FETE Complete Illustrative Solutions Fall 2010

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1 FETE Complete Illustrative Solutions Fall Learnin Objectives: 1. Modern Corporate Financial Theory Learnin Outcomes: (1b) Calculate the cost o capital or a venture or a irm usin the most appropriate method or iven circumstances and justiy the choice o method. Sources: Financial Theory and Corporate Policy, Copeland, Weston, Shastri, 4th Edition, 005, Chapter 15, Ps Commentary on Question: This question was taken straiht out o the Copeland and Weston text Chapter 15 (ps ) however we chaned the numbers. The purpose o this question was to test i the candidates understood the concept o WACC. Many candidates had diiculty with this in prior years. The question required calculations to be perormed to allow us to understand i the candidates understood the concept. The expectation was that candidates who understood these concepts would have little diiculty with this question. Most o the candidates did very well on this question. However, or part (c) below, the common error was to oret that cost o equity capital will increase with hiher leverae. Had we estimated WACC usin old cost o equity capital o 15%, we would have calculated (1-.5)*.06 * *.7 = 8.5% and we would have accepted the project. Solution: (a) Calculate: (i) The company s current weihted averae cost o capital. Usin CAPM: kt = R + [E(Rm) - R]BL =.06 + [ ] x.5 = 10.50% Thereore, the weihted averae cost o capital is: WACC = (1-rc)R B/(b+S) + ks S/(b+S) = (1-0.5) x 0.06 x x 0.85 = 9.375% FETE Fall 010 Solutions Pae 1

2 1. Continued OR Usin M-M: WACC = p(1 - tc B/(B + S)) p = R + [E(Rm) - R]BU BU = BL/(1+(1-tc) B/S) =.5/(1+(1-.5)*.15/.85) = p=.06 + [ ] x.4595 = % WAC =.10136*(1-.5*.15) = 9.38% (ii) The current cost o equity. Usin CAPM: k s = R + [E(R m ) - R ]B L =.06 + [ ] x.5 = 10.50% OR Usin M-M: ks = p + (1-tc)(p - R)B/S p = R + [E(Rm) - R]BU BU = BL/(1+(1-tc) B/S) =.5/(1+(1-.5)*.15/.85) = p=.06 + [ ] x.4595 = % ks = *( )*(.15/.85) =.1050 (b) Calculate the new weihted averae cost o capital. Usin CAPM: WACC = (1-tc)R B/(B+S) + ks S/(B+S) ks = k + new Bl x (Rm-R) BU = BL / (1 + (1 - tc) x (B / S) ) = = 50% / ( * 15%/85%) = 45.95% BL = (1+(1-tc) x (B/S)) x BU = (1 + 50% x 30%/70%) x 45.95% = 55.8% ks =.06 + [ ] x.558 = 11.0% WACC=(1-.5) x 0.06 x (0.3) x 0.7 = 8.61% FETE Fall 010 Solutions Pae

3 1. Continued OR Usin M-M: WACC = p(1 - tc B/(B + S)) This assumes p calculated in 1a. I not: p = R + [E(Rm) - R]BU BU = BL/(1+(1-tc) B/S) =.5/(1+(1-.5)*.15/.85) = p=.06 + [ ] x.4595 = % WACC = 10.14% x [ x.3 ] = 8.61% (c) Recommend whether or not the company should invest in the project, and justiy your recommendation. Since 8.50% < 8.61% The new project with its 8.5% rate o return will not be acceptable even i the irm increases its debt to total assets rom 15% to 30%. Need to incorporate hiher cost o equity capital due to hiher leverae. FETE Fall 010 Solutions Pae 3

4 . Learnin Objectives: 1. Modern Corporate Financial Theory Learnin Outcomes: (1h) Recommend a speciic leal orm o oranization and justiy the choices. (1j) Identiy sources o aency costs and explain methods to address them. Sources: FET-16-08: Financial Markets and Corporate Stratey, Grinblatt & Titman, nd Edition, Ch. 18, How Manaerial Incentives Aect Financial Decisions FET : Meinson, W. L., Corporate Finance Tehory, Ch., Ownership, Control and Compensation Commentary on Question: The question was derived rom study notes FET and FET This question was about hal recall and hal comprehension. This question covers learnin outcomes (1h), and (1j) which have rarely been tested. The perormance o the candidates on this question was very ood. Candidates enerally did well on part (a). A common mistake was to reverse the advantaes o S-Corp and C-Corp. Some candidates did not justiy the recommendation. Also many candidates discussed tax advantaes in the recommendation but did not list it in the advantaes. Part (b)(i) was enerally answered well and or (b)(ii) the question asked or a list, but candidates enerally ocused on 3 bi cateories without any urther elaboration. For part (b)(iii) most o the candidates ave internal mechanisms rather than external so they did not do well. Note that the answer below shows all the pieces that were acceptable, however it only took slihtly less than hal o the bullet points below to receive ull credit. Solution: (a) For this venture that you plan on undertakin: (i) List the advantaes and disadvantaes o choosin each o the orms. 1. Partnership Advantaes: No leal distinction each can execute contracts Partnership income is taxed only once, at the personal level Allows a lare number o people to pool expertise and capital FETE Fall 010 Solutions Pae 4

5 . Continued Partnerships are competitive in knowlede intensive business which is the type o business an actuarial irm would be Business isn t automatically terminated ollowin the death or retirement o a partner i there is a partnership areement Cheaper to establish (compared to corporations) Disadvantaes: Limited lie Limited access to capital (restricted to retained proits and personal borrowins); cannot do an IPO Unlimited personal liability they are not willin to lose personal property No leal distinction each can execute contracts, but each is personally liable or all debts o the partnership. Limited Partnership Advantaes: Allows a lare number o people to pool expertise and capital Partnerships are competitive in knowlede intensive business which is the type o business an actuarial irm would be Business isn t automatically terminated ollowin the death or retirement o a partner i is a partnership areement Partnership income is taxed only once, at the personal level Cheaper to establish (compared to corporations) Disadvantaes: Costly to establish Diicult to monitor the eneral partner Illiquid Limited access to capital (restricted to retained proits and personal borrowins); cannot do an IPO At least one partner must have unlimited liability nobody is willin to lose personal property 3. S-corporation Advantaes: Taxed as partners once, at personal level Limited liability Flexible can easily become a C-corp once the 35 shareholder ceilin is crossed Perpetual lie eature can prosper or many enerations Separate economic entity can contract individually with manaers, suppliers, customers, etc FETE Fall 010 Solutions Pae 5

6 . Continued Disadvantaes: Restricted inancin - can only have a sinle class o equity outstandin Hih transaction costs Expensive to create Most states levy annual corporate charter taxes Listin ees Reportin expenses with SEC e.. printin, mailin, leal, auditin 4. C-corporation Advantaes: Limited liability Perpetual lie eature can prosper or many enerations Separate economic entity can contract individually with manaers, suppliers, customers, etc Stable / has lots o access to capital; can issue securities, borrow money rom creditors, issue stock Ownership structure can be continually chaned Disadvantaes: Taxed twice, at personal and company level; there s concern over ater-tax proits Hih transaction costs Expensive to create Most states levy annual corporate charter taxes Listin ees Reportin expenses with SEC e.. printin, mailin, leal, auditin Governance problems (i they eventually become bi enouh) Separation o ownership and control interests between manaers and shareholders are usually not alined Collective action problem (i they eventually become bi enouh) Best interest o roup or action to be taken (to monitor and discipline manaement), but it is in no individual roup member s rational sel interest to precipitate action FETE Fall 010 Solutions Pae 6

7 . Continued (ii) Recommend and justiy which orm you should choose. (b) (i) S-corp They are only seven (meet the 35 max ceilin requirement) Limited liability Taxed only once In Limited Partnership, passive partners cannot have an active role Can easily convert an S-corp into a C-corp i the venture is successul (IPO) Describe potential conlicts o interest with your older amily members as shareholders o the company that you and your collee riends may have rom investin the assets o the irm. You and your collee riends (as manaers) may choose to: 1. Make investments that it their own expertise so that they become invaluable I desire to remain on the job is lare Diicult to replace in the uture Rely on implicit contracts & personal relationships in their business dealins, makin it more diicult or potential replacements to complete the deals they initiate/private consumption/diversion o resources. Dierent time horizons Make investments that pay o early, even when they hurt in the lon run Havin avorable inancial results in the short run may allow manaers to raise capital at more avorable rates 3. Dierent risk tolerances, e.. Make investments that minimize the manaer s risk Increase the scope o the irm to reduce chances o bankruptcy, even when it is suboptimal (i.e. they expand aster than they should) Increase scope o irm to increase their compensation, even when it is suboptimal Make investments that are too conservative, not value-maximizin to reduce chance o bankruptcy FETE Fall 010 Solutions Pae 7

