Answer choice A in Problem 26, Practice Examination 8, should be 2.

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1 Course FM Manual by Dr. Krzysztof Ostaszewski, FSA, CFA, MAAA December 008 Edition Errata Posted May 8, 009 The beginning of Problem 3 in Practice Examination 3 should read: You are given the following liability and possible assets investment portfolios cash flow streams: Time 3 Liability Portfolio Portfolio Portfolio 3 = 0% in Portfolio + 90% in Portfolio Portfolio 4 = 50% in Portfolio + 50% in Portfolio Assume end of year cash flows and a yield to maturity of 5%. Posted May 8, 009 Answer choice A in Problem 6, Practice Examination 8, should be. Posted May 8, 009 The third sentence of the solution of Problem 35 in Practice Examination 8 should be: The present value of the income produced by the underlying during the upcoming year is 60e Posted May 8, 009 In Problem 33, Practice Examination 8, answer choice B should be $894.37, and the last displayed formula in the solution should be: S = $90 e $ Posted May 8, 009 Problem 4 in Practice Examination should be: Mr. Richard Carkosheek buys a one-year bank Certificate of Deposit that pays an effective annual rate of return of %, on which 5% income tax must be paid. If the inflation rate is 5% annual effective, what is the annual after-tax real rate of return? A..86% B..9% C..94% D..97% E. 3.8%

2 Since 5% tax must be paid on % interest, only 75% of the %, i.e., 9%, is left to Mr. Carkosheek. The real after tax rate of return is therefore %..05 Answer E. In Practice Examination 4, Problem, answer choice E should be 500. In Practice Examination 4, Problem 0, the force of interest formula should be ( ), and the answer choices should be: 0. t A. 065 B. 067 C. 069 D. 07 E. 073 In Practice Examination 4, Problem 6, answer choice E should be: E. k vn δ In Practice Examination 4, Problem, answer choice was missing. It should be 0J E.. ( ) 0 The correct answer choice in Problem 4, Practice Examination 4, is E. In Problem 4, Practice Examination 3, answer D should be 05, and the last two sentences of the solution should be: The accumulated value of the payments is Therefore, 000 = ( X 85) s 0 8% +0( Is) 0 8% = ( X 85) s 0 8% +0 s 0 0 8% = 0.08 ( ) = X 85 = ( X 85) s 0 8% +5 s 0 8% 0 ( ) = ( ) s 0 8% +5.08s 0 8% 0 = Xs 0 8% 85s 0 8% +35s 0 8% 50 = Xs 0 8% + 50s 0 8% 50.

3 Answer D. X = 50 s 0 8% In Problem, Practice Examination 3, answer D should be M + L K L K last sentence of the solution should be From this a 8 = a + a 7 i a a 7, or M = L + K + i ( K L), based on which we find i = M + L K. K L Answer D. and the In Problem 34, Practice Examination, the last formula in the solution should be In Problem 8, Practice Examination 3, answer choices should be: A. 3 + n In the solution and B. 3 n + 3 D. v n = + 3 = 3 = 3 i = 3 + n. n C. 3 n E = 3 +, 3 n

4 Posted May 4, 009 In Problem 4, Practice Examination 3, the correct answer choice is D, not E. The calculation given actually shows D to be the answer. Posted April 9, 009 The last formula in the solution of Problem 6, Practice Examination 7, should be: β = f u f d u d = ( ) 0 S Posted April 9, 009 In Problem 5 in Practice Examination 6, the first sentence The two shares of the inheritance are equal at time 0, and therefore they are also equal at time. is both incorrect and irrelevant, so it should be removed. Posted April 9, 009 In Problem 8, Practice Examination 3, the balance of the margin account at the close of business on Thursday should be calculated as: = instead of = Posted April 6, 009 Answer choices in Problem 4, Practice Examination 4, should be: A. a a 7 6 B. a a 7 6 C. a a 7 6 a 4 s 4 s 3 + d D. a 7 a 6 s 3 a E. a 7 a 6 s 3 + a Posted April 6, 009 The first sentence of the solution of Problem 9 in Practice Examination 3 should be: The first statement tells us that e 7.7δ =. Posted April 5, 009 Problem 33 in Practice Examination 6 should be: Spring 006 Casualty Actuarial Society Course 8 Examination, Problem No., also Dr. Ostaszewski s online exercise No. 07 posted May, 009 You are given the following information: The 6-month (annual) interest rates in Japan and the United States are 3% and 6%, respectively, with continuous compounding.

