Subject CM2 Revision Notes For the 2019 exams Interest rate and credit risk models Booklet 7 covering Chapter 18 The term structure of interest rates Chapter 19 Credit risk The Actuarial Education CompanyCONTENTS: Contents Page Links to the Course Notes and Syllabus 2 Overview 4 Core Reading 6 Past Exam Questions 28 ‘Solutions to Past Exam Questions 58 Factsheet 128 Copyright agreement Allof this material is copyright. The copyright belongs to Institute and Faculty Education Ltd, 2 subsidiary ofthe institute and Faculty of Actuaries. "The material is sold to you for your own exclusive use. You may not hire ‘ul, lend, give, sell transmit electronicaly, store electronically or photocopy ‘any part ofil, You must take care of your material to ensure itis not used or by anyone at any time, Legal action will be taken if these terms are infringed. In addition, we may ‘800k fo take disciplinary action through the profession or through your employer, These concitions remain n force after you have finished using the course. © 1FE: 2019 Examinations Page +LINKS TO THE COURSE NOTES AND SYLLABUS Matarlal covered in this booklet Chapter 18 The term structure of interest rates Chapter 19 Credit risk ‘These chapter numbers refer to the 2019 sation of the ActEd Course Notes. ‘Syllabus objectives covered In this booklet 4, Asset valuations: 45 45a 462 453 484 495 456 457 Page 2 Models of the term etructures of intorest rates (Chapter 16) Explain the principal concepts and terms underiying the theory of aterm structure of interest rates. Describe the desirable characteristics of models for the term- structure of interest rates. ‘Apply the term structure of interest rates to modelling various ‘cash fons, including caleuating the sensitivity of the value to changes in the term structure. Deserlbe, as a computational tool, the risk-neutral approach to the pricing of zero-coupon bonds and interest-rate derivatives for a general one-factor diftsion madel forthe risk-free rate of interest. Describe, as computational tool, the approach using state- price defators to the prcing of zero-coupon bonds and inerest-rata derivatives for a general one-tactor diffusion rode! for the risk-free rate of interest Demonstrate an awareness of the Vasicek, Cox-ingersoll Rass and Hull-White modds for the term-structure of interest rates. Discuss the limitations of these one-factor models and show fan awareness of how these issues can be addressed. IF: 2019 Bearinations46 464 462 463 464 485 486 ‘Simple madels for credit risk (Chapter 19) Define the terms credit event and recavery rate. Describe the different approaches to modeling credit risk: structural models, reduced form models, intensity-based models. Demonstrate @ knowledge and understanding of the Merton model Demonstrate @ knowledge and understanding of a two-state ‘mode! for credit ratings with a constant transition intensity. Describe how the two-state model can be generalised to the Jarrow ando-Tumbuil model for credit ratings. Describe how the two-state model can be generalised to Incorporate a stochastic transition intensity. OIFE: 2019 Examinations Page 3OVERVIEW ‘This bookiet is conoemed with stochastic models ofthe term structure of Intarest rates and credit risk. These topics, which are based on the ‘pplication of the Brownian motion and slochastc calculus material in Chapters 9 and 10, are covered in Chapters 18 and 19 of the Course Notes respectively. Both of these topics are examined very frequently. Term structure models of interest rates ‘Section 4 of Chapter 18 sets out the notation to be used in the discussion of term structure models and then defines a number of interest rates and related concepts. Historically, the examiners have been keen on requiting you to manipulate these definitons, by, for example, giving you an expression for the spot rate ‘and asking you to derive the correspondirg expression forthe instantaneous forward rate. Section 2 next sets cut a bookwork list of desirable properties of term structure models of interest rates. This list can be used to generate ideas ‘regarding the likely features, advantages/disadvantages andlor merits or otherwise of particular term Structure models, such as the one-factor models, namely: + the Vasioek mode! 1+ the Coxingersoll-Ross model + the Hullnite model ‘The matarial in Chapter 18 provides the examiners with ample scope for some quite demanding manipulation questions, which might also draw on the properties and results of stochastic celculus as set out in Chapters 9 and 70. In fact, past exam papers have included some quite iffcut, ‘questions on this topic Pages (© IFE: 2019 ExaminationsCredit risk ‘This chapter discusses the madelling ofthe credit risk inherent in risky ‘corporate bonds. ‘The first two short sections outline the definitions of credit events and recovery rates and the three basic types of credit risk model, namely: 4. structural models 2, reduced form models 3. intensiiy-based models. Section 3 then provides a brief description of the Merton model, which is 2 simple example of a structure! model. This has proved a popular topic for ‘exam questions on ered risk. ‘Section 4 next considers a two-state intensity-based mode! for credit ratings and derives risk-neulval pricing formulae for a zero-coupon bond - one of Which is based on the risk-neutral transition intensity, In Section § this approach is then generalised to the mutiple-state Jarrow, Lando and Tumball (JLT) model, which is similar to the multipie-state models you may have come across in olher subjects. Whilst two-state intensity- ‘based mode's have been examined frequently, via both bookwork and ‘numerical questions, the mulliple-state model has come up less often. ‘Section 6 closes the chapter wit a brief discussion about using stochastic, rather than daterministc, transition probabilities. © FE: 2019 Examinations Page 5CORE READING All of the Core Reading for the topics covered in this booklet is contained in this section. ‘The text given in Arial Bold font is Core Reading. Chaptor 18 ~ The term structure of intorest rates ‘Tho Core Reading in this chapter written by Timothy Johnson. adepted from the course notes. Interest rate modelling isthe most important topic in derivative pricing Interest rate derivatives account for around 80% of the value of dorivative contracts outstanding, mainly swaps and credit derivatives used to support the securitisation of debt portfolios. More philosophically, is the fact that at any ene time there will be a multitude fof contracts (bonds) written on the same underlying (an interest rate), andl derivative pricing s principally concerned with pricing derivatives coherently. ‘The multitude of traded instruments leads to the first challenge in Interest rate modelling: the multitude of definitions of interest rates. ‘The basic debt instrument isthe discount bond (or, equivalently, the zero-coupon bond). This is an asset that will pay one unit of currency at time T and {s traded at time ¢eo is a continuously compounded rate, r(t,7) ("force of interest’), such that: 1 eerie GR (tT) = ‘The continuously compounded bond yield is cafculated as: InP(LT) eT) s Fixing t=0 and plotting yield, R(0,7) or r(0,7), against maturity, 7, {gives the yield curve which