8 . Continued (ii) List and explain ways in which compensation or you and your collee riends may be desined to better alin your interests with the interests o your older amily members as shareholders o the company. 1. Cash bonus Reward i business unit perormance exceeds a certain level. Stock options Riht to purchase stock at a ixed price, or a iven number o years expires value-less i stock price doesn t rise hih enouh to make the options in the money 3. Deerred cash or stock options/ olden handcus Payment received i he or she remains with the company or a iven period o time they raise the price a competitor has to pay to entice one o a irm s best employees 4. Golden parachute Cash payment made i executive loses their job; minimizes oppositions to takeovers that would be beneicial to stakeholders 5. Stock appreciation riht / phantom stock / alin interests w/shareholders/ share based compensation Give manaers cash payments that mirror those received by shareholders Minimizes stock ownership dilution 6. Relative perormance contract Determine compensation accordin to how well the executive s irm perorms relative to some benchmark Eliminates risks that are beyond the manaer s control 7. Earnins based compensation (iii) Explain other external mechanisms that can be used as a means o corporate overnance or the oranization. 1. Disciplinary takeovers rom another oranization; crude but eective. Rise in power and assertiveness o institutional investors, i.e. the older amily members Oppose manaement-sponsored takeover deense initiatives/ proxy ihts Challene unwarranted rants o stock options/executive perks Demandin perormance rom poorly perormin manaement teams FETE Fall 010 Solutions Pae 8

9 3. Learnin Objectives: 1. Modern Corporate Financial Theory 5. Eicient Markets & Inormation Theory Learnin Outcomes: (1c) Evaluate various proitability measures includin IRR, NPV and ROE, etc. (5a) Deine capital market eiciency and the value o inormation. Sources: Financial Theory and Corporate Policy, Copeland, Chapter, ps. 7-8 and Chapter 10 ps Commentary on Question: This was an application question. The purpose o this question was to demonstrate an understandin o the value o inormation. The value o inormation is a critical component in the concept o capital market eiciency. The candidates did very well on this question overall. However, areas where the candidates did poorly were: Overall: Not appropriately relectin capital requirements (i.e. time value o money) In part (a): Not rejectin the risks appropriately In part (b): Not relectin the ull period o the license Solution: (a) Calculate the value o the inormation provided by the sotware on each contract underwritten. V( No prediction (probability 50%) Test provides no predictive value Value = 1 (same as V( Bad risk (probability 5%) Probability Value Expected value o a bad risk = 0 * * * (-3) = -1 We should reject the bad risk FETE Fall 010 Solutions Pae 9

10 3. Continued Good risk (probability 5%) Probability Value Expected value o a ood risk = 0.5 * * * (-3) = 3 We should accept the ood risks V() =.5 x x x 3 = 1.5 Value o inormation = 0.5 (b) Recommend whether Saniovese Lie should acquire the three-year license. Without underwritin sotware Year 1 Year Year 3 Total Income Income at end o each year. NPV o income at hurdle rate = $7.40 With underwritin sotware # o cases proit / case Total 1 No Prediction 6 1 Good Risk 3 3 Bad Risk 3-1 -> Reject Year 1 Year Year 3 Total Income Income at end o each year. NPV o income at hurdle rate = $34.5 and subtract $1 or licensin ee to et $33.5 Recommend purchasin the sotware since NPV with ino is reater than NPV without ino Alternative Solution: NPV o inormation value less licensin ee = 1*0.5*(1/ /1.15^ + 1/1.15^3) 1 = 5.85 Since NVP is reater than 0 then accept sotware. FETE Fall 010 Solutions Pae 10

11 4. Learnin Objectives:. Corporate Financial Applications Learnin Outcomes: (e) Apply real options analysis to recommend and evaluate irm decisions on capital utilization. Sources: Copeland, Weston, Shastri, Financial Theory and Corporate Policy, 4th Edition, Ch. 9: Multi-period Capital Budetin under Uncertainty: Real Options Analysis Commentary on Question: This question asked candidates to state the three common assumptions used in real options and then apply their understandin towards a calculation o the value o a project or an option to expand. Overall, most candidates understood what was required o them to earn most o the credit or this question. For part (a), many candidates listed the 3 assumptions, but did not describe the main components o each, which was required to earn ull credit. Also, a small number o candidates listed MAD but did not expand this to Marketed Asset Disclaimer. For part (b), several candidates did not round this iure as directed in the question. For part (c), several candidates did not indicate whether it would be optimal to exercise the option, which would have demonstrated the level o analysis expected or this part o the question. Candidates were not expected to determine whether to exercise the option once it was already determined to be optimal at node H but were not penalized or this additional step. Candidates were enerally split in terms o whether they used discrete or continuous discountin, and althouh the discrete method would have been preerred, candidates were not penalized or usin this approach. Solution: (a) Describe all assumptions that you will use to price this real option. Assumption 1 MAD marketed asset disclaimer Can use the present value o the project (assumin no lexibility to make uture decisions) as a twin security FETE Fall 010 Solutions Pae 11

12 4. Continued Assumption No arbitrae assumption Replicatin portolio correctly values the real option continent on the underlyin Assumption 3 Reardless o the pattern o expected cash lows, we can use recombinin binomial trees to model the evolution o a project over time Properly anticipated prices luctuate randomly/random walk (b) Calculate q, the risk neutral probability, to two decimal places. To calculate the value o the project with the option, we need to calculate q, or the risk-neutral probability, calculated as: q = (1 + r d)/(u d) = ( /1.) / (1. 1/1.) ~ 0.59 (rounded to decimal places) (c) Calculate each o the ollowin, assumin the expansion option is available at each uture stae o the project: (i) The value at each uture stae At each node o the new tree (below), the value o the project is calculated as: Value = Max (Value x , Value without expandin) H F V 0 =? I G J FETE Fall 010 Solutions Pae 1

13 4. Continued The values at each node are: H: Max (43 x , 43) = so expand at H I: Max (300 x , 300) = 300 so do not expand at I J: Max (08.33 x , 08.33) = so do not expand at J F: Max (360 x , q + 300(1-q) ) 1 + r = G: Max (50 x , 50) = (371.60) (50) V 0 = = = (ii) The current value o the project. The value o the project with the option to expand is $307. (iii) The percentae chane in value o the project. The percentae increase in the value o the project is 307 / =.33%. FETE Fall 010 Solutions Pae 13

14 5. Learnin Objectives:. Corporate Financial Applications Learnin Outcomes: () Describe the process, methods and eects o a potential acquisition or reinsurance o a business includin its eect on capital structure, return on equity, price/earnins multiples and share price. Sources: FET : Real and Illusory Value Creation by Insurance Companies, D. Babbel and C. Merrill, Journal o Risk and Insurance, 005, Vol. 7, No.1, 1-1 FET : Tiller & Tiller, Lie, Health and Annuity Reinsurance, Ch. 5: Advanced Methods o Reinsurance Commentary on Question: This question asks the candidate to structure a unds-withheld or modiied coinsurance deal, with particular attention to the capital manaement eects o the structure. This situation requently occurs in practice when companies seek to manae products with heavy capital strain that are sellin well, such as universal lie with secondary (no-lapse) uarantees. The candidates tended to do well on parts (a) and (b). For part (c), see note in italics. Candidate responses were mixed. Solution: (a) List reasons or maintainin assets with the cedin company in a reinsurance structure. Larer sums create concern or the reserve security and reinsurance credit worthiness. Initial transaction requently involves the transer o assets that match the liabilities rather than cash. Leavin the assets with the cedin company allows the reinsurer to take advantae o the avorable existin portolio without incurrin capital ains and losses. Transer o ownership o assets or sellin assets or cash may create unnecessary and untimely capital ains and losses or the cedin company, creatin statutory capital and tax issues. Given the short timerame or execution o most transactions, it may be uneasible to purchase the volume o assets needed i cash were used. Potential Recapture at a later date is more easily accomplished i the assets stay with the cedin company. I cedin company retains a percentae o business, it may be advantaeous to leave all the assets or the underlyin business in a sinle portolio, alinin the cedin company s and reinsurer s views o credited and earned rates. The cedin company may want to continue to manae the assets or a ee.. FETE Fall 010 Solutions Pae 14

