Rural Finance and Risk Management - Report Example

Student’s Name: Instructor’s Name: Course Code & Name: Date of Submission: Question 1 a: i. Estimation of joint probability P (Z1S1) = 0.6X0.6= 0.36 P (Z1S2) = 0.6X0.3= 0.18 P (Z1S3) = 0.6X0.1= 0.06 P (Z2S1) = 0.3X0.6= 0.18 P (Z2S2) = 0.3X0.2= 0.06 P (Z2S3) = 0.3X0.2= 0.06 P (Z3S1) = 0.1X0.1= 0.01 P (Z3S2) = 0.1X0.3= 0.03 P (Z3S3) = 0.1X0.6= 0.06 ii. Marginal probability Marginal probability (PZ1) =P (S1 and Z1) + p(S2 and Z1) + p(S3 and Z1) P (Z1) = (0.6X0.6) + (0.3X0.6) + (0.1X0.1) = 0.55 P (Z2) = (0.6X0.3) + (0.3X0.2) + (0.1X0.3) = 0.27 P (Z3) = (0.6X0.1) + (0.3X0.2) + (0.1X0.6) = 0.18 iii. Posterior probability P(SiǀZk) = P(Zk and Si) / P(Zk) P (S1/Z1) = =0.655 P (S2/Z1) = = 0.327 P (S3/Z1) = = 0.018 P (S1/Z2) = =0.667 P (S2/Z2) = = 0.222 P (S3/Z2) = = 0.111 P (S1/Z3) = =0.333 P (S2/Z3) = = 0.333 P (S3/Z3) = = 0.333 iv. Bayes theorem A B C D E F G H I 1 State Prior Likelihoods P(Zk/Si) Joint Probabilities P(Si and Zk) 2 Sn P(Sn) Z1 Z2 Z3 Z1 Z2 Z3 3 S1 0.6 0.6 0.3 0.1 0.36 0.18 0.06 4 S2 0.3 0.6 0.2 0.2 0.18 0.06 0.06 5 S3 0.1 0.1 0.3 0.6 0.01 0.03 0.06 6 Check 1.0 Marginal P(Zk) 0.55 0.27 0.18 7 Posterior P(S1/Zk) 0.655 0.667 0.333 8 P(S2/Zk) 0.327 0.222 0.333 9 P(S3/Zk) 0.018 0.111 0.333 10 Check 1.00 1.00 1.00 Question 1 b: i. Joint probability tree ii. Posterior probability Where; S1- Not infected S2- Slightly infected S3- Seriously infected Z1-Unlikely Z2-Possible Z3-Probable Question 2 a. Jeff’s expected wealth next year if he chooses not to purchase insurance E(R) = P (Wealth in occurrence of earthquake) + P (Overall Wealth without earthquake) = 100,000 (0.1) + 0.9 (300000) =$280,000 b. Determination of Jeff’s risk attitude U (W) = W0.5 U (W) = 00.5=0 U (W) = 100, 0000.5 = 316.23 U (W) = 300, 0000.5 = 547.72 Jeff is risk averse this is because he has diminishing marginal utility of wealth (Kharasch & Rosow, 2013). c. Theoretical estimation of risk premium E (U) = 0.1(316.23) + 0.9 (547.72) = 524.57 The approximate premium is found by subtracting wealth at the point blue line cuts the x-axis by wealth where red line cuts the x-axis (Kharasch & Rosow, 2013). Approximate Risk premium = 275,000˜-265,000˜ = 15,000 The risk premium will be 15,000 since certain expected wealth of 265,000 gives Jeff the same utility as uncertain wealth of 275,000. Question 3 a. Expected percentage return on the stock and the standard deviation Probability Return Expected return (PXR) 0.5 15 7.5 0.3 18 5.4 0.2 6 1.2 Expected return 14.1% b. Standard deviation σ = σ2= 0.5(15-14.1)2+0.3 (18-14.1)2 + 0.2(6-14.1)2 = 18.09 σ = =4.35 c. Weighted average return on the portfolio Weighted average return = W1R1+W2R2 = + = 13.28% d. The standard deviation for the portfolio. The standard deviation measures the risk of security therefore standard deviation of risk free security is zero (Hardaker, 2004). σp = + =3.48 The risk reduces due to diversification. e. The budget line in the risk-returned trade-off Rp = Rf + ()β 13.28 = 10 + () 3.48 Slope; () = =0.94 The slope represents premium that is market return minus risk-free rate over market risk (). The security market line is positively sloped (Hardaker, 2004). Question 4: State of nature Dry Mild Wet Alternative 1 – Wheat 300 600 -300 Alternative 2 – Corn 400 500 -200 Alternative 3 – Barley 200 300 -100 Alternative 4 – Lettuce -200 100 200 a. Maximin Minimum -300,-200, -100, -200 Maximum -100 Maximin criteria: Alternative 3 – Barley- Wet (-100) b. Maximax Maximum 600, 500, 300,200 Maximum 600 Maximax criteria: Alternative 1 – Wheat - Mild (600) c. Minimax Calculation of regret matrix State of nature Dry Mild Wet Alternative 1 – Wheat 300-400 600-600 -300-200 Alternative 2 – Corn 400-400 500-600 -200-200 Alternative 3 – Barley 200-400 300-600 -100-200 Alternative 4 – Lettuce -200-400 100-600 200-200 Regret matrix State of nature Dry Mild Wet Alternative 1 – Wheat -100 0 -500 Alternative 2 – Corn 0 -100 -400 Alternative 3 – Barley -200 -300 -300 Alternative 4 – Lettuce -600 -500 0 Minimum = -500, -400, -300, -600 Maximum -300 Minimax regret criteria: Alternative 3 – Barley- Either in mild or wet state because of minimum regret of -300 d. Hurwitz alpha criteria Ii = αXi + (1 - α) Yi Alternative 1 – Wheat: (0.6)600 - (0.4)300 = 240 Alternative 2 – Corn: (0.6)500 - (0.4)200 = 220 Alternative 3 – Barley: (0.6)300 - (0.4)100 = 140 Alternative 4 – Lettuce: (0.6)200 - (0.4)200 = 40 The appropriate option is Alternative 1 – Wheat with 240. e. Principle of insufficient reason: State of nature Dry Mild Wet Avarage Alternative 1 – Wheat 300 600 -300 200 Alternative 2 – Corn 400 500 -200 233.33 Alternative 3 – Barley 200 300 -100 133.33 Alternative 4 – Lettuce -200 100 200 33.33 The appropriate option is Alternative 2 – Corn with average of 233.33. Question 5: a. Minimum acceptable return level (Rmin) Invested fund =$600,000 Loan =$45,000 Rmin =x100% = 7.5% b. safety-first ratio The safety-first ratio enables the investor to choose crop that yields optimal return. This is achieved by calculation of safety-first ratio then selecting crop with the highest ratio (Roy, 1952). This is shown below; SFRatio = Tomato = = 1 Corn = = 0.976 Potato = = 1.136 Banana = = 0.849 Lettuce = = 1.129 According to Roy’s rule, the optimal crop is the one with highest ratio. Therefore, the best crop is potato with 1.136. c. Telser’s rule Telser’s rule enables the investor to choose crop with its probability of ruin less than the set percentage. α is obtained from the z distribution table for K values. Telser’s rule is applied as shown below; Crop E(Ri) K α Tomato 16 1 15.87% Corn 14 0.976 16.35% Potato 10 1.136 12.71% Lettuce 12 0.849 19.77% Capsicum 18 1.129 12.92% According to Telser’s rule as shown in the table above, potato gives the optimal result since it has the highest expected return and its probability of ruin (12.71%) that does not exceed 15% (12.71 %< 15%). d. Expected return of the potato The expected return for the potato is as shown below; SFRatio = 1.136= = 2.4992+7.5 10.0% The same crop is optimum in both Roy’s and Telser’s rule. Therefore, the expected return will remain at 10%. Question 6: a. Decision basing on E-V Rule EV (Wheat) = [200x0.2+400x0.2+600x0.1+800x0.2+1000x0.3] = 640 EV (Barely) = [200x0.3+400x0.1+600x0.2+800x0.3+1000x0.1] = 560 EV (Corn) = [200x0.2+400x0.2+600x0.3+800x0.1+1000x0.2] = 580 The best crop to invest in is wheat since it yields higher return according to EV rule with EV of 640. b. First Stochastic Dominance F(x) = P (A< x) Outcome Wheat: (x) Barley: (x) Corn: (x) 200 40 60 40 400 80 40 80 600 60 120 180 800 160 240 80 1000 300 100 200 Where; X= Expected return for each outcome First stochastic dominance function is given as; Wheat; F(x) =0.3x Barely; F(x) =0.3x Corn; F(x) =0.4x Outcome Wheat: F(x) Barley: F(x) Corn: F(x) 200 12 18 16 400 24 12 32 600 18 36 72 800 48 72 32 1000 90 30 80 Below is CDF’s graph showing outcome against variation of expected outcome; Where; Series 1: wheat Series 2: Barley Series 3: Corn Reference: Azencott, R., Beri, A., & Timofeyev, I. (2011). 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