odd times.  Especially as the broker/bank on the other side of the trade can't immediately go to the market to close out the risk as they often prefer to do. In this case the jump is actually is large but not that large (~4%).  Note this trade price is near the close price on the day before. The system I use shows a trade that evening for 5 shares near the price on the graph.  If you called me after I was done with work and tried to buy 5 shares I'd quote you a bad price too.