15 5. Continued (b) Evaluate the advantaes and disadvantaes to Rioja o the approach in (a). Identiy approach is Modiied Coinsurance. Advantaes Applicable to all plans o insurance Cedin company avoids the necessity o liquidatin or transerrin ownership o assets to the reinsurer when an inorce block is involved Allows the cedin company to retain control o the investment policy or the block o business (especially ood or PAR) Modiied Coinsurance eliminates the reserve credit problem ound in coinsurance i the reinsurer is not licensed in the cedin company s state o domicile Reinsurer may preer Modiied Coinsurance to avoid the necessity o manain assets Disadvantaes Some ind it complicated to administer because o the Modiied Coinsurance adjustment Special transactions are required in the case o surrenders and death Transer o assets back to the reinsurer in the event o treaty termination can create exposure to capital losses or the cedin company Transer o the initial Modiication Coinsurance adjustment to the cedin company can create problems or the reinsurer (c) Calculate the coinsurance percentae (%) required such that the ree surplus set aside is adequate to support sales or one year. This may be solved in a variety o ways: The candidate should either explicitly or implicitly identiy the ollowin: Gain From Operations + Surplus = 0 Note: Full marks were iven or clear set up o this ormula either in a table or alebraic orm. Many candidates made the mistake o settin the ormula as Gain From Operations = Surplus Extendin the above ormula and reconizin that the chane in inorce reserve is oset by investment income (based on the inormation provided that Actual investment income is as expected) and that ceded entries are oset by a Modiied Coinsurance adjustment the ollowin ormula is derived: Prem*coins +Res(t-1)*Allow*Coins +Prem*Allow*coins+Prem*Coins- ResChNB+ResChIn+expenses-Surplus=0 FETE Fall 010 Solutions Pae 15

16 5. Continued Then solvin or the coinsurance % coins=(reschnb+expenses-surplus-prem)/(res(t-1)*allow +Prem*Allow+ResCHNB-Prem) Providin an answer o 3% (d) Describe conditions which should be included in a treaty in order to ain acceptance by the reulators in Rioja s country o domicile. The reinsurer must have an actual obliation to pay beneits should experience reach a certain level. Gains to the reinsurer must be based on the actual experience o the reinsurance. No event, such as insolvency or manaement chane, should automatically terminate the reinsurance in orce. However, reinsurance may be terminated due to a certain level o earnins bein attained or i a warranty is violated. Inorce reinsurance cannot be terminated unilaterally by the reinsurer, except or nonpayment o premiums. Interest paid or credited via the reinsurance should be reasonable in relation to the investment markets o the assets involved. The cedin company should not be orced to pay back losses except or voluntary termination. The relevant siniicant risks o the underlyin policies should be transerred to the reinsurer i they are deemed relevant includin: The candidate could have attained additional credit or listin risks that could be deemed siniicant such as: - Capital loss - Disintermediation - Asset deault risks FETE Fall 010 Solutions Pae 16

17 6. Learnin Objectives: 3. Derivatives and Pricin Learnin Outcomes: (3a) Deine the cash low characteristics o complex derivatives includin exotic options, interest rate derivatives, swaps and other non-traditional derivatives. Sources: Hull, J.C., Options Futures & Other Derivatives, 7th Edition, 008 Ch. 19: Basic Numerical Procedures Ch. 4: Exotic Options Ch. 6: More on Models and Numerical Procedures (6.1, 6., 6.3 only) Commentary on Question: This question tested the pricin an exotic option involvin asset exchane. The candidates needed to demonstrate they knew how the price is aected by (i) the presence o storae cost, and (ii) the chane in correlation coeicient between the two underlyin assets Most candidates did not et that the option value at time 0 is U 0 e -qut + V 0 e -qv T N(d1) - U 0 e -qut N(d). Instead many candidates derived the option value at time 0 usin U 0 + V 0 e -qv T N(d1) - U 0 e -qut N(d), which is only riht when assumin qu = 0. In eneral part (b)(i) was very straiht orward and the candidates did well. Some common mistakes included: Only calculatin the value o the exchane option in part (b)(i), but did not ive the inal value o the contract Usin 1600 and 150 as initial value, rather than 4800 and 5000 Used 4Su and 3Sv instead o Su and Sv In part (b)(ii), most candidates knew that storae cost is similar to dividend, but only a ew candidates knew that it s actually neative (i.e.: qu < 0, qv < 0 ). Many candidates discuss the eects o storae cost in terms o e -qut, but very ew candidates discussed how the relative value o qu, qv aects N(d1) and N(d). In eneral, candidates did very well in part (c). Solution: (a) Derive the value o this contract at issue; deine all symbols. The value o option at expiration is iven as max(u T, V T ) It can be written as U T + max(v T - U T, 0) The option max(v T - U T, 0) has a value only i V T > U T, where subscript implies values are at time T FETE Fall 010 Solutions Pae 17

18 6. Continued The option value at time 0 is e -rt E(U T )+ V 0 e -qv T N(d 1 ) - U 0 e -qut N(d ) where E(U T ) is the expected value o U T in a risk neutral world but E(U T ) = e (r - qu)t U 0 Thereore the option value at time 0 is U 0 e -qut + V 0 e -qv T N(d 1 ) - U 0 e -qut N(d ) d 1 =[ ln(v 0 /U 0 ) + (qu - qv + s /)T ] / (st 0.5 ), d = d 1 - st 0.5 s = s u + s v - ρ s u s v where U 0, V 0 are the assets prices at time 0, qu, qv are the dividend yield o assets U and V respectively s u and s v are volatilities o U and V, respectively, and ρ is the correlation coeicient (b) (i) Calculate the price o a 1-year European option which pays the price o 4 ounces o old or 3 ounces o platinum, whichever is reater. Assume U=old, V=platinum, Inputs are U 0 = 4*150 = 5000, V 0 = 3*1600 = 4800, T = 1, qu = qv = 0, s u = 0., s v = 0.5, ρ = 0.7 s = s u + s v - ρ s u s v = *0.7*0.*0.5 = s = d 1 =[ ln(4800/5000) / ] / = d = = N(d 1 ) = , N(d ) = = 4800* * = option price = = OR Could have also assumed U=platinum V=old, Then inputs are U0 = 3*1600 = 4800, V0 = 4*150 = 5000, d1 =[ ln(5000/4800) / ] / = d = = N(d1) = 0.64, N(d) = \0 = 5000* *0.554 = \option price = = (ii) Describe the eect on the option price i the assets require storae at nonzero cost (without urther calculation). I storae costs exist then qu < 0, qv < 0. Storae cost is essentially a neative dividend yield. The option value = U 0 e -qut + V 0 e -qv T N(d 1 ) - U 0 e -qut N(d ) Thereore option value will increase by a actor o e -qut. i qu=qv. FETE Fall 010 Solutions Pae 18

19 6. Continued The relative value o qu, qv will determine the chane in value o N(d1) and N(d). I qu > qv, then d1 will increase and hence d also will increase I qu = qv, then d 1 will not chane and hence d also will not chane I qu < qv, then d1 will decrease and hence d also will decrease But overall, the option value will increase. (c) Describe how you could make money. Since s = s u + s v - ρ s u s v I ρ is hiher than assumed, the actual volatility o the option will be lower Actual lower volatility means trader's option is over priced To make money, can short sell the trader's option and use the proceeds to invest in the synthetic option by purchasin N(d1) o platinum (V) and N(-d) o old (U) FETE Fall 010 Solutions Pae 19

20 7. Learnin Objectives: 3. Derivatives and Pricin Learnin Outcomes: (3b) Evaluate the risk/return characteristics o complex derivatives. Sources: Hull, J.C., Options Futures & Other Derivatives, 7th Edition, 008 Ch. 7: Swaps Commentary on Question: The question tested the candidates knowlede o an interest rate swap transaction and also the correct theoretical valuation o the swap ater issue. The question also demanded the candidates understand the interest rate swap transaction and whether or not it would be an eective asset liability manaement tool. For part (a), ew candidates received ull credit or this part. Frequently answers were vaue and didn't provide suicient explanation o a company borrowin where it has comparative advantae (e.., loatin) and then swappin to pay the kind o rate (e.., ixed) that it really wants. Also, some candidates did not answer the question as asked but answered based on currency swaps (sometimes points could be awarded, thouh). For part (b), many candidates received ull credit. A common error, however, was assumin that the ain rom the swap had to be split evenly, and so an incorrect swap ixed rate or incorrect swap loatin rate (or both) would be used to orce an evenly split ain. A diaram o the transactions would have been helpul but was not necessary or ull credit. Partial credit was awarded or correctly calculatin total ain even i the answer was otherwise incorrect. It was not uncommon or candidates to et the net eect correct or each party but then not calculate rate reduction or each company (as asked in the question). For part (c), the readin provides two approaches to valuin the swap. One o them, usin a orward rate areement approach, is untenable because it requires knowin the actual 6-month LIBOR rate at the valuation date and that inormation is not provided. The appropriate valuation approach is Valuation in Term o Bond Prices. Nearly every candidate ailed to reconize that the value o the loatin piece o the swap was equal to the notional at the valuation date (and so required no calculations). Few understood how to use the swap rate to determine the -yr zero rate, which is needed or correct discountin. Also, a common error in valuin B(ix) was to not use the ixed rate o the swap entered into (i.e., 7.68%) as the ixed payment, but instead use the swap rate (or bid rate or oer rate) at the valuation date. Most candidates knew that the V(swap) = B(loat) B(ix), and most candidates demonstrated correctly how to use the result o the V(swap) as calculated. FETE Fall 010 Solutions Pae 0

