Estimation of Joint, Marginal and Posterior Probabilities, and Bayes Theorem - Assignment Example

EMPIRICAL EXERCISE NAME: COURSE TITLE LECTURER: DATE OF SUBMISSION: Question One: a. Estimation of Joint, marginal and posterior probabilities, and Bayes’ theorem The use of probability in risk management is crucial since farmers are able to predict the chances of an event that might affect their crops positively or negatively (Ross, 2014). There are various types of probabilities which include joint probability; shows chances of two events happening at the same time, marginal probability; elaborates the chances of subsets of randomly selected variables i.e. likely and unlikely occurrence, posterior probability; shows chances of an event happen because another event happened (Pitman, 2009). i. Joint probability; P (Z1S1) =0.4X0.7= 0.28 P (Z1S2) =0.4X0.6= 0.24 P (Z1S3) =0.4X0.2= 0.08 P (Z2S1) =0.4X0.2= 0.08 P (Z2S2) =0.4X0.2= 0.08 P (Z2S3) =0.4X0.3= 0.12 P (Z3S1) =0.2X0.1= 0.02 P (Z3S2) =0.2X0.2= 0.04 P (Z3S3) =0.2X0.5= 0.10 ii. Marginal probability; Marginal probability (PZ1) =P (S1 and Z1) + P (S2 and Z1) + P (S3 and Z1) P (Z1) = (0.4 X 0.7)+(0.4 X 0.6)+(0.2 X 0.2)=0.56 P (Z2) = (0.4X0.2)+(0.4X0.2)+(0.2X0.3)= 0.22 P (Z3) = (0.4X0.1) + (0.4X0.2) + (0.2X0.5) = 0.22 iii. Posterior probability; Z1 Z2 Z3 P(S1/Zk) = =0.500 = = 0.429 = = 0.143 P(S2/Zk) = =0.364 = = 0.364 = = 0.545 P(S3/Zk) = =0.091 = = 0.182 = = 0.455 Check =1.000 =1.000 =1.000 iv. Bayes’ Theorem; A B C D E F G H I 1 State Prior Probability P(Zk/Si) Joint Probabilities(P(Si and Zk)) 2 Sn P(Sn) Z1 Z2 Z3 Z1 Z2 Z3 3 S1 0.4 0.7 0.2 0.1 0.28 0.08 0.04 4 S2 0.4 0.6 0.2 0.2 0.24 0.08 0.08 5 S3 0.2 0.2 0.3 0.5 0.04 0.06 0.1 6 Check 1.0 Marginal Probability P(Zk) 0.56 0.22 0.22 7 Posterior Probability P(S1/Zk) 0.500 0.429 0.143 8 P(S2/Zk) 0.364 0.364 0.545 9 P(S3/Zk) 0.091 0.182 0.455 10 Check 1.0 1.0 1.0 b. Probability tree Joint probability; Posterior probability; Question Two a. The return of the three alternatives Alternative I Investment The option for alternative one is to save the whole amount in safe money market fund earning 2% (quarterly). Compounded Amount = P (1+i/n) nt P = principal = 500,000 i=rate = 2% n=Times in a year = 3 t = time or years. = 500,000 (1+0.02) 3*3 =$597,546.28 Interest earned = 597,546.28 -500,000 = $97,546.3 Alternative II Investment Return of Tomato Year 1 Year 2 Year 3 Revenue Dry * P=(10,500*0.6)= 6,300 6,300 6,300 Revenue Wet * P=(13,300*0.4)= 5,320 5,320 5,320 Total 11,620 11,620 11,620 Total income = 11,620*3 – 10,000*3 = $4,860 Return from invested fund in safe money market fund (i=5%): i = 5% Fund available to invest in Safe money market fund = 500,000 – 30,000 = 470,000 Amount at the end of three years = 470,000 (1+0.05) 3 = 544083.8 Income = 544083.8–470,000 = $74,083.8 Total Alternative II return = Tomato income + Safe money market fund income = 4,860 + 74,083.8 = $78,943.8 Alternative III Investment Return from wheat Year 1 Year 2 Year 3 Revenue Dry * P= (7,500*0.6) = 4,500 4,500 4,500 Revenue Wet * P= (6,800*0.4) = 2,720 2,720 2,720 Total 7,220 7,220 7,220 Total income = 7,220*3 – 6,500*3= $2,160 Return from safe money market: i= 5% Fund available to invest in Safe money market fund = 500,000 – 6,500*3 = 480,500 Amount at the end of three years = 480,500 (1+0.05) 3 = 556238 Income earned = $556238–$480,500 = $75,738.8 Total income for alternative III = Wheat