71. A certain stock costs 40 today and will pay an annual dividend of 6 for the next 4 years. An investor wishes to purchase a 4-year prepaid forward contract for this stock. The first dividend will be paid one year from today and the last dividend will be paid just prior to delivery of the stock. Assume an annual effective interest rate of 5%. Calculate the price of the prepaid forward contract. (A) 12.85 (B) 13.16 (C) 17.29 (D) 18.72 (E) 21.28 72. CornGrower is going to sell com in one year. In order to lock in a fixed selling price, CornGrower buys a put option and sells a call option on each bushel, each with the same strike price and the same one-year expiration date. The current price of corn is 3.59 per bushel, and the net premium that CornGrower pays now to lock in the future price is 0. 10 per bushel. The continuously compounded risk-free interest rate is 4%. Calculate the fixed selling price per bushel one year from now. (A) 3.49 (B) 3.63 (C) 3.69 (D) 3.74 (E) 3.84 IFM-01-18 Page 37 of 105 73. The current price of a non-dividend-paying stock is 100. The annual effective risk-free interest rate is 4%, and there are no transaction costs, The stock's two-year forward price is mispriced at 108, so to exploit this mispricing, an investor can short a share of the stock for 100 and simultaneously take a long position in a two-year forward contract. The investor can then invest the 100 at the risk-free rate, and finally buy back the share of stock at the forward price after two years. Determine which term best describes this strategy. (A) Hedging (B) Immunization (C) Arbitrage (D) Paylater (E) Diversification3. An insurance company sells single premium deferred annuity contracts with return linked to a stock index, the time-r value of one unit of which is denoted by S(r). The contracts offer a minimum guarantee return rate of g%. At time 0, a single premium of amount a is paid by the policyholder, and at x y% is deducted by the insurance company. Thus, at the contract maturity date, 7, the insurance company will pay the policyholder * * (1 - y%) * Max[S(7)/S(0). (1 + 8% )']. You are given the following information: (i) The contract will mature in one year. (ii) The minimum guarantee rate of return, g%, is 3%. (iii) Dividends are incorporated in the stock index. That is, the stock index is constructed with all stock dividends reinvested. (iv) S(0) = 100. (v) The price of a one-year European put option, with strike price of $103, on the stock index is $15.21. Determine y%, so that the insurance company does not make or lose money on this contract. (A) 12.8%. (B) 13.0% (C) 13.2% (D) 13.4% (E) 13.6%. IFM-01-18 Page 42 of 105 4. For a two-period binomial model, you are given: (i) Each period is one year. (ii) The current price for a nondividend-paying stock is 20. (iii) # = 1.2840, where u is one plus the rate of capital gain on the stock per period if the stock price goes up. (iv) d =0.8607, where d is one plus the rate of capital loss on the stock per period if the stock price goes down. (v) The continuously compounded risk-free interest rate is 5%. Calculate the price of an American call option on the stock with a strike price of 22.- Exercises 7.1 Compute image, kernel and rank of A=|4 2 8 9, - Problems 7.2 Verify that a system of linear equations can indeed written in ma- trix form. Moreover show that each equation Ax = b represents a system of linear equations. 7.3 Prove Lemma 7.3 and Theorem 7.4. 7.4 Let A = be an m xn matrix. (1) Define matrix Try that switches rows a; and a';. (2) Define matrix T,(a) that multiplies row a, by a. (3) Define matrix Ti-(a) that adds row a, multiplied by a to row a;. For each of these matrices argue why these are invertible and state their respective inverse matrices. HINT: Use the results from Exercise 4.14 to construct these matrices. 7.5 Prove Lemma 7.8. 7.6 Prove Theorem 7.9. Use a so called constructive proof. In this case this means to pro- vide an algorithm that transforms every input matrix A into row reduce echelon form by means of elementary row operations. De- scribe such an algorithm (in words or pseudo-code)