APS 1002 Financial Engineering May 25 2009Name_______________________________________Student number_______________________________Exam 1Instructions: Closed notes and book. Non-programmable calculator permitted.Problem 1 (30 points, 10 points per part)Part 1: Suppose $1 dollar were invested in 1776 at 6.6% interest compounded

yearly. Approximately how much would that investment be worth today? Choose one:(a) $500,000 (b) $1,750,000 (c) 1,000,000 (d) $2,000,000.

Part 2: Consider two 5-year bonds (with face values of $100 for both): one has a9% coupon and currently sells for 101.00, the other has a coupon of 7% and currentlysells for 93.20. What is the price of a synthetic 5 year 0-coupon bond? Choose one:(a) $50.75 (b) $65.90 (c) $68.00 (d) $57.85.

Part 3: What is the effective annual rate for 18% compounded quarterly? Chooseone: (a) 19.56% (b) 19.77% (c) 18.99% (d) 19.25%?

Problem 2 (40 points, 10 points for each part)Consider the four bonds having annual payments as shown in the table below.end of year payment Bond A Bond B Bond C Bond Dyear 1 100 50 0 01000year 2 100 50 0 0year 3 1001000 501000 01000 0

Suppose the yield to maturity of all the bonds is 15%.(a) Determine the price of bonds A and C.(b) Which bond is most sensitive to a change in yield? and why?(c) Suppose that you owe $2,000 at then end of 2 years. Concern about interest

rate risk suggests that a portfolio consisting of the bonds and the obligation should beimmunized. If VA,VB,Vc,Vd are the total value of bonds of types A,B,C, and D tobuy,respectively, write down the equations that represent the immunization. (Do notsolve the equations, but explain what each equation does.)

(d) In order to immunize the portfolio, you decide to use bond C and one otherbond. Which other bond should you choose? Then, find the total amounts of each ofthe bonds to purchase for the immunization.

Problem 3 (20 points)Suppose that a bank receives the following liability scheduleyear 1 year 2 year 312,000 18,000 20,000

i.e. the bank needs to pay 12,000 at the end of the first

year, 18,000 at the end of the second year, and 20,000 at the end of the third year.

The bank wishes to use the three bonds below to form a portfolio that will generatethe required cash to meet the liabilities. All bonds have face value of a 100 and thecoupons are annual (with one coupon per year). For example, one unit of Bond 2costs 99 now and the holder will receive 3.5 after 1 year and then 3.5 plus the facevalue of 100 at the end of the second year.

Bond 1 2 3Price 102 99 98Coupon 5 3.5 3.5Maturity year 1 2 3

Formulate an optimization model that can be used to find the lowest cost bondportfolio consisting of bonds 1,2, and 3 above that will meet the liabilities. In yourmodel you wish to allow carry over of money to future periods [i.e. you may generatemore cash than needed to meet a liability] (the money should be carried over at 0%interest earned i.e. assume you earn no interest for extra cash carried over time).(Note: you must write out the detailed model showing all coefficients and variables inall constraints and objective function, explain all decision variables. DO NOT WRITE AGENERAL MODEL, Do not solve the model.)

Problem 4 (10 points)You wish to invest in 3 securities S, B, and M.The expected returns for each

security are S 10.73%,B 7.37%,M 6.27%. The standard deviations of returnsof the securities are S 16.67%,B 10.55%,M 3.40%. The correlations of thereturns between securities are given in the following table:

ij S B MS 1 0.2199 0.0366B 1 -0.0545M 1

Formulate an appropriate optimization model whose solution will give the optimalportfolio weightings for stocks, bonds, and money market (optimal in the sense offinding the minimum variance portfolio that achieves a desired level of return). Theformulation must write out all of terms of the objective and the constraints with allcoefficients taking on their appropriate numerical values! Define all decisions variablesas well. DO NOT WRITE A GENERAL MODEL. And of course, you do not have tosolve the model!!