FMAT3888 Initiatives in Financial Mathematics Semester 2, 2021 Interdisciplinary Mission: Portfolio Optimisation with Market Data Below we provide some questions for the interdisciplinary mission. You may possibly well possibly also are wanting to assemble some adjustment for some formulation of the questions, for instance, but not restricted to, adjusting the numbers in pink or maintain in thoughts more classes for the dynamic optimisation, in divulge to love more animated results. If that’s the case, please truly feel free to pause so. Setup. Truly each and each asset class in the spreadsheet provided on canvas, e.g., Dev. Equities and Hedge Funds, may possibly well also simply hang plenty of diversified sources. For the simplicity of evaluation and presentation, without loss of generality we assume each and each asset class behaves admire a single asset and admits some designate process. We are succesful of work on six asset classes, including Cash (Asset Class 1), Dev. Equities (DEQ, Asset Class 2), Australian Equities (AEQ, Asset Class 3), Emerging Market Equities (EMEQ, Asset Class 4) Australian Mounted Hobby (AFI, Asset Class 5) and Dev. Gov. Bonds (DGB, Asset Class 6). Let Si = (Sit down)t∈N be the designate process for Asset Class i, i = 1, 2, 3, 4, 5, 6. Here time t is in months and thus Sit down is the designate of Asset Class i at the end of the t-th month. For i = 1, 2, 3, 4, 5, 6, assume the dynamics of the designate of Asset Class i satisfies Sit down+1 = S i t · eX i t , t = 0, 1, 2, . . . . (1) DenoteXt = (X 1 t , X 2 t , …, X 6 t ). AssumeX0,X1,X2, . . . are i.i.d., and each and each admits multivariate same outdated distribution with point out a = (a1, a2, …, a6) ∈ R6 and covariance matrix B = (bij)i,j=1,2,3,4,5,6. For i = 1, 2, 3, 4, 5, 6, denote αit the monthly return of Asset Class i in the t-th month. By (1), αit = Sit down Sit down−1 − 1 = eXit − 1 =⇒ Xit = ln(1 + αit). Expose that the realised monthly returns αit for these asset classes since January 2001 to April 2021 are provided in the spreadsheet. 1 Parameter Estimation Q1. Estimate the parameters ai and bij for i, j = 1, 2, 3, 4, 5, 6 utilizing market data for the 2 time intervals: (A) from 1/1/2007 to 31/12/2010, (B) from 1/1/2011 to 31/12/2014. Q2. For n = 1, 2, . . . , by (1) the return for Asset Class i from the beginning of the t-th month to the beginning of the (t+ n)-th month is given by αit,n = Sit down+n−1 Sit down−1 − 1 = exp ( t+n−1∑ k=t Xik ) − 1. Gift that αit,n d = eY i − 1, i = 1, 2, 3, 4, 5, 6 (2) the set (Y 1, Y 2, …, Y 6) admits multivariate same outdated distribution with point out na and covariance matrix nB. Q3. Let the random vector R(1) := (R (1) 1 , R (1) 2 , …, R (1) 6 ) (resp. R (2) := (R (2) 1 , R (2) 2 , …, R (2) 6 )) model the joint annual (resp. two-twelve months) returns for six Asset Classes. For k = 1, 2 denote µ (k) i := E [ R (k) i ] , c (k) ij := Cov ( R (k) i , R (k) j ) , ρ (k) ij := c (k) ij√ c (k) ii √ c (k) jj , i, j = 1, 2, 3, 4, 5, 6. Utilize the results in Q1 and Q2 to compute/estimate µ (k) i , c (k) ij , ρ (k) ij for i, j = 1, 2, 3, 4, 5, 6 and k = 1, 2 for the 2 time intervals (A) and (B) from Q1. Comment: Here in the above we use lognormal distribution (as an substitute of same outdated distribution) to model the annual and two-twelve months returns R(1) and R(2). Gaze (2). The motive being that we may possibly admire the return charge to be above −1 (why?). For computational cause, it may possibly possibly possibly well also very neatly be less complicated to use same outdated distribution as an substitute for R(1) and R(2) in Q4 and Q6 later (must you use judge to use exponential utility). (Please check if right here’s so or not.) You may possibly well possibly also simply pause so if right here’s so. Approximating lognormal by same outdated: Recall that ex− 1 ≈ x when x ≈ 0. Hence for Y ∼ N(µ, σ2), if Y ≈ 0 with huge likelihood, i.e., when µ, σ2 ≈ 0 (why?), then with huge likelihood