5 The spot price of the yen is $ The futures price for a contract deliverable in 6 months is $ Calculation the present value of riskless arbitrage profit available, per yen. A. $ B. $ C. $ D. $ E. No arbitrage is possible The arbitrage-free futures price is $ e ( ) $ This means that a six-month futures price in the market of $0.009 requires a payment of $0.009 $ e ( ) $0.009 $ = $ in excess of the arbitrage-free price upon contract maturity. By buying the underlying with borrowed funds and shorting the futures, one can collect this difference as a riskless profit upon contract maturity. The present value of that profit is e 0.06 $0.009 $ e ( ) $ Answer B. Posted April 5, 009 Problem 33 in Practice Examination 9 should be: You are given the following: A stock is currently selling for $ per share. The stock is not expected to pay any dividends over the next 3 years. The annual risk-free rate with continuous compounding is 3%. If the three-year forward price is $3, calculate the present value of a riskless arbitrage profit available. A. $0.00 B. $0.55 C. $0.90 D. $0.98 E. $.07 The arbitrage-free forward price is $ e $4.07. Since the price in the market is $3, there is a riskless arbitrage profit of available. The stock can be purchased on the forward for $3, while selling it short now for $ and investing the proceeds in a riskfree bond will result in covering the short with the shares bought on the forward, and receiving $4.07 $3 = $.07 in three years. In today s money, this is worth $.07 e $0.98. Answer D. Posted April, 009 The answer choices for Problem 8 in Practice Examination 4 should be: A. 368 PLN B. 35 PLN C. 300 PLN D. 68 PLN E. 5 PLN Also, the last two sentences of the solution should be:

6 Mr. Carkosheek s payoff upon option exercise is 3 PLN per share, for a total of 300 PLN. His profit is PLN. Answer E. Posted April, 009 Problem 3 in Practice Examination should be reworded and read as follows: Mr. Roman Carkosheek has developed keen interest in the shares of Pszczochoszczysz, a Polish agricultural conglomerate traded on the Warsaw Stock Exchange. Current price is 500 Polish Złoties (PLN) per share. The stock does not pay any dividends. The following one-year European options on Pszczochoszczysz are currently available: Exercise price Put price Call price After studying these prices, Mr. Carkosheek concludes that the 50 exercise price option is priced correctly, but the 500 exercise price option presents an arbitrage opportunity for him. He decides to take advantage of an arbitrage opportunity, by creating a portfolio consisting of 000 shares of Pszczochoszczysz (short or long, as needed), a risk-free oneyear bond, and options with 500 exercise price. Calculate the amount of the riskless profit this portfolio will earn over one year. Assume that all conditions for put-call parity to hold are satisfied for the 50 exercise price options, and that Mr. Carkosheek can borrow and lend at exactly the risk-free rate prevailing in the market. A. No arbitrage is possible B. 37,888 C.,440 D. 0,395 E.,045 Let i be the risk-free interest rate for the next year. Using the put-call parity for the 50 exercise price options we get = , + i and this results in 50 i = %. We know that these options are priced correctly, so that this is the correct one-year riskfree rate. For the 500 exercise price option, the stock price is 500, while the present value of the exercise price plus a call minus a put is worth Thus the stock is cheap in relation to a portfolio of present value of exercise price

