Part 1 - Uncertainty and utility
07 October 2025 14:38
Uncertainty and utility
Different investment opportunities exist with different risk-return characteristics
- The general assumption is that investors will like higher returns and dislike higher risk
- And of investor preference for more vs less)
Expected utility
- Suppose end of period wealth is a random variable Wi with n possible incomes, i = (1,n)
- Probability of each possible outcome Wi is pi
- Utility from each possible outcome Wi is U(Wi)
○ Wealth is tangible assets and utility is the measure of what these things do for one's wellbeing (also impacted by several
other factors)
○ Therefore U depends on W
- Then:
○ I.e. work out the utility of each wealth outcome; work out the probability of each outcome and then add them up
- The expected utility of wealth IS NOT THE SAME as the utility of expected wealth which would be U(E[W])
○ Which is why we work out the utility of wealth then multiply by the probabilities
EXAMPLE
- To find the most risky investments, calculate the expected outcomes/returns for each investment; find the standard deviation,
and that which has the highest SD is the riskiest = B
- Assume the utility function: U(W) = 4W - (1/10) W2
○ e.g. if outcome is 18, U(W) = 4*18 - (1/10) * 324 = 39.6
- So expected utility for investment C:
○ E[U]Inv C = 39.6*(1/4) + 38.4*(1/4) + 33.6*(1/4) + 25.6*(1/4) = 34.3
○ E[U]Inv A = 36.3
○ E[U]Inv B = 26.98
The utility function
- This utility function (quadratic form utility function) is credible up to the point where W = 20 because above this point, output
utility begins to drop
- In order to be credible, the utility function needs to continually rise and never drop or level out given we assume investors
always prefer more to less
Finance Page 1
, ○ The shape of the function is concave down increasing:
Concave down: positive slope but the slope reduces as we go left to right
Increasing: the overall value of the function is rising
EXAMPLE CTD.
The ranking of investments (any investments we make according to expected utility) remains unchanged under linear
transformation if:
- A constant is added to the utility function
- Or the utility function is scaled by a positive constant
○ In the example above, we would choose investment A then C then B and we would still have the same preference for
investments if the utility function were transformed as: a + bU(W)
Risk aversion and fair lottery
A 'fair lottery' is defined as one that has expected value (wealth) of zero
- A risk-averse individual is defined as one who would not accept a fair lottery
○ This implies a concave down increasing utility function with respect to wealth
In the simple example:
○ Parameters for a two-outcome game and fair lottery
W = k1 with probability p
W = k2 with probability 1-p
E(W) = pk1 + (1-p) k2 = 0
k1 /k2 = -(1-p)/p or p = -k2 /(k1 -k2 )
- Tossing a fair coin: p = 1/2 and k1 = -k2 = $100, say
- A risk-averse individual would not take a gamble with equal probabilities of winning or losing the same amount (i.e. fair coin
toss)
○ Given the expected value of the fair game is 0
WHY?
- If RA investor starts at the dot, change in wealth will be +$100 or -$100
○ In terms of wealth, the change is neutral (win vs lose is same amount)
○ BUT because of the concave-down increasing shape of the utility function; the loss in utility they would experience if
they lost the game would be greater than the gain in utility if they were to win the game
Thus, a RA investor will not take a fair game due to the asymmetry in regards to utility
Utility functions
More is preferred to less
- Utility is strictly monotone increasing with W
- So marginal utility is positive
○ U'(W) > 0 means we want U to increase as W increases
Circular d here is used for partial differentiation of a function which may be a function of more than one variable
given that utility may depend on other factors, not just wealth
- But for risk-averse investors
○ The condition of diminishing marginal utility means the function is concave down
Example
You could pay to participate in a gambling game which involves tossing a fair coin i.e. p = 0.5 of 'heads'
- Gamble of receiving:
○ £16 for a 'head'
○ Or £4 for 'tails'
- So E[W] = 0.5 (£16) + 0.5 (£4) = £10
- If the cost of playing = £10, net E[W] - £10 = 0 i.e. showing it is a fair game/fair lottery
- The amount an individual would be willing to pay to play the game depends on the risk appetite:
A risk-neutral person would take on a gamble in a fair game and would likely be willing to pay £10 to play