gives Information on the term structure, how interest rates for different maturities are related. Typically, the yield curve increases with maturity, reflecting uncertainty about farfuture rates, Howaver, if current rates are unusually high, the yield curve can be downward sloping, and is inverted. ‘Thore are various theorles explaining the shape of the yleld curve. The ‘expectations theory argues that the long-term rate [s determined puraly by current and future expected short-term rates, so that tho expected final value of investing in a sequence of short-term bonds aquals the final value of wealth from investing in long-term bonds. © 1FE: 2019 Examinations Page‘The market sogmentation theory argues that different agents in the ‘market have different objectives: pension funds determine longer-term talus, market makers determine chort term rates, and businesses dotermine medium-term rates, which ars all determined by the supply and demand of debt for these different market segments, “The liquidity preference theory argues that lenders want to lend short term while borrowers wish to borrow long term, and so forward rates are higher than expected future zero rates (and yield curves are upward sloping). ‘The short or instantaneous rate, r(t), is the interest rate charged today {for avery short period (le overnight). This is defined (equivalently) as: He) = r(st+d)e Rt 3) sine enet.t00) ‘The short rate is offen the basis of somre interest rate models; however, itwill not generate, on its own, discourt bond prices. ‘The forward rate, F(0,t,7) if discretely compounded and £(0,t,7) if continuously compounded, relates to a loan starting at time f, for the fixed forward rate, the forward rate, repaid at maturity, 7. It involves ‘three times, the time at which the forward rate agreement is entered Into (typically 0), the start time of the forward rate, ¢ and the maturity of the forward rate agreement, T. ‘The law of one-pricelthe no-arbitrage principle, implies: Pon VF Fosry=(ZOn) a Pages (© 1FE: 2019 Examinations10 For continuously compounded forward rates: (r10,7)—r(0,9)7. Tt = 10,0) 11 So, in the limit tT , we get the instantaneous forward rate: = 10,17) 27 200-7) £00,7)=(0,7)+7 a Finer) 12 Hence, the fundamental theorem of calculus tells us that: PUT)= orf frtaa 13 Equilibrium models start with a theory about the economy, such that interest rates revert to some long-run average, are positive or their volatility is constant or geometric. Based on the model for (typically) the short rate, the implications for the pricing of assets is worked out. Examples of equilibrium models are Rendleman and Bartter, Vasicek and Cox-Ingersoll-Ross. Being based on ‘economic fundamentals’, equilibrium models rarely reproduce observed term structures. This is unsatisfactory. No-arbitrage models use the term structure as an Input and are formulated to adhere to the no-arbitrage principle. An example of a no- arbitrage model is the Hull-White (one- and two-factor). (© IFE: 2010 Examinations Page 844 We will assume that the short rate Is driven by an ito diffusion: ry = ME ra)e + (C1 IG, wire Wj isa Wiener process under the martingale measure. 15 We define the bank, or money-market, account process as: dB = FB at, By =1 and: om 16 According to the standard theory, all discounted assets must be martingales under the martingale measure, of: PUT) pg [4 B wl] PUT) =Eo| Buri] PuT= tole val] ‘Compare with: 1 nen =o Jecayau} Pogo 10 © IFE: 2019 Examinations17 The difference between P(¢,7) and B, is captured by noting: Plt, woo{ feta} its a molf {fron we nol 18 We now assume that the discount bond price, P(t,7), is some deterministic function of the short-rate process, 1: PT} tn) «(Pt -e(ownam) = 87*[a(attd)-natandat] By applying Ito's Lemma to the function gft,r,) we have: PUT) _ g-1{ 2a(tan) , doltsre) APO C26 yon o{ 2a } B (2agee Na MNS Gre eer —raltr) alt 181 2 enya, om 419 In order for this to be a martingale, we require that: Se er) natend=d EO Bee alt) withthe boundary condition that P(T,T) = o(Tser)=4 forall ry. (© IFE: 2019 Examinations Page 1120 Vasicek assumes that: Or, = aunt + ot, forconstants a>0, y and, o 24 Hore 1 represents the ‘mean’ lavel of the short rate. Ifthe short rate {grows (driven by the stochastic term) the drift becomes negative, pulling the rate back to j1. The speed ofthe ‘reversion’ is determined, by a If @ Is high, the reversion will be very quick. 22 This yiolds the partial differential equation: Balt), 2 ety) Oe oat) =0 23 To establish the form of g(t.7): Uhlenbeck process: (t,7) recall that for the Ornstein- rea nelte pont +eforenai, ‘and s0, integrating again: 24 This implies that [ rds Is normally distibuted with conditional mean: a e{frs aedealee) Page 12 @IFE: 2018 ExaminationsUsing ito isometry: + r veri vont fe[Fortiet a a we can see that: snl] 25 Since f r,ds is normally distributed, using the moment generating 4 function ofa normal, N(.¥*), random variable: e[e*] ja onthe? by putting t=~1, we can say that: P(.T)=Eq fos vss] (orsde-al-e")) S(t ds =exp| In general, by setting: 2-1) and: ate (u-Ze)een-ro9-Eeren (© IFE: 2019 Examinations Poge 13wo have: persone 26 In Vaslcak’s model (and Hull:-White, below) interest rates are not tretly positive. This assumption Is not Ideal fora shortrate model. CIR use the Feller, or square root mean reverting process which is postive (it tan instantaneously touch 0 but immed ately re-bounds) = aly~ nat rafal for constants @>0, 4>0 and, c. 27 The associated PDE is: aalt.nd) , 211) oy py 4 Paltirdd + aan) +3 or, — Hattsn)=0 at Oh Again, PUT HET with bfey=-— 20 =. (+ ayer - +27 and: Page 14 © IFE: 2019 Examinations28 It turns out that these values for a and b are not that different from those in Vasicok’s model. ‘The distribution of », is given by a ‘non-central chi-squared’ Aistribution. This is a ‘fattalled’ distribution. InP(tT) T-t at, T)~bitT Tat Then the yield curves coming out of the Vasicek model are of three (related) type: Yiela 30 Recall that the duration, D, of an asset, whose interest rate dependent price is given by B, Is defined AB AB Wo po -Dsy where y is the instrument's yield. @IFE: 2019 Examinations Page 15In the context of Interest rate models, this is equivalent PUL) er DPT) = -B(ETIP UT) and there Is a connection between this duration D , and the funetion b. 31 The Hull:White model is an extension of Vasicek where the mean revorsion level, 11, is a deterministic function of tt ote = alnlt)—nidtroai, for constants a>0 and 32 This yields a PDE similar to Vasicek, and so we can start by ‘guessing’ that P(tT) = o%'7H8UM = 9(t,n) and'so: Aglttd) , Bt) eer) a A FEM) 92 — egltsn) = net 2 OTD GB 07 rats) = 0 (ab(t,T)—b'(t,7)-1) +267) (tr Ol aot Tydte LeeNe? This is essentially Vasicek but we have: b= fooo(f ato y a(tsT)= | (-auls)b(s.7)+407H"(s.7)) ds Page 18 © FE: 2019 Examinations33 The advantage of this model over Vasicek is that y(t) can be chosen to reproduce (as closely as possible) the exact yield curve, rather than the restricted forms of the Vasicek model. 34 The properties of those models are summarised below: 7 >0 | Dietabation Model | Dynami See foralle | oF % Vasicek | dr, =alunn)dt + oot, No | Normal cin | dnetr-nyitsoyiirt, [vee | Nomconl Hale White. | dq=aluty—hidtsoai, | No—_| Normal Vasicek i jr Nomcentral ites | erm atttatsediant, |e | Stttrey There are analytic solutions for Pt,T) and option prices for each of these four models. 