SOCIETY OF ACTUARIES Quantitative Finance and Investment Advanced Exam Exam QFIADV MORNING SESSION

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1 SOCIETY OF ACTUARIES Exam Exam QFIADV MORNING SESSION Dae: Thursday, Ocober 30, 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 wrienanswer 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 QFIADV. 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 QFIADV-Fron Cover Schaumburg, IL

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3 **BEGINNING OF EXAMINATION** 1. (5 poins) You are he Chief Risk Officer of an insurance company which sells a variey of life insurance producs. Your company uses OTC derivaives o hedge and ransfer various risks. You are currenly in he process of idenifying and measuring counerpary risk. (a) (1 poin) Define: (i) (ii) (iii) Counerpary risk Counerpary exposure Poenial fuure exposure (PFE) (b) (c) (d) (1 poin) Describe wo miigans ha reduce counerpary risk, and explain how each reduces his risk. (2 poins) Describe four main specificaions of a PFE measuremen model. (1 poin) Describe wo main uses of a PFE model. Exam QFIADV Fall GO ON TO NEXT PAGE

4 2. (7 poins) Your company has been using he Black-Scholes model o price variable annuiies. Your boss hinks his model gives a good approximaion of marke prices, bu you disagree. He wans you o explain why he consan volailiy erm in he Black- Scholes model is no appropriae o model he volailiy of a sock price. (a) (1 poin) Explain he concep of volailiy smile. (b) (i) (1 poin) Idenify he empirical fac abou equiy volailiy smiles ha is well illusraed in he graph below monh implied volailiy for sock ABC (Spo price is 1884) Implied volailiy Srike price Calculaed using call opions Calculaed using pu opions Exam QFIADV Fall GO ON TO NEXT PAGE

5 2. Coninued (ii) (1 poin) Idenify he empirical fac abou equiy volailiy smiles ha is well illusraed in he graph below. 17-day and 1-year implied volailiy for sock ABC (calculaed on April 30, 2014) 35.00% 30.00% Implied volailiy 25.00% 20.00% 15.00% 10.00% 5.00% 0.00% 1,550 1,600 1,650 1,700 1,750 1,800 1,850 1,900 1,950 Srike price Expiry: May 17, 2014 Expiry: April 30, 2015 Your eam is using he Black-Scholes model o hedge heir porfolio of variable annuiies invesed in an index wih value S a ime. Since you raised concerns abou he model s consan volailiy assumpion, hey hesiae beween hedging porfolios I and II presened below. For a given variable annuiy wih embedded opion value ψ a ime, he hedging porfolio is given by =Δ SS +Δ BB +Δ XX, Where ΔS, ΔB and Δ X are he quaniies invesed in he index S, he money marke B and he sraddle opion X on he index, respecively. In boh porfolios, Δ B = ψ ΔSS Δ XX. The oher quaniies are indicaed in he able. QUESTION CONTINUED ON NEXT PAGE Exam QFIADV Fall GO ON TO NEXT PAGE

6 2. Coninued Porfolio I II Δ S ψ S ψ Δ S X X S Δ X ψ σ 0 X σ (c) (d) (e) (2 poins) Describe he hedging sraegy of each porfolio, highlighing poenial problems relaed o volailiy hedging, if any. (1 poin) Recommend one of he hedging porfolios. (1 poin) Sugges improvemens o Porfolio II. Exam QFIADV Fall GO ON TO NEXT PAGE

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8 3. (7 poins) You are he CEO of he ABC Life Insurance Company (ABC). Alhough ABC has radiionally been a marke leader in produc design, is new producs have been poorly received repeaedly over he las hree years. You believe ha a reason for his may be he Produc Design Commiee (PDC), which designs ABC s new producs. For he pas hree years, he PDC has consised of en acuaries who have been wih ABC for heir enire career. During ha ime hey were asked o provide expeced ranges of new sales and only provided an accurae range (meaning ha acual sales were wihin he specified range) abou 20% of he ime. This informaion was no shared wih he commiee. (a) (2 poins) Idenify and explain how commiee bias applies o he PDC and causes he PDC o repeaedly design poor producs. One of your underwriers, who has a background in behavioral finance, suggess ha crowds should be used insead of commiees. (b) (2 poins) (i) (ii) (1 poin) Idenify and explain characerisics of crowds ha differ from he PDC. (1 poin) Recommend changes o he PDC o ake advanage of hese crowd characerisics. You decide o implemen your underwrier s suggesion by allowing employees o anonymously make predicions abou sales, rewarding hem wih bonuses based on he accuracy of heir predicion. You observe he following: 1. Employees geing numerous small bonuses are happier han hose employees receiving one large bonus, even if he large bonus is greaer han he sum of he small bonuses. 2. Employees ofen predic he occurrence of exremely unlikely sales oucomes. 3. Employees ofen deposi he earned bonuses ino bank accouns insead of invesing hem ino he equiy marke. (c) (3 poins) Explain each of he above observaions in he conex of Prospec Theory. Exam QFIADV Fall GO ON TO NEXT PAGE

