When investing, is the risk/reward tradeoff linear?
The relationship is not linear, and depends on a lot of factors. The term you're looking for is efficient frontier, the optimal rate of return for a given level of risk. The goal is to be on the efficient frontier, meaning that for the given level of risk, you're receiving the greatest possible rate of return (reward). http://www.investopedia.com/terms/e/efficientfrontier.asp If a market is efficient then risk/reward should be linear.  In simple markets like stocks and bonds, everyone thinks the same way and the risk/reward calculation is simple, so everyone can have an accurate idea of the risk/reward ratio, unless the company has serious undisclosed problems. But in other markets like derivatives and mortgage bonds, few people understand what they're buying so the risks remain hidden. Someone might think a company will do well, so they buy an derivative on that company. But no one understands risk/reward calculations on derivatives, so the risk/reward on the derivative could be way off the price on the derivative. The risk-reward relation depends on what you are changing. In the most cases people ask about, it is not linear but I will give examples of both. Nonlinear case 1:  As you diversify your portfolio, the firm-specific risks of various stocks cancel each other out without necessarily affecting the expected return of the portfolio.  Reduction in risk without any loss in returns--very nonlinear.  Nonlinear case 2: If you are changing the weights in your portfolio to move along the efficient frontier, then you the risk-reward relation is a hyperbola, which is nonlinear. Nonlinear case 3: If you are changing the weights in your portfolio to move away from the efficient frontier, then you increase risk without adding a fully compensatory amount of return.  There could be many paths along the risk-reward plane, but generally it will not be linear in the sense that it will not be on the same line as your initial, efficient, portfolio and your savings account. Linear case 1: The most common sense in which we think of the risk-reward relation being linear is when the thing you are changing is the size of your investment.  If you take money out of savings to put in your fully diversified portfolio without changing the relative weights, your expected returns will increase linearly. Linear case 2: If you believe the CAPM, then the expected return of an asset stock is linearly proportional to the market risk of the firm.  If you could change the market risk of a single asset without changing anything else, then you would linearly change its expected return. The general rule about the risk/reward relation is this: If you are changing the size of your investment, the relation is linear.  If you are changing its composition, the relation is nonlinear Ditto Bill and I upvoted his answer. But let me add a bit. If everyone knew exactly what the risk was for every investment, then prices would be bid up or down until every stock (or bond or derivative or whatever) was valued at exactly risk times potential profit. (Or more precisely, integral of risk times potential profit.) If company A was 100% guaranteed to make $1 million profit this year, while company B had 50% change to make a $2 million profit and 50% to make $0, and every investor in the world knew that, then I'd expect the total price of all shares of the two stocks to stabilize at the same value. The catch to that, though, is that no one really knows the risk. The risk isn't like, we're going to roll a die and if it comes up even the company makes $1 million and if it comes up odd the company makes $0, so we could calculate the exact probability. The risk comes from lack of information. Will consumers want to buy this new product? How many? What are they willing to pay? How capable is the new CEO? Etc. It's very hard to calculate probabilities on these things. How can you precisely calculate the probability that unforeseen events will occur? So in real life prices are muddled. The risk/reward ratio should be roughly sort of approximately linear, but that's about the most one can say.
"Is playing the lottery a wise investment? --Probably not.  Is playing the lottery an investment at all? --Probably not though I'll make a remark on that further below.  Does it make any sense to play the lottery in order to improve your total asset allocation? --If you follow the theory of the Black Swan, it actually might.  Let me elaborate. The Black Swan theory says that events that we consider extremely improbable can have an extreme impact. So extreme, in fact, that its value would massively outweigh the combined value of all impacts of all probable events together. In statistical terms, we are speaking about events on the outer limits of the common probablity distribution, so called outliers that have a high impact.  Example: If you invest $2000 on the stock market today, stay invested for 20 years, and reinvest all earnings, it is probable within a 66% confidence interval that you will have an 8 % expected return (ER) per year on average, giving you a total of roughly $9300. That's very much simplified, of course, the actual number can be very different depending on the deviations from the ER and when they happen. Now let's take the same $2000 and buy weekly lottery tickets for 20 years. For the sake of simplicity I will forgo an NPV calculation and assume one ticket costs roughly $2. If you should win, which would be an entirely improbable event, your winnings would by far outweigh your ER from investing the same amount.  When making models that should be mathematically solvable, these outliers are usually not taken into consideration. Standard portfolio management (PM) theory is only working within so called confidence intervals up to 99% - everything else just wouldn't be practical. In other words, if there is not at least a 1% probability a certain outcome will happen, we'll ignore it. In practice, most analysts take even smaller confidence intervals, so they ignore even more.  