Risk & Risk Aversion

Front 1

simple prospect

Back 1

an investment opportunity in whicha certain initial wealth is placed at risk, and there are only two possible outcomes.

Front 2

speculation

Back 2

The assumption of considerable business risk in obtaining commensurate gain.

Front 3

commensurate gain

Back 3

Positive risk premium - an expected profit greater than the risk-free alternative.

Front 4

considerable risk

Back 4

The risk is sufficient to affect the decision.

Front 5

gamble

Back 5

To bet or wager on an uncertain outcome.

Front 6

gamble vs. speculation - economically speaking

Back 6

A gamble is the assumption of risk for no purpose but enjoyment of the risk itself, whereas speculation is undertaken in spite of the risk involved because one perceives a favorable risk-return trade-off.

Front 7

fair game

Back 7

A prospect that has zero risk premium.

Front 8

risk averse

Back 8

A risk averse person will reject investment portfolios that are fair games or worse.

Front 9

utility function

Back 9

U = E(r) - .005*A*Sigma_Sq

Front 10

A

Back 10

Index of investor's risk aversion.

Front 11

certainty equivalent rate

Back 11

The rate that risk-free investments would need to offer with certainty to be considered equally attractive as the risky portfolio.

Front 12

risk-neutral

Back 12

A risk-neutral investor judges risky prospects solely by their expected rates of return.

Front 13

mean-variance (M-V) criterion

Back 13

E(r_a) >= E(r_b)
and
sigma_a <= sigma_b
and
at least one inequality is strict (rules out equality).

Front 14

hedging

Back 14

Investing in an asset with a payoff pattern that offsets exposure to a particular source of risk.

Front 15

covariance

Back 15

Measures how much the returns on two risky assets move in tandem.

Front 16

Cov(r_a,r_b)

Back 16

SUM_s[ Pr(s)*
[r_a(s) - E(r_a)] *
[r_b(s) - E(r_b)]

Front 17

correlation coefficient

Back 17

Scales the covariance to a value between -1 and +1.

Front 18

sigma_sq_P

(P = portfolio)

Back 18

w_sq_1 * sigma_sq_1 +
w_sq_2 * sigma_sq_2 +
2*w_1*w_2 * Cov(r1,r2)

Front 19

median

Back 19

The outcome value that exceeds the outcome values for half the population and is exceed by the other half.

Front 20

mode

Back 20

The outcome with the highest probability.

Front 21

mean absolute deviation (MAD)

Back 21

SUM( P(s) * |r(s) - E(r)|

Front 22

second central moment around the mean

Back 22

Variance

Front 23

skewness (3rd central moment)

Back 23

Asymmetry of a dist'n
(+) right skewness
(-) left skewness

M3 = SUM(P(s)*[r(s) - E(r)]^3)

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moments

Back 24

1st = reward

2nd & higher = uncertainty of reward

Even = likelihood of extreme values

Odd = asymmetry

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utility value

Back 25

U = E(r) - b0*Sigma_sq + b1*M3 - b2*M4 + b3*M5 -...

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Samuelson's "Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances, and Higher Moments" proves:

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1. The importance of all moments beyond the variance is much smaller than that of the expected value and variance. Disregarding moments higher than the variance will not affect portfolio choices.

2. The variance is as important as the mean to investor welfare

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Major assumption of Samuelson's "Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances, and Higher Moments"

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Compactness - if the risk can be controlled by the investor. Equivalent to the continuity of stock prices.

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alternate assumption to normal dist'n assumption - lognormal

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Continuously compounded annual rate of return (r) is normally dist'd.

r_e = effective annual rate = e^r - 1

r_e is dist'd lognormally

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r_e(t) for short holding periods

Back 29

e^(rt) - 1 for short time (t)

mean and variance are proportional to mean and variance of annual, continuously compounded ROR on stock and to the time interval (t).

Front 30

expected utility function

Back 30

E[U(W)] = pU(W1) + (1-p)U(W2)