SOLUTIONS. Solution. The liabilities are deterministic and their value in one year will be $ = $3.542 billion dollars.

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1 Illinois State University, Mathematics 483, Fall 2014 Test No. 1, Tuesday, September 23, 2014 SOLUTIONS 1. You are the investment actuary for a life insurance company. Your company s assets are invested in a bond portfolio with the value of $3 billion dollars, and a stock portfolio of $1 billion dollars. The liabilities of your company are life and annuity reserves of $3.5 billion dollars. Over the next year, liabilities will increase by 1.2%, the current risk-free rate. Over the same period, the stock portfolio return will follow a normal distribution with mean 8.2% and standard deviation of 15%, and the bond portfolio will follow a normal distribution with mean 3% and standard deviation of 5%, and the correlation of these two returns is 0.25, and their joint probability distribution is bivariate normal. Over the next year, there will be no new premiums and no new liabilities, all growth of both assets and liabilities comes from the existing asset-liability portfolio. Find the probability that the return on capital the company will exceed the expected return of the stock portfolio. The liabilities are deterministic and their value in one year will be $ = $3.542 billion dollars. Let us write R S for the random rate of return of the stock portfolio, and R B for the random rate of the bond portfolio. Then the random value of the asset portfolio at the year-end will be (in billions of dollars) 3 ( 1+ R B ) +1 ( 1+ R S ). Since the joint probability distribution of R S and R B is standard normal, the random value of the asset portfolio at the year-end is normal with mean (in billions of dollars) E( 3 ( 1+ R B ) +1 ( 1+ R S )) = E( R B ) +1 E R S and the standard deviation ( ( ) +1 ( 1+ R S )) = Var( R B +1 R S ) = Var 3 1+ R B ( ) = % % = 4.172, = Var( 3 R B +1 R S ) = 9Var( R B ) + Var( R S ) + 6Cov( R B, R S ) = = = = The random value of the surplus, let us call it C (C stands for capital) at the year-end is a normal random variable with mean = 0.63 and standard deviation of (in billions of dollars), and the random rate of return on surplus, let us call it R C ) is ( ( ) 3.5 ) R C = C 3+1 ( 3+1) 3.5 Initial surplus = 2C 1,

2 so that R C is normal with mean ( ) = 2E( C) 1 = = 0.26, E R C and standard deviation of ( ) = Var( 2C 1) = 2 Var( C) Var R C The probability that the return on capital the company will exceed the expected return of the stock portfolio therefore is (note that we write Z for a standard normal random variable) R Pr( R C > 0.082) = Pr C > = ( ) = Pr Z > Spring 2014 Casualty Actuarial Society Course 9 Examination, Problem No. 5 Given the following information for constructing an investment portfolio: Asset Expected Return Standard deviation A 8% 20% B 10% 25% The risk-free rate is 2%. The correlation coefficient between asset A and asset B is The investor has a risk-aversion coefficient of 5. Calculate the expected rate of return and standard deviation of the optimal complete portfolio. Comment from K.O.: Assume that the utility function is U ( R,σ ) = E( R) 0.5Aσ 2, where R is the random rate of return, A is a coefficient of risk aversion (expressed as a decimal, i.e., 5% is written as 0.05, not treated as 5, and this is ), and σ is the standard deviation of R. The optimal portfolio consisting of only Asset A and Asset B has allocation to Asset A of ( E( R A ) r F )σ 2 B ( E( R B ) r F )Cov( R A, R B ) w A = ( E( R A ) r F )σ 2 B + ( E( R B ) r F )σ 2 A ( E( R B ) r F + E( R A ) r F )Cov( R A, R B ) = ( 8% 2% ) ( 10% 2% ) ( 0.4) = ( 8% 2% ) ( 10% 2% ) (( 8% 2% ) +10% 2% ) ( 0.4) and allocation to Asset B w B = 1 w A = That optimal risky (i.e., with no risk-free asset included) portfolio has expected return ( ) = = , E R P while its variance is