Income + Safe money market fund income = 2,160 + 75,738.8 = $77,898.8 b. Preferred Alternative The preferred alternative is ranked according utility derived from each alternative (Dunis, Laws & Naïm, 2003). The utility function is given as W0.5 representing risk averse i.e. fear to take risks. The following shows the satisfaction derived from each alternative. W (U) = (W) 0.5 Where W is total wealth at end of the three-year period; Alternative I = (500000+97,546.28)0.5 = 773.01 Alternative II = (500,000+78,943.8)0.5 = 760.88 Alternative III = (500,000+77,898.8)0.5 = 760.20 Investing in safe money market fund provides the highest utility in maximising wealth (773.01). Also, it has no risk to changes in weather condition. The alternative is suitable for a risk aversive individual (Dunis, Laws & Naïm, 2003). Question Three State of nature Alternatives Dry Mild Wet Barley 500 200 -200 Corn 200 300 -200 Potato 300 100 -100 Wheat -200 100 300 i. Maximin Minimum; -200,-200, -100, -200 Maximum; -100 Gunnedah should choose to plant Potato with condition that it is wet because of maximum of minimum of -100. v. Maximax Maximum; 500, 300, 300,300 Maximum; 500 Gunnedah should choose to plant Barley with probability of being dry because it gives maximum of maximum of 500 compared to the other three alternatives. vi. Minimax Regret matrix table; State of nature ALTERNATIVES Dry Mild Wet Barely 500-500 200-300 -200-300 Corn 200-500 300-300 -200-300 Potato 300-500 100-300 -100-200 Wheat -200-500 100-300 300-300 Regret matrix State of nature ALTERNATIVE Dry Mild Wet Barley 0 -100 -500 Corn -300 0 -500 Potato -200 -200 -400 Wheat -700 -200 0 The minimum values are Barley (wet) -500, Corn (wet)-500, Potato (wet)-400, Wheat (Dry) -700 Maximum= -400 Gunnedah should choose planting potato with likelihood of wet state because it gives maximum regret of -400. vii. Hurwitz alpha criteria The Hurwitz alpha criteria is given by Ii = αXi + (1 - α) Yi Barley: Ii = (0.4)500-(0.6)200= 80 Corn: Ii = (0.4)300-(0.6)200= 0 Potato: Ii = (0.4)300 - (0.6)100 = 60 Wheat: Ii = (0.4)300 - (0.6)200 = 0 Gunnedah should choose planting Barley because it gives the highest value of 80. viii. Principle of insufficient reason STATE OF NATURE ALTERNATIVE Dry Mild Wet Average Barley 500 200 -200 166.67 Corn 200 300 -200 100 Potato 300 100 -100 100 Wheat -200 100 300 66.67 According to the principle of insufficient reason calculation Gunnedah should choose planting barley since it has the highest average of 166.7. Question Four i. Minimum acceptable return level (R min) The loan amount marks the level of minimum acceptable return since the money does not belong to the farmer. Invested fund =$600,000 Loan =$30,000 R min = x100% = 5% ii. 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 (Dunis, Laws & Naïm, 2003). This is shown below; SFRatio = Banana SFRatio = = 1.059 Capsicum SFRatio = = 1.05 Corn SFRatio = = 1.82 Potato SFRatio = = 0.94 Tomato SFRatio = = 1.18 According to Roy’s rule, the alternative with highest ratio is optimum (Hardaker et al, 2015). Corn has the highest ratio of 1.82. iii. Telser’s rule The farmer use telser’s rule to choose crop to plant using the probability of ruin compared to predetermined possibility. The crop with less than predetermined possibility is chosen to provide low risk or high return (DeFusco, Ivanov & Karels, 2009). Alpha (α) values are obtained from z table using