# Higher the risk, higher the return should be, why?

Can anyone please tell me if the expected return is the same, why the rational investor would always choose less riskier investment?

Investment A: 10% expected return, 10% chance of losing investment Investment B: 10% expected return, 5% chance of losing investment It’s obvious any rational investor will choose investment B because it has less risk.

Why would you take more risk, if the expected return is the same. For example… 1) Flip a coin. If it comes up heads, you win \$100. If not, you win nothing and never get to play again. 2) Play Russian roulette with three bullets in a six chamber gun. If you aren’t blown to smithereens, you get \$100. If you are. You win nothing and never get to play again. In money terms, both give you the same expected payment. But one is riskier. Now, why would a rational person prefer game 1 to game 2???

So you are saying no one would bet on the flip of a coin. I dont think so. There are people who would bet. There are people who would risk losing \$50 for gaining \$50.

sameer, you’re missing bchadwick’s point. Game 1 is less riskier because you don’t risk DYING. Just read over both our posts a few times, sleep on it, and hopefully tomorrow you’ll see why less risk for the same return is better. Edit: This is a really important concept on a fundamental level. You need to understand how the efficient frontier works and why it’s efficient in L1. I think if you go over those readings in the books and it should come to you.

That example is not correct. Even if there is 1/1000 chance of dying for winning winning \$1000, people might not opt for it. But central to the whole finance is the assumption that investors are risk averse. I am asking does that always hold? So you do a survey in your office today - asking people are they willing to bet \$10 on the flip of a coin. Money would double if they call right and would become zero otherwise. Finance theory says that such bet would not be able to extract a cent from anyone’s pocket. But in practise people would take that bet. Not all but probably many.

The idea of the “economically rational person” is not the same as real human beings. In practice, there are people who are “risk loving.” These are the sorts of people who do extreme sports where there is a real chance of dying, or people who are compulsive gamblers, etc… So yes, there are people that get an extra thrill out of taking risks, just because they are risks. And in fact, these are people that would rather have the chance of winning \$1000 on a coin toss than simply be handed \$500. However, if you ask people if they’d rather have \$1000 or \$0 based on a coin toss or have \$500, the vast majority will opt for the \$500. In order to induce them to do the coin toss, the expected value on the coin toss needs to be more than the \$500 they could take guaranteed. That difference is called the “risk premium”. Interestingly, psychological experiments show that if you tell someone that they have a chance of owing \$1000 or \$0 on a coin toss or have to pay a fixed and guaranteed fine of \$500, many more people will opt for the coin toss. So it seems that the way we process gains is different than the way we process losses. On average, it seems that a loss of \$10 brings as much distress as a gain of \$20 brings joy. So most people really are risk averse. Taking unnecessary risks is not considered rational, because it’s fairly senseless expose yourself to risks that are not necessary. Economic analysts would say that risk loving people - though they exist - are not really rational human beings, because they are exposing themselves to unnecessary risks and making themselves more vulnerable than they need to be, but something in their brains just gives them the juice to want to do it anyway. And a lot of them try prop trading. And for the CFA exam, if you are managing money for other people, you are expected to do it in a rational way, which means, you don’t expose them to risks unless it is consistent with the investor policy statement of risks you are allowed to take, and within those limits, you’re going to be rewarded for getting the highest return possible. This basically boils down to finding the most risk efficient portfolio possible given the available constraints.

^Thanks. “On average, it seems that a loss of \$10 brings as much distress as a gain of \$20 brings joy. So most people really are risk averse” Thats seems to be a reasonable explaination for risk averseness. Surprisingly I have not come across this in CFA or for that matter in any textbook. My hunch for risk averseness was “marginal utility of money”. So people can very well bet \$10 on flip of a coin but almost no one would bet his entire wealth. But I did not come across a piece of literature which backed this. Hence posted this question.

The level 3 curriculum covers this.

Yes, risk aversion is a natural consequence of a declining marginal utility curve. It comes from the fact that if you have set of probabilistic outcomes, each paying some quantity of money X, then the expected utility of an uncertain outcome is less than the utility of the expected (i.e. average) outcome. Mathematically, if U(X) is the utility of having X dollars, then, if the utility curve is upward sloping (more is better) and concave downward (the next dollar is better, but not quite as satisfying as the last dollar), then: E( U(X) ) <= U( E(X) ) i.e. the expected utility of the outcome is less than the utility of the expected outcome, except in the case where the outcome is certain, in which case they are the same.