9 4. (6 poins) You are an invesmen acuary a Happy Life, an insurance company in he U.S. Recenly, he CFO has become more ineresed in adding alernaive asses o he company s invesmen porfolio. (a) (2 poins) Describe briefly he four major feaures of alernaive invesmens. The CFO s main goals in adding an alernaive asse class o he invesmen porfolio are he following: (i) (ii) (iii) (iv) Higher overall reurns Willingness o ransfer a larger proporion of he porfolio o long-erm invesmens Moderae increase o risk Good background and supporive informaion abou he asse class An analys in your invesmen deparmen has suggesed privae equiy funds as he believes his asse class is no as well-esablished as some of he ohers, and he poenial benefis may be worh exploring. (b) (c) (2 poins) Evaluae wheher an invesmen in privae equiies mees each of he four goals. (2 poins) Propose a differen alernaive asse class ha aligns wih he CFO goals. Exam QFIADV Fall GO ON TO NEXT PAGE

10 5. (7 poins) You are considering he following ineres rae models for implemenaion a your life insurance company: Lognormal Forward-LIBOR Model (LFM) A G2++ shor rae model (a) (2 poins) Compare and conras he wo models on: (i) (ii) How hey model ineres rae dynamics How hey can be calibraed o marke insrumens You are planning o hedge ineres sensiive insurance liabiliies using swapions wih various expiraions and underlying mauriies. (b) (1 poin) Recommend one of he wo models and jusify he choice. Your manager has decided o use he Lognormal Forward-LIBOR Model. The following able shows he enor srucure and he associaed ATM caple lognormal implied volailiies: Forward raes Fixing in Years( T i ) ATM caple lognormal implied volailiy F () F() % F () % F () % F () % Your manager would like you o parameerize he volailiy surface by using a piecewise consan funcion ha is defined in he following able: Insan. Vols [ 0, T ] [ T, T ] [ T, T ] F() 1 η F () 2 η 2 η F () 3 η 3 η 2 η 1 (c) (1 poin) Calculae η 3 based on he above ables. 2 3 Exam QFIADV Fall GO ON TO NEXT PAGE

11 5. Coninued Now your manager would like you o consider a parameric approach for he volailiy surface. The following equaion will be used o parameerize he volailiy surface: bt ( 1 ) ( ( 1 ) ) i σ i() =Φ i a Ti + d e + c (d) (e) (1 poin) Describe he feaures of his parameerizaion. (2 poins) Describe how you would calibrae his model o caple prices. Exam QFIADV Fall GO ON TO NEXT PAGE

12 6. (8 poins) You are a corporae invesmen acuary a Trading Life Co. researching fixedincome invesmen sraegies in he Energy secor. You are considering rades on bonds issued by Sauce Peroleum Co. and Pai Solar Co. (a) (1 poin) Lis and explain wo reasons why invesors ener negaive basis rade. You are given he following informaion abou four bonds issued by Sauce Peroleum Co.: Mauriy (years) Bond spread (bps) Defaul probabiliy 3% 4% 4% 5% Expeced recovery rae 70% 70% 60% 60% (b) (c) (d) (1 poin) Calculae he CDS Spread (assuming a simplified one-sep ime period approach) and Bond-CDS basis for each bond. (1 poin) Idenify he bond ha offers he bes negaive basis rade arbirage opporuniy and describe he sraegy. (1 poin) Idenify he bond ha offers he bes posiive basis rade arbirage opporuniy and describe he sraegy. You are given he following informaion for a Pai Solar Co. bond and he risk-free yield curve: Mauriy (years) 3 Face value 1000 Annual coupon 4% 1-year spo risk-free rae 1% 2-year spo risk-free rae 2% 3-year spo risk-free rae 2% Z-spread 280bps (e) (f) (2 poins) Calculae he marke price of he 3-year Pai Solar Co. bond. (2 poins) Calculae he par asse swap spread of he 3-year Pai Solar Co. bond. Exam QFIADV Fall GO ON TO NEXT PAGE

