Certainty equivalent of Fair Gamble

Discussion in 'CM2' started by Vince, Feb 18, 2020.

  1. Vince

    Vince Member

    Hi,

    I was having trouble understanding the following statement and after some investigating I think there is a mistake: Chapter 2, Section 3.2
    "Consider the certainty equivalent of a gamble. For a risk-averse individual this is higher than the actual likelihood of the outcome."

    I have assumed that the 'actual likelihood of the outcome' is the expected outcome of the gamble, ie. Sum of (probabilities x gamble returns).
    I think this should read, "For a risk-averse individual this is lower than the actual likelihood of the outcome."

    E.g. a risk-averse individual and a fair gamble; the 'actual likelihood of the outcome' is 0, but the certainty equivalent is negative, hence lower.

    Am I correct in this line of thought?
     
    Last edited by a moderator: Feb 18, 2020
  2. Aditya jain

    Aditya jain Member

    Hey,

    Actual likelihood of the outcome refers to the actual probability of the outcome and not the expected outcome of the gamble.
     
  3. Vince

    Vince Member

    Thank you for your reply but I'm still not sure I follow what you mean by 'actual probability of the outcome'.

    A value of wealth is needed to compare to the certainty equivalent of the gamble, Cx (say), and the only sensible value I can conceive is then the expected outcome ('outcome' being initial wealth plus gamble), but this will still be higher than a certainty equivalent of [initial wealth + gamble].

    Alternatively, it doesn't make sense to be comparing actual probabilities to Cx, a value of wealth, and if the 'outcome' is the expected utility, then this also cannot be sensibly compared to Cx.
     
  4. John Potter

    John Potter ActEd Tutor Staff Member

    Vince,

    I don't think the original wording is very good, it seems to be comparing a monetary amount with a probability. So, I think all subsequent attempts to improve the wording are a good idea (thanks to Aditya for trying!).

    But what it's getting at is what you originally talk about. If everybody was risk-neutral, the certainty equivalent (CE) would just be summing the prob of an event*cost of event happening. eg, a fair gamble has a CE of 0. Toss a coin, win £10, lose £10, who cares? Take it or leave it. But we are risk-averse, especially at higher amounts. Toss a coin, win £1000, lose £1000, who cares? I do! I haven't got £1000 to lose. So I would pay to get out of this gamble.

    It's a similar concept on all other situations that the risk-averse investor would pay more to get out of taking the risk. Once you've understood that concept (which I think you have already) you don't need to get hung up on higher or lower etc. because it's probably just a matter of semantics and whether we are looking at the positive or negative side.

    Eg let's say a risk-neutral investor would happily pay £5 to avoid taking gamble X whilst a risk-averse investor would pay £10.
    £10 is higher than £5. But they are paying it. So, should we say that -£5 is higher than -£10? Technically, the CE is the latter but let's not worry too much about it,

    Good luck!
    John
     

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