3 ( ) = Var R P ( 0.4) , and its standard deviation, Var( R P ), is With utility function U ( R,σ ) = E R (as opposed to risk-free asset, the other part of the portfolio) is y * = E ( R P ) r F A σ P Based on this, the expected return of the optimal complete portfolio is ( ) 0.02 = %. The standard deviation of the optimal complete portfolio is = %. ( ) 0.5Aσ 2, optimal allocation to the risky portfolio 3. Spring 2000 Casualty Actuarial Society Course 8 Examination, Problem No. 16 Assets A, B, and C have the following prices as of January 1, 2000: Asset A: $90.70 Asset B: $ Asset C: $ The assets have the following risk-free cash flows (given in their entirety): Asset June 30, 2000 December 31, 2000 June 30, 2001 A $0.00 $ $0.00 B $8.00 $ $0.00 C $5.00 $ 5.00 $ Calculate, as of January 1, 2000, the sum of the following nominal annual, compounded semiannually, spot rates: from time 0 to time 0.50 years: s 0.50, from time 0 to time 1 year: s 1, and from time 0 to time 1.5 years: s 1.5. Base on the information about the bond A s 1 = 100, so that s 1 = %. Based on the information about the bond B = s 0.5 This gives s 1 2.

4 8 s 0.5 = s %. Based on the information about Bond C = 5 1+ s s s 1.5 This gives s 1.5 = s s %. 1 The sum of the three is approximately 28.88% April 2014 Casualty Actuarial Society Course 9 Examination, Problem No. 9b Companies A and B both ant to borrow $50 million for 10 years and are offered the following annual borrowing rates: Floating Fixed Company A LIBOR 0.1% 4.1% Company B LIBOR + 0.9% 5.4% Devise a swap agreement that is mutually beneficial to both companies and creates a 0.06% gain to a financial institution serving as an intermediary. Note from K.O.: To obtain a unique solution, and to simplify the solution, assume that all floating payments made within the swap are LIBOR (i.e., any floating payments to Company A, or Company B, or to the financial intermediary). In the floating rate borrowing, Company A has an absolute advantage of (LIBOR + 0.9%) (LIBOR 0.1%) = 1.0% and in fixed rate borrowing, Company A has an absolute advantage of 5.4% 4.1% = 1.3%. Therefore, Company A has comparative advantage in fixed rate borrowing, because its absolute advantage in that area is greater. Thus, Company A should borrow fixed, Company B should borrow floating, and they should swap. Thus Company A borrows at 4.1%, and Company B borrows at LIBOR + 0.9%, and they swap. The difference of absolute advantages 1.3% 1% = 0.3% is the total gain that can be had split equally between the two companies. Note that 0.06% of it goes to the financial intermediary, so only half of 0.30% 0.06% = 0.24%, i.e., 0.12%, goes to each of the two companies. As a result of the swap, Company A will pay fixed for its loan, receive fixed from the swap, and pay floating. It is common to make the floating

5 payment in a swap to be exactly LIBOR (you do not have to assume this, but such assumption simplifies things). Its floating borrowing rate is LIBOR 0.1%, and with gain of 0.12%, it would end up paying LIBOR 0.1% 0.12% = LIBOR 0.22%, but since we assume that the floating payment of the swap is LIBOR, Company A has to receive the 0.22% in the fixed payment, so the fixed payment to Company A has to be 4.1% % = 4.32%. If the financial intermediary receives 0.06% gain from each company, the payments would be 4.1% 4.32% 4.35% ß Company A ß Financial Intermediary ß Company B à à à LIBOR + 0.9% LIBOR LIBOR 0.03% But it is simpler to net the 0.03% payments between Financial Intermediary and Company B and have 4.1% 4.32% 4.38% ß Company A ß Financial Intermediary ß Company B à à à LIBOR + 0.9% LIBOR LIBOR This is, of course, exactly equivalent, and now all floating payments are LIBOR. The net result to Company A is that it pays LIBOR + 4.1% 4.32% = LIBOR 0.22% versus regular cost of LIBOR 0.1%, a gain of 0.12%, and to Company B the net result is that it pays LIBOR + 0.9% LIBOR % = 5.28% versus regular cost of 5.4%, a gain of 0.12%. 5. Spring 2008 Casualty Actuarial Society Course 8 Examination, Problem No. 13 A corporate bond has the following features on the morning of January 28, 2009: One year of maturity. 11% semi-annual coupon. 15% promised yield-to-maturity, nominal annual, compounded semiannually. Due to financial turmoil during the morning of January 28, 2009, investors believe the following: Bondholders will receive 80% of par value at maturity. All remaining coupon payments will be made in full. The bond s price will drop by 15% by market close on January 28, Calculate the expected yield-to-maturity expressed as a nominal annual rate compounded semiannually if the bond is purchased at market close on January 28, Before the turmoil, this bond s price (per $100 of principal) is After turmoil, the principal repaid at maturity will be $80 (per $100 of original principal) and the new price is 85% of 96.41, i.e., Let us write i ( 2) for the new nominal annual rate compounded semiannually once the bond is purchased at market close on January 28, We have

