MORNING SESSION. Date: Wednesday, May 4, 2016 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES

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1 SOCIETY OF ACTUARIES Exam QFICORE MORNING SESSION Dae: Wednesday, May 4, 016 Tme: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Insrucons 1. Ths examnaon has a oal of 100 pons. I consss of a mornng sesson (worh 60 pons) and an afernoon sesson (worh 40 pons). a) The mornng sesson consss of 11 quesons numbered 1 hrough 11. b) The afernoon sesson consss of 6 quesons numbered 1 hrough 17. The pons for each queson are ndcaed a he begnnng of he queson.. Falure o sop wrng afer me s called wll resul n he dsqualfcaon of your answers or furher dscplnary acon. 3. Whle every aemp s made o avod defecve quesons, somemes hey do occur. If you beleve a queson s defecve, he supervsor or procor canno gve you any gudance beyond he nsrucons on he exam bookle. Wren-Answer Insrucons 1. Wre your canddae number a he op of each shee. Your name mus no appear.. Wre on only one sde of a shee. Sar each queson on a fresh shee. On each shee, wre he number of he queson ha you are answerng. Do no answer more han one queson on a sngle shee. 3. The answer should be confned o he queson as se. 4. When you are asked o calculae, show all your work ncludng any applcable formulas. When you are asked o recommend, provde proper jusfcaon supporng your recommendaon. 5. When you fnsh, nser all your wren-answer shees no he Essay Answer Envelope. Be sure o hand n all your answer shees because hey canno be acceped laer. Seal he envelope and wre your canddae number n he space provded on he ousde of he envelope. Check he approprae box o ndcae mornng or afernoon sesson for Exam QFICORE. 6. Be sure your wren-answer envelope s sgned because f s no, your examnaon wll no be graded. Tournez le caher d examen pour la verson françase. 016 by he Socey of Acuares Prned n he U.S.A. 475 N. Marngale Road Exam QFICORE-Fron Cover Schaumburg, IL

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3 **BEGINNING OF EXAMINATION** 1. (5 pons) Consder he sock prce ha S 1.4S1 wh probably S1 wh probably 0.7 where 1,, wh S The rsk free rae for each perod s %. (a) ( pons) Deermne f each of he followng s a marngale. Jusfy your answer. () () under he real-world measure. () Z ES I under he real-world measure. Consder a pu opon on hs sock wh a srke prce 10 and expraon a =. (b) (c) ( pons) Consruc a porfolo o replcae he opon, usng he sock, rskfree borrowng and lendng, and assumng he absence of arbrage. (1 pon) Descrbe suaons n whch rsk-neural prcng s no vald. Exam QFICORE Sprng GO ON TO NEXT PAGE

4 . (6 pons) Consder an economy of wo counres, A and B from he perspecve of an nvesor n counry A. The currency exchange spo rae un of B s currency) follows: X (expressed n number of uns of A s currency per dx P Xd XdWX. X A non-dvdend-payng sock D whch s raded n counry B (expressed n B s currency) follows: dd D d dw, P D D D P P where x, x, D and D are consans wh x 0 and D 0, WX and WD are P- measure Wener processes wh nsananeous correlaon (.e. =). A Counry A Savngs Accoun M avalable n counry A (expressed n A s currency) follows: dm M A A rd,wherer s consan. A A B Counry B Savngs Accoun M avalable n counry B (expressed n B s currency) follows: dm M B B rd,wherer s consan. B B B Le N be he Counry B Savngs Accoun expressed n he Counry A currency. Le F DX be he sock D prce expressed n he Counry A currency. Assume ha he markes are complee. (a) (1 pon) Derve he dynamcs of B N under he P-measure. Exam QFICORE Sprng GO ON TO NEXT PAGE

5 . Connued (b) (1 pon) Derve he dynamcs of X under Counry A s marngale measure (Q ) (c) (1 pon) Derve he dynamcs of F under Counry A s marngale measure (Q ) (d) (1 pon) Derve he dynamcs of D under Counry A s marngale measure (Q ) Le G 0 T T T 0 has he payoff: G X D K srke prce. denoe he value of he forward conrac a me, whch a me T n he currency of counry A, wh K he 0 T (e) ( pons) Derve he far srke prce, K*, a me 0 of hs forward conrac. Exam QFICORE Sprng GO ON TO NEXT PAGE