21 7. Continued Solution: (a) Describe and critique the Comparative Advantae arument to explain the popularity o interest rate swaps in capital inancin. The arument is called the comparative-advantae arument. Some companies have a comparative advantae when borrowin in ixed-rate markets. Others have a comparative advantae in loatin rate markets. To obtain a new loan, it makes sense or a company to o to the market where it has a comparative advantae. As a result, the company may borrow loatin (ixed) when it wants ixed (loatin). A swap is then used to transorm a ixed (loatin) rate loan into a loatin (ixed) rate loan. Criticism: The terms o loatin rates bonds are likely to be reneotiated in the event that a credit ratin downrade occurs. In this way, the company payin the ixed rate in the swap doesn t necessarily lock in their cost o unds. (b) Show that a swap rate o 7.36% between Malbec and Valpolicella would lower the borrowin costs or both companies, and calculate the rate reduction or each company. Malbec should borrow ixed at 8.00% because it has comparative advantae (CA) at ixed rate Enter into a swap to receive ixed at 7.36% Pay loatin rate o LIBOR (=L) Malbec s net inancin cost is to pay loatin = 8.00% % + L = L % Malbec s rate reduction usin swap = Usual loatin borrowin cost Floatin cost w/ swap = (L %) (L %) = 0.66% Valpolicella should borrow loatin at L % because it has CA at loatin rate Enter into a swap to pay ixed 7.36%, and Receive loatin o L Valpolicella s net inancin cost is to pay ixed = L % +7.36% - L = 9.7% Valpolicella s rate reduction usin swap = Usual ixed borrowin cost Fixed cost w/ swap = 9.70% - 9.7% = 0.43% (Note, as a check: 0.66% % = 1.09% = (9.70% %) [(L+1.91%) (L+1.30%)].) FETE Fall 010 Solutions Pae 1

22 7. Continued (c) Evaluate whether or not the swap has proved to be an eective asset liability manaement tool or Valpolicella. The swap is eective i the ain in the swap osets the loss in the project. Gain in swap = (value o swap at t=1) (value o swap at t=0) At time 0, the V(swap) = 0. At time 1, B(Floatin) = Notional = 10,000,000, because a payment was just made At time 1, B(Fix) = discountin o remainin 4 ixed payments Need yr zero rate to value B(Fix): The two year swap rate is the averae o the bid and oer rates, 8.17% = (8.15% %)/ The two year swap rate is equal to the two year par yield, thus, or 100 notional, 4.085*[e^(-.5*7.9%) + e^(-1*8%) + e^(-1.5*8.1%)] *e^(-* yr zero) = 100 So yr zero = 8.01% At time 1, B(Fix) = 10,000,000*[3.68% (e^(-.5*7.9%)+e^(-1*8.0%)+e^(- 1.5*8.1%)+e^(-*8.01%))+e^(-*8.01%)] B(Fix) = 9,85,61.64 V(swap) = B(Floatin) - B(Fixed) = 147, V(swap) osets about 77% o loss in project value, thereore the swap was mostly eective FETE Fall 010 Solutions Pae

23 8. Learnin Objectives: 3. Derivatives and Pricin 4. Financial Modelin Learnin Outcomes: (3) Demonstrate mastery o option pricin techniques and theory or equity and interest rate derivatives. (4c) Deine and apply the concepts o martinale, market price o risk and measures in sinle and multiple state variable contexts. Sources: Hull, J.C., Options Futures & Other Derivatives, 7 th Edition, 009 Chapter 1, Wiener Processes and Ito s Lemma Chapter 7, Martinales and Measures Commentary on Question: This question asks candidates to use the market price o risk, Ito s lemma process, and martinale stochastic process rom Chapter 1 and Chapter 7 o Hull s book. Part (a): Many o the candidates received ull credit or this question. Most candidates were able to calculate the market price o risks or both processes, and identiied that they are equal. Some candidates simply wrote down the correct market price o risk, without showin intermediate calculations or explanations why both are equal. In all these cases, we ranted the candidate ull credit. Part (b): About hal the candidates received ull credit or this question. Most candidates were able to at least write down Ito's Lemma. A lot o candidates were also able to ive the ull proo. Some candidates took a short cut by usin the direct result o Ito's lemma under the condition G = ln S, and substitutin the mean and standard deviation into the result. In such cases, we ave only partial credit. Part (c): Many o the candidates received ull credit or this part. Part (d): Very ew o the candidates received ull credit or this question. Some candidates approached the question by takin the lon route - to use irst principle to ind the dierential equal or d(/), and substitutin in parts. In these approaches, the alebra turns out to be messy. FETE Fall 010 Solutions Pae 3

24 8. Continued Part (e): Many o the candidates received ull credit or this question. We awarded ull credit as lon as we saw the mentionin o "zero drit", or "drit less" process. Part (): Many o the candidates received ull credit or this question. Solution: (a) Identiy the market price o risk in the processes above. d dt dz (ormula 7.7 o Hull s) r (ormula 7.8 o Hull s) Since is the same or derivatives with the same source, Market price o risk = (b) Demonstrate the ollowin usin Itô s lemma. 1 dln ( r ) dt dz 1 dln ( r ) dt dz G G 1 G G Ito s lemma dg a b dt b dz x t x (ormula 1.1 o Hulls) x Let F= ln, then F 1 F F 1, 0,, a r, b t 1 d ln df d ln r r 0 dt dz dt dz FETE Fall 010 Solutions Pae 4

25 FETE Fall 010 Solutions Pae 5 8. Continued Let G= ln, then b r a G t G G,, 1 0,, 1 dz dt r dg d ln dz dt r d ln (c) Show that ln d dt dz Simply use (b) and ind the dierence rom (b). d d d d ln ln ) ln (ln ) (ln dz dt r dz dt r 1 1 dz dt ) ( ) ( (d) Demonstrate the ollowin usin Itô s lemma iven : H dh Hdz Let H=e X, then X= ln( /) and ) (, ) (, 0,, X X b a e X H t H e X H

26 8. Continued d dh ( X ) ( e ( x e dz Hdz 1 ) 0 e X X ) dt e dz H dz (e) Explain, in words, why H is a martinale Because there is no drit term / 0. Martinale is a zero-drit stochastic process () Calculate the expected price o H in one week i the current price o is $1 and the current price o is $4. Since this is a martinale process, expectation o uture chane is / is zero, so the best estimate is the current price, which is 1/4 = 3. FETE Fall 010 Solutions Pae 6

27 9. Learnin Objectives: 4. Financial Modelin Learnin Outcomes: (4a) Describe and evaluate equity and interest rate models. Sources: Hull, J.C., Options Futures & Other Derivatives, 7th Edition, 008 Ch. 8: Interest Rate Derivatives: The Standard Market Models Commentary on Question: This question asks candidates to demonstrate the application o Black s model on caplet price to derive implied volatility. Part (a) required simple knowlede and understandin o how a caplet works. Part (b) required the candidates to use ormulas provided to calculate the implied volatility. Overall, this was expected to be a airly basic question. For part (a), this question was made overly complicated by many candidates tryin to solve it rom irst principles; the 1 point value indicated that candidates should be done in about 3 minutes. Less than hal o the candidates ot the correct answer to this part. Because the price o a cap can be decomposed into the sum o caplet prices, the price o the caplet is simply the dierence between the two observed cap prices. Some candidates incorrectly included additional discount actors or totaled all the observed prices. For part (b), the key to this question was identiyin that the normal distribution calls could be simpliied into 1 call that can be back solved or. Most candidates understood how to set up the equation to solve or the vol. Those that ot the ormulas correct many times did not substitute the values correctly or make interpretations o the results. Many candidates struled with the correct value o T[k+1] and the simpliyin relationship between d1 and d. Candidates enerally did not do well on this part, thinkin that it was impossible to solve and overlooked the identity with the normal distribution which allowed it to be solved analytically. Solution: (a) Calculate the time T 1 caplet price. Caplet = Price o Cap[1] Price o Cap[.75] = = 4.5 (b) Calculate the implied volatility or the T 1 caplet or Black s model. Model Parameters are: T.75 T 1.5 P 0, T 1.03 L F R.03 1 k k1 k k1 k k FETE Fall 010 Solutions Pae 7

