Risk to Reward Ratio Calculation
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 implemented this in MatLab about 10 years back. You just calculate your conditional variance of the required assets (x_i), use matrix multiplication on the correlation matrix (rho_i_j) from the same asset (this could be a point of research but unless you are using extreme conditions on the VaR it makes little difference) then apply a standard Markiowitz optimisation approach. You can then just use simple Sharpe ratio (marginal return over conditional risk) at every point on the efficient frontier. Then choose the maximum Sharpe ratio point. 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 Annualize quarterly returns:      AR = (1+QR)^4  Where *QR* is a decimal return, e.g. 0.05.  Standard deviations are similar:      Annual SD = SD * sqrt(T)  If you have quarterly deviations, *T=4*, if you have daily, *T=252*, etc.  As an aside, for work with money riding on it, it is *not* okay to aggregate standard deviations if there's autocorrelation amongst observations at a smaller time scale. Volatility is often quoted this way and that's fine, but it is dangerous to do any sort of risk management with this and you'll require more due diligence. It's a good enough approximation for napkin math, though. "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." The Sharpe ratio is, perhaps, the method you are looking for.  That said, not really sure beta is a meaningful metric, as there are plenty of safe bets to be made on volatile stocks (and, conversely, unsafe bets to be made on non-volatile ones). "Of course, which is why you need to have a scoring function / utility function for the ""filters"", i.e. Are you going to value it by rate of accuracy hor by a metric where wins = +2, losses = -1, such that it uses a criteria like that to decide whether or not a filter adds value, (some even use a compound effect i.e. wins = 2+e^(1+w) where w is the consecutive wins).  A metric like the above would capture the trade off between predictive power and profit. Also some traders watch their Max DD very carefully so they may be very risk averse." "Old question I know, but I have some thoughts to share. Your title and question say two different things.  ""Better off"" should mean maximizing your ex-ante utility.  Most of your question seems to describe maximizing your expected return, as do the simulation exercises here.  Those are two different things because risk is implicitly ignored by what you call ""the pure mathematical answer.""  The expected return on your investments needs to exceed the cost of your debt because interest you pay is risk-free while your investments are risky. To solve this problem, consider the portfolio problem where paying down debt is the risk-free asset and consider the set of optimal solutions.  You will get a capital allocation line between the solution where you put everything into paying down debt and the optimal/tangent portfolio from the set of risky assets.   In order to determine where on that line someone is, you must know their utility function and risk parameters.  You also must know the parameters of the investable universe, which we don't." "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." "The key point to answer the question is to consider risk aversion. Assume I suggest a game to you: Throw a coin and if you win, you get $5, if you lose nothing happens. Will you play the game? Of course, you will - you have nothing to lose! What if I suggest this: If you win, you get $10,000,005 and if you lose you must pay $10,000,000 (I also accept cars, houses, spouses, and kidneys as payment). While the expected value of the second game is the same as for the first, if you lose the second game you are more or less doomed to spend the rest of your life in poverty or not even have a rest of your life. Therefore, you will not wish to play the second game. Well, maybe you do - but probably only if you are very, very rich and can easily afford a loss (even if you had $11,000,000 you won't be as happy with a possible raise to $21,000,005 as you'd be unhappy with dropping to a mere $1,000,000, so you'd still not like to play).  Some model this by taking logarithms: If your capital grows from $500 to $1000 or from $1000 to $2000, in both cases it doubles, hence is considered the same ""personal gain"", effectively. And, voíla, the logartithm of your capital grows by the same amount in both cases. This refelcts that a rich man will not be as happy about finding a $10 note as a poor man will be about finding a nickel.  