35 Bearing in mind that the purpose of interest rate models Is to price jrest rate derivatives, there are some short-comings of short-rate models: ‘© Single factor short-rate models mean that all maturities behave in the same way - thera is ne independence. This essentially means they are useless for pricing swaptions (but OK for capsitioora). © Thore is little consistency in valuation between the models. © They are difficult to calibrate. (© IFE: 2019 Examinations Page 17136 Ono-factor models, such a8 Vasicek and CIR, have certain limitations. with whleh It is important to be familiar. First, we look at historical intorest rate data we can see that changes in the prices of bonds with different terms to maturity are not perfectly correlated as one would expect to see ifa one-factor model was correct. Sometimes we even see, for example, that short-dated bonds fall In price while long-dated bonds go up. Recent research has suggested that around three factors, rather than one, are required to ‘capture most of the randomness in bonds of different durations. ‘Socond, itwe look at the long run of historical data we find that there have been sustained periods of both hich and low interest rates with porlods of both high and low volatility. Again, those are features which bre difficult to capture without introducing more random factors into a model. ‘This issue Is especially Important for two types of problem in Insurance: the pricing and hedging of Ieng-dated insurance contracts with interest rate guarantees; and assetdiability modelling and long- term risk management. ‘One-factor models do, nevertheless have their place as tools for tho valuation of simple liabilities with no option characteristics; or short- term, straightforward derivatives contracts. 37 For other problems itis appropriate to make use of models which have more than one source of randomness: so-called multiactor models, Hull-White is not really a multifactor model, the jx and a parameters are deterministic and aid fitting. A multifactor version of Vasicek would involve a multidimensional driving Wiener process and possibly stochastic 1. 38 Today, ‘market models’ have superseded short-rate models for situations where the correlation betwoon differont maturity rates is critical, such as the pricing of swaptions. These treat each maturity instrument (such as forward rate) as a cistinet object, correlated to other similar assets using a multidimensional Wiener process in a no- arbitrage set-up based on the ‘state price deflator’ approach. Page 18 © 1FE: 2019 Bearinations39 Define a traded asset, based on the (traded) discount bonds P(s,¢) and P(s,T): x(9)= 1(P(s.0-Pt7)) The price of a traded asset divided by another traded asset must be a martingale under the measure associated with the numeraire. 40 So, labelling Q” as the measure associated with using P(s,7) as the numeraire, we have: Xs) PT xo. F(s,t,T) = a 9] Eq [F(tt.)] Under Q” , the forward rates, F(s,t,7) are martingales, and: F(5,t,T)=¢(s,0F(s,t,.T)qWo Note that we use tas the parameter for the volatility, rather than T, because the forward matures at t and does not exist in (¢,7). Chapter 19~ Credit risk ‘The Core Reading in this chapter Is adapted from course notes written by Timothy Johnson. 41 Cradit risk exists when a party may default on its obligations. Credit risk is usually ignored with respect to payments by a sovereign government in its own currency, but needs to be accommodated for if an obligation is met in a currency issued by a third-party (such as ‘corporate obligations, obligations by a government in a currency it does not control}. (© IFE: 2019 Examinations Page 1942 Aeredit ioss only oxists if the counter-party defaults and the contract has value. In a forward or swap contract, both longireceiving and short/paying partias are exposed to a c-edit risk, since either party ‘could default if the market moves against them. ‘For options and bonds, the purchaser of the optionibond is exposed to default by the swriterlissuer, but they do not have an obligation to the writerfissuer. Credit risk Is calculated as an expected oss: Expocted Loss = Exposure x Probability of Default x Loss Given Default All the parameters have an implicit time dependence. The Loss Given Default (LGD) Is the percentage of the exposure that will be lost on a default, the recovery rate is the reciprocal of the LGD (Recovery Rate 4100%-L.GD}. Usually some value can be recovered when a counter- party defaults. 43 Credit risk changes with the market and good practice Is to assess both current and potential exposure. The current exposure is the ‘current market value of the asset, the future exposure should be based ‘on a wide range of future scenarios, wih different default probabilities. 44 Crodit events, which might result in a failure to meet an obligation (defined for the purposes of credit derivatives), includ + actions that are associated with benkruptey or insolvency laws ‘+ downgrade by ‘Nationally Recognised Statistical Rating ‘Organisations’, (NRSROs such as Moody's, S&P and Fitch) = failure to pay + repudiation / moratorium ‘+ restructuring ~ when the terms of the obligation are altered so as to make the new terms less attractive to the debt holder, such as a reduction In the interest rate, re-scheduling, change in principal, ‘change in the level of seniority. Page 20 @1FE: 2019 Examinations45 Structural, or firm-value, models are used to represent a firm's assets and llabilities and define a mechanism for default. Typically, default ‘occurs when a stochastic variable (or process) hits a barrior representing default. The main example of a structural model Is the Merton model. These models deliver an explicit link between a firm's default and the economic conditions and provide a sound basis for estimating default correlations amongst difforent firms. Tho disadvantage is identifying the correct model and estimating its paramoters. 46 Reduced-form models do not attempt to deliver a representation of a firm, like structural models do, Rather they are statistical models that use observed data, both macro and micro, and so can usually be ‘fitted’ to dat ‘The market statistics most commonly used are the credit ratings issued by NRSROs. The credit rating agencies will have used detailed data specific to the issuing corporate entity when setting their rating. ‘They will also regularly review the data to ensure that the rating remains appropriate and will re-rate the bond either up or down as necessary. Default is no longer tied to the firm value falling below a threshold- level, as In structural models. Rather, default occurs according to ‘some exogenous hazard rate process. 