28 9. Continued The ormula or the d values and their simpliication is: ln F / R Tk / 3 d1 T 4 d d T d 1 k 1 1Nd N d 1 k From the Equation iven and substitution: Caplet L kp 0, Tk 1Fk N d N d1 1 Nd From the Normal Table we approximate d1 and solve or Implied Volatility d / % FETE Fall 010 Solutions Pae 8

29 10. Learnin Objectives: 1. Modern Corporate Financial Theory Learnin Outcomes: (1d) Deine and compare risk metrics used to quantiy economic capital and describe their limitations. (1e) Apply the concept o economic capital and describe methodoloies or allocatin capital within a inancial oranization. Sources: Investment Guarantees, Hardy, 003 Ch. 9: Risk Measures (pp only) Capital Allocation by Percentile Layer, by N. Bodo Commentary on Question: The determination and allocation o dierent types o capital are important to risk manaement at an insurance company. The question tested candidates knowlede o the relationship between required surplus and ree surplus as well as a correct application o the Bodo alorithm. This question proved to be diicult, but a ew candidates were, at a minimum, able to calculate required surplus, ree surplus, losses which pierced the ree surplus, and correspondin conditional exceedence probabilities o those losses. Many candidates expressed concern over the independence o the two distributions, claimin that it would have been too lon a question. However, while this was a lon question as asked it could be answered under the conditions presented. We remind the candidates to answer the question as asked and i they eel a question is too lon to try to demonstrate understandin o the concepts bein tested. The candidates should also know that it was not our intent to absorb hue amounts o time in arithmetic. There were our key concepts required or the successul completion o the question: Free surplus covers company losses in excess o the required capital level. Knowlede and understandin o how to calculate CTE. The problem asks only to allocate the ree surplus usin the capital allocation by layers method; the Bodo text example allocates all required surplus. The capital allocation by layers method requires determinin the worst possible losses that pierce the amount o capital in question and then workin to the next worst, etc. This worst-down approach is clearly seen in the Bodo text. Since the ree surplus was the top layer o capital, we expected the candidates to bein to allocate startin with the reatest losses and workin down similar to the text. We ound that very ew candidates used this approach and oten times started rom the bottom and incorrectly had only one event piercin that level o capital. FETE Fall 010 Solutions Pae 9

30 10. Continued o Candidates who successully answered the question tended to identiy the above requirements to obtain a tractable problem setup: - The ree surplus covers company losses at or just below $13 million. - Then the worst losses, beinnin with $13 million loss (derived rom UL $3 million + Term $10 million), were those that pierced ree surplus and are used to determine the layers and subsequent allocations and conditional exceedence probabilities. o Candidates who had diiculty with the allocation by layers calculations tended to mischaracterize the relationship between ree surplus and the risks/losses that pierced it. This most oten led to layers o losses and conditional exceedence probabilities oriinatin with the $0 UL and Term losses usin the smallest losses which did not pierce ree surplus. Additionally assumin when the exceedence probabilities or the lowest level were calculated, they did not consider all the events that would pierce this level o capital. o Overall, the candidates had diiculty with this question. Some o the this was attributed to the question bein very lon as asked. However, we also believe there was a eneral lack o understandin o how to allocate by capital layers. Because o the overall scores on averae bein very low, this question ultimately had little eect on a candidates overall success or ailure. o The answer below is a complete answer to the question. In order to et ull credit on this question you would have only needed to write enouh to demonstrate mastery o the our key concepts above. Solution: Calculate the total amount o ree capital or Carmenére Lie and allocate the ree capital to each o the two products by the allocation-by-layers method. Required Capital = 137.5% * CTE(97) A review o the loss distributions clearly shows that all combinations o the Term $10 million, $5 million, and $4 million losses are contained in the larest 3%, with a sliht adjustment. Top 0.8% losses = Term $10 million + UL $x million Rane rom $13 million to $10 million Expected value = $10 million + E(UL losses) = $10.54 million Next 0.9% losses = Term $5 million + UL $x million Rane rom $8 million to $5 million Expected value = $5.54 million FETE Fall 010 Solutions Pae 30

31 10. Continued Next 1.3% losses = Term $4 million + UL $x million Rane rom $7 million to $4 million Expected value = $4.54 million Thereore the preliminary CTE(97) = [ 0.8%* %* %*4.54 ] / 3% = $6.154 million O the Term $ million losses, Term $ million + UL $3 million = $5 million (4.5% * 0.8% =.036%) and Term $ million + UL $.6 million = $4.6 million (4.5% * 0.9% =.0405%) are actually in the top 3% replacin the same amount o the Term $4 million + UL $0 loss This adjustment is worth 0.036%*(5-4) %*(4.6-4) / 3% = /3% = 0.00 million Thus the CTE(97) = = million And required surplus = * = million And ree surplus = = million The allocation-by-layers approach is to allocate this $4.511 million which means losses exceedin $8.489 million are those that pierce the ree surplus. Event 1 = Maximum Possible Loss which penetrates into ree surplus = $13 million = Term $10 million + UL $3 million Event = Next larest loss = $1.6 million = Term $10 million + UL $.6 million And similarly: Event 3 = $11. million = Term $10 million + UL $1. million, Event 4 = $11 million = Term $10 million + UL $1 million, Event 5 = $10.5 million Event 6 = 10.5 million Event 7 = 10 million Event 8 = 8 million = Term $5 million + UL $3 million does not pierce ree surplus Thus, there are 7 layers. FETE Fall 010 Solutions Pae 31

32 10. Continued Layer 1 = $13 million - $1.6 million = $400,000 Layer = $1.6 million - $11. million = $1,400,000 Layer 3 = $11. million - $11 million = $00,000 Layer 4 = $11 million - $10.5 million = $500,000 Layer 5 = $ $10.5 = $50,000 Layer 6 = $ $10. = $50,000 Layer 7 = $10 - $8.489 = $1,511,000 Probabilities: P(Event i) = P(Term loss contained in Event i) * P(UL loss contained in Event i) P(Event 1) = * = (Cumulative probability = ) P(Event ) = * = (Cumulative probability = ) P(Event 3) = * = (Cumulative probability = ) P(Event 4) = * = (Cumulative = ) P(Event 5) = * = (Cumulative = ) P(Event 6) = * 0.38 = (Cumulative = ) P(Event 7) = * 0.44 = (Cumulative = 0.008) Only Event 1 penetrates Layer 1 and thereore receives 100% o surplus: Term(1) = Allocation to Term = $400,000 * (share o Term in $13 million loss / Total loss o $13 million) = $400,000 * (10/13) = $308,000 Events 1 and both penetrate Layer and thereore share in the $1.4 million in relation to their share o the losses and the conditional exceedence probabilities: Term() = Allocation to Term = $1.4 million * (Term share o Layer 1) * (Conditional exceedence probability or Layer 1) + (Term share o Layer ) * (Conditional exceedence probability or Layer ) = $1,400,000 * [(10/13)*( / ) + (10/1.6)*( / )] = $1,095,000 Similarly, Term(3) = Allocation to Term = $00,000 * (Term share o Layer 1) * (Conditional exceedence probability or Layer 1) + (Term share o Layer ) * (Conditional exceedence probability or Layer ) + (Term share o Layer 3) * (Conditional exceedence probability or Layer 3) = $00,000 * [(10/13)*( /0.0004) + (10/1.6)*( /0.0004) + (10/11.)*( /0.0004)] = $166,000 FETE Fall 010 Solutions Pae 3

33 10. Continued Term(4) = $500,000 * [(10/13)*( /0.0006) + (10/1.6)*( /0.0006) + (10/11.)*( /0.0006) + (10/11)*(0.0004/0.0006)] = $439,000 Term(5) = $50,000 * [(10/13)*( / ) + (10/1.6)*( / ) + (10/11.)*( / ) + (10/11)*(0.0004/ ) + (10/10.5)*(0.0006/ )] = $30,000 Term(6) = $50,000 * [(10/13)*( / ) + (10/1.6)*( / ) + (10/11.)*( / ) + (10/11)*(0.0004/ ) + (10/10.5)*(0.0006/ ) + (10/10.5)*(0.0006/ )] = $40,000 Term(7) = $1,511,000 * [(10/13)*( /0.008) + (10/1.6)*( /0.008) + (10/11.)*( /0.008) + (10/11)*(0.0004/0.008) + (10/10.5)*(0.0006/0.008) + (10/10.5)*(0.0006/0.008) + (10/10)*(0.0006/0.008)] = $1,476,000 Term allocation = Sum Term(1) throuh Term(7) = million UL allocation = million million = million FETE Fall 010 Solutions Pae 33