The effect of an insurance is that you replace an uncertain event of great damage with a certain event of little damage. Of course, the insurance company plays the same game, with roles swapped - so why do they play? One point is that they play the game very often, which tends to nivel the risks - unless you do something stupid and insure all inhabitants of San Francisco (and nobody else) against eqarthquakes. But also they have enough capital that they can afford to lose the game. In a fair situation, i.e. when the insurance costs just as much as damage cost multiplied with probability of damage, a rational you would eagerly buy the insurance because of risk aversion. Therefore, the insurance will in effect be able to charge more than the statistically fair price and many will still (gnawingly) buy it, and that's how they make a living. The decision how much more one is willing to accept as insurance cost is also a matter of whether you can afford a loss of the insured item easily, with regrets, barely, or not all." "I think you're confusing risk analysis (that is what you quoted as ""Taleb Distribution"") with arguments against taking risks altogether. You need to understand that not taking a risk - is by itself a risk. You can lose money by not investing it, because of the very same Taleb Distribution: an unpredictable catastrophic event. Take an example of keeping cash in your house and not investing it anywhere. In the 1998 default of the Russian Federation, people lost money by not investing it. Why? Because had they invested the money - they would have the investments/properties, but since they only had cash - it became worthless overnight. There's no argument for or against investing on its own. The arguments are always related to the investment goals and the risk analysis. You're looking for something that doesn't exist." what is the mechanism by which they   make money on the funds that I have in   my account? Risk drives TD Ameritrade to look for profits, Turukawa's storytelling about 100,000$ and 500$ is trivial. The risk consists of credit risk, asset-liability risk and profit risk. The third, based on Pareto Principle, explains the loss-harvesting. The pareto distribution is used in all kind of decentralized systems such as Web, business and -- if I am not totally wrong -- the profit risk is a thing that some authorities require firms to investigate, hopefully someone could explain you more about it. You can visualize the distribution with rpareto(n, shape, scale) in R Statistics -program (free). Wikipedia's a bit populist description: In the financial services industry,   this concept is known as profit risk,   where 20% or fewer of a company's   customers are generating positive   income while 80% or more are costing   the company money. Read more about it here and about the risk here. Risk in finance is defined as standard deviation of returns. This is a measure of size of your returns, both negative and positive. Since the mean return is positive (at least for the stock market and fixed income), if you double the standard deviation your mean return also doubles along with it. In this way you are compensated by the market for taking on more risk. It looks like you need a lot more education on the subject. I suggest you pick up a book on investing and portfolio management to get a first idea. Dividend yields are currently way below 5% on blue chips. Unlike coupons from fixed income instruments (which, in the same risk category, pay a lot less), dividend yields are not guaranteed and neither is the invested principal amount.  In either case, your calculation is far away from reality.  Sure, there are investments (such as the mentioned direct investments in companies or housings in emerging economies) that can potentially earn you two digit percentage returns. Just remember: risk always goes both ways. A higher earning potential means higher loss potential. Also, a direct investment is a lot less liquid than an investment on a publicly quoted high turnover market place. If you suddenly need money, you really don't want to be pressed to sell real estate in an emerging market (keyword: bid ask spread). My advice: the money that you can set aside for the long term (10 years plus), invest it in stock ETFs, globally. Everything else should be invested in bond funds or even deposits, depending on when you will need the access. As others have pointed out, consider getting professional advice. Factoring in the odds of winning (1 in 292.2 million), the expected return on a $2 ticket is between $1.79 at the high end, and $1.35 at the low end.  Your odds of winning are drastically reduced if you don't play (/s), and your expected return depends on what your choose to do with that $2. "There may well be several such graphs, I expect googling will turn them up; but the definition of risk is actually quite important here. My definition of risk might not be quite the same as yours, so the relative risk factors would be different. For example: in general, stocks are more risky than bonds.  But owning common shares in a blue-chip company might well be less risky than owning bonds from a company teetering on the edge of bankruptcy, and no single risk number can really capture that. Another example: while I can put all my money in short-term deposits, and it is pretty ""safe"", if it grows at 1% so that my investment portfolio cannot fund my retirement, then I have a risk that I will run out of money before I shuffle off this mortal coil.  How to capture that ""risk"" in a single number? So you will need to better define your parameters before you can prepare a visual aid. Good Luck"