Please answer b and c.
Homework 1471: Quantitative / Optimization Methods in Finance (Due date: 17/12/21) Final Project (Part I) Surname, Name: Student ID: Instructions: • You can do this assignment in groups of at most 4. Please submit no more than one submission per group. • Please submit your final project as a report (as a word or pdf file). And try to explain and justify everything that you have done, clearly pointing your assumptions and answers. Do not submit just excel files or output of your software. • Please use Learn@WU Assignments tool to submit your work. Question 1 (15 points) Suppose today is November 30, 2052. A pension fund will need to cover the following stream of liabilities over the subsequent four years (in million dollars): To cover these liabilities, the pension fund intends to use a portfolio comprised of the following 14 US treasury notes: (a) Compute the dirty (full) price of each of the above 14 bonds. For consistency, assume today is November 30, 2052. (b) Formulate a linear programming model to find the lowest-cost dedicated portfolio that covers the stream of liabilities. To eliminate the possibility of any interest risk, assume a 0% reinvestment rate on cash balances carried from one date to the next. Assume no short sales are allowed. What is the cost of your portfolio? What is the composition of your portfolio? (c) Use the linear programming sensitivity information from part (b) to determine the term structure of interest rates implied by the portfolio. 1 2 Question 2 (35 points) Let y denote the current year. A municipality sends you the following liability stream (in million dollars): (a) Determine the current term structure of treasury rates, and find the present value, dollar duration, and dollar convexity of the stream of liabilities. Please explain the main steps (interest rates, discount factors, compounding, etc.) followed in your calculations. You can find current data on numerous websites such as http://finance.yahoo.com/bonds http://fixedincome.fidelity.com/fi/FILanding (b) Identify at least 30 fixed-income assets that are suitable for a dedicated portfolio. Use assets that are considered risk-free, e.g., US government noncallable treasury bonds, treasury bills, or treasury notes. Display a succinct summary of the main characteristics of the bonds you chose (prices, coupon rates, maturity dates). (c) Formulate a linear programming model to find the lowest-cost dedicated portfolio that covers the stream of liabilities. To eliminate the possibility of any interest risk, assume a 0% reinvestment rate on cash balances carried from one date to the next. Assume no short sales are allowed. What is the cost of your portfolio? What is the composition of your portfolio? (d) Use the linear programming sensitivity information to determine the term structure of interest rates implied by the portfolio. Use a plot to compare it with the current term structure of treasury rates. (e) Formulate a linear programming model to find the lowest-cost portfolio that matches the present value, dollar duration, and dollar convexity of the stream of liabilities. Assume no short sales are allowed. What is the cost of your portfolio? How much would you save by using this immunization strategy instead of dedication? Is your portfolio immunized against non-parallel shifts in the term structure? Explain why or why not. (f) Combine a cash matching strategy for the liabilities during the first three years and an immunization strategy based on present value, duration and convexity for the liabilities during the last five years. Compare the cost of this portfolio with the cost of the two previous portfolios. (g) The municipality would like you to make a second bid: What is your best dedicated portfolio of risk-free bonds you can create if short sales are allowed? Did you find arbitrage opportunities? Did you take into consideration the bid–ask spread? Did you set limits on the transaction amounts? Discuss the practical feasibility of your solution. http://finance.yahoo.com/bonds http://fixedincome.fidelity.com/fi/FILanding Sheet1 Question 1 a)Coupon MaturityClean PriceAccrued interest Dirty priceDays since last payment=0.8328767123 3.50%5/31/53101.5631.480103.043Days between payments 0.50%5/31/53100.1880.209100.397 2.00%11/30/53101.7460.847102.593 0.25%11/30/53100.0780.104100.182 2.25%5/31/54102.9410.965103.906 0.25%5/31/54100.0230.104100.127 2.125%11/30/54103.6560.917104.573 0.25%11/30/54100.0160.104100.120 2.215%5/31/55104.4610.964105.425 1.375%11/30/55103.0310.590103.621 3.25%5/31/56109.7381.485111.223 1.75%5/31/56104.5700.762105.332 2.75%11/30/56108.8791.247110.126 0.875%11/30/56101.5160.370101.886 b)year 0.511.522.533.54price B1101.75101.563 B2100.25100.188 B31101101.746 B40.125100.125100.078 B51.1251.125101.125102.941 B60.1250.125100.125100.023 B71.06251.06251.0625101.0625103.656 B80.1250.1250.125100.125100.016 B91.06251.06251.06251.0625101.0625104.461 B100.68750.68750.68750.68750.6875100.6875103.031 B111.6251.6251.6251.6251.6251.625101.625109.738 B120.8750.8750.8750.8750.8750.875100.875104.57 B131.3751.3751.3751.3751.3751.3751.375101.375108.879 B140.43750.43750.43750.43750.43750.43750.4375100.4375101.516 First Constraint: 101,75x1 + 100,25x2 + 1x3 + 0,125x4 + 1,125x5 + 