The University of Sydney School of Mathematics and Statistics FMAT3888 Projects in Financial Mathematics Semester 2, 2021 Interdisciplinary Project: Portfolio Optimisation with Market Data Below we provide some questions for the interdisciplinary project. You may want to make some adjustment for some parts of the questions, for example, but not limited to, adjusting the numbers in red or consider more periods for the dynamic optimisation, in order to have more interesting results. In that case, please feel free to do so. Setup. In reality each asset class in the spreadsheet provided on canvas, e.g., Dev. Equities and Hedge Funds, may contain several different assets. For the simplicity of analysis and presentation, without loss of generality we assume each asset class behaves like a single asset and admits some price process. We will 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 Fixed Interest (AFI, Asset Class 5) and Dev. Gov. Bonds (DGB, Asset Class 6). Let Si = (Sit)t∈N be the price process for Asset Class i, i = 1, 2, 3, 4, 5, 6. Here time t is in months and thus Sit is the price 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 price of Asset Class i satisfies Sit+1 = S i t · eX i t , t = 0, 1, 2, . . . . (1) Denote Xt = (X 1 t , X 2 t , ..., X 6 t ). Assume X0,X1,X2, . . . are i.i.d., and each admits multivariate normal distribution with mean 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 Sit−1 − 1 = eXit − 1 =⇒ Xit = ln(1 + αit). Note 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 using market data for the two 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+n−1 Sit−1 − 1 = exp ( t+n−1∑ k=t Xik ) − 1. Show that αit,n d = eY i − 1, i = 1, 2, 3, 4, 5, 6 (2) where (Y 1, Y 2, ..., Y 6) admits multivariate normal distribution with mean 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-year) 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. Use 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 two time intervals (A) and (B) from Q1. Remark: Here in the above we use lognormal distribution (instead of normal distribution) to model the annual and two-year returns R(1) and R(2). See (2). The reason is that we would like the return rate to be above −1 (why?). For computational purpose, it might be easier to use normal distribution instead for R(1) and R(2) in Q4 and Q6 later (if you use decide to use exponential utility). (Please check if that is the case or not.) You may do so if that is the case. Approximating lognormal by normal: Recall that ex− 1 ≈ x when x ≈ 0. Hence for Y ∼ N(µ, σ2), if Y ≈ 0 with large probability, i.e., when µ, σ2 ≈ 0 (why?), then with large probability eY − 1 ≈ Y and thus eY − 1 would behaves like a normal random variable for the most of the time. In this case it is reasonable to approximate eY − 1 by a normal random variable. One naive approach for the approximation is to use Y . However, as E[eY − 1] 6= E[Y ] and Var[eY − 1] 6= Var[Y ], it may be better to do moment matching and to approximate using normal distribution with mean E[eY − 1] and variance Var[eY − 1]. The same applies to the multivariate case. 2 Static Portfolio Optimisation Q4. Consider an investor who statically invests all her wealth in these four asset classes for two years. Answer the following questions for both cases where the estimation is based on two sets of market data, i.e., for time intervals (A) and (B) from Q1. (a) Solve the utility maximisation problem: max E[U(R(2)w)] subject to w1 + w2 + w3 + w4 + w5 + w6 = 1, where 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 corresponding to the two data sets. (c) Compare your result from (a) (with data set (B)) with the realised return on her portfolio using the market data for the period from 1/1/2015 to 31/12/2016. Q5. Under the setup of Q4, answer the following questions for both cases where the estimation is based on the time intervals (A) and (B) from Q1. (a) Find the efficient frontier for the market (Cash, DEG, AEQ, EMEQ, AFI, DGB) in the plane (σ, µ) using 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) Find the portfolio with the minimum variance which yields at least 12% for the expected return. To this end, solve the optimisation problem: min wTCw subject to w1µ1 + w2µ2 + w3µ3 + w4µ4 + w5µ5 + w6µ6 ≥ 0.12, w1 + w2 + w3 + w4 + w5 + w6 = 1, where 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 corresponding to the two data sets. (d) Compare your result from (b) (with data set (B)) with the realised return on her portfolio using the market data for the period 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. Consider an investor who invests all her wealth in these four asset classes for two years, during which she will adjust her portfolio weights at the beginning of the second year. For k = 1, 2, denote ξk := (ξk1 , ξ k 2 , ..., ξ k 6 ) the returns of the four asset classes for the k-th year. Note 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 at the beginning of the first year (resp. second year). Then the return of the profolio over the two-year investment period is given by G(w,u) = (1 + ξ1w)(1 + ξ2u)− 1. (why?) Suppose the investor believes that parameters estimated using the data set (B) are valid. Assume short selling is not allowed. Answer the following questions. (a) Solve the utility maximisation problem: max E[U(G(w,u))] subject to w1 + w2 + ...+ w6 = 1, u1 + u2 + ...+ u6 = 1 where U(x) = −e−γx with γ = 1. Note u = u(ξ1) may depend on the realisation of ξ1. (b) Compare your result with that for Q4(a). Q7. Under the setup of Q6, answer the following questions. (a) Solve the portfolio optimisation problem: min Var[G(w,u)] subject to E[G(w,u)] ≥ 0.12, w1 + w2 + ...+ w6 = 1, u1 + u2 + ...+ u6 = 1. Note here the control u = u(ξ1) may depend on the realisation of ξ1. (b) Compare your result with that for Q5(b) (b) Comment on the differences/similarities of your results from Q6 and Q7. 3