7 invested in a risk-free bond, long call and short put, while the two positions are supposed to be equivalent. Mr. Carkosheek should be long 000 shares of Pszczochoszczysz, the cheaper of the two, by buying them at 500 per share. He should also short the expensive one to create an arbitrage. In order to do so, he must borrow the present value of exercise price on 000 shares, i.e., 500,000, sell call options on 000 shares for 30,890 and buy puts on 000 shares for 0,90. His net cash expenditure will be 500, ,000 30, , In one year, he will own 000 shares, he will owe 500,000 and will be short 000 calls with strike price 500 and will be long 000 puts with strike price The option combination is equivalent to being short forward with exercise price 500, so he will sell his shares for 500, receiving 500,000, exactly enough to pay off the loan. The net position will be exactly zero. On a net basis, he will be paid upfront, and not have to pay anything at the end of the year. The cash he receives upfront will accumulate to Answer E. Posted April 5, 009 The last formula in the solution of Problem in Practice Examination 5 should be: 0.05( 0, a 5 4% a 8 5% ) Posted April 5, 009 Problem 3 in Practice Examination and its solution should read: 3. Brydykyne, a Ukrainian mining company, is selling coal, which is equivalent in its quality and all properties to coal from Central Appalachia. Brydykyne has a cost of $00 per short ton, and has a contract to sell short tons in one year. Current price of coal is $0 per short ton. Brydykyne is certain to produce the coal as required, but the company is concerned about uncertainty of the price at which they will sell their product. The one-year risk-free interest rate in U.S. dollars is 3%, effective annual. After Brydykyne sells its coal, it has to pay a tax of 40% on all profits, with interest on the coal s cost included in the expenses, while if it has a loss, there is not tax payable, and no tax refund is available. If Brydykyne is able to enter into a forward now at the arbiragefree forward price, for a fee of $0,000 payable at the expiration of the contract, find the after-tax profit it will have. A. $5800 B. $90000 C. $0000 D. $ E. $ The arbitrage-free forward price is $0.03 = $ Thus Brydykyne will sell short tons at that price, and the profit on that sale alone will be:

8 ( 0.4) $04.03 $ = $6,800. Aftex tax Cost accumulated with interest However, the company also has to pay the fee of $0,000 at the time when they receive that profit on the sale, so the actual profit is $5,800. Answer A. Posted April 5, 009 The text of Problem 34 in Practice Examination 3 should read: Your company holds an option-free bond that is free from default risk. The bond has a five-year term to maturity with 8% coupons paid annually. The par value of a unit of this bond is $ The annual yield to maturity on the bond is 7.5%. Find the Macaulay duration of the bond. Posted April 4, 009 The last formula in the solution of Problem, Practice Examination 4, should be: v n t + + v n t = + v n t v ( ) = + v n t d. Posted April 4, 009 In Practice Examination, Problem 8, answer choice C should be 395. Posted April 3, 3009 In Practice Examination, Problem 3, the answer choices should be: A.,346 B. 3,65 C. 5,93 D. 7,396 E. 8, They were missing. Also, in Problem 3, "transaction" was misspelled as "transation. Finally, in Problem 7 the corporate income tax rate stated in the problem is 60%, not 0%. Posted February, 009 In the discussion of forwards and futures on bonds on page 58, the sentence: The underlying in this case is a zero-coupon bond maturing at time n + k, issued at time n, which effectively is a unit monetary amount paid at time n + k, so that its present value is P( n + k). nhould be The underlying in this case is a zero-coupon bond maturing at time n + k, issued at time k, which effectively is a unit monetary amount paid at time n + k, so that its present value is P( n + k). The bond considered is issued at time n not k.

9 Posted November, 008 Problem 35 in Practice Exam 3 should be: Spring 00 Casualty Actuarial Society Course 8 Examination, Problem No. 3a (some explanations and multiple choice answers added) You are given the following: A stock is currently selling for $0 per share. The stock is not expected to pay any dividends over the next 3 years. The annual risk-free rate with continuous compounding is 5%. If the three-year forward price is $3, calculate the present value of a riskless arbitrage profit available. A. $0.00 B. $0. C. $0.0 D. $0.4 E. $3.00 The arbitrage-free forward price is $0 e $3.4. Since the price in the market is $3, there is a riskless arbitrage profit of $0.4 available at the time forward is exercised. The present value is ( $0 e $3)e $0 $9.80 = $0.0. Answer C. Posted November 4, 008 Only 30 out of 35 questions in Practice Examination 8 were reproduced in the questions for this exam, while all 35 were listed in solutions. This is an error. Please include questions 3-35 among the examinations questions. Posted November 3, 008 The solution of Problem 3 in Practice Examination 5 should be: Based on put-call parity applied to a three-year period we have 50 e e 0.08 ( ) 60e = 4 P, where P is the put price. Based on this P = 60e e e 0.6 ( ) Answer B. Solution previously given had typos in the number of years for discounting of payments of dividends and exercise price. Posted October 8, 008 The last sentence of Problem 4 in Practice Examination 5 should be: Under the following reinvestment interest rate movement scenarios effective //04, what best describes the insurance company s profit (or loss) as of /3/07 after the liability is paid?

10 The date //04 was mistakenly typed as /3/04 Posted October 3, 008 There are two exercises.3. The second one should be renamed.4, and the subsequent exercises numbers should be increased by one.