eY − 1 ≈ Y and thus eY − 1 would behaves admire a same outdated random variable for the as a rule. In this case it’s life like to approximate eY − 1 by a same outdated random variable. One naive methodology for the approximation is to use Y . Alternatively, as E[eY − 1] ̸= E[Y ] and Var[eY − 1] ̸= Var[Y ], it may possibly possibly possibly well also very neatly be better to pause 2d matching and to approximate utilizing same outdated distribution with point out E[eY − 1] and variance Var[eY − 1]. The same applies to the multivariate case. 2 Static Portfolio Optimisation Q4. Steal into myth an investor who statically invests all her wealth in these six asset classes for 2 years. Acknowledge the following questions for each and each cases the set the estimation is in accordance to 2 sets of market data, i.e., for time intervals (A) and (B) from Q1. (a) Resolve the utility maximisation anguish: max E[U(R(2)w)] arena to w1 + w2 + w3 + w4 + w5 + w6 = 1, the set w = (w1, w2, w3, w4, w5, w6) T is the vector of weights, and U(x) = −e−γx with γ = 1. (b) Comment on the differences of your results comparable to the 2 data sets. (c) Study your consequence from (a) (with data house (B)) with the realised return on her portfolio utilizing the market data for the duration from 1/1/2015 to 31/12/2016. Q5. Below the setup of Q4, reply the following questions for each and each cases the set the estimation is in accordance to the time intervals (A) and (B) from Q1. (a) Acquire the ambiance pleasant frontier for the market (Cash, DEG, AEQ, EMEQ, AFI, DGB) in the airplane (σ, µ) utilizing the estimated parameters µi := µ (2) i , cij := c (2) ij , ρij := ρ (2) ij for i, j = 1, 2, 3, 4, 5, 6. (b) Acquire the portfolio with the minimal variance which yields a minimal of 12% for the expected return. To this end, solve the optimisation anguish: min wTCw arena to w1µ1 + w2µ2 + w3µ3 + w4µ4 + w5µ5 + w6µ6 ≥ 0.12, w1 + w2 + w3 + w4 + w5 + w6 = 1, the set w = (w1, w2, w3, w4, w5, w6) T is the vector of weights and C = [cij ] is the covariance matrix for R (2). 2 (c) Comment on the differences of your results comparable to the 2 data sets. (d) Study your consequence from (b) (with data house (B)) with the realised return on her portfolio utilizing the market data for the duration from 1/1/2015 to 31/12/2016. (e) Comment on the differences/similarities of your results from Q4 and Q5. 3 Dynamic Portfolio Optimisation Q6. Steal into myth an investor who invests all her wealth in these six asset classes for 2 years, in which she’s going to adjust her portfolio weights initially of the 2d twelve months. For k = 1, 2, denote ξk := (ξk1 , ξ k 2 , …, ξ k 6 ) the returns of the six asset classes for the k-th twelve months. Expose that ξ1 and ξ2 are i.i.d. copies of R(1). Let w = (w1, w2, …, w6) T (resp u = (u1, u2, …, u6) T ) be the portfolio weights initially of the predominant twelve months (resp. 2d twelve months). Then the return of the profolio over the 2-twelve months funding duration is given by G(w,u) = (1 + ξ1w)(1 + ξ2u)− 1. (why?) Assert the investor believes that parameters estimated utilizing the info house (B) are decent. Mediate short selling just isn’t allowed. Acknowledge the following questions. (a) Resolve the utility maximisation anguish: max E[U(G(w,u))] arena to w1 + w2 + …+ w6 = 1, u1 + u2 + …+ u6 = 1 the set U(x) = −e−γx with γ = 1. Expose u = u(ξ1) may possibly well also simply rely on the realisation of ξ1. (b) Study your consequence with that for Q4(a). Q7. Below the setup of Q6, reply the following questions. (a) Resolve the portfolio optimisation anguish: min Var[G(w,u)] arena to E[G(w,u)] ≥ 0.12, w1 + w2 + …+ w6 = 1, u1 + u2 + …+ u6 = 1. Expose right here the maintain watch over u = u(ξ1) may possibly well also simply rely on the realisation of ξ1. (b) Study your consequence with that for Q5(b) (b) Comment on the differences/similarities of your results from Q6 and Q7.
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