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Rule 1 of investing is that the more risk you're willing to take, the higher your potential for reward. Bonds are low-risk and come with maybe 1-2% returns, stocks much higher risk and maybe 5-15% returns if you know a little, but if you're really adventurous (and can afford it), go into venture capital for sometimes over 100% return a year. But is the risk/reward relationship linear?

If I opt for an investment with double the risk, am I expected to get double the return? If not, what's the relationship between risk and reward usually, roughly speaking?

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    A key point is that higher risk doesn't automatically come with higher expected rewards - it depends on the quality of investment you make. The relationship you describe is true (although not linear) for optimal investments - in real life is quite difficult to find the right 'higher rewards' investments. – Aganju Jun 25 '17 at 17:59
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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

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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.

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    When it comes to derivatives, things do get strange. I've become a fan of call spreads, those that return, say 5X on a 30% stock move over 12 months. Yet, it seems that 30% move is far more likely than 1 in 5 or 20 % – JTP - Apologise to Monica Jun 25 '17 at 18:59
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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

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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.

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