34 11. Learnin Objectives:. Corporate Financial Applications Learnin Outcomes: (b) Describe the process, methods and uses o inancial reinsurance (surplus relie) and recommend a structure that is appropriate or a iven set o circumstances. (c) Describe the process, methods and uses o insurance securitizations and recommend a structure that is appropriate or a iven set o circumstances. Sources: FET : Securitization o Lie Insurance Assets and Liabilities P. 3-13, 40-4 FET : Tiller & Tiller, Lie, health and Annuity Reinsurance, Ch. 5: Advanced Methods o Reinsurance P Commentary on Question: This is a recall, application and analysis interated question askin candidates to utilize the concept o inancial reinsurance and securitization to deal with surplus strain problems. In eneral, part (b) o this question was answered well. Many candidates could demonstrate their understandin o inancial reinsurance and securitization structure. However, very ew candidates answered part (c)(iii) and part (d) correctly. Part (c) iii) is a very simple application o option theory. Most candidates did not realize part (d) is related to part (a) to (c) and just repeated part (b). Solution: (a) Determine the reduction in irst year surplus strain in Solution 1. Beore reinsurance ater tax surplus strain = - (Gross Premium + Inv. Inc - Commission - Expenses - Death Beneits - Chane in Reserves) * (1-tax rate) = - ( )* (1-30%) = 1568 Ater reinsurance Net premium (NP) = 4000 * (1-30%) = 800 Net Investment Income (NI) = 80 ModCo adjustments (MA) = 400 * 30% = 70 Commissions (C) = 400 * (1-30%) = 1680 Allowance (A) = 15% * 4000 * 30% = 180 FETE Fall 010 Solutions Pae 34

35 11. Continued Expense (E) = 100 * (1-30%) = 840 Death Beneit (DB) = 30 * (1 30%) = 4 Chane in Reserves (R) = 400 Surplus strain ater tax = - (NP + NI + MA + A - C - E - DB R) * (1 tax rate (30%)) = 955 Reduction in ater tax surplus strain = = 613 (b) Briely describe the economic rationale or securitization. Creation o new classes o securities Facilitate risk manaement Add liquidity to inancial market Improve market eiciency and capital utilization Unlock embedded proits in a block o business Reduce cost o capital Increase ROE, improve other operatin measures (c) Speciic to Carmenére Lie s Solution, (i) Explain how the securitization o the Term Lie in-orce block would address Carmenére Lie s Universal Lie surplus strain problem. Securitization takes the Term Lie cash lows promised to Carmenére Lie in the uture and sells them or $ 7 million Securitization costs o 6.7 million ives net o 0.3 million used to reduce surplus strain (ii) Describe the cash lows (I, II, III, IV) and role each party plays. (I) - Carmenére Lie makes payments equal to the expected mortality costs under the Term Lie block to Rosé Capital (SPV) (II) - Carmenére Lie will receive payments rom Rosé Capital based on actual mortality experiences under the block. FETE Fall 010 Solutions Pae 35

36 11. Continued (III) - Rosé Capital would issue mortality notes to investors (IV) - Investors pay proceeds to Rose Capital or mortality notes (iii) Derive a unction or the option payo in terms o CM (Cumulative Mortality) and D (Actual deaths in the indexed pool o 010). Option payo lower strike piece = MAX (CM - 15% D, 0) Option payo upper strike piece = MAX (CM - 150% D, 0) Option payo = (MAX (CM -15%D, 0) MAX (CM -150% D, 0) / 5%D (d) Outline or 3 advantaes or both solutions and recommend a solution or Carmenére Lie. Mortality securities uncorrelated with inancial markets. Ratin aencies tend to view securitization avorably. Cost o reinsurance is better or insiniicant deals. Reinsurance takes less time to implement. Net decrease in surplus strain is bier or the reinsurance (613K vs. 300 K) or securitization. The reinsurance option can be done in a shorter period o time. Hence, Solution 1 is the recommendation. FETE Fall 010 Solutions Pae 36

37 1. Learnin Objectives: 1. Modern Corporate Financial Theory Learnin Outcomes: (1) Identiy reulatory capital requirements and describe how they aect decisions. Sources: A Comparative Analysis o US, Canadian, and Solvency II Capital Adequacy Requirements in Lie Insurance, Sharara, Hardy and Saunders. Commentary on Question: This question tested the new syllabus material by Shahara, Hardy and Saunders. The question asked or only the components reardin investment risk. Additionally, we did not want the candidates doin a brain dump o everythin out o that study note, hence we reduced the scope o the question and only ave credit or that which addressed the question. A sizable number o candidates misinterpreted the valuation o investment risk to apply to the valuation o the liabilities. Most candidates answered part (b)(i) well. Most candidates did not answer part (b)(ii) in the manner expected, ivin implications o Solvency II rather than actions companies would take to manae the risks. Solution: (a) Describe how investment risk is assessed by the capital adequacy requirements o the Canadian and U.S. reulatory capital reimes. Address: (i) Valuation methodoloies Canada: Assets held or tradin (HFT) or Available or Sale (AFS) AFS held to maturity Realized capital ains/losses not in available capital Unrealized Gains/losses included in available capital US: Assets held at market value, amortized cost, equity method, Book Value (cost) IMR = Moderates impact rom interest related ains AVR = Smooths impact rom credit deaults and equity risk related ains/losses FETE Fall 010 Solutions Pae 37

38 1. Continued (ii) Capital ormulas Canada: C1 Risk Risk o loss resultin rom on-balance sheet asset deault and continencies in respect o o-balance sheet exposure and related loss o income; and the loss o market value o equities and related reduction o income Factor based approach US: C0 - Asset Risk Ailiates C1cs Unailiated Common Stock C10 Asset Risk Other C3a Interest Rate Risk and Market Risk Factor based approach (iii) Diversiication beneits (b) (i) Canada: No diversiication beneits US: Diversiication o Asset Risk other with Interest Risk and Market Risk Compare Solvency II investment risk reulatory requirements in Canada and the U.S. Valuation: Assets marked to market or marked to model i unavailable Assets valued on an economic basis. Capital requirement: Assess market risks includin: Interest rate risk, equity, credit spread, orex, property risk SCR = solvency capital requirement = net impact o TSBR due to assumption chane rom best estimate to capital requirement Total SCR = SQRT[sum over i,j (correlation i,j) * SCR i * SCRj] Company can employ internal model to et reduced capital or Capital market hedin prorams. Siniicant diversiication beneit under Solvency II FETE Fall 010 Solutions Pae 38

39 1. Continued (ii) Describe the impact o Solvency II on Canadian and U.S. insurer decisions on portolio allocation. Interest rate risk less or US and Canada since havin ixed actors tied to liabilities No reinvestment in Canadian model Canadian ramework has incentive to invest in riskier assets and arbitrae the reulatory environment US ramework is risk insensitive since liabilities are valued at ixed rate Solvency II delinks liability and asset valuation FETE Fall 010 Solutions Pae 39

40 13. Learnin Objectives: 1. Modern Corporate Financial Theory Learnin Outcomes: (1d) Deine and compare risk metrics used to quantiy economic capital and describe their limitations. Sources: Investment Guarantees, Hardy, 003, Ch. 8: Dynamic hedin or separate account uarantees Ch. 9: Risk Measures (pp only) Commentary on Question: This question was pulled directly out o the Hardy text (Chapter 9) but the numbers were chaned to make the option o into the money more oten than normal. Only ormulas on ormula sheet were needed to solve parts (a) and (b). The rane was provided or part (b) so that in the event part (b) was too diicult or done incorrectly, the answer could be used to ive some commentary to part (c). Overall candidates did airly well on this question. Generally or parts (a) and (b) the candidates either did it completely correct or completely incorrect with a handul that ot partial credit. Some candidates even ot the correct answer without usin the ormulas on the ormula sheet directly but instead derived the ormula. The candidates who did the question wron usually did not understand the variables in the ormula. For part (c) only a handul o candidates ot siniicant points. Solution: (a) Calculate the probability that the GMMB matures in the money. ξ = 1 - Φ [ (lo G/S0 n(μ + lo (1 m))) / ( n σ) ] Formula 9.4 Question asked or probability o GMMB maturin in the money so need to use 1 - ξ G/S0 =.8 n=10 μ =.005 m=.0015 σ =.06 = Φ [ (ln(.8) 10 ( lo ( ))) / ( 10 * 0.06) ] = Φ(-0.978) = FETE Fall 010 Solutions Pae 40