13 7. (7 poins) You are he Chief Invesmen Officer of ABC Life. You manage he long duraion invesmen grade bond porfolio of he company. Your mandae is o minimize he Tracking Error Volailiy (TEV) compared o he U.S. Invesmen Grade (IG) credi index. The following bonds are lised in your porfolio wih bes esimae bid-ask spreads: Bond Opion Adjused Bid-Ask spread Benchmark Bond Spread Duraion (basis poins) (Yes\No) A 5 50 No B 5 37 Yes C No Bonds A and B have been consisenly quoed by dealers every business day for he las year while Bond C has no been quoed for hree monhs. You are also given he following: Adjusmen Facor = 1.6 Non-quoed Adjusmen Facor = 1.08 (a) (1 poin) Calculae he Liquidiy Cos Score (LCS) for each bond. You have been alered by he Chief Risk Officer of a poenial sovereign crisis which may cause credi and liquidiy concerns. You anicipae marke illiquidiy and ha rading aciviy will significanly slow down for he nex few monhs. (b) (2 poins) Assess he poenial impac of hese marke condiions on he bonds LCS. You have been asked o esablish a sraegy o survive he prediced adverse marke condiions while saying rue o your mandae and considering cos as a key componen. You have he following choices on how o srucure your porfolio: (i) (ii) (iii) A porfolio rebalanced o manage liquidiy on he basis of issue size, volume and ime of issue. A porfolio rebalanced such ha he argeed porfolio LCS is lower han ha of he Invesmen Grade index LCS. A porfolio rebalanced such ha he argeed porfolio LCS equals ha of he Invesmen Grade index LCS. (c) (4 poins) Assess he appropriaeness of each sraegy and recommend which one o use. Exam QFIADV Fall GO ON TO NEXT PAGE

14 8. (6 poins) The porfolio you manage has exposure o Canadian ineres raes. You have decided o use principal componen analysis (PCA) o assess he ineres rae risk. You are using random variables x1,, x5 o represen changes in he 6-monh, 1-year, 5- year, 10-year, and 20-year raes, respecively. For purposes of he PCA, you have ranformed he random variables x i ino sandardized daa as follows: xi μ xi z i = σ x i You are denoing he principal componens by yi = βi 1z1+ + βi5z5, where z1,..., z 5 are deermined as above. You are given he eigenvecors of he esimaed covariance marix of z, and he corresponding eigenvalues, in he able below: Eigenvalues λ Eigenvecors i β i β i2 β i3 β i4 β i5 (a) (b) (1 poin) Deermine which of he principal componens above are necessary in order o explain a leas 90% of he variabiliy of he erm srucure of ineres raes. (2 poins) Compue he facor srucure for each of he principal componens you deermined in (a). You have deermined ha he mos adverse scenario for your porfolio is a downward parallel shif in he erm srucure. (c) (1 poin) Idenify which componen from your principal componen analysis corresponds o your porfolio s adverse scenario. Exam QFIADV Fall GO ON TO NEXT PAGE

15 8. Coninued You are given he following informaion: Mauriy i 6-Monh 1-Year 5-Year 10-Year 20-Year Curren Ineres Rae 0.93% 1.56% 2.51% 3.00% 3.54% Sandard Deviaion of x i 0.43% 0.41% 0.25% 0.17% 0.12% Average of x i (d) (2 poins) Compue he wo-sandard-deviaion erm srucure movemen relaed o your porfolio s adverse scenario. Exam QFIADV Fall GO ON TO NEXT PAGE

16 9. (7 poins) You are an invesmen acuary a Longeviy Life and are considering ineres rae models for pricing ineres sensiive producs. (a) (b) (c) (1 poin) Describe he primary shorfall associaed wih using a one-facor model. (1 poin) Describe he componens of he G2++ model and explain why hey lead o efficien procedures for pricing zero coupon bonds and caples. (3 poins) Idenify hree poenial approaches ha can be used o calibrae a wofacor G2++ model o real-marke volailiy daa. For each, ouline key consideraions. You decide o develop a G2++ model, however your manager is concerned abou using such a model due o he possibiliy of negaive raes. The uncondiional expecaion of he ime 10 insananeous shor rae is 2% and he corresponding sandard deviaion of he insananeous shor rae is 1%. (d) (1 poin) Calculae he risk-neural probabiliy of a negaive shor rae a ime 10. (e) (1 poin) Ouline a brief response o your manager on he possibiliy of negaive raes. **END OF EXAMINATION** Exam QFIADV Fall STOP

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