Measuring Risk Aversion
See how to put a number on fear of risk. One slider makes the fear bigger or smaller.
The bigger the fear, the more the sure thing is worth. A gamble averaging 50 won gets sold for a sure 💰 money of 30 won. The 20 won shaved off is the risk premium. The 📏 ruler that measures this fear is the formula below.
For example, one friend quits a coin-flip game right away for a sure 30 won. Another friend holds out for 45 won. The second friend is simply less afraid.
u ( x ) = x 1 − γ 1 − γ u(x) = \frac{x^{1-\gamma}}{1-\gamma} u ( x ) = 1 − γ x 1 − γ r A ( x ) = − u ′ ′ ( x ) u ′ ( x ) = γ x r_A(x) = -\frac{u''(x)}{u'(x)} = \frac{\gamma}{x} r A ( x ) = − u ′ ( x ) u ′′ ( x ) = x γ Part of these learning paths
Examples & Exercises
In CRRA utility, γ = 1 \gamma = 1 γ = 1 u ( x ) = ln x u(x) = \ln x u ( x ) = ln x
For the 50-50 lottery ( 150 , 50 ) (150, 50) ( 150 , 50 ) r A ( 100 ) r_A(100) r A ( 100 )
ln c = 1 2 ln 150 + 1 2 ln 50 = ln 150 ⋅ 50 \ln c = \tfrac{1}{2}\ln 150 + \tfrac{1}{2}\ln 50 = \ln\sqrt{150 \cdot 50} ln c = 2 1 ln 150 + 2 1 ln 50 = ln 150 ⋅ 50 r A ( x ) = − u ′ ′ ( x ) u ′ ( x ) = 1 x r_A(x) = -\frac{u''(x)}{u'(x)} = \frac{1}{x} r A ( x ) = − u ′ ( x ) u ′′ ( x ) = x 1 γ = 1 \gamma = 1 γ = 1 ( 150 , 50 ) (150, 50) ( 150 , 50 ) Show proof
From ln c = 1 2 ln 150 + 1 2 ln 50 = ln 150 ⋅ 50 = ln 7500 \ln c = \frac{1}{2}\ln 150 + \frac{1}{2}\ln 50 = \ln\sqrt{150 \cdot 50} = \ln\sqrt{7500} ln c = 2 1 ln 150 + 2 1 ln 50 = ln 150 ⋅ 50 = ln 7500 c = 7500 ≈ 86.60 c = \sqrt{7500} \approx 86.60 c = 7500 ≈ 86.60 100 − 86.60 = 13.40 100 - 86.60 = 13.40 100 − 86.60 = 13.40
With u ( x ) = ln x u(x) = \ln x u ( x ) = ln x u ′ ( x ) = 1 / x u'(x) = 1/x u ′ ( x ) = 1/ x u ′ ′ ( x ) = − 1 / x 2 u''(x) = -1/x^2 u ′′ ( x ) = − 1/ x 2 r A ( x ) = − u ′ ′ ( x ) / u ′ ( x ) = 1 / x r_A(x) = -u''(x)/u'(x) = 1/x r A ( x ) = − u ′′ ( x ) / u ′ ( x ) = 1/ x r A ( 100 ) = 0.01 r_A(100) = 0.01 r A ( 100 ) = 0.01
This matches the CRRA formula r A ( x ) = γ / x r_A(x) = \gamma/x r A ( x ) = γ / x γ = 1 \gamma = 1 γ = 1
Show that the CRRA consumer is risk neutral at γ = 0 \gamma = 0 γ = 0 ( 150 , 50 ) (150, 50) ( 150 , 50 )
u ( x ) = x 1 − 0 1 − 0 = x u(x) = \frac{x^{1-0}}{1-0} = x u ( x ) = 1 − 0 x 1 − 0 = x γ = 0 \gamma = 0 γ = 0 Show proof
At γ = 0 \gamma = 0 γ = 0 u ( x ) = x 1 / 1 = x u(x) = x^{1}/1 = x u ( x ) = x 1 /1 = x U ( L ) = 1 2 ⋅ 150 + 1 2 ⋅ 50 = 100 = E [ L ] U(L) = \frac{1}{2} \cdot 150 + \frac{1}{2} \cdot 50 = 100 = \mathbb{E}[L] U ( L ) = 2 1 ⋅ 150 + 2 1 ⋅ 50 = 100 = E [ L ]
The certainty equivalent is c = U ( L ) = 100 c = U(L) = 100 c = U ( L ) = 100 100 − 100 = 0 100 - 100 = 0 100 − 100 = 0 r A ( x ) = γ / x = 0 r_A(x) = \gamma/x = 0 r A ( x ) = γ / x = 0
In the figure the normalized curve coincides with the chord and the Jensen gap vanishes.
7. Choice under Uncertainty