41 13. Continued (b) Calculate the 95% Value at Risk (VaR) or the present value o the GMMB liability at inception, showin that it alls within the rane $4 to $48. VaR95% = (G F0 exp(- za* n* σ + n(μ + lo (1 m))e^(-rn) (ormula 9.7) G = 00 F0 =50 za = n=10 σ =.06 μ =.005 m=.0015 e^(-rn) = e(-.05*10/1) =.6065 VaR95% = (00 50 exp(-1.644* 10 * ( lo (1.0015))*.6065 = (c) Outline the points you would make in response to this comment. 1. The CEO is riht about PV MC, = 41. (eqn 8.14 in Hardy).. But risk is non-diversiiable, so number o policies is not very relevant. Lots o small policies maturin at the same time represent same risk as one bi one. 3. The CTE shows siniicant additional risk i the 5% worst case transpires - $15 over the expected MC; 4. And i the MC isn t heded, it will be worse, because the MC income alls when the uarantee moves into the money. 5. Option price also demonstrates that this is not a ree option. 6. Better to hede the option at, approximately, the option price, to mitiate risk and minimize economic capital requirement. FETE Fall 010 Solutions Pae 41

42 14. Learnin Objectives:. Corporate Financial Applications Learnin Outcomes: (d) Evaluate alternate options or utilizin capital and recommend the most appropriate use in a iven situation. Sources: Copeland, Weston, Shastri, Financial Theory and Corporate Policy, 4 th Edition, Ch. 15: Capital Structure and the Cost o Capital: Theory and Evidence Commentary on Question: This question was desined to examine WACC in a more advanced context. Part (a) was a pretty straiht orward calculation or most candidates. Part (b) relied on knowlede o equation (15.67) or parts (i) and (ii) VL(V) = VU(V) + Tc*B(V) DC(V) = VU(V) + Tc*B + Tc*B pb*tc*b avb*pb and equation (15.19) in part (iii) WACC = p * (1-t*B/(B+S)). For part (b)(iii), many candidates used the WACC ormula (1-t)*Kb*B/(B+S) + Ks*S/(B+S). This ormula is acceptable and correct. However, it requires the candidate to recalculate Ks and Kb or each debt level, which many ailed to do correctly. Solution: (a) Determine the value o Petit Verdot Inc. Equation (15.) rom the text cost o equity or an all-equity irm (p) = r + beta * market risk premium p = 4% + * 3% = 10% Lon term value o irm = S = EBIT * (1 - tax rate) / p S = (1-35%) * 10 million / 10% = 65 million (b) Calculate under the our dierent debt scenarios: (i) Expected value o the tax beneit rom the deductibility o interest payments Tax beneit rom deductibility o interest = debt value * tax rate * (1 - PV o $1 continent on uture business disruption), or in mathematical equation orm Tc*B(V)= Tc*B pb*tc*b = Tc*B * (1-pB). 6.5 million debt level: 6.5 million * 35% * (1-0.) = 1.80, million debt level: 8.5 million * 35% * (1-0.5) =,31, million debt level: 10.5 million * 35% * (1-0.30) =,57, million debt level: 1.5 million * 35% * (1-0.35) =,843,750 FETE Fall 010 Solutions Pae 4

43 14. Continued Note: The ollowin alternative solution was used by several candidates and was acceptable or ull marks, which is the Tc*B part o the expected solution. This is the deinition o tax beneit rom equation (15.6) instead o (15.67). Tax beneit rom deductibility o interest = debt value * tax rate, or in mathematical equation orm Tc*B(V)= Tc*B. 6.5 million debt level: 6.5 million * 35% =.75 million 8.5 million debt level: 8.5 million * 35% =.975 million 10.5 million debt level: 10.5 million * 35% = million 1.5 million debt level: 1.5 million * 35% = million Althouh the alternative is acceptable or part (b)(i), the expected solution is required in part (c) or ull marks. A candidate that submits the alternative or (b)(i) would need to recalculate the remainin parts o expected solution, namely multiplyin the alternative solution by (1-pB), to et ull marks in part (c) below. A common error in this part was multiplyin the cost o debt by the debt value. (ii) Expected present value o the business disruption costs Use avb*pb rom equation (15.67) Present value o disruption cost = PV o $1 * disruption cost = avb*pb 6.5 million debt level: 75,000 * 0. = 55, million debt level: 300,000 * 0.5 = 75, million debt level: 35,000 * 0.30 = 97, million debt level: 400,000 * 0.35 = 140,000 (iii) Petit Verdot s weihted averae cost o capital. Use weihted averae cost o capital = p * (1 - tax rate * D / (D + S)) 6.5 million debt level: 10% * (1-35% * 6.5/ ( )) = 9.68% 8.5 million debt level: 10% * (1-35% * 8.5/ ( )) = 9.60% 10.5 million debt level: 10% * (1-35% * 10.5 / ( )) = 9.51% 1.5 million debt level: 10% * (1-35% * 1.5 / ( )) = 9.44% FETE Fall 010 Solutions Pae 43

44 14. Continued (c) Evaluate the beneits o the our debt scenarios in terms o optimal capital structure and recommend the best alternative or Petit Verdot. The Net Beneit is the increase in value, or a portion o equation (15.67) Tc*B pb*tc*b avb*pb Net beneit = tax beneit - PV disruption cost = Tc*B pb*tc*b avb*pb = Tc*B(V) DC(V) Hihest net beneit = optimal capital structure 6.5 million debt level: 1,80,000-55,000 = 1,765, million debt level:,31,50-75,000 =,156, million debt level:,57,500-97,500 =,475, million debt level:,843, ,000 =,703,750 Debt level 1.5 million has the hihest net beneit Thereore, 1.5 million debt level has the optimal capital structure FETE Fall 010 Solutions Pae 44

45 15. Learnin Objectives: 5. Eicient Markets & Inormation Theory Learnin Outcomes: (5e) Deine the elements o a ame, includin inormation sets, etc., Nash equilibrium, mixed strateies; explain the prisoners' dilemma and other special cases o a twoperson, two-state, sinle period ame, and explain the qualitative implications o repeated ames. Sources: FET : An Introduction to Applicable Game Theory, by Gibbons, R., Journal o Economic Perspectives, Winter 1997, pp Commentary on Question: The question draws on material presented in Study Note FET by Robert Gibbons. In that Note, Gibbons presents a number o ame tree illustrations or his example problems. On pae 134, he includes both a pararaph describin the payos in the problem and a ame tree diaram. The ame tree diaram includes payos at the end o each branch and uses the convention that the upper position payo applies to the player irst to act while the lower position payo applies to the second player. He continues to use this convention on pae 143 in a ame tree diaram that is similar to the one used in this question. This question assumes amiliarity with the example on pae 143 o the study note. A candidate amiliar with that example is expected to be able to create the normal orm payo table associated with that diaram. The candidates overall did very well on this question. Solution: (a) Show the normal orm representation o this ame. Player GNI Player CB A' B' A 0,10 0,0 B 0, 0 0, 10 R 10,30 10, 30 (GNI, CB) FETE Fall 010 Solutions Pae 45

46 15. Continued (b) Deine a Nash Equilibrium. A Nash Equilibrium is a combination o strateies where each player's stratey is the best response to the other player's stratey. (c) Identiy all pure-stratey Nash Equilibria or this ame. (A, A') and (R, B') are Nash equilibria. (d) Calculate CB s expected payo or A and B. E[A'] = p ( 10) + (1 - p) 0 = 0-10p E[B'] = p( 0 ) + (1-p) 10 = 10-10p (e) Determine the impact on CB s stratey. The expected payo o A' is always hiher than the expected payo o B'; thereore CB will always choose A'. () Determine the impact, i any, to the Nash Equilibria. (A, A') is still a Nash Equilibrium. FETE Fall 010 Solutions Pae 46

47 16. Learnin Objectives: 3. Derivatives and Pricin Learnin Outcomes: (3) Demonstrate mastery o option pricin techniques and theory or equity and interest rate derivatives. (3) Identiy limitations o each option pricin technique. Sources: Hull, J.C., Options Futures & Other Derivatives, 7 th Edition, 008 Chapter 17: The Greek Letters Commentary on Question: This question covers the Greeks, which is essential or hedin, at a basic level, core material or this exam. Candidates are asked or the basic deinitions and assessments about how they behave in dierent market conditions. Candidates did part (a) and (b) airly well. Many candidates did not answer parts (c) and (e). Many candidates treated Gamma in the same way as Delta in part (d), which costs them points. Part (e) intended to make the point that Greeks are ood only or small chanes and all the Greeks will chane as the underlyin chanes, includin Gamma. Solution: (a) Deine in words the ive major Greeks in the equity option world. Delta: Rate o chane o the option price with respect to the price o underlyin asset Gamma: Rate o chane o delta with respect to the price o underlyin asset Vea: Rate o chane o the value o the option with respect to the volatility o the underlyin asset Rho: Rate o chane o the value o the option with respect to the risk ree interest rate Theta: Rate o chane o the value o the option with respect to the passae o time (b) Determine whether these Greeks are positive, neative or zero or stocks, European calls, and European puts Stock Calls Puts Delta Gamma Theta Vea Rho FETE Fall 010 Solutions Pae 47