447 An intensity-based model Isa particular type of continuous-timo reduced-form model. It typically models the jumps between different states (usualy credit ratings) using transition intensities. The disadvantage of reduced-form models is that they sometimes lack the clarity of structural models, OIFE: 2019 Examinations Page 2148 Classical finance defines the value of « firm F(t) as the sum ofits dobt, B(¢) and equity £(¢), so: F()=8()+E(0 ingle zero-coupon bond with face value of Lshich matures at time T- Since debt is sanior to equity, the value of equity at maturity: E(1)=mox{FiT)-L.0} and so the value of a firm's equity is a call option on the value of the firm with a strike of the debt. Consequontiy, E(O) = F(go(d)—Lea{a,) a with: k + (r+ SoRMT | ENS NN, gyageon FI where op is the volatility of the firm value. The value of the debt today is F(t)—E(t). 49 Unfortunately, F(t) (and of ) are unobservable, since they depend on the market's assessment of B(t)- However, if we assume that the value of the firm and the equity both follow geometric Brownian motion, and E(t) =f(F(t)) thon by Ito: E(t) , 1, y ZEW Ett) E(t) = | HF OE fod F(t) Seay (tt ore sem (t)(uedt + oW;) Page 22 (© FE: 2019 Examinations50 ‘Comparing terms leaves: 2 (0) F(t) E(t) = oft) o6F(}0(a4) @ Using equations (1) and (2) we have: oe (t)= o¢F()0(c;) (E(t) +L" 04d) oeE(t) E(t)+ Le 0(a,) ‘This is not trivial since op is used to-calculate 02. Recall that, in the Black-Scholes world, @(d) Is the risk-noutral probability that a call option will be exercised, that Is it expires in the money. In this context, this means that: 1-O(da) = O(-d2) Is the risk-neutral probability that the firm is in default at time T (but not that thas dofaulted in (0,7) and then recovered). This method provides a rough estimate of the probability of default. ‘One limitation is that the default probability Is given In the abstract, risk-neutral world, The real-world probability can be derived using: where the real-world drift, up , replaces the risi-less drift. However, dp is not observable, (OIF: 2019 Examinations Page 2554 In Europa, before the Reformation of the Catholic Church in the sixteenth century, the charging of interest was only permissible as @ ‘Compensation for tha riak that tha lender took on: this is captured in the opening observation of the Black-Scholes pape’ It should not be possible to make « risk-less profit. 52 Consider a lender lending a sum of money, L. The lender is concerned. that the borrower will not default, a hopafuily rare event, and wi ‘eventually pay back the loan. Poisson worked out that ifthe rate of a rare event occurring was 4 then the chance of there being k raro ‘events in n time poriods was given by: 53 Say tho londer assesses that the borrower will default ata rate of 4 defaults a day ~ known as the hazard rate — and the loan will last T days. The lender might also assume that they will get all their monoy back, providing the borrower makes no defaults in the T days, and nothing ifthe borrower makes one or more defaults. On this basis the lender's mathematica: expectation of the value of the oan Evalue of loan] = P{No default)xL + P(Oofault)x0 Using the Law of Rare Events, the probability of no defaults is given when k=0, so: ane Evalue of oar']= xh + P(Befault) x0 We can ignore the second expression, since itis zero, then: {value of loan] =Le*” Page 24 (© FE: 2019 Examinations56 ‘So, the lander is handing over L with the expectation of only getting Le ¢ T related fo the distribution of { r(uyou. i [Total 15} Poge 32 (©IFE: 2019 Examinations‘Subject C78, September 2008, Question 10 Stale how the price at time 1 of a zero-coupon bond paying £1 at T {Gonoted by B(t,7))is related to: (2) spotrate curve (6) instantaneous forward rate curve (©) instantaneous tisk free rate Define all notation used. 4 Subject C78, September 2008, Question 11 () Describe the Merton mode! for assessing credit isk. a ‘A company has just issued a zero-coupon bond of nominal value £8m with rmaturly of one year. The value of the assets of the company is £10.008m and this value is expected to grow at an average of 10% per annum Compound with an annual volatity of 20%. The company is expected to be wound up after one year when the assels wil be used to pay off the bondholders with the remainder being distributed to the equity holders. Shares in the company ate currently traded at a market capitalisation of £2,9428m, (i) Estimate the riskefroe rate of interest in the market to within 1% pa, Stating any additional assumptions that you make, to} {Total 18} ‘Subject C78, Apri! 2009, Question 6 Describe three different approaches to modelling credit risk. ro) © FE: 2019 Examinations Page 83‘Subject C78, Apri! 2009, Question 9 The zero-coupon hand market is assumed to be arbitrage-oe and ‘complete. Consider the folowing model for the instantaneous forward rate process: ott) tT ats o(6.T 98M, where (W;it2 0) is a standard Brownien motion with respect to the risk: noutral probability measure Q () State how the price of a zero-coupon bond is related to the Instantaneous forward rate. By Using Ho calculus, itis possible to prove that the dynamics for the zero- ‘coupon bond price are given under @ asfollowe: aUAT) i Bry mT STM, where: (tT) : 2 (t)~ fat, s)as-+} J ics) Stt.T)=-F othe (0) Explain the relationship betwoen a and o under the condition that the ‘bond market is complete. Give reasons for your answer. (6) {Total 8) Poge 34 (© IFE: 2018 ExaminationsSubject CT8, September 2009, Question 6 (Describe the two-state modol for credit ratings. a In two-state model a zero-coupon defaultable bond is due to redeem at par in two years’ time. I default occurs the recovery rate is 3. The continuously-compounded risk-free rate of return is r. Under the probability measure P;, the default intensity is constant and equal to A and the wore Of af ae +0220] otherwise, (i) Show that P, is an equivalent martingale measure fer this model. [3] A derivative contract pays $1,000 after two years if and only ifthe bond has defaulted. GH) (@) Determine a constant portfolio in the defaultable bond and cash which replicates the derivative. (©) Calculate the fal price for the derivative. 6 relates to the fact stated in part (i. [3] (v) Explain how your answer to (i Total 15) ‘Subject C78, April 2010, Question 4 Consider the following stochastic difforontial equation for the Instantaneous Fisk free rato (also reforred to as the short-ate): ant)=a(o—r(0) dt ca Its soltion is given by: HU) = tp exn(-at) + (1~exp(-at)) + exp(-at)f,oxptasia, (© IFE: 2019 Examinations Page 350 ‘You may algo use the fact that for T >t: exp (al =A) oT 3 ff oydu =2(7 + (10-0)2 2 fF (-oxm(-ai7-s)} aM, () Derve the price at time ¢ of a zero-ccupon bond with maturity T. [10] {W) (a). State the main drawback of such a mode! for the short-rate. (0) State the name and stochastic diforential equation of an alternative ‘mode! forthe short-rate that Is not subject to the drawback. [2] (Total 12} ‘Subject CTE, April 2010, Question 6 (Describe the two-state mods! for credit ratings under the real world measure, @ (i) Explain how the two-state model is generalised in the Jarrow-Lando- Turnbull model ie Frotal 12] ‘Subject C78, October 2010, Question 3 Discuss whether onevactor models are good models for the short rate of Interest (instantaneous risk-free rate). Include discussion of extensions that may be conskiered to improve the modcl. Iilustrale your discussion by defining and refering to particular models 119) Page 26, © IFE; 2019 Examinations11 Subject CT8, October 2010, Question 4 () In the context of eredit risk for defaultable bonds: (2) Give three examples of a credit event (©) Give three examples of an outcome of a default (6), Define the recovery rat. nm (i). Describe the two-state model for ereit ratings. 