MORNING SESSION. Date: Tuesday, October 28, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES

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1 SOCIETY OF ACTUARIES Quaniaive Finance and Invesmen Core Exam QFICORE MORNING SESSION Dae: Tuesday, Ocober 28, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Insrucions 1. This examinaion has a oal of 100 poins. I consiss of a morning session (worh 60 poins) and an afernoon session (worh 40 poins). a) The morning session consiss of 9 quesions numbered 1 hrough 9. b) The afernoon session consiss of 6 quesions numbered 10 hrough 15. The poins for each quesion are indicaed a he beginning of he quesion. 2. Failure o sop wriing afer ime is called will resul in he disqualificaion of your answers or furher disciplinary acion. 3. While every aemp is made o avoid defecive quesions, someimes hey do occur. If you believe a quesion is defecive, he supervisor or procor canno give you any guidance beyond he insrucions on he exam bookle. Wrien-Answer Insrucions 1. Wrie your candidae number a he op of each shee. Your name mus no appear. 2. Wrie on only one side of a shee. Sar each quesion on a fresh shee. On each shee, wrie he number of he quesion ha you are answering. Do no answer more han one quesion on a single shee. 3. The answer should be confined o he quesion as se. 4. When you are asked o calculae, show all your work including any applicable formulas. When you are asked o recommend, provide proper jusificaion supporing your recommendaion. 5. When you finish, inser all your wrien-answer shees ino he Essay Answer Envelope. Be sure o hand in all your answer shees because hey canno be acceped laer. Seal he envelope and wrie your candidae number in he space provided on he ouside of he envelope. Check he appropriae box o indicae morning or afernoon session for Exam QFICORE. 6. Be sure your wrien-answer envelope is signed because if i is no, your examinaion will no be graded. Tournez le cahier d examen pour la version française by he Sociey of Acuaries Prined in he U.S.A. 475 N. Maringale Road Exam QFICORE-Fron Cover Schaumburg, IL

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4 **BEGINNING OF EXAMINATION** 1. (7 poins) You are an invesmen acuary invesigaing he usefulness of a meanrevering model based on he process defined by: X dx = α X d + dw where α > 0 and W is a Wiener process. The iniial value of he process is X = x. 0 0 (a) (1.5 poins) Verify ha saisfies he inegral equaion: X = x 0 e α + e α X 0 e αs dw S based on he funcion F = X e α and using Io s lemma. (b) (3 poins) Derive, using par (a), closed-form expressions for: (i) (ii) Expeced value, Variance, Var( X ) E ( X ) (c) (0.5 poins) Explain why he process { X } is a zero mean revering process and inerpre he parameer α. You read he following commen: We live in a world primarily driven by random jumps, and ools designed for random walks address he wrong problem. We should be using fracal (scalable) models. (d) (2 poins) Compare he key characerisics of Guassian (nonscalable) and fracal (scalable) models, and heir effeciveness in assessing financial sraegies. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

5 2. (6 poins) The following 2 binomial rees show he possible values for a sochasic process a imes 0, 1, and 2 for wo differen probabiliy measures P and Q ha apply o he same ree: Probabiliy Measure P Tree Values (Transiion probabiliies are indicaed nex o he corresponding arrows) = 0 = 1 = Probabiliy Measure Q Tree Values (Transiion probabiliies are indicaed nex o he corresponding arrows) = 0 = 1 = (a) (b) (1 poin) Explain why P and Q are equivalen measures. (2.5 poins) Deermine under which measure P or Q he process is a maringale. (c) (1.5 poins) Calculae he one-period sandard deviaion from ime = 0 o = 1 under measure P. (d) (1 poin) Calculae he value of he Radon-Nikodym derivaive of Q wih respec o P for ime 1. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

6 3. (5 poins) (a) (1 poin) Define and explain he following conceps associaed wih implied volailiy: (i) (ii) Volailiy smile Volailiy surface (b) (2 poins) Compare he advanages and disadvanages of using sochasic volailiy vs. consan volailiy in he Black-Scholes model. You are asked o model a porfolio of equiy opions and currency opions. The following GARCH (1, 1) model is used o se he volailiy for an opion pricing model ha is based on a lognormal disribuion. σ = + αμ + βσ where α = 10%, β = 70%, and w = 0.3% , n w n 1 n 1 You observe ha μ0 = μ1 = 15%. (c) 2 2 (2 poins) Calculae he change in σ 2 when σ 0 is decreased from 8% o 7%. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

7 4. (6 poins) You are given he ask o model wo correlaed asse variables ha are simulaed using he same random process. They follow he following sochasic differenial equaions: dx = 0.15d dW ds = 0.05S d S dw X 0 = S0 = 1 W is a Wiener process. ( ) (a) (2 poins) Calculae E ln( S )/ using Io s lemma. (b) (1 poin) Explain why ln( E( S ))/ is greaer han he answer from (a). (c) (1 poin) Calculae he 95% confidence inerval of X 5. (d) (2 poins) Calculae S, 5 given X 5 = 3. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