6 = 1+ i ( 2) + 1+ i ( 2) 2. By entering N = 2, PV = 81.95, PMT = 5.5, FV = 80, we obtain I/Y = 5.555, and i ( 2) is twice that, i.e., 11.11%. 6. An investor enters into a long position in one stock market index futures contract. Each index unit currently costs One futures contract is defined on 250 units. The initial margin is 10%, and the maintenance margin is 80% of initial margin account balance. The investor receives interest at the rate of 5% annual effective on margin balance. Available funds may be withdrawn weekly, and required margin call must be met with the same frequency. If during the first week the price drops to 1450, calculate the payment investor will be required to pay at the end of week 1. The initial value of the contract is = 375,000. Thus the initial margin deposit is 37,500. This will, by the end of the week, grow to , , The change in the value of the position is ( ) 250 = This results in the margin account balance of = The position value is then = 362,500, and that the required margin is now ,500 = Thus the investor must deposit 30,000 25, = 4, to replenish the margin account. 7. Mr. Carkosheek owns a television set factory in Minsk. The current price of the factory is 800 million Polish złoties (PLN), but the factory will require an immediate expenditure of PLN 340 million, to upgrade it to the new environmental standards imposed by the European Union, and that liability is unconditionally imposed on its owner, Mr. Carkosheek, as of now, so that his net worth in the factory is only PLN 460 million. Mr. Carkosheek buys an option to sell the factory in one year for PLN 400 million at a cost of PLN 9.1 million. The risk-free interest rate in Poland is 3% and it remains at that level for a year. After one year, the market price of the factory falls sharply due to Asian competition, and Mr. Carkosheek exercises the option. Find Mr. Carkosheek s profit over the year, as of the year s end.

7 Mr. Carkosheek starts the year with net worth of PLN 460 million. He spends PLN 9.1 million on the option purchase, and that cost, accumulated with interest to the end of the year is PLN = million. By including that interest cost, we effectively assume that the option was purchased with borrowed money, but we have to do this, because there is no other source of cash. He receives PLN 400 million from the sale of the factory, and after paying off the loan used for the purchase of the option he is left with = million of Polish złoties. His profit, in millions of PLN, is = You are given the following spot rates (effective annual) as of January 1, 2009: Year Spot Rate % % % EIB Company agrees to a forward rate agreement, in which it will lend $1,000,000 to Levin Enterprises on January 1, 2011, and be paid back that principal amount of $1,000,000 with interest at the fixed rate of 5% on December 31, Calculate the value of that forward rate agreement to EIB Company as of January 1, EIB will has a cash outflow of $1,000,000 on January 1, 2011, two years from January 1, 2009, and a cash inflow of $1,050,000 on December 31, 2011, three years from January 1, The value of those flows as of January 1, 2009 is $ $ The value as of January 1, 2010 is $ $ A share of Czyszyczysko company trades at the Warsaw stock exchange and it currently sells for 4000 Polish złoties (PLN). A one-year European put with exercise price of PLN 4000 sells for PLN 199. A one-year European call with exercise price of PLN 4172 also sells for PLN 199. The current risk free interest rate is 8% per annum continuously compounded. Assume that investor has no transaction costs, and only pays a premium for any option purchased, and receives premium for any option written. Find the range of prices of one share of Czyszyczysko one year from now for which purchasing a collar consisting of the two options, for which the prices are given, produces a profit. Let S denote the price of one share of Czyszyczysko at options expiration. A collar is created by purchasing a put option and writing a call option, with the put exercise price below the call exercise price. The collar in this problem consists of long put with exercise price of 4000, and short call with exercise price of As the two options have equal

