Essay About 00.3Pass And Function F Of N Variables

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Homework 2 MF6401. What is the meaning of Lagrange Multiplier? Ans The Lagrange multipliers method is one of methods for solving constrained extrema problems. Assume that we have a function f of n variables {y = f (x1,, x n)} and we would like to find the value of x1,, xn that will maximize or minimize this function subject to constraint g (x1,…, xn). The Lagrange multipliers method is based on setting up the new function (the Lagrange function)                        L(x1,…, x n,λ) = f(x1,…, x n)  +λg (x1,…, x n) where λ is an additional variable called the Lagrange multiplier. From the Lagrange equation, the conditions for a critical point are                                        L´ x1 = f´x1 +λg´ x1                                        L´ x2 = f´x2 +λg´ x2                                                …                                        L´ xn = f´xn +λg´ xn                                        L´ λ = g (x1,…, xn)Rearrange the first n equations and set it equal to zero as                                                                        λ or    = λ[pic 3][pic 4]The Lagrange multipliers can interpret that the value of optimal f will change λ unit if the exogenous variable (g(x)) change 1 unit. This show the marginal effect of changing in exogenous variable affect to the value of optimal objective variable (f). If the Lagrange multipliers are too high, the value of f will change more than the constraint which the Lagrange multiplier is low.3. What is the optimum point of investment? If the utility function is quadratic utility function and

ReturnStandard Deviation[pic 5]Investment 10.150.2-0.2Investment 20.080.1Begin with the efficient Frontier (Calculate from excel), we separate weight of investment into 11 groups. PortfolioWeight 1Weight 2Expected Return(%)S.D.(port%)1018120.10.98.70.77830.20.89.40.67240.30.710.10.68250.40.610.80.80860.50.511.51.0570.60.412.21.40880.70.312.91.88290.80.213.62.472100.90.114.33.1781110154[pic 6]After that we create the quadratic utility function as and assume that a = 0 (it doesn’t play any role). .However, we have to create the conditions of b and c to support the theory.[pic 7][pic 8]ConditionReasonb>0 and c<0Utility has positive relationship with return but negative relationship with S.D. (risk).b+ 2cRp > 0Positive marginal Utility ( first diff > 0)2c < 0 Relative Risk aversion (second diff < 0 )We find that the optimum point of investment depend on preference of investor (b and c). If you change value of b and c, the optimum point will change as table below b = 0.3 and c = -0.7PortfolioExpected ReturnS.D.Expected Utilityb+2c*RpConditionCondition10.080.010.019450.188Passa0Assume20.0870.007780.020759330.1782Passb > 00.3Pass30.0940.006720.0219831890.1684Passc < 0-0.7Pass40.1010.006820.0231267410.1586Passb+2c*Rp > 0Table 50.1080.008080.02418950.1488Pass2c < 0-1.4Pass60.1150.01050.0251653250.139Pass70.1220.014080.0260424280.1292Pass80.1290.018820.0268033650.1194Pass90.1360.024720.0274250450.1096Pass100.1430.031780.0278787220.0998Pass110.150.040.028130.09Passb = 0.3 and c = -0.6PortfolioExpected ReturnS.D.Expected Utilityb+2c*RpCondition10.080.010.02010.204Pass20.0870.007780.0215222830.1956Pass30.0940.006720.0228713050.1872Pass40.1010.006820.0241514930.1788Pass50.1080.008080.0253624280.1704Pass60.1150.01050.026498850.162Pass70.1220.014080.0275506520.1536Pass80.1290.018820.0285028850.1452Pass90.1360.024720.0293357530.1368Pass100.1430.031780.0300246190.1284Pass110.150.040.030540.12Pass