That's the reason, though, why no object that would fall within the realms of this outer limit is an investment in terms of the PM theory. Or at least not a recommendable one. Having said all that, it still might improve your position if you add a lottery ticket to the mix. The Black Swan theory specifically does not only apply to the risk side of things, but also on the chance side. So, while standard PM theory would not consider the lottery ticket an investment, thus not accept it into the asset allocation, the Black Swan theory would appreciate the fact that there is minimal chance of huge success.  Still, in terms of valuation, it follows the PM theory. The lottery ticket, while it could be part of some ""investment balance sheet"", would have to be written off to 0 immediately and no expected value would be attached to it. Consequently, such an investment or gamble only makes sense if your other, safe investments give you so much income that you can easily afford it really without having to give up anything else in your life. In other words, you have to consider it money thrown out of the window.  So, while from a psychological perspective it makes sense that especially poorer people will buy a lottery ticket, as Eric very well explained, it is actually the wealthier who should consider doing so. If anyone. :)" "The previous answers make valid points regarding the risks, and why you can't reasonably compare trading for profit/loss to a roll of the die. This answer looks at the math instead. Your assumption: I have an equal probability to make a profit or a loss. Is incorrect, for the reasons stated in other answers. However, the answer to your question: Can I also assume that probabilistically speaking, a trader cannot do worst than random? Is ""yes"". But only because the question is flawed. Consequently it's throwing people in all directions with their answers. But quite simply, in a truly random environment the worst case scenario, no matter how improbable, is that you lose over and over again until you have nothing left. This can happen in sequential rolls of the dice AND in trading securities/bonds/whatever. You could guess wrong for every roll of the die AND all of your stock picks could become worthless. Both outcomes result in $0 (assuming you do not gamble with credit). Tell me, which $0 is ""worse""? Given the infinite number of plays that ""random"" implies, the chance of losing your entire bankroll exists in both scenarios, and that is enough by itself to make neither option ""worse"" than the other. Of course, the opposite is also true. You could only pick winners, with an unlimited upside potential, but again that could happen with either dice rolls or stock picks. It's just highly improbable. my chances cannot be worse than random and if my trading system has an edge that is greater than the percentage of the transaction that is transaction cost, then I am probabilistically likely to make a profit? Nope. This is where it all falls apart. Just because your chances of losing it all are similarly improbable, does not make you more likely to win with one method or the other. Regression to the mean, when given infinite, truly random outcomes, makes it impossible to ""have an edge"". Also, ""probabilistically"" isn't a word, but ""probably"" is." Bad question.  Even people choosing a 401k strategy have to choose risk over reward.  But looking at this simply, if you take a 10 year chart of Gold, you will not find any other stock or commodity that could have been invested in that would have provided you with the same results with as little volatility over this time period.  http://www.kitco.com/LFgif/au3650nys.gif "Years ago I wrote an article Risk, Reward, Coin Flipping which explains from a 'game theory' perspective how diversifying works to minimize standard deviation in one's returns.  It's long and tedious, not easy to summarize, but it holds up well, I'm pleased with how the analogy does its job.  Update - the above is too ""link-only"", written over 5 years ago. The article I wrote offers a mathematical approach via an understandable example of coin flipping. With just 2 options, a 'head' is a 10% loss, while a 'tail' is a 30% gain. This actually represents the market fairly well as it results in a 10% average gain and 28% standard deviation for just 2 flips. The article shows how by 'diversifying', choosing to make multiple smaller bets, the average 10% stays the same, but the standard deviation is brought down dramatically, 7.6% when we use a sample experiment with 7 coins." So I did some queries on Google Scholar, and the term of art academics seem to use is target date fund. I notice divided opinions among academics on the matter. W. Pfau gave a nice set of citations of papers with which he disagrees, so I'll start with them. In 1969, Paul Sameulson published the paper Lifetime Portfolio Selection By Dynamic Stochaistic Programming, which found that there's no mathematical foundation for an age based risk tolerance. There seems to be a fundamental quibble relating to present value of future wages; if they are stable and uncorrelated with the market, one analysis suggests the optimal lifecycle investment should start at roughly 300 percent of your portfolio in stocks (via crazy borrowing). Other people point out that if your wages are correlated with stock returns, allocations to stock as low as 20 percent might be optimal. So theory isn't helping much. Perhaps with the advent of computers we can find some kind of empirical data. Robert Shiller authored a study on lifecycle funds when they were proposed for personal Social Security accounts. Lifecycle strategies fare poorly in his historical simulation: Moreover, with these life cycle portfolios, relatively little