8 5. (9 poins) Given he following informaion you are asked o esimae he price C of a European call opion on a defaul-free zero-coupon bond using he Black-76 pricing formula: The mauriy of he bond is U years; The expiraion of he opion is T years where T is much shorer han U; K = srike price of he opion; The price a ime of a defaul-free zero-coupon bond mauring a ime m, P ( m,, ) saisfies he SDE below: dp (, m) = r() d+ σ (, m) dx, where P(, m) o r() is he risk-free rae a ime ; o σ ( m, ) is volailiy of (, ) P m; and o X is a Wiener process. The Black-76 pricing formula gives he price of he opion as: C = P 0, T FN d KN d, where ( )( ( 1) ( 2) ) o F = E P( T, U) is he forward price a T of he bond mauring a U; F V ln + K 2 d =, d = d V ; V ln P TU, ; and o o V = variance of ( ( )) o boh he aformenioned expecaion and variance are under he equivalen maringale measure. ( ) PU, You now need o derive formulas for F and V. Firs you define G() =. You PT (, ) F = E G T P T, T = 1. noe ha ( ) and V = variance of ln ( G( T )) since ( ) (a) (1.5 poins) Derive he SDE of ln ( G() ). (b) (2.5 poins) Explain how you can use Girsanov heorem o claim ha here exiss a maringale measure under which he drif of G() vanishes. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

9 5. Coninued (c) (3 poins) (i) Derive he SDE of G() under he maringale measure in (b). (ii) Deermine he drif-shif needed o creae he maringale measure in (b). (d) (2 poins) (i) Express F in erms of P ( 0, T) and P( 0, U ). (ii) Express V as an inegral involving ( T, ) and ( U, ) σ σ. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

10 6. (9 poins) Your CFO asked you o invesigae a European asse S. Afer some research, you found ou he asse follows a special model where he probabiliy of an annual upmovemen decreases every year for 5 years: S 0 = 1 S = 1.2 S 1 wih probabiliy of 0.9 /10 (for =1, 2, 3, 4, and 5) 0.95 S 1 wih probabiliy of /10 (for =1, 2, 3, 4, and 5) 1.2 wih probabiliy of 0.3 (for 6 ) S wih probabiliy of 0.7 (for 6 ) S 1 S is he value of S a ime in Euro. An exoic opion C pays ln( S ) a = 6 only if S moves up in he firs year. (a) Le Z represen he number of annual up-movemens of asse S for 6 : (i) (1 poin) Derive he price of S a = 6 as a funcion of Z. (ii) (3 poins) Calculae he expeced payou of C. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

11 6. Coninued Assume he exchange rae X in Dollar per Euro is deermined by he following sochasic differenial equaion: (,, ) dx = i i X d + σ X dw us euro X 0 = 1.3 Where i us, is he U.S. inflaion rae from 1 o, i euro, is he inflaion rae in Europe from 1 o. i us, i euro, σ 1 3% 4% 20% 2 3% i euro,2 20% 3 3% i euro,3 20% W is a Wiener process. Assume X and S are independen and S is radable only a he end of each year. Le A be a simple dollar reurn of he invesmen in S over he period 1 o. (b) (i) (2 poins) Derive a formula for EA ( 3) in erms of i euro,2 and/or i euro,3. (ii) (3 poins) Calculae i euro,2 and i euro,3 such ha he process A is a maringale. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

12 7. (6 poins) You are responsible for a moderae-sized foundaion. You are given he following invesmen guidelines: Avoid concenraion risk Inves in a broad equiy marke (a) (1 poin) Deermine, given he guidelines, which of he following indices is he mos appropriae o serve as he equiy benchmark o evaluae invesmen performance. Jusify your answer. Dow Jones Indusrial Average Index S&P MidCap 400 Index Value Line Arihmeic Composie Index Now suppose ha he foundaion has given you a mandae o manage an index fund ha racks he S&P 500 index. To consruc your index fund, you mus choose one of he following mehods: Full replicaion Opimizaion (b) (1 poin) Recommend he mos appropriae mehod. Jusify your answer. Now suppose ha your mandae is o inves wih acive managers. (c) (2 poins) Describe he advanages and disadvanages of he following choices: (i) (ii) (iii) Hiring a single manager in eiher a growh or value syle (bu no boh). Hiring one growh manager and one value manager. Hiring one manager wih a marke-oriened syle. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

