Preference Toward RiskRisk PremiumIndifference Curve between Expected Value and Standard DeviationReducing Risk: DiversificationCite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 1 1 Preference Toward Risk 14.01 Principles of Microeconomics, Fall 2007 Chia-Hui Chen September 26, 2007 Lecture 9 Preference Toward Risk, Risk Premium, Indifference Curves, and Reducing Risk Outline 1. Chap 5: Preference Toward Risk 2. Chap 5: Risk Premium 3. Chap 5: Indifference Curve 4. Chap 5: Reducing Risk: Diversification 1 Preference Toward Risk - Risk Averse / Neu-tral / Seeking (Loving) Three different kinds of behaviors: Risk Averse (Figure 1) • Facing two payoffs with the same expected value, prefer the less risky one. • Diminishing marginal utility of income. • Relation between the utility of expected value and expected utility u(E(x)) > E(u(x)). Example. u(x) = ln x. Risk Neutral (Figure 2) • Facing two payoffs with the same expected value, feel indifferent. • Linear marginal utility of income. • Relation between the utility of expected value and expected utility u(E(x)) = E(u(x)).Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 2 2 Risk Premium u(x) 3 2.5 2 1.5 1 0.5 0 x 0 1 2 3 4 5 6 7 8 9 10 Figure 1: The Utility Function of Risk Averse. Example. u(x) = x. Risk Seeking (Figure 3) • Facing two payoffs with the same expected value, prefer the riskier one. • Increasing marginal utility of income. • Relation between the utility of expected value and expected utility u(E(x)) < E(u(x)). Example. u(x) = x 2 . 2 Risk Premium Risk premium. The maximum amount of money that a risk-averse person would pay to avoid taking a risk. Example (Job Choice). Assume that a risk-averse pe rson whose utility function corresponds with the curve in Figure 4 has two possible incomes.u(x) 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 x u(x) 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 x Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 3 2 Risk Premium Figure 2: The Utility Function of Risk Neutral. Figure 3: The Utility Function of Risk Seeking.Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 4 3 Indifference Curve between Expected Value and Standard Deviation u(x) 20 18 16 14 12 10 8 6 4 2 0 0 5 10 15 20 25 30 x Figure 4: Risk Premium: A Utility Function. • His income I might be 10 with probability 0.5 and 30 with probability 0.5. Then the ex pected value of income I is: E(1) = 10 0.5 + 30 0.5 = 20,× ×with an exp e c ted utility: E(u(I)) = u(10) 0.5 + u(30) 0.5 = 10 0.5 + 18 0.5 = 14.× × × ×′ ′ • If we offer him a fixed income I , I = 16, then his expected utility is: E(u(I ′ )) = u(16) × 1 = 14 × 1 = 14. One can see that ′ E(u(I)) = E(u(I )). ′However, E(I) − E(I ) = 4. This means the person is willing to give up a value of 4 in exchange for a riskless income. Thus, the risk premium is ′ Risk P remium = E(I) − E(I ) = 20 − 16 = 4. 3 Indifference Curve between Expected Value and Standard Deviation The indifference curve we discussed befo re is about the quantities of two different goods, now we consider the indifference curve about expected value and standard deviation (Figure 5).Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 5 3 Indifference Curve between Expected Value and Standard Deviation E x 1330 1320 1310 1300 1290 400 500 600 700 800 900 1000 1280 1270 1260 1250 1240 σ Figure 5: Indifference Curve between Expected Value and Standard Deviation. Probability 0.5 Probability 0.5 Job 1 900 1600 Job 2 625 2025 Table 1: The Income and Probability of Two Jobs. Example (Job choice). Suppose one has the following utility function u(x) = √x and two job choices (see Table 1). Calculate expected utilities: E(u(x1)) = 0.5 ×√900 + 0.5 ×√1600 = 35, E(u(x2)) = 0.5 ×√625 + 0.5 ×√2025 = 35. Thus, these two jobs give the person the same utility level, i.e. they are on a same indifference curve. In order to plot the indifference curve, we should calculate their expected values and standard deviations. E(x1) = 1 250 σ(x1) = 4 94 E(x2) = 1 325 σ(x2) = 9 90 Job 2 has higher expected value of income but it is riskier. (Figure 5) Compare Figure 6 and Figur e 7. The former is mo re risk averse since one must compensate more for more risk.E x 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 σ 20 18 16 14 12 x E10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 σ 6 3 Indifference Curve between Expected Value and Standard Deviation Figure 6: Indifference Curve between Expected Value and Standard Deviation, Larger Slope. Figure 7: Indifference Curve between Expected Value and Standard Deviation, Smaller Slope. Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 7 4 Reducing Risk: Diversification 4 Reducing Risk: Diversification Diversification. Reducing risk by allocating resources to different activities whose outcomes are not closely related. Example (Selling air conditioner and heater). Suppose that the