4 ‘Two companies have zoro-coupon defaultable bonds in issue. Bond A has £2m nominal in Issue. Bond B has £3m nominal in issue. Both bonds redeem in exactly 2 years’ time. Under 2 risk-neutral measure, each bond defaults (not necessarily independent) at 2 constant rato, Both bonds have a 60% recovery rate. Ass ‘+ a continuously compounded risk-free rate of interest of 3% pa + the issue of Bond A is priced at £1.6m + the issue of Bond B is priced at £2.2m (i) Evaluate the two default rales (under arisk-neviret measure). 4] ‘There is also a traded derivative security, D, priced at £52 which pays £100 ator 2 years if (and only if) atleast one of the bonds default, (iv) @) Determine a hedging portfolio for the security which pays £100 after 2years if and only if both bonds default by considering fixed portfolios consisting of bond A, bond 8 and security D and a risk free zero-coupon bond paying £100 at redemption in exactly 2 years, (©) Calculate the fair price for the security that pays £100 if and only if both bonds default ‘61 (Total 23} (© IFE: 2019 Examinations Page 9742, Subject C78, April 2011, Question 9 In an oxtoncion of the Merton madel, a very highly geared company has two tlers of debt, a senior debt and a junior debt. Both consist of zero-coupon bonds payable In three years’ time. The senior debt is paid before the junior debt Lot F; be the value of the company at time f, Ly the nominal of the senior dob and Lz the nominal ofthe junior debi 4) (@) State the value of the senior debt at maturity (©) Deduee the value ofthe junior debt at maturity. ol ‘The current gross value of the company (6 £3.2m. The nominal of the senior debt ig £1.2m and that of the junior debt is €2m. The continuously- ‘compounded risk-free rate is 4% per annum, the volatility of the value of the company is 30% per annum and the price’ of £100 nominal of the senior bond is £88.26, (i) Calculate the theoretical price of £109 nominal ofthe junior debt [6], [Total 10}, 13° Subject C78, April 2011, Question 10 Lot (t,7) be the price at ime fof a zero-coupon bond paying £1 at time T., 4 be the short ate of interest, P be the real-world probablity measure and @ be the risk-neutral probability measure, (Write down two equations for the price of a zero-coupon bond, one of Which uses the risk-neutral approae’ to pricing and the other of which uses th slato-price-defator approach to pricing. 2 (State the Stochastic Differential Ecuation (SDE) of the short rate % ‘under Q for the Vasicek mode! and the general type of process this SDE represents. el (i) Solve the SDE for the short rate ftom part (i ro () Deduce the form of the distribution cf the zero-coupon bond price under this model 2) Total 121, Page 38 ‘@IFE: 2019 Examinations14. Subject CT8, September 2011, Question S (List the desirable characteristics of a model for the term structure of inlerest rates. 4) (@ Write down the stochastic differential equation for the short rate under @ in the Hull-White model. 0 (ii) Indicate whether or not the Hul-White mode! shows the characteristics listed in i (Total 9} 48 Subject CT8, September 2011, Question 11 () Draw 2 diagram to ilustrate the Jarrow-Lando-Tumbull model for credit default, defining any notation used. 4 ‘A model was proposed for a country’s sovereign debt as follows: ‘Tho economy is in one of three states: 1 (good), 2 (bad), 3 (default). Transition intensities 4, are constant and as follows: Agate Ang <0, py = 0.25, yg =0.75, dy, =0 forall j and yy = Jeg It follows thal, if p(t) Is the probability that the economy is in stale 7 at time ¢ then: a) +0.25p2(0) ond Pall. p(t)~ pate Sot At) = 2p4t)~ pall) Wi @) stows (0) Derive a similar equation for F defined by K(t)=2ex(t)+ pelt). 2] ‘Suppose that this country's economy Is in State 2 at time 0. (© IFE: 2019 Examinations Page 986 ” (i) Find the probabiity that it isin default at time 2 4) ‘Assume a continuously-compounded risk-iree rate of 2% per annum and @ recovery rata of 60% (wv) (@) Deduce the price under this model for a zero-coupon bond in this ‘country with a redemption value of 100 and @ redemption date in ‘wo years" timo, (0) Calculate the credit spread. 31 rota! 13} Subjact C78, April 2072, Question 6 () Write down a stochastic differentia! equation for the short rate r(t) for the Vasicek model fo i) State the type of process of which the Vasicek model is 2 particular example. m (i) Solve the stochastic diferential equation in () 8 Gv) State the distribution of r(t) for t given. a (¥) Derive the expected value and the second moment of r(t) for f given. 8 (vi) Outline the main drawback of the Vasicek model fy Frotat 12 Subject CT8, April 2012, Question 9 () Describe the two-state model for credit ratings and its generalisation to the Jarrow ando-Tumbull model. 4) Companies A and 8 are joint investors in a high-risk project to build a new space plane, Each of the two companies’ zero-coupon bonds are modelled according to a two-siale model, Company A's bonds have a recovery rale of 5, = 60%, while Company B's have a recovery rate of J =50%. All bonds mature in nine months. Company A's bonds have a current price of $82 per $100 nominal, Company B's bonds have a current price of $79 per $100 nominal, The continuously-compounded risk-free rate is 1.5% pa. Page 40 © FE: 2018 Examinations18 19 20 (i) Calculate the implied risk-neutral default intensities 4, and ag, ‘assuming that they are constant. al ‘A competitor to the space plane project now starts to sell a derivative ‘socurity which pays $100,000 at the end of nine months if and only if both companies defauil within the nine months (a double-defaul). The current price for the derivative Is $7,800. (ii) Caiculate the implied risk-neutral probabilly of a double-default and the corresponding constant rate. ai (Ww) Calculate the maximum price for this derivative, by considering the maximum possible double-default rate. ‘41 {otal 14] Subject GTB, October 2012, Question 5 ‘State eight desirable characteristics of a term structure model 8 ‘Subject C78, October 2012, Question 6 {)) State the stochastic differential equations for the short rate r(t) in the ‘Vasicak model and the Cox-ngersoll-Ross model. 2 (8) Explain the impact of a movement in the short rate on the volatility term in both models. 2) Total 4] Subject C78, October 2012, Question & {)_ Describe the Merton model for pricing a defaultable bond. 41 ‘A very-highly-geared company, Risky plo, has issued zero-coupon bonds payable in three years’ time for a nominal amount of £3,200m, A Black-Scholes model for the value of the company is adopted. (i) Derive an expression forthe value of the debt. 3 ‘The current gross value of the company is £6,979m. The continuously. ‘compounded risk-free interastrale is 2% pa and the price of £100 nominal of the bond is £82,603. (© IFE: 2019 Bxarsinatons Page tt2 ‘An insurance company Is offering defauit insurance on Risky plc. They wil ‘charge @ premium of £55,000 for a contract which pays £1m at the end of three yoars if Risky ple defaults (fi) Discuss whether there is an arbitrage opportunity. (4) [Total 14] ‘Subject CT8, April 2013, Question 8 (part) Describe three limitations of one-factor term structure models 5) ‘Subject CT8, Aprit 2013, Question 10 (amended) (i) Describe the two-state model for credi: defaults. 