MORNING SESSION. Date: Wednesday, October 30, 2013 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES

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1 SOCIETY OF ACTUARIES Quaniaive Finance and Invesmens Core Exam QFI CORE MORNING SESSION Dae: Wednesday, Ocober 30, 013 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 11 quesions numbered 1 hrough 11. b) The afernoon session consiss of 6 quesions numbered 1 hrough 17. The poins for each quesion are indicaed a he beginning of he quesion.. 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.. 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 since 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 QFI Core. 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. 013 by he Sociey of Acuaries Prined in he U.S.A. 475 N. Maringale Road Exam QFI Core-Fron Cover Schaumburg, IL

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3 **BEGINNING OF EXAMINATION** 1. (4 poins) (b) (c) (1.5 poins) Describe each of he following: acual volailiy, implied volailiy and realized volailiy. Idenify how hey differ in erms of ime period, number of associaed ime scales, and abiliy o be measured. (1.5 poins) You wish o esimae 60-day fuure volailiy using hisorical daa. Describe ARCH and EWMA including heir advanages and disadvanages. (1 poin) Idenify and specify a common model ha combines he feaures of ARCH and EWMA. Exam QFI Core Fall GO ONTO NEXT PAGE Quaniaive Finance and Invesmens Core

4 . (7 poins) You are given he following: Sock price S is modeled as a geomeric Brownian moion as follows: δ ds = μ+ S d+ δs dw where o W is a Wiener process o μ and δ are posiive consans Discouned sock price process D is defined as, o r is a posiive consan wih r < μ. Se r, where f ( S T ) is he payoff of a derivaive conrac ha maures a ime T. (1 poin) Sae he Girsanov s Theorem. (b) ( poins) Show ha D is no a maringale. (c) ( poins) Demonsrae how o use Girsanov s Theorem o conver D maringale. ino a (d) ( poins) Explain how you can consruc a replicaing sraegy for he derivaive. Exam QFI Core Fall GO ONTO NEXT PAGE Quaniaive Finance and Invesmens Core

5 3. (7 poins) Le W be a Brownian moion and ( ) T σ (, ) 0 Io-inegral for he process σ ( S, ). I T = S dw be a well defined (1 poin) Sae he condiions on he process ( S, ) defined. (b) (1 poin) Deermine he mean and variance of I ( T ). σ for which ( ) (c) (1 poin) Deermine wheher or no WdW is an Io-inegral. T 0 I T is well (d) ( poins) Suppose ha F is a soluion of he sochasic differenial equaion: Le V V ( F, T) df = σ F dw = be a wice coninuously differeniable funcion and define a new process Y by Show ha equaion. r (, ) Y = e V F Y is a maringale if and only if (, ) V F saisfies he parial differenial V F rv 0 V σ + = F (e) ( poins) Le T > 0,, and suppose ha V saisfies he equaion in (d) for < T, while for = T we impose he boundary condiion: (, ) g( F) V F T = where g is some given funcion. r( T ) Show hav( F, ) = e E[ g( F ) I ]. T Exam QFI Core Fall GO ONTO NEXT PAGE Quaniaive Finance and Invesmens Core

6 4. (6 poins) For a specific mandae, you need o produce monhly ineres rae scenarios in a real world conex. You have seleced an AR(1) model: r = φ + φ + ε 0 1r 1 wih ε having a normal disribuion wih mean 0 and variance of σ. ( ) To deermine he parameers se of he model θ { φ,, 0 φ1 σ } = you have access o monhly hisorical daa of he ineres rae ( r1, r,, r T ). (1.5 poins) Deermine he mean and variance of he uncondiional disribuion of r., ha you will need o maximize o obain he MLE parameers from all he daa provided. (b) (1.5 poins) Develop he log-likelihood funcion, ln f ( r,, rt r1, θ ) (c) (d) (.5 poins) Deermine he sysem of algebraic equaions ha needs o be solved o ge he MLE parameers for he AR(1) model. (0.5 poins) Calculae he half-life measure of he speed of mean-reversion if he θ = 0.015, 0.7, esimaed parameers are { } Exam QFI Core Fall GO ONTO NEXT PAGE Quaniaive Finance and Invesmens Core

7 5. (4 poins) Suppose ha he quarerly log earnings w of airline company Easke follows he following model: 4 4 ( ) ( ) w = (1 B) 1 B y = ( B) B a a follows a normal disribuion wih mean 0 and variance of σ. 5 ( σ) = ( a 1) ( σ 1) The las 5 log earnings, residuals and volailiies are summarized in he following able: ime y a σ (b) (0.5 poins) Idenify he periodiciy and he parameers of he ime series. (1.5 poins) Calculae he 1-sep ahead predicion of he log earnings, assuming ha he forecas origin is 50. (c) ( poins) Deermine he Auocorrelaion Funcion (ACF) of w. Exam QFI Core Fall GO ONTO NEXT PAGE Quaniaive Finance and Invesmens Core

8 6. (5 poins) Consider he Vasicek spo rae model, α( μ ) parameers α,, dr = r d + σdw, where he μ and σ are known posiive consans. W is a Wiener process. (4.5 poins) Show for < sha ( s ) E r r = μ+ ( r μ) e α s ( ) σ ( s ) Var rs r 1 e α = α (b) (0.5 poins) Describe wha hese wo equaions imply for he condiional mean and variance of spo raes as s. Exam QFI Core Fall GO ONTO NEXT PAGE Quaniaive Finance and Invesmens Core

