Question24 Which of the following statements regarding risk-averse investors is true? They only accept risky investments if they have higher positive risk premiums They have negative risk aversion coefficients They only invest in risk-free assets They would always choose a lower risk investment over a higher risk investment They would always choose the investment with the highest return ResetMaximum marks: 3 Flag question undefined

Question

BlackTom AI · Accepted Answer

Risk-averse investors require a positive compensation for bearing risk: a risk premium. Key concepts: risk aversion (concave utility functions), risk premium (expected return minus risk-free return required to accept a gamble), and the Arrow–Pratt measure (positive for risk-averse individuals).

Step-by-step reasoning:
1. Define risk premium π = E[R_risky] − R_f. A risk-averse investor accepts a risky asset only if the expected return exceeds the risk-free return by at least π ≥ 0 so that expected utility of the risky prospect ≥ utility of the certain payoff.
2. Utility comparison (general): EU(risky) ≥ U(wealth*(1+R_f)) ⇔ Accept risky. For concave U(·) (risk aversion), this typically requires a positive π.
3. Numerical illustration: starting wealth W = 100, risk-free return R_f = 2% → certain wealth = 102 → U_certain = sqrt(102) = 10.0995 (using U(W)=√W as example of concave utility). Consider a risky asset with two equally likely outcomes giving wealth 120 and 90 (expected wealth = 105, so E[R_risky] = 5%, π = 3%). EU_risky = 0.5·√120 + 0.5·√90 = 0.5·10.954 + 0.5·9.487 = 10.2205. Since 10.2205 > 10.0995, the investor accepts the risky asset because the positive risk premium compensates for risk.
4. If π were smaller (or negative), EU_risky could fall below U_certain and the investor would reject the risky asset.

Why the correct answer is correct:
- "They only accept risky investments if they have higher positive risk premiums" matches the definition: risk-averse agents demand positive compensation (π>0) to bear risk.

Why other choices are incorrect:
- "They have negative risk aversion coefficients": false—risk-averse individuals have positive Arrow–Pratt coefficients.
- "They only invest in risk-free assets": false—risk-averse investors can hold risky assets if adequately compensated.
- "They would always choose a lower risk investment over a higher risk investment": false—choice depends on risk-return trade-off, not risk level alone.
- "They would always choose the investment with the highest return": false—returns must be evaluated net of risk; highest nominal return may not compensate for higher risk.

Conclusion: The correct answer is "They only accept risky investments if they have higher positive risk premiums," which matches the provided answer.

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