4) Company A’s bonds are modelled according to a two-state model. ‘Company A has two zero-coupon bonds in issue, both with a recovery rate of 5 = 60%. Bond 1 matures In one year Bond 2 in two years’ time, Bond 4 has a continuously-compounded credk spread of 4%, Bond 2 has a ontinuously-compounded credit spread of 5%. The continuously- ‘compounded risk-free rate is 1.5% pa, (i) (@) Caleulate the price per $100 nominal of each bons. (©) Deduce the implied risk-neutral probabilities of no default in one year and in two years’ time, (8) (ii) Determine the implied values of the default intensities, assu they ate constant for each of the two years, that a (Total 13} ‘Subject CT8, September 2013, Question & () Write down @ stochastic eifferential equation for the short rate r in the Vasicek model defining any notation weed. 1 (i) Ust the desirable and undesirable features of this model for the term Structure of interest rates. tl {a) Solve the stochastic differential equation from your answer to part (i. (©) Comment on the stat ical propeties of rp, T> a {Total 12} Page 42 © IFE: 2019 Examinations24 Subject CT8, September 2013, Question 10 (Describe the Merton mode! for pricing a bond subjact to default risk [4] Avery highly geared company ~ XYZ ple ~ has Issued zero-coupon bonds payable in four years' time. The debt is a nominal $120m. (@) Give expressions for the value of the debt in four years! time and today, ‘adopting a Black-Scholes model for the velue of XYZ ple. aI The curent gioss value of XYZ ple is $180m. The continuously compounded Ttisk-ee interest rate is 2% pa and the continuousy- ‘compounded credit spread on the bond is 4.5% pa. (i) Cateuiate the pries of the bond today. ty {iv) Estimate, to the nearest 1%, the implied volatility of the value of XYZ. [3] (¥)_ Determine the implied risk-neutral probability of defaut. BI {Total 15}, 25 Subject CT8, April 2014, Question 5 Consider the following model forthe short rate r oy, = un dt +002 where 42 and o are fixed parameters and Z is a standard Brownian motion. (@ Comment on the suitabiity ofthis mode! for the short rate. 4 ‘An altomative model for the short rate is the Vasicek model i, alu at +2 (i) Derive an expression for IT ruydu 8 (i) State the distebution of IF r(u)ae a {Total 14) © IFE: 2019 Examinations Page 4326 Subject CT8, April 2014, Question 10 ‘A company has twn zer-counon bonds in issue. Bond A redeams in 1 year ‘and the current price of £100 nominal is £92.60. Bond C redeems in 2 years and the current prica of £100 nominal is £74.72. ‘The continuously-compounded risk-free rate is 2.5% pa for the next two years, () Write down the formula for the general zero-coupon bond price in the two-state model for credit ratings, defring al the terms used. 21 (i) Determine the implied cisk-neutral probability of default for Bond A, ‘assuming this madel holds, and a reccvery rate of 50% for Bond A. [3] 1 Bond A defaults then Bond C automatically defaults with a recovery rate of zero, whereas if Bond A does not default then Bond C may stil default in the ‘second year, but with a recovery rate of 50%. (i) Modify your answer to part () to give a formula for the current price of Bond C. 8 (iv) Calculate the risk-neutral probability of default for Bond C. fo ‘Total 17} 27 Subject CT8, September 2014, Question 9 ‘A company has issued a ‘oan in the form of a zero-coupon bond which redeems in one year from now. The bond is priced at £92.78 per £100 nominal and the recovery rate in the event of 3 default is assumed to be ‘50%. ‘The continuously-compounded risk-Tee rate for one year is 3% ps. (0 Write down the formula for the bond price under the two-state model, dofining all the terms used Bi (i). Calculate the risk-neutral probabil that the bond defaults. a Assume that the Merton model holds and that the annual volatility of the ‘company’s total assets Is 13%, Give an exprassion for the risk-neutral probability that the company defaults, defining any other terms you use. 8) W) Caleulate the ratio of nominal loan to total asset value, assuming that the risk-neutral default probabil is tre same 2s calculated in (i). [2] Page 44 © 1FE: 2019 Examinations28 29 (Y) Calculate the ratio of loan value to total asset value and hence determine the percentage of total assets represented by equity value at time zero. 8) [Total 13} ‘Subject C78, April 2015, Question 9 {i) Outline the three types of credit risk model 8) (Describe how the Merton model can be used to estimate the risk-neutral probabilty of defaut gi Let r be the constant continuously compounded risk-free rate and 5 be the ‘constant recovery rate for a defaullable zero-coupon bond in a two-state mode! for credit rating with a deterministic transition intensity (i) State the formuta forthe bond price 0 {@) Determine the risk-neutral default intensity if he zero-coupon bond price is oven by: BLT) Of s-(1-8){t-ern[-7?-Py/6})] Rl () Calculate the fair price of an insurance contract which pays £1,200,000 after two years ifthe bond defaults in the frst year and the continuously ‘compounded risk-free rate is 2% pa. fe (Total 11] ‘Subject C78, April 2015, Question 10 ‘There are two risk-free zero-coupon bonds trading in a market, Bond X and Bond Y. The shortate of interest, 1, folows a Vasicek model: a, = al ~7 yal +00, where Wi isa standard Brownian motion. (Write down the formula for the price of a rsk-ree zero-coupon band at lime £,, with bond maturity st ime T , under the Vasicek model, {3] © 1FE: 2019 Examinations Page 4530 In this market the parameters for the Vasicek model are a =0.5, w= 4% and = 10%. Tho shortwate at time 0 1p, is 2% pa. Bond X matures at time 1, and Bond Y matures at time 3. Both bonds are for a nominal value of $100, (il) Calculate the fair price of Bond x. @ Bond Y has a fair price at time 0 of $90. (ii) Derive the market-implied risk-free spot rate of interest with meturlty 3 years. 2 (iv) Derive the market-implied risk-free forward rate of interest from time 1 to time 3. 2 [Total 10}, Subject CT8, October 2015, Question ¢ (Describe the Merton model for credit risk 5 A company Is about to issue a zero-coupon bond which will deem in T = 5 years at £12.3 billion. The value of current issued share capital is £12.5092, billion and the company has na other debt. The continuously compounded riskefree rate of interest is 5% pa and the volatility of the company's gross asset value is assumed to be 20% pa, ‘Assume that the share price will not change on issue and that the ‘assumptions of the Black-Scholes model apply. (i) Write down the relationship between the current equity value of the ‘company and F;, the final gross value of the company's assets. [1] (ii) Estimate Fo,, the gross value of the company’s assets just after the ‘bond issue, using the Black-Scholes formula and interpolation. im (iv) Determine the corresponding credit spread on the loan. a Total 15} Page 46 @IFE: 2019 Examinatons31 Subject CTS, October 2015, Question 10 Consider a market withthe following properties: Fw) [aan | Ron [ce = 5 = __ | £160.00 2 © | 20% [etoz02 4% | #9416 | (@) | ei06.18 3% | forse | 30% | @ (| #8604 [35% [2115.09 where: © ristime, © F(s.f) 1s the forward rate al time 0 from time s to time ¢ © B{s.t) is the price of a zero-coupon bond at time $ maturing at time £ with a nominal value of £100, '¢ Rs.) is the spot rate of interest at time s for the period s to ¢ © Off) isthe value of a cash account at time ¢ () Catcutate values of (2), (0), (c) and (4) in the table above. (1 ‘At time 0 an investor buys £4,000 nominal of zero-coupon bonds maturing at time 2, and £2,000 nominal of zero-coupon bonds maluring af time 4.” At time 1 interest rate expectations have changed as set out below. Fit= 1.9, 8% 2% o% (i) Calculate the foss the investor will make if she sells the bonds at time 1. 