Hello guys. I have trouble in understanding the utility function of Risk averse investors and risk loving investors. It is stated that the utility function for risk averse investor is concave with a diminishing marginal utility of wealth. Can someone explain me why this is the case? Also why is the function convex for risk seeking investor? The explanation provided in CFAI and schweser materials goes over the top of my head!
Thanks in advance.
Risk seeking - (Straight from Schweser) Lets take the classic example of a guy who is struggling to pay his tuition fees and with the 10$ he has he goes to get a lottery ticket to get money. His tuition fees needed is lets say 25K$.
If he gets 10$ jackpot, he isnt going to be happy… If he gets anything less than a significant amount he is not happy… He is a risk seeking individual and so you can see the utility function is convex here…
Risk Averse - This is what most of us are and explanation should be simple. You are happy if your return is low if you need to take less risk. But with higher returns, your utility also increases steadily till it flattens out… Just like your Suppl-Demand curve!! After a certain level, it makes no changes to your satisfaction…
I hope this helps…
see if this helps…
risk-averse: concave shape represents diminishing marginal utility of wealth which you already know - think of it as the gain from $1 increase in wealth is less than the ‘pain’ from $1 decrease in wealth. Therefoe you want to protect your money - hence risk averse. ur afraid to lose money becaue pain of potentially losing is higher than potentially winning.
risk-seeking: convex - gain $1 increase in wealth is higher than ‘pain’ you feel from losing the same amount. You feel high betting/gambling whatever…you like risk.
Don’t think I can explain better…maybe someone else can.
Thanks all for the replies (especially davidkcfaprep. You’ve nailed it). Can’t believe this eluded my understanding! I’m pretty clear now 
I think risk-aversion means that the investor is only likely to invest when she sees a +ve risk premium . That means the bet must pay off more than the certain payoff ( otherwise known as risk-free rate) by a probabilty-weighted margin .
A typical lottery offers outsize rewards at very very low probability levels , making the expected return quite low compared to the risk free alternative i.e. the risk-premium is negative
The risk seeking individual will take the bet especially when the risk premium is negative . So for example the certain bet which pays off little is not attractive to them because it has a low or zero risk premium .
In other words risk-seeking individuals do not consider probabilities or perhaps ignore them .
The risk-averse individual takes probabilities into account and will prefer the certain bet over the uncertain one when the payoffs are probability-equivalent . Also she will take her risk-aversion parameter into account . Even when there is probability-equivalence , she has a utility function which weights risk higher ( i.e. more negative ) than return to a degree , so she is likely to prefer the certain bet over the uncertain one up to a point where the reward of the risky bet exceeds a function-multiple of the risk. Theorists have proposed quadratic, log and variable utility functions among many others
The risk-neutral individual is indifferent between the certain equivalent bet and the uncertain one when the probability-equivalence is same. He will go or not go for the zero risk premium case with indiffference