Utility

Question 1

Utility

Answer 1

The sense of happiness, welfare or psychic benefit that one gets from wealth level or rate of return (or other economic goods)

Question 2

Axioms of utility theory

Answer 2

1. Investors prefer more to less
2. Investors behave rationally and make consistent rational decisions
3. Investors appreciate certainty: they would rather take a certainty equivalent amount today over a chance at a higher but uncertain amount.

Question 3

Risk-loving characteristics

Answer 3

• enters a fair game of chance at a price higher than the expected value.
• convex utility curve: next unit of wealth will increase investors utility more than the last unit did.
• u’(w) > 0
• u’’(w) > 0 -> MU increases at a faster rate for higher levels of wealth.
• E[U(W)] > U[E(W)]
  Expected utility of wealth is greater than utility of expected wealth

Question 4

Risk-averse characteristics

Answer 4

-concave u fn
- utility added from each additional unit of wealth diminishes as wealth increases
- u’(w) > 0
- u’’(w) < 0

Question 5

Utility fns with ARAs and RRA

Answer 5

all risk averse investors
Log -> decreasing ARA -> constant RRA
Quadratic -> increasing ARA -> increasing RRA
Power -> decreasing ARA -> constant RRA
Exponential -> constant ARA -> decreasing RRA

Question 6

ARA

Answer 6

In monetary terms, the absolute amount of money one would invest in risky assets

ARA>0
- reduce their monetary investment in risky assets as wealth increases
ARA<0
- increases their monetary investment in risky assets as wealth increases
ARA=0
- monetary investment in risky assets stays the same -> constant absolute risk aversion

Question 7

RRA

Answer 7

Relative risk aversion: the percentage of wealth that one commits to risky assets

RRA >0
- reduce their % investment in risky assets as wealth increases -> increasing relative risk aversion
RRA < 0
- increase their % investment in risky assets as wealth increases -> decreasing relative risk aversion
RRA =0
- % stays constant as wealth increases -> constant absolute risk aversion

Question 8

ARA formula

Answer 8

= -U’’(W)/U’(W)

Question 9

RRA formula

Answer 9

= -WU’’(W)/U’(W)

Question 10

Log function

Answer 10

U(W) = ln(W)
ARA => decreasing
RRA => constant

Question 11

Quadratic fn

Answer 11

U(W) = a + bW - cW^2
ARA = increasing
RRA = increasing

Question 12

Power fn

Answer 12

U(W) = -a + W^(1-b)
ARA = decreasing
RRA = constant

Question 13

Exponential fn

Answer 13

U(W) = 1 - e^(-bW)
ARA = constant
RRA = decreasing