8 Poge a?32 “The investor decides to koap the bonds rather than saling them at ime 1, (i) Comment on whetin tke investor can restructure her portfolio to recover her loss if interest rates remain unchanged. a Total 9} ‘Subject CT8, Apri! 2016, Question 9 (Draw a diagram to Hlustrate the Jartow-Lando-Tumbull mode! for crecit default, defining any notation used, 14) Consider a three-stete eredit model for a company in discrete time. The states are Healthy (H), Unhealthy (U) or Defauited (0), Transition probablities from state 7 to state j, py, are constant: Pau = 0-4 Pay = 0.08 Pup = 0.02 Pup =0.3 oj =9 forall jxD Denote the probabilly that the company is in state / at time { (years) as pO) ‘A-company is in the Healthy state a time 0, (i) Cateuiate po(2), fe the probebilty that the company Is in the Defeuit slate at time 2. a The company issues a zero-coupon bond al time 0, with maturity at time 2 and nominal value £100. The continuously compounded risk-free rate of interest fs 4% per annum. ‘Assume that the bond returns Its nominal value at time 2 if the company is not in default, or x% of its nominal valus at time 2 if the company is in default. ‘The falr price of the bond at time 0 is £87.63 (i) Calculate the value of x, the assumed percentage recovery on default. a) Page 46 @1FE: 2019 Examinations(fv) Calculate the credit spread on the bond. i (¥) Comment on the impact on the current price of the bond if it retumed x% ofits nominal value al the time of default rather than at time 2. _{1] [Total 10] 33. Subject C78, April 2016, Question 10 In the Vasicek model, the short rate of interest under the risk-neutral Probability measure is given by: ‘ ya0+e™ (-0)eafo"aw, where k. @, ¢>0 and Wis @ standard Brownian mation Consider the reiated process: R= Jrads a where 4 isthe shor rate defined above () Show that Ry has a normal distribution with mean and variance given by: tee E[R]=01+(-0) == and: ek a ri] © IFE: 2019 Examinations Poge 49Lot P(0,1) be the price al time 0 of a zero-coupon bond with redemption date t>0. (i) Show that, under the Vasicek model: -atrj ee Progyme 2 8 (8) Show, by usin the resus rom parts) and (i), that P(O,t)= alte here BY) = and: At)= 0) Zoro] @ we (0) Stato tho main drawback of the abcve model forthe tem sinuture of interost rates. ol {Total +4) 34 Subject C78, October 2016, Question § (Write down the properties of the follewing two models for interest rates: (a) the one-factor Vasicek modet (b) the CoxingersoltRoss mode! [You are not required to give any formulae for the models] 4 Page 60 @IFE: 2019 Examinations35 The Vasicek torm structure model is described by the following stochastic differential equation: ar j=alb—n)at+oum, vith intial value f) and a,8,0°>0 (Show, by solving the Vasicek stochastic ciferential equation, that: nage soft ofo* aw, 3 (i) Determine the expectation, the variance and the distribution ofthe short vale 8] [Total 11] ‘Subject C78, October 2016, Question 10 ‘A company has issued zero-coupon bonds paysble in five years’ time for & ‘nominal amount of £100m. The company has also issued 1 milion non- dividend-paying shares. A Black-Scholes model for the value of the ‘company is adopted (Derive an expression forthe value of the debt at time 0 using the Merton ‘model, in terms of the total value of the company and the value of a call option: 4] The current total value of the company is £200m, The continuously compounded risk-free interest rate is 1% per annum. ‘The current arbittage-free prices of options on the company’s shares, with ‘maturiy in five years! time and a strike price of £100, are as follows: © putoption = £17.30 © call option = £27.55 (Calculate, using put-call parity, the value of the zero-coupon bonds per £400 nominal fo} @ IFE: 2019 Examinations Page st36 ‘The volatility ofthe tolal value of the company is 17% per annum, (i) Determine the approximate ehange In the share rina and the bond price that would arise from a £1m increase in the total value of the ‘company. [int: consider the delta of an appropriate option.| ry iv) Comment on the relative change in the share and bond prices in part (i). 2 (©) Comment, without carrying out any calculations, on how the relative change in part (ji) would differ if the total value’of the company was lower. 1 (Total 14} ‘Subject CT8, Apri! 2017, Question & (i) Uist the desirable characteristics of a term structure model. 8 Let B(¢,7) be the price al time f>0 of a zero-coupon bond which pays a value of 1 when it matures at time T. Lot F(t, 5.7) be the forward rate at time tfor 2 deposit starting at time S>t and expiring at time T > S Consider the following two Investment strategies implemented at ime t A | Attime t Purchase one zero-coupon bend maturing al time 7. ‘Continue fo hold the bond to tme T. B | Attime £ Purchase a=oF"STIT-S) zero-coupon bonds maturing at time S0. Page 6339 Under this model, the shor rate 7 follows normal distribution with mean Bty)= ger +PUI-e%) and variance: a FG) = Z-0 Var (); 3a ) (i) Assess, using the information provided above, whether the model {generates interest rates that are mearvreverting and, f 80, the value to Which they revert. 2 {v) Assess, using the information provided above, the relevance of the parameter a to any mean reversion. 2 Total 11] ‘Subject C78, September 2017, Question 9 Consider the Merton madel for credit risk, ‘Assume that a firm has Issued a zero-ccupon bond maturing in five years’ time with maturity value €100m, and thal the current value of the firm's assets is €110m, Further assume that the estimated volatiy of the finn's assets is 25% per ‘annum and the tisk-free rate of interest is 2% per annum continuously ‘compounded. (i) Show thet the current value of the debt ofthe firm is €76.68m. 6 (i) Cateulate the yield fo maturlty of the debt. 3 i) Calculate the credit spread on the debt i [otal 40} Page 64 @IFE: 2019 ExaminationsSOLUTIONS TO PAST EXAM QUESTIONS ‘Subject CT, April 2008, Question 4 (The two-state model for credit ratings ‘Assimple model can be set up, in continuous time, with two states for a bond: 4. N= not previously defaulted 2. D =defaulted, If the transition intensity ftom Nv to D, under the real-world measure P, is constant and is denoted by 7, this model can be represented as: omer a : | D ‘As well a6 the transition intensity 2 under P, there is an altemative probability measure Q with a different intensity 4. Under this model itis ‘assumed that the defaul-tree interest rate term structure is deterministic with r()=r fora t (i) Formula for zero-coupon bond price