6 3. (9 pons) Suppose ha he forward neres rae F; T and he equy sochasc dfferenal equaon (SDE) dynamcs below: 0 s 0 df ; T m, T d, T dx wh F 0; T F 0, ds r S d S dw wh S 0 s 0 S follow he where X W 1 Z wh W and Z beng ndependen Brownan moons under he rsk neural measure ; 1 1, s 0 are consan parameers; and,, T, mt T sds. n Le a P, T where n : 1 maury T a me. PT,,1 n, s he zero coupon bond prce wh (a) (1 pon) Derve Gven ha d a n erms of d and dx. n : a S n : d log A d B dw C dz (b) (3.5 pons) Derve expressons for A, B, and C. (c) ( pons) Derve d a n : S based on your answer for par (b). We would lke o prce an opon wh a payoff equal o max a ST maury me T.,0 a opon : Tn (d) (.5 pons) () () Derve he equvalen marngale measure for evaluang he expecaon of he payoff of hs opon usng Grsanov Theorem and all he prevous resuls. Descrbe how he expecaon of hs payoff can be evaluaed usng he resuls n par (d)(). Exam QFICORE Sprng GO ON TO NEXT PAGE

7 4. (4 pons) You are gven he followng wo sochasc dfferenal equaons for X and Y respecvely:, dx Yd X dw dy X d dv where W and V are ndependen sandard Wener processes. 1 Y Le Z W V and F X e. (a) (1.5 pons) Derve he sochasc dfferenal equaon for F n erms of Z. (b) (1 pon) Calculae he mean and varance of Z Z (c) (1.5 pons) Calculae he mean of Z Z Z s for 0 s and s 0. for 0 and s 0. Exam QFICORE Sprng GO ON TO NEXT PAGE

8 5. (7 pons) Le W be a sandard Wener process, and T be a fxed me n he fuure. Defne a paron 0, 1,..., n of he nerval 0,T such ha T for 0,1,..., n. n You are gven he followng: I. W W W 1 II. W h E We he for any 0,1,,..., n 1, where T h n III. T WT T Ee 1 e 1 X IV. Expeced value of a lognormal dsrbuon s Ee X N (, ) Noe ha n1 W E e W 0 n1 j W W W j Ee W E e W W j 0 j e where (a) (b) (1.5 pons) Calculae (.5 pons) Calculae E e W n1 W 0. E e W e T n1 W W T 1 0. Exam QFICORE Sprng GO ON TO NEXT PAGE

9 5. Connued T Ws WT (c) ( pons) Show ha e dw e 1by provng ha W T 0 T converges o e 1 n mean square convergence. s S T n1 0 e W W s T T Ws WT W W (d) (1 pon) Show ha e dw 1 0 S e by provng ha de e dw usng Io s Lemma. Exam QFICORE Sprng GO ON TO NEXT PAGE

10 6. (5 pons) Company ABC has sgnfcan equy holdngs n Company XYZ. ABC s consderng hedgng s poson by radng equy opons on XYZ s sock. Curren marke condons are as below: Yelds of non-coupon payng Treasury secures wh dfferen maures: (All raes are annual connuous.) Term Yeld (%) 1 monh 3 monh 6 monh 1 year year 3 year 5 year 7 year 10 year 0 year Impled volaly of a-he-money (ATM) equy opons on XYZ s sock wh dfferen maury: (All raes are annualzed.) 1-monh 3-monh 1-year 5-year 15-year ATM volaly 0% 1% % 3% 4% The relaonshp beween he mpled volaly and he moneyness gven erm o maury can be descrbed usng he followng equaon: (, ) =(1, ) + 1 () where = s he rao of fxed srke prce K o he underlyng sock prce S s he erm o maury of he opon () s volaly adjusmen, expressed as a funcon of erm o maury and () >0 Exam QFICORE Sprng GO ON TO NEXT PAGE

11 6. Connued ABC can borrow funds a 50 bass pon spread over he Treasury rae wh same erm of maury. ABC can loan ou s fund a 30 bass pon spread over he Treasury rae wh same erm of maury. ABC operaes s busness n a counry where ncome axes are leved on nvesmen ncomes ncludng dvdend, neres and capal gans. The bd-ask spread n he marke s 1%. By regulaon, marke parcpans are only allowed o shor-sell he socks when he mos recen movemen beween raded prces was upward. A large corporaon TDD s n a merger negoaon wh company XYZ. If he merger s successful, all XYZ shares wll be bough back and shares of he new eny wll be ssued. (a) (.5 pons) Descrbe fve dfferences n he acual marke condons above from he Black-Scholes assumpons. Informaon on a European call opon on XYZ sock s gven as below: The opon now has 1 year lef before expraon Underlyng XYZ sock prce a ssue = 100 Curren XYZ sock prce = 110 The rao deermned a ssue = 1.1 The volaly adjusmen scheme s gven as follow: 1-monh 3-monh 1-year 5-year 15-year Volaly adjusmen () 11% 1% 13% 14% 15% There are wo choces of volaly o use for calculang he opon value: I. The one-year volaly II. Arhmec average of he volales unl expraon (b) (c) (1.5 pons) Calculae he volaly under he above wo assumpons, respecvely. (1 pon) Deermne whch volaly should be used o prce he opon and sae why. Exam QFICORE Sprng GO ON TO NEXT PAGE