K values as shown below; Crop E(Ri) K Α Banana 14 1.06 14.46% Capsicum 12 1.05 14.69% Corn 9 1.82 3.44% Potato 10 0.94 17.36% Tomato 16 1.18 11.90% Potato is the only crop that recorded a higher probability of ruin (17.36) than predetermined value (15%), but the best alternative is corn since it has the lowest (3.44) as compared to the other three that passed Telser’s rule test. iv. Expected return of Corn The expected return for corn is as shown below; SFRatio = 1.818 = = 4 + 5 = 9% The combination of Roy’s and Telser’s rule picks Corn to give optimum value and thus expected return remains at 9% (Cresswell, Burke & Pardo, 2006). Question Five a. Decision basing on E-V Rule Corn EV = (200x0.2) + (400x0.3) + (600x0.4) + (800x0.1) = 480 Potato EV = (200x0.2) + (400x0.4) + (600x0.2) + (800x0.2) = 480 Wheat EV = (200x0.1) + (400x0.3) + (600x0.2) + (800x0.2) = 420 The E-V rule states that the highest value represents the alternative with optimal return (Bierman, Bonini & Hausman, 1991). Therefore, calculation above shows that potato and corn are the best alternative to choose with EV of 480. a. First Stochastic Dominance According to Alexeyev (2000), the formula of first stochastic dominance is given by F(x) = P (A< x) Outcome Corn: (x) Potato: (x) Wheat: (x) 200 40 40 20 400 120 160 120 600 240 120 120 800 80 160 160 Assume the variance for the three crops are; 203.96 for corn, 188.68 for potato and 188.68 for Wheat. The weight of variance of each crop is given by; Corn = = 0.35 Potato = =0.32 Wheat = = 0.32 X= Expected return The second stochastic dominance function is given as; Corn; F(x) =0.35x Potato; F(x) =0.32x Wheat; F(x) =0.32x The second stochastic dominance is derived from the table above as shown below; Outcome Corn: F(x) Potato: F(x) Wheat: F(x) 200 14 12.8 6.4 400 42 51.2 38.4 600 84 38.4 38.4 800 28 51.2 51.2 The CDF’s Graph The second stochastic dominance represented in the graph above shows that wheat and potato had similar behaviour. On the other hand, Corn showed higher variation making it the best alternative (Alexeyev, 2000). Reference: Alexeyev, V. (2000). Quantitative analysis. Honolulu: University Press of the Pacific. Bierman, H., Bonini, C., & Hausman, W. (1991). Quantitative analysis for business decisions. Homewood, Ill. [u.a.]: Irwin. Cresswell, A. M., Burke, G. B., & Pardo, T. (2006). Advancing return on investment, analysis for government IT: a public value framework. Center for Technology in Government, University at Albany, SUNY. DeFusco, R., Ivanov, S., & Karels, G. (2009). The exchange traded funds’ pricing deviation: analysis and forecasts. Journal Of Economics And Finance, 35(2), 181-197. Dunis, C., Laws, J., & Naïm, P. (2003). Applied quantitative methods for trading and investment. Chichester, England: John Wiley. Hardaker, J.B., Lien, G., Anderson, J.R. and Huirne, R.B. (2015). Coping with Risk in Agriculture: Applied Decision Analysis. Third Edition. CABI Publishing, London. Kharasch, E., & Rosow, C. (2013). Assessing the Utility of the Utility Function. Anesthesiology, 119(3), 504-506. Lombardi, M., & Calzolari, G. (2009). Indirect estimation of -stable stochastic volatility models. Computational Statistics & Data Analysis, 53(6), 2298-2308. Pitman, J. (2009). Probability. Beijing: World Publishing Corporation. Ross, S. (2014). Introduction to probability models. Amsterdam ; Boston [etc.]: Academic Press. Read More