8 premiums, this collar has zero cost. If we write S for the stock price, then the payoff of the long put position is 4000 S if S < 4000, 0 otherwise, and the payoff of the short call position is 0 if S < 4172, 4172 S otherwise. By adding the two payoffs we obtain the payoff of the collar 4000 S if S < 4000, 0 if 4000 S < S if S As there is no cost, the payoff is the profit. Only 4000 S is positive for S < 4000, while for other ranges of prices the payoff is zero or negative. 10. Spring 2009 Casualty Actuarial Society Course 8 Examination, Problem No. 29 A fund manager has a well-diversified portfolio that mirrors the performance of the S&P 500 and is worth $315 million. The portfolio manager would like to buy insurance against a reduction of more than 7% in the value of the portfolio over the next 6 months. Given the following information: The value of the S&P 500 is 900. The dividend yield on both the S&P 500 and the portfolio is 3%. The risk-free rate is 3% per annum. The volatility of the index is 15% per annum. Calculate the cost of insurance if the fund manager buys European put options. Assume all interest rates are continuously compounded The portfolio effectively consists of = units of the S&P 500 index. 900 The portfolio will be protected against a drop of more than 7% by buying that number of put options on the S&P 500 index with exercise price of 900 ( ) = 837. Since the option purchased is European, we can use the Black-Scholes formula to calculate the cost of insurance. The price of one such put option, by the Black-Scholes formula, is P = Ke rt N ( d 2 ) Se δt N ( d 1 ), where ln d 1 = Se δt Ke rt σ σ 2 t t = 900e ln 837e ,

9 d 2 = d 1 σ t , so that and ( ) = 1 N ( d 1 ) = , ( ) = 1 N ( d 2 ) = , N d 1 N d 2 P = 837e e , and for the needed of such options, the total cost is times that, or approximately Suppose that a portfolio P is worth $125 million and the S&P 500 Index is at The value of the portfolio mirrors the value of the index, and each option contract is for $100 times the index. What type of options and how many of these options should be purchased to provide protection against the value of the portfolio falling below $100 million in one year? If the portfolio falls from $125 million to $100 million, this represents a 20% decline in value. We want an option contract, which will protect the entire portfolio from dropping more than 20% than its current value. This will be achieved by a portfolio of puts with the same total exercise price, for all contracts combined, as the value of the total portfolio after a 20% decline. Each option corresponds to $100,000 in the current value of the portfolio, so for $125 million we need 1250 option contracts. The contracts must be for a 20% decline from the current 1000 value of the index, i.e., for exercise price of Spring 2013 Casualty Actuarial Society Course 8 Examination, Problem No. 1(a) Given the following information about investment options: U = E( r C ) 0.5Aσ 2 C. The risk-free rate of return is 3%. The risk premium on the risky portfolio is 5%. The reward-to-variability ratio of the risky portfolio is The aversion parameter is 2. Calculate the certainty equivalent rate of the risky portfolio. The expected rate of return of the risky portfolio is (risk-free rate of return plus the riskpremium) 3% + 5% = 8%. The reward-to-variability (Sharpe) ratio of the risky portfolio is =, σ P and this implies that

10 σ P = 0.25 = = Therefore, the utility of this portfolio (which is also the certainty-equivalent) U = = = 0.04.

SOLUTIONS 913,

SOLUTIONS 913, Illinois State University, Mathematics 483, Fall 2014 Test No. 3, Tuesday, December 2, 2014 SOLUTIONS 1. Spring 2013 Casualty Actuarial Society Course 9 Examination, Problem No. 7 Given the following information