0,125x6 + 1,0625x7 0,125x8 + 1,0625x9 + 0,6875x10 + 1,625x11 + 0,875x12+ 1,375x13 + 0,4375x14 - s1 = 12000000 Second Constraint: 101x3 + 100,125x4 + 1,125x5 + 0,125x6 + 1,0625x7 0,125x8 + 1,0625x9 + 0,6875x10 + 1,625x11 + 0,875x12+ 1,375x13 + 0,4375x14 + s1 - s2 = 10000000 . . . As treasury notes pay interest on a semi-annual basis, we assume that coupons are paid on 1st January and 1st July. The full price of a bond is the sum of the clean price and the accrued interest. Accrued interest is the total interest accumulated on a bond since its last coupon date. The dirty price on 30th November 2052 implies that the last coupon is paid on 1st July. The calculation of accrued interest includes the last coupon date but excludes the value date. Therefore, 152 days have passed since the last coupon payment. Accrued interest = Face Value x (coupon/number of payments per year) x (days since last payment/days between payments) 5 Quadratic Programming: Theory and Algorithms 5.1 Quadratic Programming A quadratic program is an optimization problem whose objective is to minimize or maximize a quadratic function subject to a finite set of linear equality and inequality constraints. By flipping signs if necessary, a quadratic program can be written in the generic form: min x 1 2x TQx+ cTx s.t. Ax = b Dx ≥ d (5.1) for some vectors and matrices c ∈ Rn, b ∈ Rm, d ∈ Rp, A ∈ Rm×n, D ∈ Rp×n, Q ∈ Rn×n. As observed in Chapter 1 we may assume that Q is a symmetric matrix. The term quadratic programming model is also used to refer to a quadratic program. We will use these terms interchangeably throughout the book. Quadratic programming models arise in a variety of practical contexts. The seminal mean–variance model of Markowitz and most of its variants for portfolio selection are quadratic programs as we illustrate in Example 5.1 below and discuss in full detail in Chapter 6. The popular ordinary least-squares and lasso estimation procedures in linear regression are also quadratic programs. Quadratic programs are also often solved as subproblems in the solution of more general nonlinear optimization problems. Observe that the constraint set in (5.1) is convex since it is a system of linear inequalities. Furthermore, the objective function of (5.1) is convex when Q is a positive semidefinite matrix. Throughout this chapter we assume that Q is symmetric and positive semidefinite. Therefore problem (5.1) is a convex program. A quadratic programming model is in standard form if it is written as follows: min x 1 2x TQx+ cTx s.t. Ax = b x ≥ 0. (5.2) Example 5.1 (Asset allocation) Assume the one-year returns of the asset classes large stocks, small stocks, and bonds have the following correlations and standard deviations: 72 Quadratic Programming: Theory and Algorithms Large Small Bonds Standard deviation Large 1 0.6 0.2 0.12 Small 0.6 1 0.5 0.20 Bonds 0.2 0.5 1 0.05 Determine the asset allocation of minimum risk, that is, find a portfolio com- prised of these three asset classes whose return has the lowest standard deviation. Assume the portfolio can only hold long positions in each of the asset classes. This problem can be formulated as a quadratic programming model. To that end, first construct the covariance matrix V of asset returns: this is the matrix whose (i, j) entry is the covariance of asset i and asset j; that is, ρij · σi · σj . Using matrix notation and ‘◦’ to denote the componentwise product of matrices, the covariance matrix can be computed as V = ⎡⎣ 1 0.6 0.20.6 1 0.5 0.2 0.5 1 ⎤⎦ ◦ ⎡⎣0.120.20 0.05 ⎤⎦ [0.12 0.20 0.05] = ⎡⎣ 1 0.6 0.20.6 1 0.5 0.2 0.5 1 ⎤⎦ ◦ ⎡⎣0.0144 0.024 0.0060.024 0.04 0.01 0.006 0.01 0.0025 ⎤⎦ = ⎡⎣0.0144 0.0144 0.00120.0144 0.04 0.005 0.0012 0.005 0.0025 ⎤⎦ . We are now ready to describe the quadratic programming formulation for the above asset allocation problem. (A more detailed discussion is given in Chapter 6.) Quadratic programming model for asset allocation Variables: xi: percentage of the portfolio invested in asset i for i = 1, 2, 3. Objective (minimize the variance of the portfolio return): min x xTVx = min x1,x2,x3 ( 0.0144x21 + 0.04x 2 2 + 0.0025x 2 3 + 0.0288x1x2 + 0.0024x1x3 + 0.01x2x3 ) Constraints: x1 + x2 + x3 = 1 (percentages add up to one) x1, x2, x3 ≥ 0 (long-only positions). Observe that even in this small example the quadratic objective is much more concise and easier to write using matrix notation. We now discuss the special case of a convex quadratic program without con- straints. As Example 5.3 below illustrates, this kind of model arises naturally in the ordinary least-squares procedure. 5.1 Quadratic Programming 73 Consider a quadratic program without constraints: min x 1 2x TQx+ cTx. (5.3) The optimality conditions in this case are as follows. Theorem 5.2 Let c ∈ Rn, Q ∈ Rn×n and assume Q is symmetric and positive semidefinite. If (5.3) is bounded, then it attains its minimum. Furthermore, a point x ∈ Rn is an optimal solution to (5.3) if and only if Qx+ c = 0. (5.4) When Q is positive definite, the problem (5.3) has the unique minimizer x = −Q−1c. When Q is positive semidefinite but not positive definite, the matrix Q is singular and the problem (5.3) is either unbounded or has multiple solutions. Example 5.3 (Ordinary least squares) Assume (xi, yi), for i = 1, .