13 7. Coninued You are analyzing he performance of an acive manager whose objecive is invesing for growh and income, wih an orienaion o mid-cap socks wihin he universe of U.S.- domiciled companies. You can choose from he following indices for use in a reurnsbased syle analysis: The S&P/Vanguard 500 Growh and Value indices, which have a large-cap orienaion. The Russell 2000 Growh and Value indices, which have a small-cap orienaion. The Russell 1000 Growh and Value indices, which include large-cap and mid-cap shares. The Russell Top 200 Growh and Value indices, which ogeher represen he 200 larges marke-cap securiies in he Russell 1000 Index. The Russell Midcap Growh and Value indices, which ogeher represen he 800 smalles marke-cap issues in he Russell 1000 Index. Your assisan recommends he S&P/Vanguard 500 Growh and Value indices and he Russell 2000 Growh and Value indices for he syle analysis. (d) (e) (1 poin) Criique your assisan s selecion. (1 poin) Recommend a more appropriae selecion of indices. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

14 8. (7 poins) LMN Life specializes in selling fixed deferred annuiy producs wih no marke value adjusmens. The company currenly has $9.6 billion of asses backing $8 billion of liabiliies. You have been asked o recommend an opimal asse allocaion sraegy using he liabiliy cash flow as a benchmark. You have he choice of he following four sraegic asse allocaions (SAAs): Sraegic Asse Allocaion ( ) m Expeced Reurn Sandard Deviaion of E R Reurn( σ ) m Asse Duraion Liabiliy Duraion 1 9% 17.3% % 15.0% % 10.0% % 5.0% 3 5 Porfolio Composiion SAA 1 SAA 2 SAA 3 SAA 4 Equiies 30% 25% 20% 15% Corporae Bonds (NAIC 2) 40% 45% 55% 65% Srucured Asses (NAIC 5) 30% 30% 25% 20% Toal 100% 100% 100% 100% You are given he following invesmen objecives and consrains: Invesmen objecives: based on he liabiliy s required reurn o mainain policy persisency, you are asked o arge a 6% minimum risk-adjused reurn wih 1% arge racking risk. You are also given he following uiliy funcion o deermine risk-adjused expeced reurns: 2 ( ) 0.01σ E R m m Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

15 8. Coninued Sauory consrains: he company arges o mainain he curren S&P raing of A by minimizing he use of Risk-Based Capial (RBC). The followings are he curren RBC charges applicable o his block of business: RBC Capial C-1 Risk Charge (Pre-ax) NAIC 1 0.4% NAIC 2 1.3% NAIC 3 4.6% NAIC 4 10% NAIC 5 23% NAIC 6 30% Oher 30% RBC Capial C-3 ALM Risk Charge The Sauory capial assessmen concludes ha he C-3 facor under he sochasic capial calculaion is always smaller han he 0.6 facor prescribed by he NAIC. (a) (2 poins) (i) (ii) Explain wha racking risk is and he meaning of he arge racking risk specified in he invesmen objecives, assuming normaliy of excess porfolio reurns. Sugges ways o achieve a lower racking risk. (b) (3 poins) (i) (ii) Recommend one of he four sraegic asse allocaions solely based on he invesmen objecives and consrains lised above. Criique your recommendaion in par (i) from he ineres rae risk perspecive. (c) (2 poins) Recommend wo marke insrumens for hedging he ineres rae risk inheren in he recommendaion under par (b). Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

16 9. (5 poins) The Organic Cauliflower Grower Associaion of Greenminded Valley (OCGAGV) represens he ineres of is member producers. I adverises he producers, sabilizes he income of is members and promoes healhy living and environmenal value. They creaed a coningency fund o sabilize he revenue of he members. Members are expeced o conribue o he fund, according o heir income, during profiable years. In reurn, during non-profiable or less profiable years, he fund will provide suppor paymens o he members in relaion o heir needs. This fund has no planned mauriy. This associaion also encourages among heir members he use of he mos ecological mehods. As long as eiher fund is adminisered solely for he benefi of is members, his associaion is no subjec o ax. (a) (2 poins) Develop a suiable invesmen policy for he coningency fund. Lis all elemens. You are given a universe of possible asses: Symbol Socks μ σ DDP Deep dig peroleum 10% 35% ST Science TV 8% 30% AIR African Indian Rice 7% 25% Symbol Bonds Raing Coupon Price Yield o mauriy Term Duraion GP Greenpeace A 5.00% % UST U.S. Treasury AAA 3.00% % HT Here Telecom BB 7.00% % Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

17 9. Coninued And he correlaion marix among hem and wih he cauliflower producion (OCGAGV): ρ DDP ST AIR GP UST HT OCGAGV DDP 1.00 ST AIR GP UST HT OCGAGV (b) (c) (2 poins) Discuss he appropriaeness of each asse for he coningency fund. (1 poin) Discuss he appropriaeness of overweighing bonds relaive o equiies o suppor he liabiliies: (i) In he presence of a defici (ii) In he presence of a surplus **END OF EXAMINATION** Exam QFICORE Fall STOP Quaniaive Finance and Invesmen Core

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