Disclaimer Although the final examination is very similar in terms of both content and structure to this mock examination, the solutions provided here are guides only and they do not include all steps and thus they are not completed solutions. You will need to provide all steps during the examination in order to receive full marks. Background A client is seeking your advice on the composition of his portfolio, which consists of two firms, namely John Group and Alice Partner. After some preliminary investigations, your team has discovered that productivity of both firms can be written mathematically as: 1 1 y (K, L) = 2K 2 L 2 (1) where K and L denote capital and labour inputs, respectively. Interestingly, both firms also share the same budget constraint, namely: K + L = 1. (2) Furthermore, let rJ and rA denote the returns of John Group and Alice Partner, respectively with J and A denote their respective returns. The estimated returns and variances are functions of their productivities based on a sample size of 20 observations. Specifically, J = 2 1 + 3(y 1) 3 2 (3) and 1 A = 1 + 2(y 1) 2 (4) with 1 J = 2y + (y 1) 2 1 (5) A = (y 1)2 + 2 (6) and 1 where J and A denote the estimated returns of John Group and Alice Partner, respectively. Similarly, J and A denote the estimated standard deviations of John Group and Alice Partner, respectively. Question 1. Show that the production function as defined in equation (2) can be expressed as 1 1 y (L) = 2 (1 L) 2 L 2 (7) under the budget constraint. Solution: K + L = 1 K = 1 L, substitute this into the production function gives: 1 1 y =2K 2 L 2 1 1 =2 (1 L) 2 L 2 Question 2. Derive the First Order Necessary condition to maximize production function as defined in equation (7). Solution: The First Order Necessary Condition is 1 (1 L) 2 dy =0 dL 1 1 1 L 2 + (1 L) 2 L 2 =0 Question 3. what is the optimal mix of inputs and what is the optimal level of output (or productivity) based on the condition as derived in Question 2? 2 Solution: 1 1 dy 21 1 2 + (1 L) 2 L 2 =0 = (1 L) L dL L=L L =(1 L ) 1 L = 2 K =1 L = 1 2 1 1 y(K, L) =2 2 2 =1 Question 4. What are the expected returns and variances of both firms under the assumption of rationality? That is, assuming both firms will produce at the optimal level of productivity. Solution: 1 2 J = 1 J = A = 1 A = 2 Question 5. Under the assumptions of normality and rationality, what is the probability to observe negative returns from John Group? What is the expected loss at the 5% level? What can you advise based on these predictions? Solution: Let rJ denotes the return for John Group. \u0012 \u0013 rJ J 1 P (rJ 0) =P J 2 \u0012 \u0013 rJ J 1 =1 P < J 2 =1 0.6915 =0.3085. 3 Let x be the threshold such that P (rJ x) = 0.05, therefore \u0012 P rJ J x J J J \u0013 = 0.05 Note that for a standard normal distribution P (X 1.64) = 0.9495 and P (X 1.65) = 0.9505, therefore P (X 1.64) = 0.0505 and P (X 1.65) = 0.0495. Hence 1.65 x J 1.64 J 1.15 x 1.14. So the expected loss at the 5% level is between 1.14% and 1.15%. Question 6. Under the assumptions of normality and rationality, what is the probability to observe negative returns from Alice Partner? What is the expected loss at the 5% level? What can you advise based on these predictions? Solution: Similar to the previous answer, the loss at 5% level is x such that x A 1.64 A x1 1.64 1.65 2 2.3 x 2.28 1.65 So the expected loss at the 5% level is between 2.28% and 2.3%. Question 7. Are the returns of both firms statistically different from zero under the assumption of normality and rationality? Please state the hypotheses clearly and show all steps in oyur testing procedure as well as the specific distribution you used. 4 Solution: The hypotheses for John Group are: H0 :J = 0 H1 :J 6= 0. The test statistic is J J Zr = N J 20 = 2 = 5 2.236. The hypotheses for Alice Partner are: H0 :A = 0 H1 :A 6= 0. The test statistic is A A Zr = N A 20 = 2 = 5 2.236. From the table, the critical value for a t(19) is 2.093 at the 5% level, bear in mind that this is a two sided test. Since |Zr | > |Zcv |, we reject the null that the average return is not statistically different from zero. Question 8. Are the returns statistically different between the two firms under the d ( assumptions of normality, rationality and Cov J , A ) = 1? Please 5 state the hypotheses clearly and show all steps in your testing procedure as well as the specific distribution you used. Solution: The hypotheses in this case are: H0 :J = A H1 :J 6= A . The test statistic is N (J A ) Zr = p 2 A + 2 2Cov ( J , A ) J 0.5 20 = 1+4+2 r 5 = 7 0.8452. The test statistic is unlikely to exceed the critical value at any reasonable level. Therefore, we do not have enough evidence to reject the null. d (rJ , rA ) = 1, what are the optimal Question 9. Under the assumption that Cov weights for the two assets, namely, John Group and Alice Partner, if your client were to invest in the two firms? What is the expected profit/loss? Solution: The expected return of the portfolio is rp (w1 , w2 ) = w1 J + w2 A where w1 and w2 are the weights for John Group and Alice Partner, respectively, with w2 = 1 w1 . The risk of the portfolio is d J , rA ). p2 (w1 , w2 ) = w12 J2 + w22 A2 + 2w1 w2 Cov(r By solving the optimisation of L (w1 , w2 ) = rp (w1 , w2 ) p2 (w1 , w2 ) we 6 get w1 = 0.6786 w2 = 0.3214. rp = 0.6607 p2 = 0.4375. with Question 10. Assuming the two firms are no longer rational and produce at the level inconsistent with the optimal output as derived previously. Specifically, assuming that both firms are now producing at y = 2. What are the optimal weights for the two assets in this case? What is the expected profit/loss? Solution: For y = 2 7 2 J = 4 J = A = 3 A = 3. By following the same procedure as the previous question, we get w1 = 0.3796 w2 = 0.6204 rp = 3.1898 p2 = 5.2986. with Although the expected return is a higher when firms do not produce at the optimal level, the risk associated with the portfolio is also higher. 7