9 7. (6 poins) (1 poin) Describe he reasons for and agains yield curve fiing using a onefacor ineres rae model. You are doing yield curve fiing work wih a Ho & Lee spo ineres rae model. The process for he risk-neural spo rae is dr = η () d + cdx, where he sandard deviaion of he spo rae process, c, is consan, and he drif rae η is ime dependen. The soluion of he bond pricing equaion for a zero-coupon bond is T (,; ) Z r T = e where AT ( ; ) = η ( s)( T sds ) + c ( T ) ( ; ) ( ) A T rt (b) (3 poins) Show ha he funcional form for η () mus be η () c ( ) log Z M ( ; ) * * * = in order for zero coupon bonds for all mauriies o have he correc value. where *, o be fied, is oday s dae, and * ZM ( ; ) are he discoun facors in he marke. (c) ( poins) Express A( T ; ) in erms of * * M Z ( ; T), Z ( ; ),, T and c. M Exam QFI Core Fall GO ONTO NEXT PAGE Quaniaive Finance and Invesmens Core

10 8. (4 poins) ( poins) Describe how he following derivaives can be used o manage he duraion of a bond porfolio and commen on heir advanages and disadvanages. (i) (ii) (iii) Ineres Rae Fuures Ineres Rae Swaps Ineres Rae Opions You are given he following informaion: A porfolio consiss of bonds A, B and C: Bond Duraion Marke Value A 5 1 million B 7 million C million The arge dollar duraion of he porfolio is 4.3 million. The cheapes o deliver bond underlying he ineres rae fuures conrac has he following properies: The price is 100,000 The duraion is 6, and The conversion facor is 1.1. (b) ( poins) Calculae he number of fuures conracs he porfolio manager should buy or sell o achieve he arge duraion. Exam QFI Core Fall GO ONTO NEXT PAGE Quaniaive Finance and Invesmens Core

11 9. (7 poins) Bull & Bear Life Insurance Company inroduces an Equiy Indexed Annuiy (EIA) wih 10-year mauriy. The annual crediing rae is equal o: Min(Max(3%, ACRC), Max(0.5%, IR)), where: IR is 1-year oal reurn on S&P 500 index from he previous anniversary; and ACRC (Annual Crediing Rae Cap) is deermined in he beginning of each anniversary. Invesmen Sraegy = 1-year zero-coupon bond Ineres Rae = % fla across all he erm srucure (coninuous compounding) S&P 500 Dividend rae = 3% (coninuous compounding) S&P year implied volailiy = 0% Targe profi margin and expenses combined are 0.9% per year. No early wihdrawal is allowed. You may approximae opion pricing using Greeks. (b) (c) (1 poin) Calculae he saic hedging budge for he firs year o achieve he arge profi margin and expenses. (1 poin) Recommend a saic hedging sraegy for his crediing rae policy. (3 poins) Deermine he firs year cap of he crediing rae (ACRC) given he profi margin and expenses. The company is considering invesing in he 10-year Treasury bond o pick up addiional yield and adding a feaure ha allows early wihdrawals wih no penalies or adjusmens. (d) ( poins) Assess he disinermediaion risk and recommend a risk miigaion sraegy. Exam QFI Core Fall GO ONTO NEXT PAGE Quaniaive Finance and Invesmens Core

12 10. (6 poins) JNH offers a diversified porfolio of producs from paricipaing whole life, universal life and variable universal life o fixed and variable annuiies. A new CIO, Mr. Yao, has recenly heard abou porfolio segmenaion and is debaing wheher he should adop his approach for JNH. (b) (1 poin) Describe he porfolio segmenaion approach. (1.5 poins) Recommend and jusify your recommendaion o Mr. Yao o adop he porfolio segmenaion approach. The asse allocaion of he paricipaing insurance segmen is as follows: Asse Classes Curren Allocaion Permissible Range Cash and Shor Equivalens % 1%-5% Public Bond 30% 6%-34% Privae Placemen 30% 6%-34% Morgage 0% 16%-4% Equiies 18% 14%-% The paricipaing insurance segmen has an overall implici annual reurn guaranee of 5%. The porfolio yield is a 5.6% currenly. Mr. Yao would like o perform a acical asse allocaion shif o express his views on he capial markes. He expecs credi spread o widen. Therefore, he plans o emporarily re-allocae asses from privae placemen o public bond as much as permied. A 1% allocaion o public bond from privae placemen is expeced o resul in an 8 bps reducion in he overall porfolio yield. There is also a 0 bps ransacion cos for such a acical adjusmen. (c) (d) (1.5 poins) Recommend a acical asse allocaion for he paricipaing insurance segmen ha incorporaes Mr. Yao s plan and he segmen s objecives. (1 poin) Evaluae wha addiional informaion should be considered before implemening a acical asse allocaion for JNH. Mr. Yao wans o use fixed income Exchange Traded Funds (ETFs) o perform he reallocaion, bu worries abou he ETF prices. (e) (1 poin) Describe hree facors ha affec he price a which an ETF rades. Exam QFI Core Fall GO ONTO NEXT PAGE Quaniaive Finance and Invesmens Core

13 11. (4 poins) Your firm, which operaes sricly in he bond universe, has received a mandae o manage $100 M of asses from a pension plan. Your managing parner has given you he ask o develop he benchmark ha will be used o rack he performance of his mandae. (1 poin) Describe he consideraions ha come in play when selecing a benchmark. I has been decided ha he performance of he mandae/porfolio will be benchmarked on index ABC. (b) ( poins) Lis and describe he various sraegies ha can be used when creaing a fixed income porfolio based on an index. The head of your firm has saed ha, based on he excepional abiliies of is saff, he porfolio should be managed acively. (c) (1 poin) Idenify he advanages and disadvanages of acive managemen of fixed-income porfolios. **END OF EXAMINATION** Exam QFI Core Fall STOP Quaniaive Finance and Invesmens Core

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