INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 05 h November 007 Subjec CT8 Financial Economics Time allowed: Three Hours (14.30 17.30 Hrs) Toal Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1) Do no wrie your name anywhere on he answer shee/s. You have only o wrie your Candidae Number on each answer shee/s. ) Mark allocaions are shown in brackes. 3) Aemp all quesions, beginning your answer o each quesion on a separae shee. However, answers o objecive ype quesions could be wrien on he same shee. 4) Fasen your answer shee/s ogeher in numerical order of quesions. This, you may complee immediaely afer expiry of he examinaion ime. 5) In addiion o his paper you should have available Graph paper, Acuarial Tables and an elecronic calculaor. Professional Conduc: I is brough o your noice ha in accordance wih provisions conained in he Professional Conduc Sandards, If any candidae is found copying or involved in any oher form of malpracice, during or in connecion wih he examinaion, Disciplinary acion will be aken agains he candidae which may include expulsion or suspension from he membership of ASI. Candidaes are advised ha a reasonable sandard of handwriing legibiliy is expeced by he examiners and ha candidaes may be penalized if undue effor is required by he examiners o inerpre scrips. AT THE END OF THE EXAMINATION Please reurn your answer shee/s and his quesion paper o he supervisor separaely.
Q. 1) i) Ouline he cenral role ha he inflaion model s wihin he Wilkie Model ii) The Wilkie Model proposes an AR process for he coninuously-compounded rae of inflaion, I() ha can be wrien as: I()=a + b(-1)+e() Where e()~n(0, σ ) and a, b are consans, wih 1<b<1. (a) Derive an expression for he long-erm average inflaion rae in erms of a and b Explain an economic jusificaion for using an AR process o model inflaion (c) Explain why a model of he above form would no be suiable for share prices (1.5) (4.5) [11] Q. ) The following expression for he price of a discoun securiy is obained by fiing he Cox- Ingersoll-Ross shor-rae model o curren marke prices: P(,T)=exp[A(,T) B(,T)*r ] Where A(,T)= ln [ [10*exp{0.5(T-)}]/[9 exp{0.5(t-)}+1] ] And B(,T)=40[exp{0.5(T-)}-1]/[9 exp{0.5(t-)}+1] (i) Show ha P(,T) is a soluion of : P P P + 0.5rσ rp + a( μ r ) = 0 r r (9) (ii) Following from he above, deduce he values of a, μ, and σ (iii) Derive a formula for he spo rae R(0,T) and deduce he limiing value of he spo rae as T---> infiniy [15] Q. 3) A cerain share pays a dividend every quarer which is 1% of he share price immediaely before he ex-dividend dae. Immediaely before one such ex-dividend dae, he share price is Rs 100. The risk-free ineres rae is 5% per annum. Page of 5
i) Deduce wheher i migh be advanageous o exercise eiher of following American pu opions immediaely: (a) expiry dae in one monh, exercise price=rs 150 expiry dae in hree monhs, exercise price=rs 140 (4) ii) An American pu opion should be exercise immediaely if he underlying share below a criical value S c. No dividends are paid on his share. price falls Explain wih reasons, how he value of S c depends on : (a) (c) he risk-free ineres rae he exercise price of he opion he volailiy of he share price [8] Q. 4) An insiuional fund divides is asses beween an equiy index fund and a bond index fund. The mean and sandard deviaion of he annual reurn from each fund and he correlaion beween reurns is given below: Mean(μ) Sd Dev(σ) ρ EQUITY ρ BOND EQUITY FUND 0.10 0.0 1 0.6 BOND FUND 0.07 0.10 1 The fund can borrow or lend a he risk-free annual rae of 0.05. (i) (ii) Wha is he opimal spli beween he equiy index and bond index funds? Deduce he gradien of he ransformaion line passing hrough his opimal porfolio. (7) [9] Q. 5) Consider a muliple sae, one-period pricing model in which here are wo asses X and Y ha provide payoffs in each of he nex-period saes as follows. Sae Probabiliy of sae occurance X Y Good 4% 1.65.0 Bad 58% 1.0 1.10 Marke price 1.35 1.50 (i) Find he sae prices for boh he good and bad saes. (ii) Find he expeced reurn on each asse. (5) [7] Page 3 of 5
Q. 6) The ime decay (hea) of a European call opion on a share which pays no dividend is given by: 1 σs exp( z ) c θ c = = Er exp( r( T )) Φ( y) π ( T ) Where 1 z= ln( S / E) + ( r + σ ) τ / σ T and y=z-σ ( T ) i) Use pu-call pariy o derive he hea of a European pu opion on he same sock, wih he same expiry dae and exercise price. ii) Derive he limiing values of hea, for boh pus and calls, for deep in-he-money and deep ou-of-he-money opions. iii) The following parameers apply o a European pu opion: S=Rs 100, E=Rs 15, sigma=0.0, T-=1, r=0.05. Esimae by how much he premium of he opion would change in one day if he share price sayed he same. Q. 7) i) Defining all he symbols you use, wrie down for he Arbirage Pricing Theory: (a) he assumed relaionship beween he reurn on a risky asse and he value of N economic facors denoed by F i (where i =1,, N) he formula for he expeced risk premium on he asse, saing he saisical requiremens for his formula o hold. [8] ii) Use he formula you gave in (i) above o show ha he Capial Asse Pricing Model can be hough of as a special case of he Arbirage pricing Theory iii) The annual reurns on asses A and B can be fully explained by he following equaions involving wo economic facors F 1 and F, where E[F 1 ]= E[F ] =0 R A =0.13+ 3F 1 +F R B =0.07+ F 1 +F (a) derive he wo porfolios which exacly follow he movemens in each of he economic facors. if he annual risk-free reurn is 0.05, deduce he expeced reurn on each of hese porfolios. Page 4 of 5
(c) hence verify ha he formula you saed in (i) above gives he correc expeced risk premiums on asses A and B. [1] Q. 8) Lis he hree main ypes of muli-facor asse reurn models and describe he differences in heir approach. [5] Q. 9) i) Sae he assumpions underlying he Black-Scholes opion pricing formula and discuss how realisic hey are. (6) ii) An invesmen bank has wrien a number, N, of European call opions on a non-dividend paying sock wih srike price Rs 140, curren sock price Rs 10, ime o expiry of 6 monhs and an assumed coninuously-compounded ineres rae of 5% p.a. The bank is dela-hedging he opion posiion assuming he Black-Scholes framework holds. I has 00,000 shares of he sock and is shor Rs 190,00,000 in cash. (a) By using he hedging posiion and he Black-Scholes formula for he value of he opion, derive wo equaions saisfied by N and σ, he bank s assumed volailiy. Esimae σ by inerpolaion. (5) (c) Deduce he value of N. [15] Q. 10) i) Use a binomial ree o illusrae and deermine he value a European pu opion using he following parameers: Time o mauriy: 3 monhs Time inerval for ree: 1 monh Risk free rae: 10% p.a., compounded coninuously Curren sock price: Rs 100 Opion exercise price: Rs 110 Dividend: Rs, immediaely afer monh wo Probabiliy of upward move 50% An upward or downward movemen always resuls in a change of 10% (up or down). (hin: he share price drops by Rs afer he dividend is paid a =) (6) ii) Describe in words (no formulae) wha he dela of a derivaive is. iii) Esimae he dela of he pu opion for he second ime sep using he ree in your answer o (i). [10] ********************* Page 5 of 5