48 16. Continued (c) For each o the above scenarios, determine whether a positive or neative delta, amma, or vea in your tradin book would produce a proit. Positions Scenario Market Conditions Delta Gamma Vea 1 Swit Downward Price - Increases + Increases + Increases Movement; Risin Implied Gain Gain Gain Volatility No Price Movement; Fallin Implied Volatility Does Not Matter Does Not Matter - Increases Gain (d) Estimate the value chane in your tradin book at closin today. Option Units Delta Gamma Theta/Day Vea Stock Price Movement Chane In Volatility Call Totals Put Totals Stock Total New delta = delta + amma * price chane = * (-4) = Averae delta = ( )/ = Chane o value = *(-4) * = (e) Explain how the value o your tradin book will chane i Tempranillo Corp. stock alls 50%. The stock holdin will lose 50%. The Puts will ain at a rate approachin 50%; the short calls will have essentially no price chane as they re so ar OTM. As the put $delta is less than the stock the book is underheded, so the position will lose money. Or The total delta position is neative indicates that the book makes money or a small movement in the stock price. Delta and other Greeks will chane as the stock moves. The book is net lon in notional ( ) = 00, so the book will lose money or a bi drop in the stock price. FETE Fall 010 Solutions Pae 48

49 17. Learnin Objectives: 4. Financial Modelin Learnin Outcomes: (4a) Describe and evaluate equity and interest rate models. (4e) (4) Recommend an equity or interest rate model or a iven situation. Describe how option pricin models can be modiied or alternative techniques that can be used to deal with option pricin techniques limitations. Sources: Hull, J.C., Options Futures & Other Derivatives, 7th Edition, 008 Ch. 1: Estimatin Volatilities and Correlations Validation o Lon-Term Equity Return Models or Equity-Linked Guarantees by Hardy, Freeland and Till, NAAJ Vol. 10 No. 4, October 006 (Sections 1-4 only) Commentary on Question: This question is testin candidates knowlede on the modelin o volatility with common stochastic investment models. In eneral, candidates perormed well in the calculations involved in this question. However, many had diiculty with the translation o the mathematical equations to intuitive explanations. For part (a), there are three common mistakes ound in this question: Some candidates answered usin ormulas whereas the question speciically asked or words. Some candidates read the question as, What are the characteristics o each o the models. Some candidates answered by re-wordin the question. For example, The RSLN- model is a lonormal model that switches between two reimes. Part (b) proved to be a very diicult question or most candidates. Many candidates were conused as to whether p-value was ood or bad. Most candidates ailed to adjust the max lo-likelihood values or the number o parameters. Part (c)(i) is where the candidates earned the most points. For part (c)(ii), some candidates took positions or and aainst the colleaue but ailed to take a inal stand. FETE Fall 010 Solutions Pae 49

50 17. Continued Solution: (a) Describe the volatility process in words or each o these models, explainin how the process is stochastic. The GARCH model is deterministic projectin the irst period ahead and stochastic or projectin periods ater the irst. The GARCH variance is a weihted averae o lon term variance, the previous period variance and an innovation term rom the previous time period. The volatility in this model can increase suddenly, but it will tend back to the lon run value (mean reversion), similar to the way uitar strins twan and then stabilize. This model is a stochastic process because o its dependence on the previous time period. The RSLN- model has volatilities, one or each o its states. The volatility process is stochastic as it moves between the two states ollowin a Markov process. The Markov process depends on the current state rather than a historical state. The RDDD- model is identical to the RSLN- model except there is an additional drawdown term which adds pressure to the volatility to return to the previous hih-water mark whenever the volatility drops below that hih-water mark. (b) Determine which model provides the best overall it to the equity index in question, based on this inormation. Justiy your selection. GARCH: Accordin to the J-B test, the residuals are not normal so the model does not it the data. RSLN and RSDD both have p-values that pass the J-B test. The number o parameters need to be considered. The RSDD has 8, the RSLN has 6 so the RSDD is penalized more than RSLN. Consider tests: AIC and BIC BIC: Choose the hiher value o (max lo-likelihood) (.5) x (# o parameters) x ln(# values) RSLN: 663. (.5) (6) ln(379) = RSDD: (.5) (8) ln(379) = Thereore, RSLN should be chosen. FETE Fall 010 Solutions Pae 50

51 17. Continued (c) (i) (ii) Calculate the lon run volatility or this GARCH model expressed as an annual rate. Formula or lon run volatility (= ): = 0 / (1 1 ) = square root o [.0069 / ( ) ] = 6.9% To annualize this monthly value, multiply by square root o 1 = 3.97% Outline the points you would make in reply. You should state whether or not you aree with your colleaue. You should assume her calculation is correct. Volatility is important but it s oten not a ood measure o risk. GARCH volatility tends to be symmetric which understates the risk o lower returns in the let tail. Also, i the volatility is symmetric, too much emphasis is iven to the up scenarios where risk is nonexistent or the GMMB. The hih Beta in the GARCH model means the model has a slow mean reversion to the lon term volatility. The hih volatility reime within RSLN is partnered with a lower mean so there s a hiher probability o low returns which is more conservative or a GMMB. I there are loner periods in the hih volatility reime (RSLN) there could be more weiht in the let tail which is important or a GMMB. The importance o which model to use is aected by whether or not there are any hedin practices in place. The lon term volatility may not be as important as the day-to-day or month-to-month modelin. Based on these points, I would disaree with my colleaue. FETE Fall 010 Solutions Pae 51

52 18. Learnin Objectives: 4. Financial Modelin Learnin Outcomes: (4a) Describe and evaluate equity and interest rate models. Sources: Hull, J.C., Options Futures & Other Derivatives, 7th Edition, 008 Ch. 19: Basic Numerical Procedures, Ch. 8: Interest Rate Derivatives: The Standard Market Models FET : Babbel, D. & Fabozzi, F.J., Investment Manaement or Insurers, 3rd Edition, Ch 13: Problems Encountered in Valuin Interest Rate Derivatives by Y. Pierides Validation o Lon-Term Equity Return Models or Equity-Linked Guarantees by Hardy, Freeland and Till, NAAJ Vol. 10 No. 4, October 006 (Sections 1-4 only) Commentary on Question: This question is about valuin an interest rate derivative by Monte Carlo simulation and discussin the appropriateness o the simulation result or listed applications. For part (a), many candidates orot to multiply by 0.5 to et the quarterly payment. Another common mistake was to calculate the payo o a swap rather than a cap. Others tried to calculate the value o the option, or which there is no inormation. For part (b) in (i), some candidates incorrectly cited calculatin the averae o payos beore takin the present value. In (iii) some candidates incorrectly cited applyin the shock to the scenario rates directly. In (iv), most candidates correctly stated the deinition o the CTE98 but ailed to justiy their answer on appropriateness o calculation. Solution: (a) Calculate the payo o the 1-year interest rate cap with a $500 million notional usin the outcome rom this scenario. Lδ k MAX( R k RK, 0) or payo at time t k+1 (ormula 8.5 o Hull s) RK = 3% The payo at time 0, 0.5 and 0.5 is all zero since R k <= 3% At t=0.75, the payo is 500M x 0.5 x max(3.1349%-3%,0)=168,68 At t=1.00, the payo is 500M x 0.5 x max(3.047%-3%,0)=58,750 FETE Fall 010 Solutions Pae 5

53 18. Continued (b) Evaluate the appropriateness o usin these simulations or each o applications (i) throuh (iv), and, when appropriate, describe how to proceed. (i) Estimate the price o the cap. Appropriate, because o q-measure. The price is equal to the discounted expectation o its payos. (ii) Determine the error o the estimated price. Appropriate, because the standard error o the estimate is is the number o trials and ω is the standard deviation. M, where M (iii) Calculate the sensitivity o the price to movement in the short rate Not appropriate, because the rate is constant in all scenarios. Need to enerate another 1000 scenarios with a small chane in the rate. (iv) Calculate CTE(98) Not appropriate, as LMM is a q-measure model or pricin, and CTE is a p-measure risk measure. CTE(98) = take the averae o worst 0, which % o worst outcome. FETE Fall 010 Solutions Pae 53