…so what this means is that you should expect a lower utility for risky outcome than a guaranteed outcome with the same expected payoff. So if I were to tell you that you could have \$5 now, or take a coin flip that pays \$0 or \$10, and you cared only about money (and not the thrill of the game), a rational utility-maximizer should (and most people do) rational to prefer \$5. But maybe it seems that it’s easy to imagine wanting the coin flip. Let’s change that to the possibility of \$10MM or \$0 on a single coin flip (and you can only play once). Would you take \$5MM guaranteed now, or do the coin flip? If you are like most people, you’d take the \$5MM guaranteed. This is because the utility of \$5MM to an ordinary person is substantially more than 1/2 * Utility( \$10M)… i.e. the first \$5MM of the \$10MM is a lot more useful to your existence than the second \$5MM, and therefore it is not worth risking losing the first by trying to get the second. In order for you to want to take a risk instead of taking the guaranteed \$5MM, you’d have to have the chance to win *more* than \$10MM, so that the expected value of the coin flip ends up being more than \$5MM. Probably considerably more. In order to risk giving up a guaranteed \$5MM, maybe the risky bet would have to have a payoff of \$7MM before you’d feel perfectly comfortable doing it. If that’s the case, then the bet needs to have a “risk premium” of 20% ( 7/5-1=20%) because it has to pay 20% more than the riskless case in order to satisfy you that it’s a good bet. Another thing that happens is that the further you go out on the utility curve, the flatter the curve becomes. This is why having more assets tends to go along with higher ability to take risk. Losing \$10000 just doesn’t affect your utility much if you’re Bill Gates, but it does if you’re a starving student.

^^ not that this is a mind blowing concept or anything, but the fact that you wrote all of that out so eloquently (for a forum at least) made me read it. you should write books about utility bchad! truly a champion.

What’s the utility of books on utility?

Bang on chad…that’s why a contestant at the final question in ‘who wants to be a miilionaire’ would invariably opt for half a million instead of hazarding a guess even if he has narrowed his choices to 2…another angle is if he decides to try his luck and lose everything the public wud label him a stupid surely but would not say that he is courageous if he indeed wins a million

True, but there are a lot of extra things going on in the “Millionaire” game. Social pressure being the big one. Taking the money gets people teased for not having the guts to go for more, so the “guaranteed” number also has a disutility to it that you wouldn’t see if this were a private non-televised transaction. The TV show also wants to promote drama, and therefore may be selecting for more risk-loving people. It does show why people don’t tend to take the money when they’re at \$10,000, but do tend to do it more often when they’re at \$500,000.

Agree with eveything…but do you also think that this fundamental reason for risk aversion has not got enough attention in theory of finance…everywhere discussion starts with that investors are risk averse

> risk aversion has not got enough attention in theory of finance Well, it’s gotten _some_ attention, if you count those Swedes: http://www.lmgtfy.com/?q=kahneman+tversky+nobel+prize

sameeragarwal Wrote: ------------------------------------------------------- > Agree with eveything…but do you also think that > this fundamental reason for risk aversion has not > got enough attention in theory of > finance…everywhere discussion starts with that > investors are risk averse Many pricing models (e.g. Black Scholes) actually suppose that investors are risk neutral. In a risk neutral world, all assets have the same expected return, and investors are indifferent to risk. Interesting discussion, by the way.

cityboy Wrote: ------------------------------------------------------- > Many pricing models (e.g. Black Scholes) actually > suppose that investors are risk neutral. In a risk > neutral world, all assets have the same expected > return, and investors are indifferent to risk. isn’t this because BS is only useful for a limited time period (less than a year or two). in the short-term, its impossible to assess with high probability, the return of a specific asset class. thus why the risk premium on equity can only really be captured with a certain level of certainty in the long-run and why day and swing trading is assumed to be a zero-sum game…

The reason that options theory uses risk neutral pricing is that you can construct a replicating portfolio that eliminates the risk of creating the option. Therefore, to create an option, you don’t actually have to take risk, you just have to create the replicating portfolio. Now, the theory of replicating portfolios does assume that the dynamic hedging process has no transaction costs (in reality it does), and it assumes that prices are continuous (no gapping, which happens), so there is more risk involved than the theory accounts for. This is one of the reasons (among many) that average implied volatility tends to be larger than average realized volatility.