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CO 372: Portfolio Optimization Models
创建日期：2024-11-06 14:25:09
所属分类：金融Finance
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Problem Set 4

CO 372: Portfolio Optimization Models

Fall 2024

Problem Set 4

Handed out: 2024-10-23.

Due:  Fri, 2024-11-01, 4pm EDT, on Crowdmark.  Papers must be handed in on-line using the labeled dropbox on Crowdmark. The Crowdmark link will be sent out on the same day this is posted.  Each question is handed in as a separate upload.  You can either prepare your solutions electronically using, e.g., LaTeX, or else you can hand-write them and submit a scan.  In the latter case, please take care that the scan is of good quality with a white background.

Your papers may be handed in up to 24 hours late, in which case there is a 10% late penalty. Please use the late-paper dropbox on Crowdmark if you are handing in a late paper.  You may hand in some questions on time and others late.  In this case, use the on-time dropbox for the on-time questions and the late dropbox for the late questions.  However, you may not split the parts of a question (i.e., (a), (b), etc.)  between the two dropboxes.

Collaboration policy. This problem set is a solo effort.  Students are allowed to discuss question with each other in general terms including helping each other on Piazza. However, no student should share an entire solution to a question, nor should any student hand in work that entirely represents someone else’s effort.

1.  We are given a Pr3′ instance with all the usual assumptions, that is, the version of Pr3 in which there are n − 1 risky securities and a single risk-free security available whose return is rf . Assume there is a single wealth constraint.

A formula was given in lecture for a value tm  > 0 (the “market point”) such that the portfolio [tmH −1 (r(¯) − rf e);0] is optimal for Pr3′ for the choice t := tm  in Pr3′ .

(a) Turn this around: given an arbitrary tm  > 0, positive definite matrix H ∈ Sn−1, and vector r(¯) ∈ Rn−1, find a formula rf  so that the optimizer for Pr3′  is [tmH −1 (r(¯)−rf e);0].

Explain why the formula is valid (and in particular, does not involve division by 0) because of all the assumptions made.

(b) Does the formula extend to tm  = 0?  Explain.

(c) Does the rf  that emerges from the formula in (a) satisfy the inequality α0  > rf , which was assumed in lecture? Explain.

2.  Consider Pr3′ .  As usual, assume t ≥ 0 and H is positive definite.  Suppose (contrary to the usual assumption) that α0  < rf , where α0 is the return of the min-risk portfolio that uses only risky investments.

(a) The formulas from lecture show that the optimal portfolio has the form. where x(ˆ) = tx′  and xn   = 1 − teT x′ , where x′  ∈ Rn−1  depends on H , r(¯), and rf , and

that this optimal portfolio never short-sells the risk-free security. Explain.

(b) Show that there must be at least one risky security i ∈ {1, . . . , n − 1} such that every optimal portfolio (for all t ≥ 0) invests an amount ≤ 0 in that security.

3.  Pr1′ , which was presented in lecture, may be reformulated as: minxTHx/2 subject to Ax = b, r(¯)Tx − q = rp , q ≥ 0.  In other words, a slack variable q has been introduced, and a sign constraint is imposed on the slack variable. Call this formulation Pr1′ S (S

for “slack”). Assume H ∈ Sn , A ∈ Rp ×n , b ∈ Rp , r(¯) ∈ Rn.

(a) Write down the KKT conditions for Pr1′ S.  Note that there are no general-form. inequality constraints in Pr1′ S, but there is a sign constraint. Label the four parts.

Caution: The gradient equation of Pr1′ S involves vectors of length n + 1 rather than n because there are n+1 problem variables [x;q].  There are p+1 equality constraints in these n + 1 variables. Using block matrix notation, rewrite the equality constraints in terms of a (p+1)×(n+1) matrix before attempting to write the gradient equation. Similarly, the objective function should be rewritten as [corrected on 2024-10-27]:

in order for the LHS of the gradient equation to come out correctly.

(b) Argue that from any solution for the KKT conditions of Pr1′ (which were presented in lecture), a solution to the KKT conditions of Pr1′ S may be obtained.