is contributed when the allocation to stocks is high, since earnings are relatively low in the younger years. Workers contribute only a little to stocks, and do not enjoy a strong effect of   compounding, since the proceeds of the early investments are taken out of the stock   market as time goes on. Basu and Drew follow up on that assertion with a set of lifecycle strategies and their contrarian counterparts: whereas a the lifecycle plan starts high stock exposure and trails off near retirement, the contrarian ones will invest in bonds and cash early in life and move to stocks after a few years. They show that contrarian strategies have higher average returns, even at the low 25th percentile of returns. It's only at the bottom 5 or 10 percent where this is reversed. One problem with these empirical studies is isolating the effect of the glide path from rebalancing. It could be that a simple fixed allocation works plenty fine, and that selling winners and doubling down on losers is the fundamental driver of returns. Schleef and Eisinger compare lifecycle strategy with a number of fixed asset allocation schemes in Monte Carlo simulations and conclude that a 70% equity, 30% long term corp bonds does as well as all of the lifecycle funds. Finally, the earlier W Pfau paper offers a Monte Carlo simulation similar to Schleef and Eisinger, and runs final portfolio values through a utility function designed to calculate diminishing returns to more money. This seems like a good point, as the risk of your portfolio isn't all or nothing, but your first dollar is more valuable than your millionth. Pfau finds that for some risk-aversion coefficients, lifecycles offer greater utility than portfolios with fixed allocations. And Pfau does note that applying their strategies to the historical record makes a strong recommendation for 100 percent stocks in all but 5 years from 1940-2011. So maybe the best retirement allocation is good old low cost S&P index funds! "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." "First, to mention one thing - better analysis calls for analyzing a range of outcomes, not just one; assigning a probability on each, and comparing the expected values. Then moderating the choice based on risk tolerance. But now, just look at the outcome or scenario of 3% and time frame of 2 days. Let's assume your investable capital is exactly $1000 (multiply everything by 5 for $5,000, etc.). A. Buy stock: the value goes to 103; your investment goes to $1030; net return is $30, minus let's say $20 commission (you should compare these between brokers; I use one that charges 9.99 plus a trivial government fee). B. Buy an call option at 100 for $0.40 per share, with an expiration 30 days away (December 23). This is a more complicated. To evaluate this, you need to estimate the movement of the value of a 100 call, $0 in and out of the money, 30 days remaining, to the value of a 100 call, $3 in the money, 28 days remaining. That movement will vary based on the volatility of the underlying stock, an advanced topic; but there are techniques to estimate that, which become simple to use after you get the hang of it. At any rate, let's say that the expected movement of the option price in this scenario is from $0.40 to $3.20. Since you bought 2500 share options for $1000, the gain would be 2500 times 2.8 = 7000. C. Buy an call option at 102 for $0.125 per share, with an expiration 30 days away (December 23). To evaluate this, you need to estimate the movement of the value of a 102 call, $2 out of the money, 30 days remaining, to the value of a 102 call, $1 in the money, 28 days remaining. That movement will vary based on the volatility of the underlying stock, an advanced topic; but there are techniques to estimate that, which become simple to use after you get the hang of it. At any rate, let's say that the expected movement of the option price in this scenario is from $0.125 to $ 1.50. Since you bought 8000 share options for $1000, the gain would be 8000 times 1.375 = 11000. D. Same thing but starting with a 98 call. E. Same thing but starting with a 101 call expiring 60 days out. F., ... Etc. - other option choices. Again, getting the numbers right for the above is an advanced topic, one reason why brokerages warn you that options are risky (if you do your math wrong, you can lose. Even doing that math right, with a bad outcome, loses). Anyway you need to ""score"" as many options as needed to find the optimal point. But back to the first paragraph, you should then run the whole analysis on a 2% gain. Or 5%. Or 5% in 4 days instead of 2 days. Do as many as are fruitful. Assess likelihoods. Then pull the trigger and buy it. Try these techniques in simulation before diving in! Please! One last point, you don't HAVE to understand how to evaluate projected option price movements if you have software that  does that for you. I'll punt on that process, except to mention it. Get the general idea? Edit P.S. I forgot to mention that brokers need love for handling Options too. Check those commission rates in your analysis as well." If you plan to take profit at $1.00 then your profit will be $40. Then, if you set your stop at $0.88 then your loss if you get stopped will be $20.  So your Reward : Risk = 2:1.  Note, that this does not take into account brokerage in and out and any slippage from the price gapping past your stop loss. "I'll give the TLDR answer. 