this assignment is combination of math and finance
– Assignment 2 – MATH3090/7039: Financial mathematics Assignment 2 Semester I 2018 Due 3pm Thursday 10th May Weight 7 % Total marks 30 marks Final version Submission: Hardcopy to the submission box by 3pm Thursday 10th May. In addition, you will also need to submit a softcopy (scanned copy) of your assignment and your code on Blackboard. Notation: “Lx.y” refers to [Lecture x, Slide y] Assignment questions - all students 1. (8 marks) (Investment) You are considering a 4-year investment project which is expected to cost $1, 500, 000. In each year, you have decided that there are 3 possible states of the economy: good, average, and poor. In each individual year there is a 45% chance of the economy being good and a 25% chance of it being poor. Project costs are expected to be constant at $50,000 each year. You forecast the following revenues for the project: Economy Year 1 Year 2 Year 3 Year 4 Good 200,000 350,000 350,000 300,000 Average 150,000 225,000 325,000 250,000 Poor 100,000 220,000 275,000 225,000 You have arranged the following sources of funding: (i) $200,000 from a 4-year fixed interest loan whose annual loan payments are $56,630. (ii) $300,000 from a 4-year zero-coupon bond with a face value of $350,000. (iii) $250,000 from a 4-year coupon-paying bond issue whose coupon rate is 5% and face value is $250,000. (iv) $300,000 from an ordinary share issue where a dividend of $20,000 will be paid in one year and it is expected to grow at 3.5% per annum. (v) $450,000 from a preference share issue whose annual dividend will be $30,000. Should you invest in the project? (Use discrete compounding.) Note: For the part involving Newton’s method, choose y0 = 0.05 and apply one iteration. 2. (10 marks) (Yield curve modelling) Consider a binomial model of the yield curve over 3 years where y0,1 = 4%. The probability of an up movement in 1-period forward rates for year t = 2, 3 is pt = 0.25+0.2t, and 1-period forward rates can go up by a factor of u = 1.5 or down by a factor of d = 0.75. Calculate the zero-coupon bond yield curve and the implied 1-period forward rates embedded in this yield curve. MATH 3090/7039 – 1 – Roxane Foulser-Piggott – Assignment 2 – 3. (12 marks) (Yield curve and swap pricing) Assume that you observe the following yield curve for for government’s coupon paying bonds. • There are a total of 30 bonds. • For the k-th bond, k = 1, . . . , 30, the maturity is k years. • The face value is $100,000 and the coupon rate for the k-th bond, k = 1, . . . , 30, is 4%. • The prices of the bonds are given in the following table Table 1: Bond prices k prices k prices k prices 1 98,828.9817 11 89,083.9301 21 81,633.1317 2 97,812.0511 12 87,944.5962 22 81,287.3700 3 96,937.8969 13 86,976.3584 23 81,087.7608 4 96,159.9962 14 85,928.4188 24 80,919.5136 5 95,269.2339 15 84,982.5065 25 80,780.7515 6 94,353.5669 16 84,248.2589 26 80,669.7282 7 93,276.0334 17 83,540.8304 27 80,584.8196 8 92,237.8837 18 82,911.5228 28 80,524.5143 9 91,261.0455 19 82,417.0923 29 80,487.4060 10 90,214.0597 20 82,009.0742 30 80,472.1856 Assume that all the coupon payments are made annually. Use continuous compounding. a. (4 marks) Modify the Newton iteration program that you developed for Assignment 1 to compute the yield to maturity (YTM) for each bond. Submit a table similar to Table 1 with the “price” column being replaced by the “YTM” column filled with the computed YTMs. Plot YTMs vs maturities and comment. You may find the Matlab function plot useful. For the initial guess, choose 10%. Use the stopping criteria |yn+1 − yn| < 10−6. b. (4 marks) taking as input the computed ytms in part (a), implement in a matlab pro- gram to compute spot zero-coupon bond yield curve embedded in the observed coupon- paying bond yield curve, and the implied one-year forward rates. submit table 2 filled with computed values. table 2: table for question 2 (b) period spot forward 1 . . . . . . 2 . . . . . . ... ... ... 30 . . . . . . math 3090/7039 – 2 – roxane foulser-piggott – assignment 2 – c. (4 marks) suppose you enter into a 30-year vanilla fixed-for-floating swap on a notional principal of $1,000,000 where you pay the fixed rate of 6.15% and the counter-party pays the yield curve plus 1%. code in matlab a program to compute the swap value. submit a table of results, similar to the table on l5.12. assignment questions - math7039 students only no more questions will be added. math 3090/7039 – 3 – roxane foulser-piggott 10−6.="" b.="" (4="" marks)="" taking="" as="" input="" the="" computed="" ytms="" in="" part="" (a),="" implement="" in="" a="" matlab="" pro-="" gram="" to="" compute="" spot="" zero-coupon="" bond="" yield="" curve="" embedded="" in="" the="" observed="" coupon-="" paying="" bond="" yield="" curve,="" and="" the="" implied="" one-year="" forward="" rates.="" submit="" table="" 2="" filled="" with="" computed="" values.="" table="" 2:="" table="" for="" question="" 2="" (b)="" period="" spot="" forward="" 1="" .="" .="" .="" .="" .="" .="" 2="" .="" .="" .="" .="" .="" .="" ...="" ...="" ...="" 30="" .="" .="" .="" .="" .="" .="" math="" 3090/7039="" –="" 2="" –="" roxane="" foulser-piggott="" –="" assignment="" 2="" –="" c.="" (4="" marks)="" suppose="" you="" enter="" into="" a="" 30-year="" vanilla="" fixed-for-floating="" swap="" on="" a="" notional="" principal="" of="" $1,000,000="" where="" you="" pay="" the="" fixed="" rate="" of="" 6.15%="" and="" the="" counter-party="" pays="" the="" yield="" curve="" plus="" 1%.="" code="" in="" matlab="" a="" program="" to="" compute="" the="" swap="" value.="" submit="" a="" table="" of="" results,="" similar="" to="" the="" table="" on="" l5.12.="" assignment="" questions="" -="" math7039="" students="" only="" no="" more="" questions="" will="" be="" added.="" math="" 3090/7039="" –="" 3="" –="" roxane="">