(c) Argue that from any solution for the KKT conditions of Pr1′ S, a solution to the KKT conditions of Pr1′ may be obtained.

4.  (a) Write down the KKT conditions for the capital asset pricing model that appears on Slide 5 of M22.  Clearly identify four parts.  Note that the gradient equation involves the matrix H+ , not H alone.

(b) Make all the assumptions in M20–M22.  Show that if t > tm , where tm  is defined in Slide 4 of M21,then xn  = 0 in the optimizer to the problem of Slide 5 of M22. Suggested approach for a proof by contradiction:  If xn   >  0, then the KKT complementarity condition requires a certain multiplier to be 0.  If this multiplier is 0, then the KKT conditions of this problem reduce to the KKT conditions for the problem of Slide 5 in M20. But for t > tm , that optimizer to that problem is not feasible for the problem at hand.

5. The function fmincon in Matlab can solve general constrained minimization problems (although it is not always the best method). Consider the problem of maximizing the Sharpe ratio given H , r(¯), and rf , assuming a single budget constraint. This can be set up as an optimization problem.

Write the following functions in Matlab. The first takes these three values H , r(¯), and rf , and then invokes fmincon to maximize the Sharpe ratio. There is one constraint, namely, the budget constraint. It returns the optimal portfolio. Here is the header:

function  [x,errflag]  = maximize_sharpe(H,  rbar,  rf)

%  Return  the  optimizer  of  the  Sharpe  ratio  given  data  H,  rbar,  and %  rf .    If  fmincon  fails,  the  errflag=true,  else  errflag=false .

Here are some tips on fmincon.   Type  doc  fmincon  for  a  description of its input and output arguments. You need to retrieve the first three output arguments so that you can examine  exitflag in order to determine how to set  errflag.   This  is  a minimization routine, so to maximize a function, give its negative as the objective. The first argument is a function handle.  A function handle in Matlab has the form. @(x)(), where x is the argument of the function (not a variable in the surrounding code that you write).   For  example,  if  you  wanted  to minimize the function x2  − ax, where a is a coefficient, you would say

a  =  .  .  . ;

my_fhandle  =  @(x)(x^2  -  a*x);

[optx,fval,exitflag]  =  fmincon(my_fhandle,  . . .)

For maximizing the  Sharpe  ratio,  there  are  no  inequality  constraints  and  a  single equality constraint. Use the vector (1/n,..., 1/n) as the starting guess.

Write a second function that plots the efficient frontier by repeatedly maximizing the Sharpe ratio.  It should take a vector of rf ’s from the user.  For each rf , it invokes maximize_sharpe. If maximize_sharpe returns an error, then no point is plotted. In the plot statement, use the linestyle. ’-*g’, which displays a green asterisk for each point plotted and connects the asterisks with a green line. Here is the header:

function plot_efficient_frontier_sh(H,  rbar,  vector_rf)

%  Invoke maximize_sharpe  for  each  entry  of  vector_rf  (same  H  and  rbar)

%  each  time .    For  each  computed  solution  x,  work  out  the  risk

%  and  expected  return .    Plot  the  risk  and  expected  return  for

%  all  solutions  as  a  green  curve .    If  maximize_sharpe  indicates %  an  error,  then no point  is plotted .

Next, download ps4_data. mat from the website to obtain H and r(¯) .  Call your function from PS3 to computerp,min, that is, the return of themin-risk portfolio of risky stocks. I got 1.0298 from my calculation.

Next, call plot_efficient_frontier_sh(H,  rbar,  vector_rf) using for argument vector_rf: linspace(.9,1.02,30), that is, 30 equally spaced points from 0.9 to 1.02.

Try again with 30 equally spaced points from 0.9 to 1.05; a strange plot will emerge.

Hand in: listings of both functions, the requested plots (1.02 case and 1.05 case), and a trace of how you computed rp,min. Then write 1–2 sentences to answer the following question: Why did the code work poorly for the 1.05 case?

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PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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