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I am not an expert in the field. So bear with me if my terminology is bad.

I want to understand what risk premium of a portfolio is. I understand that there are different forms of risk. The basic idea seem to stem from the following basic difference. A portfolio which gives a fixed return of 5%, is considered different to a portfolio which gives a mean of 5% but has variance. Since variance is not "desired", the latter needs a premium return to match the former portfolio.

  1. Is this understanding correct?
  2. Is there any mathematical basis in saying that variance is not "desired"? Or is it purely psychological? (based on individual risk aversion, utility functions and so on).
  3. If the reasoning is purely psychological, should we logically factor in this premium when creating a portfolio?
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  1. This intuition is correct. Formally, we consider that people are risk averse which is just another way of saying that they prefer more stable to less stable cash flows. Another equivalent way of saying this is that they are disposed to sacrifice some gains on average for the added stability.

  2. The fundamental reason is indeed entirely psychological. We approximate the tendency of people to require compensation for accepting more fluctuation in their portfolio through risk aversion. In terms of utility functions, per the inequality of Jensen, that implies a concave utility function.

  3. Normal human beings tend to care about how much and what type of risk they are taking with whatever portion of their wealth they have invested in a given portfolio. This is often done using Sharp ratios: people trying to get a sense of how much on average you'd get paid to take "units" of risk, understood as volatility.

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  • $\begingroup$ Thanks for your reply. It really helps. But I still find it hard to rationalize the difference. If I get 50 cents a day for playing game A, and if I get 1$ with 50% chance of winning it for playing game B, I don't see much difference between these games, provided you can play enough times. Is there anything I am missing? Or is the risk premium just a measure of, how much you are willing to avoid stomach churns when you don't win the game on a day? $\endgroup$
    – finnewbie
    Mar 2 '20 at 4:55
  • $\begingroup$ So yes, people dislike ups and downs in their wealth and would rather avoid this. If you talk to any stockholders around the world today (20200302) they will tell you how disgusted they felt the last week or so with the rapid stock market drop. (Even though you could argue that it was just a reversal of the big gains of the previous few months). $\endgroup$
    – noob2
    Mar 2 '20 at 9:40
  • $\begingroup$ If you only care about the expected value of a lotterie, A and B are indeed identical and you should be indifferent between playing either game or no game if the cost of entry was 50 cents per day. Now, consider a third game: I toss a coin for as long as I get heads. Your payoff is 2^N where N is the number of times you got head. This game has an INFINITE expected payoff. How much would you be willing to pay to play? I'm guessing, not infinity. You want a lower price so that on average you GAIN from playing that game. Risk aversion is one possible resolution of the St-Petersburg paradox. $\endgroup$
    – Stéphane
    Mar 2 '20 at 19:30
  • $\begingroup$ This is good insight. The way I interpret this is, even though in the long run, things regress towards mean, there may be impracticality such as the time until such a convergence. So a portfolio which gives fixed return, is better, since it is very simple. Hope I am correct. This does not seem to be obvious at all, yet not many texts seem to explain this. Any better reads on this? $\endgroup$
    – finnewbie
    Mar 2 '20 at 23:17
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    $\begingroup$ I think I see why you seem to be confused: you're trying to find a reason behind risk aversion. The point I am trying to make is that we absolutely do not care why people are risk averse, we only care that they are risk averse. The reason you won't find ample discussions about this is that it's a fundamental assumption behind expected utility theory -- it's taken as a given. The only discussions you will find are about psychological experiments showing it's not rich enough to capture how people actually deal with uncertainty. $\endgroup$
    – Stéphane
    Mar 3 '20 at 4:06

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