12 7. (4 pons) Consder European opons on a non-dvdend payng sock wh srke prce, expry me T, rsk free neres rae r and volaly. You are gven he followng deny rt Ke d S d where x 1 1 x e, d1 and d are defned as n he Black Scholes formula. (a) ( pons) Show ha, a me, he hea c of a call opon s: d1 rt c S rke Nd T (b) (1 pon) Derve an expresson for he hea p of a pu opon. (c) (1 pon) Calculae lm and S 0 c lm S 0 p Exam QFICORE Sprng GO ON TO NEXT PAGE

13 8. (6 pons) The shor rae r follows a one-facor Brownan moon process wh consan drf a and consan volaly : dr a d dw,. The arbrage-free prce PT a me of a zero coupon bond maurng a me T s gven n he followng formula: where, exp (, ) ( ) ( ) PT AT T r at AT, T 6 Le F, T be he arbrage-free forward rae a me for he perod from o T. 3 (a) ( pons) Derve he formula for F, T (b) (1 pon) Show ha, df T T d dw dp rd T dw P (c) (1 pon) Show ha (d) ( pons) Prove F, T sasfes he condon relang dffuson o drf n he HJM forward rae model. Exam QFICORE Sprng GO ON TO NEXT PAGE

14 9. (4 pons) Consder a Vascek Model: 1 dr r d dx where X s a Wener process, and, and are consans wh 0. (a) ( pons) Show ha for T, 1 s r r e e e dx A forward conrac s wren on a zero-coupon bond whch has prncpal 100 and maures n 0 years. The forward conrac maures n 10 years and he conrac prce s 58. s (b) ( pons) Calculae he payoff of he forward conrac a s maury, assumng 0.1, 0.01, 0.004, and he prevalng shor rae 10 years from now s %. Exam QFICORE Sprng GO ON TO NEXT PAGE

15 10. (6 pons) You are he nvesmen manager a a prvae Unversy n he U.S., asked wh wrng an Invesmen Polcy for each of he followng wo funds: A fund supporng defned benef (DB) pensons for he faculy and admnsraon saff a he Unversy, and An endowmen o asss underprvleged sudens wh uon (a) (b) (c) (d) (1 pon) Compare and conras your fducary responsbles for each fund. (1 pon) Compare consderaons n seng reurn objecves for DB plan asses and for endowmens. (1 pon) Compare consderaons n seng rsk objecves for DB plan asses and for endowmens. ( pons) Idenfy he followng nvesmen polcy consrans for each fund: I. Lqudy requremens II. III. IV. Tme horzon Tax concerns Legal and regulaory requremens V. Unque crcumsances The DB plan s underfunded, and he Unversy has been usng nvesmen ncome from he endowmen o keep he plan fundng saus from declnng furher. (e) (1 pon) Assess wheher hs s an accepable sraegy. Exam QFICORE Sprng GO ON TO NEXT PAGE

16 11. (4 pons) You are responsble for managng he rsks of a non-agency Resdenal Morgage Back Secures (RMBS) porfolo and a Commercal Morgage Back Secures (CMBS) porfolo of a large nsurance company. (a) (1.5 pons) Compare he non-agency RMBS porfolo and he CMBS porfolo on he relave bass n erms of he followng rsks. () () () Concenraon rsk Ineres rae rsk Prepaymen rsk (b) (1 pon) Descrbe how he followng wo measures can be used n assessng rsk of CMBS. () () Deb Servce Coverage Rao (DSCR) Loan-o-Value (LTV) (c) (1.5 pons) Descrbe a deermnsc, rule-based defaul modelng framework for he CMBS porfolo, projecng DSCR and LTV o deermne one of he hree possble oucomes of he loan: Term defaul Tmely pay-off Maury defaul/loan exenson **END OF EXAMINATION** Exam QFICORE Sprng STOP

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