Finance Page 2
07 October 2025 14:38
Uncertainty and utility
Different investment opportunities exist with different risk-return characteristics
- The general assumption is that investors will like higher returns and dislike higher risk
- And of investor preference for more vs less)
Expected utility
- Suppose end of period wealth is a random variable Wi with n possible incomes, i = (1,n)
- Probability of each possible outcome Wi is pi
- Utility from each possible outcome Wi is U(Wi)
○ Wealth is tangible assets and utility is the measure of what these things do for one's wellbeing (also impacted by several
other factors)
○ Therefore U depends on W
- Then:
○ I.e. work out the utility of each wealth outcome; work out the probability of each outcome and then add them up
- The expected utility of wealth IS NOT THE SAME as the utility of expected wealth which would be U(E[W])
○ Which is why we work out the utility of wealth then multiply by the probabilities
EXAMPLE
- To find the most risky investments, calculate the expected outcomes/returns for each investment; find the standard deviation,
and that which has the highest SD is the riskiest = B
- Assume the utility function: U(W) = 4W - (1/10) W2
○ e.g. if outcome is 18, U(W) = 4*18 - (1/10) * 324 = 39.6
- So expected utility for investment C:
○ E[U]Inv C = 39.6*(1/4) + 38.4*(1/4) + 33.6*(1/4) + 25.6*(1/4) = 34.3
○ E[U]Inv A = 36.3
○ E[U]Inv B = 26.98
The utility function
- This utility function (quadratic form utility function) is credible up to the point where W = 20 because above this point, output
utility begins to drop
- In order to be credible, the utility function needs to continually rise and never drop or level out given we assume investors
always prefer more to less
Finance Page 1
, ○ The shape of the function is concave down increasing:
Concave down: positive slope but the slope reduces as we go left to right
Increasing: the overall value of the function is rising
EXAMPLE CTD.
The ranking of investments (any investments we make according to expected utility) remains unchanged under linear
transformation if:
- A constant is added to the utility function
- Or the utility function is scaled by a positive constant
○ In the example above, we would choose investment A then C then B and we would still have the same preference for
investments if the utility function were transformed as: a + bU(W)
Risk aversion and fair lottery
A 'fair lottery' is defined as one that has expected value (wealth) of zero
- A risk-averse individual is defined as one who would not accept a fair lottery
○ This implies a concave down increasing utility function with respect to wealth
In the simple example:
○ Parameters for a two-outcome game and fair lottery
W = k1 with probability p
W = k2 with probability 1-p
E(W) = pk1 + (1-p) k2 = 0
k1 /k2 = -(1-p)/p or p = -k2 /(k1 -k2 )
- Tossing a fair coin: p = 1/2 and k1 = -k2 = $100, say
- A risk-averse individual would not take a gamble with equal probabilities of winning or losing the same amount (i.e. fair coin
toss)
○ Given the expected value of the fair game is 0
WHY?
- If RA investor starts at the dot, change in wealth will be +$100 or -$100
○ In terms of wealth, the change is neutral (win vs lose is same amount)
○ BUT because of the concave-down increasing shape of the utility function; the loss in utility they would experience if
they lost the game would be greater than the gain in utility if they were to win the game
Thus, a RA investor will not take a fair game due to the asymmetry in regards to utility
Utility functions
More is preferred to less
- Utility is strictly monotone increasing with W
- So marginal utility is positive
○ U'(W) > 0 means we want U to increase as W increases
Circular d here is used for partial differentiation of a function which may be a function of more than one variable
given that utility may depend on other factors, not just wealth
- But for risk-averse investors
○ The condition of diminishing marginal utility means the function is concave down
Example
You could pay to participate in a gambling game which involves tossing a fair coin i.e. p = 0.5 of 'heads'
- Gamble of receiving:
○ £16 for a 'head'
○ Or £4 for 'tails'
- So E[W] = 0.5 (£16) + 0.5 (£4) = £10
- If the cost of playing = £10, net E[W] - £10 = 0 i.e. showing it is a fair game/fair lottery
- The amount an individual would be willing to pay to play the game depends on the risk appetite:
A risk-neutral person would take on a gamble in a fair game and would likely be willing to pay £10 to play
Finance Page 2
