Aps 502
.pdf
School
University of Toronto**We aren't endorsed by this school
Course
APS 502
Subject
Accounting
Date
Dec 18, 2024
Pages
4
Uploaded by BaronElectron9771
\! APS 502 Financial Engineering I March. 7, 2019 Nome:_\CH TR0 VWMWY Student #:__ \@S 23 8¥So0 Tnstructions: Closed book and closed notes except for one side of a 3 by 5inch index card of notes. Only a simple scientific non-financial calculator with no programming and no memory capability is allowed. Please write neatly as this will aid in providing maximum partial credit. IMPORTANT: Interpretation of the exam questions is part of the exam and so no questions will be taken DURING the exam that ask for clarification of the questions. You MUST turn in this question sheet with your answer booklet or else your exam will NOT be marked. SHOW WORK! Problem 1 (12 points, 4 points each) You will be buying a 400 square feet condo in Toronto for $600,000. You will make a $150,000 down payment and borrow $450,000 from the bank. The amount borrowed will be paid back in equal monthly payments over 25 years with a nominal interest rate of 8.5%. (a) What is the effective annual rate? (b) What is the monthly payment? (c) How much do you owe the bank immediately after the 47th payment? Problem 2 (9 points) Suppose we have the spot rates s; ans s; where i < j and the forward rate fi.;- Suppose that (1 +8:) 1+ fi < (14 s;)7. Then, specify a sequence of trades that represents an arbitrage strategy. Problem 3 (12 points, 4 points each) For each assertion below state whether it is TRUE, FALSE, or "it depends". NO explanation is required. (a) The term structure of interest rates must be always upward sloping because longer maturity bonds are riskier. — Colw * Lorows ’S\P\l (b) Bonds with higher coupon rates have more interest rate risk. &alye (c) To reduce interest rate risk, a pension fund with more assets than liabil- ities (i.e. present value of its assets is larger than present value of its liabilities) should invest in assets with longer duration than its liabilities. Problem 4 (20 points, 4 points each except (c) is 8 points)) Consider the following two bonds. Bond 1 is a one year bond with a face value of $100 with coupon rate of 7% with semi-annual coupons. Bond 2 is 5 year bond with face value of $100 with a coupon rate of 9% and has annual coupon payments. Assume that the yield is 9% for both bonds. (a) Find the price Py of Bond 1 and the price P, of Bond 2. (b) Find the MacCaulay duration D; of Bond 1 and the MacCaulay duration D, of Bond 2 by using the DEFINITION of MacCaulay duration.
(c) Suppose that you have a liability of $400 at the end of year 2, a $500 liability at the end of year 3 and a liability of $1500 at the end of year 4. Find a bond portfolio of Bond 1 and Bond 2 that will meet the liabilities but is immunized against changes in interest rate. (d) Recall that bonds with shorter duration are less sensitive to changes in interest. For the following assertion determine whether it is true or false and provide a brief explanation. "It is always better to buy as many bonds as possible with shorter duration." Problem 5 (12 points) A small pension fund has the following liabilities (in millions of dollars) year T2 8 liability | 24 | 26 | 28 For example, the pension fund has to payout 24 millions dollars at the end of year 1, 26 millions dollars after year 2, etc... The pension fund will use the following bonds to construct a bond portfolio that they will form today and hold until all Bonds expire and whose cash flows will offset the liabilities. Bond 1 2 3 4 5 Price (8) 102.44 | 99.95 | 100.02 | 102.66 87.90 Coupon (%) 5.625 | 4.75 | 4.25 5.25 0.00 Maturity (year) | 1 2 2 3 3 Rating A B A A B All bonds have a face value of §100 and coupon payments are made annually. The rating quality is a meaure of the likelihood that the issuer of the bond will pay coupon payments and face value. An ’A’ rating is better than a 'B’ rating. Formulate the best linear programming model that you can (assume any carry over of cash does NOT earn interest) that will construct the minimum cost bond portfolio that will meet the liabilities subject to that no more than 50% of the bond portfolio’s (dollar) value can be in bonds with a rating of B. Problem 6 (20 points) You are asked to price some options on ABC stock. ABC’s stock price can go up by 15 percent every year, or down by 10 percent.The risk free rate is & percent and the current price of ABC stock is §100 per share. Consider a 3 period (i.e. three year) binomial tree to model the price of ABC stock for parts (a) and (b) below. (a) (8 points) Price a European put option on ABC stock with a maturity of 3 years and a strike price of $95. (b) (8 points) Price a European call option on ABC stock with a maturity of three years with a strike price of $100 but with a knock-out value of §125
meaning that if the price of one share of ABC is $125 or higher at maturity then the option payoff becomes 0. (c) (4 points) In the binomial option pricing method why is it that we should \)‘ N A always use the risk-free interest rate to discount for every time period as opposed P»,\ e o\fu 4 A to any other interest rate? Your answer should be no more than a few sentences. 3 Nes vtk Ay \ Wrxayen) (& imbllclem'7 ('lsfpolmtS) for thr It is estimated that the t at 3 ¥ gold mine is for lease for three years. It is estimated that the current ;5 ¥ . amount of gold remaining in the mine is zg = 20,000 ounces. The price of gold s K L a\“‘\m b is currently $1728.55 per ounce. If z is the amount of gold remaining in the mine at the beginning of a year, the cost to extract z < z ounces of gold in that year is $60022/1.1¢. The interest rate is 10%. How much is this lease worth? SHOW ALL WORK including your work taking derivatives. \l\r\\-\{xB = D W ‘P\Q\/wm B \/as (5(— Zw L\