1) You can't forecast the price direction. If you get it right you got lucky. If you think you get it right consistently you are either a statistical anomaly or a victim of confirmation bias. Countless academic studies show that you can not do this. 2) You reduce volatility and, importantly, left-tail risk by going to an index tracking ETF or mutual fund. That is, Probability(Gigantic Loss) is MUCH lower in an index tracker. What's the trade off? The good thing is there is NO tradeoff. Your expected return does not go down in the same way the risk goes down! 3) Since point (1) is true, you are wasting time analysing companies. This has the opportunity cost of not earning $ from doing paid work, which can be thought of as a negative return. ""With all the successful investors (including myself on a not-infrequent basis) going for individual companies directly"" Actually, academic studies show that individual investors are the worst performers of all investors in the stock market." I researched quite a bit around this topic, and it seems that this is indeed false. Long ter asset growth does not converge to the compound interest rate of expected return. While it is true that standard deviations of annualized return decrease over time, because the asset value itself changes over time, the standard deviations of the total return actually increases.      Thus, it is wrong to say that you can take increased risk because you have a longer time horizon.  Source Foreign stocks tend to be more volatile -- higher risk trades off against higher return potential, always. The better reason for having some money in that area is that, as with bonds, it moves out-of-sync with the US markets and once you pick your preferred distribution, maintaining that balance semi-automatically takes advantage of that to improve your return-vs-risk position. I have a few percent of my total investments in an international stock index fund, and a few percent in an international REIT, both being fairly low-fee. (Low fees mean more of the money reaches you, and seems to be one of the better reasons for preferring one fund over another following the same segment of the market.) They're there because the model my investment advisor uses -- and validated with monte-carlo simulation of my specific mix -- shows that keeping them in the mix at this low level is likely to result in a better long-term outcome than if i left them out. No guarantees, but probabilities lean toward this specfic mix doing what i need. I don't pretend to be able to justify that via theory or to explain why these specific ratios work... but I understand enough about the process to trust that they are on (perhaps of many) reasonable solutions to get the best odds given my specific risk tolerance, timeline, and distaste for actively managing my money more than a few times a year. If that. "First of all, ""going risky"" doesn't mean driving to Las Vegas and playing roulette. The real meaning is that you can afford higher risk/return ratio compared to a person who will retire in the following ten years. Higher return is very important since time works for you and even several extra percent annually will make a big difference in the long run because of compound interest effect. The key is that this requires the investment to not be too risky - if you invest in a single venture and it fails you lose all the money and that's worse that some conservative investment that could yield minimum income. So you still need the investment to be relatively safe. Next, as user Chris W. Rea mentions in the comment funds and ETFs can be very risky - depending on the investment policy they can invest into some very risky ventures or into some specific industry and that poses more risk that investing into ""blue chips"" for example. So a fund or an ETF can be a good fit for you if you choose a right one." While the Vanguard paper is good, it doesn't do a very good job of explaining precisely why each level of stocks or bonds was optimal. If you'd like to read a transparent and quantitative explanation of when and why a a glide path is optimal, I'd suggest the following paper:  https://www.betterment.com/resources/how-we-construct-portfolio-allocation-advice/ (Full disclosure - I'm the author).  The answer is that the optimal risk level for any given holding period depends upon a combination of:  Using these two factors, you construct a risk-averse decision model which chooses the risk level with the best expected average outcome, where it looks only at the median and lower percentile outcomes. This produces an average which is specifically robust to downside risk. The result will look something like this:   The exact results will depend on the expected risk and return of the portfolio, and the degree of risk aversion specified.  The result is specifically valid for the case where you liquidate all of the portfolio at a specific point in time. For retirement, the glide path needs to be extended to take into account the fact that the portfolio will be liquidated gradually over time, and dynamically take into account the longevity risk of the individual.  I can't say precisely why Vanguard's path is how it is. "'Buy and Hold' Is Still a Winner: An investor who used index funds and stayed the course could have earned satisfactory returns even during the first decade of the 21st century. by By Burton G. Malkiel in The Wall Street Journal on November 18, 2010: ""The other useful technique is ""rebalancing,"" keeping the portfolio asset allocation consistent with the investor's risk tolerance. For example, suppose an investor was most comfortable choosing an initial allocation of 60% equities, 40% bonds. As stock and bond prices change, these proportions will change as well. Rebalancing involves selling some of the asset class whose share is above the desired allocation and putting the money into the other asset class. From 1996 through 1999, annually rebalancing such a portfolio improved its return by 1 and 1/3 percentage points per year versus a strategy of making no changes."" Mr. Malkiel is a professor of economics at Princeton University. This op-ed was adapted from the upcoming 10th edition of his book ""A Random Walk Down Wall Street,"" out in December by W.W. Norton. http://online.wsj.com/article/SB10001424052748703848204575608623469465624.html" "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."