f-0-0 -oo( Jer) | whee + B{G,T) is the price at time ¢ of a zero-coupon bond with redemption payment 1 at time T + 6 isthe recovery rate by the lender on the bond on defauit + 1 jgthe (assumed constant) risk-ree force of interest B(r)= + 4(e) isthe deterministic transition rate inthe risk-neutral word Q (© 1FE: 2019 Examinations Page 86‘Subject C78, April 2008, Question 10 {) Solution to tho stochastic differentia! equation “Tho solution to the equation is given by: rncr%ro)+ (1-04) -o0sfletaw, o or rT) = oe) b(t") 200" [ota @ (i) An expression for {7 rlujdu cr(u)= (0 ~ru))du-+odWa Integrating bot sides of this SDE gives: few = AT )=rlt)=abiT—0)-af" uydus of” awe Rearranging gives: j) alb—r(u)ydueafy dw FF rtuyau = 7 —9-Feery—rto) sf ame @ Equation (2) from part () gives: AT) = eT t) 40) AT) aoe 9 oa, ‘Substituting this into (3) and rearranging gives: Jf aya = 07 -0)~ sy ema, re) +2 aw alt meer -netet—b| = } 2 Poge 56 © IFE: 2019 Examinations(it) Distribution of JF e(ujetu Jf ruydulr(t) is normally distributed. (iv) Prige of a zero-coupon bond The price, B(t,7), at ime t of a zero-coupon bond providing a payment of 1 atime T is given by: BUT) =| [oo (47 tox] From part (i), we know that {7 r(u}dulr(t) is normally distributed. So: oxo(-f yas tt) is lognormatly distributed, Hence: BULT)= Fol om (-ff ux] ~o{-£9 [Uf cena] Evarof ff rope} Subject CT8, September 2008, Question 10 (a) B{t,T)= exp[-RULTYT =O} (6) 84,7) = x0) -{7At] © eun=Eoleo(-f! rine) ‘Subject CT8, September 2008, Question 14 (i) Deseribe the Merton Model ‘See Core Reading paragraph 48. (© IFE: 2019 Examinations Poge STEstimate the risk-free rate of interest ‘ha tho cauity of 2 company ean be thought of as a call option on its assets, ‘80 a Black-Scholes-type formula can be used to value the equity of the company. Here we know + the face value of the company's debt, = 8 + the value of the company's assets, Fj = 10.008 ‘+ the annual volaliy of those assets, = 0.20 ‘+ the ferm of the debt, Tf =4 + the value of the company’s equity = 2.9428 but we do not know the risk-free rate of interest, r ‘Substituing the known parameters and values into the Black-Scholes formula gives: 2.9428 =10,0080(c,)~Be-"(d,) (1) vite: 10008). (--onyaz!) gaa) oa dy = dy-0.285 We can estimate the corresponding or ‘implied’ value forthe rist-free rate r by trying different values for r in (1) above and then comparing the value ‘obtained for E, with the actual known valve of 2.9428. So, if r=0.10: (22922). (ot0+ 20.2%) x1 oak 72022 dy = dy -0.2.=1.52022 Page 58 (© FE: 2019 Examinations‘and: &, = 10,0090(d,)-8e- aia, 10.009 x (1.72022) -Be™"ax1.52022) = 10.009x0.95730--80"" x0.93577 = 2.80788 Nets a 28078 than act ae o 20428 or zal et ora 350. The kw al ne inp va ‘must be higher than 0.10. So, let's try r European call option, p This gives: 10.009" (0.15 + 340.27} «1 Jowett tn gab E, = 10.0090%04)~80"9aXd,) = 10.009 x 6(1.97022) -86"50(1.77022) = 10.008x0.97559~ 86°" 0.96165 =9,14308 3.14308 is greater than the actual value of 2.9428, which, given that p> 0 tells us thal the actual value of ¢ must be lower than 0.15, Thus: (2.9428 ~2,90788) (= 040-5 4308 -2.80788) %(0.15~0.10) 0.1201 ithe risk-tee rate Is 12% pa (to the nearest 1% pa) © IFE: 2019 Examinations Page 59Subject C78, Apri! 2009, Question 6 ‘See Core Reading paragraphs 45 fo 47. ‘Subject C78, April 2009, Question 9 () Formula for zero-coupon bond price BXt,T) Bl.T)=ex9[-[/rtudde] instantaneous forward-rate, Jo the force of interest at future time T implied by the current market prices at time ¢ Relationship betwoon a and & By definition, under the risk-neutral probability measure @, the expected: rate of drift of the underlying equals the riseiree force of return, rt). In this context, the risky and tradable asse: the zero-coupon bond, Hence, under @: Dejan + STAI, BUT) AULT) gp So, this with the general expression for 2282) given in equating this wih the general expression for SEAT? given in the question gives: T {t be Ht) = mit,T) =rlt)~faft.syds+| o,s)ds | 1 G } r (t ¥ => Jatusyds«! fottside| i 1 J Page 60 IF: 2019 Examinations‘Subject CT, September 2009, Question 6 () The two-state model for credit ratings ‘See Core Reading paragraphs 54 and 56. (i) Show P; is an equivalent martingale measure Define Z; ~e"D, where D, is the band price formula given in the question, We need to consider the expected value of Z; conditional on default having ‘happened by an eattier time s and also conditional on default not having happened by an earlier time s . £[z;[Detaut by eatiortime s]=e™tser" be worgenrt28) oD, =Z, £[Z;[No dofautby ear time &] 459°" +P (Default between t and 6) sete ttt[ (toot) 0A2-)] Po deta bntneen tans) Hoe HED (te MD) pee M2 [at e tetO-nfsloHn fro Me Horan) gry] (ats fafa rte] eta 5fr-e 89} cA] "0,22, Dg ertigrat 0-0 geet) In both cases, the expected value of Z; is its value at the earlier tme s and 80 Z, isa martingale. Therefore P; Is an equivalent martingale measure. (© IFE: 2019 Examinations Page 61(life) Replicating portfotio. Let the partials now oneiet of # unite ofthe defaultable hand and w units of cash, It the bond defauits before maturity then the value of the portfolio will be: + ye If the bond doesn’t default, then the value ofthe portfolio wil be: ovyer Setting these values equal to the payoffs trom the derivative gives the equations: 8 eye = 1000 o oe ye oo 2) “Taking (2) away from (1) gives: 4000 H5~~ 1000 = 9-40 Equation (2) then gives: wane? (GX) Fair price of derivative The inal value, atime 0, ofthe defaitabls bond i: dae [afi-o}ne] 2r[5r07* (1-5)] Page 62 @IFE: 2019 Examinations‘Tho fair price of the derivative is given by: = 10006 (1-0) (lv) Relevance of equivafent martingale measure ‘inca wo know Py can equiolont martingale moasure, we can sy thal he value of ho derivative is equal to the expected dorvative payot under this Teasure, discounted fo ime 0 atthe tak hee tale of eres a Em [xh] =1,0000"® xP (default) 0%? (no default) = 10000" (1-04) Yo ‘This s the same answer as derived in par (i(). Subject C78, April 2010, Question 4 (i). Derive the zero-coupon bond price ‘The price ofthe zoro-coupon bond at ime ¢ is oven by 2Eq[exn{-f7 wu) re] 21.7) =e] (ff uid) ‘0 ‘The question tlls us that: j-o(-aiT-b) @ 2 ZT (1-ex9(-0(7-0})) am, rus = BIT ~t)+(ri¢)~b) ‘This involves the stochastic (to) integral {7 (1~-exp(-a(T -))} a, . GFE: 2018 Examinations Page 63This integral has the form 7 f(s}dW, . were f(s)=1-exp(-a(7 ~9}). So Wo can use the reeult from Section 14 cf Chapter 9 of the Course Notes. Which states that: SF Hsram, ~W(0.f7 [Hef 2) The varianco of tis dstrbution, which wowil call vis therefore: v= [ [AerF de f° (~exp(-0t7 -9))f os = ff (1~2exp(-a(T -s)} +090 (-2a(7 -5)} os E - [s ~Zowp(-a(t-s)) show (20-5) =), (t-exp(-20t7 0) vera Seeker) (eee Par -P)) ‘So we see that the distribution of the stochastic integral is: ff (t-exp-atr-s))) ey, ~ NOV) “The distribution of {7 rlu)au is therefore: JP nuyau~N (ms?) whore: m=o{r—t)+(tt)~6)| teva) and #-Senifest emigeta Pogo se © 1FE: 2019 Examinations0 1" “The bond price we require is BUT) =E[exp(-X)], where x ~v(m,s2) Using the formula for the moment generating function (MGF) of @ normal distribution, this is: BUT) = EL exp(-X)] = My(-1) = exo[ mi-1)+4%(~1)*] = exp(-m+4?) (ie) Drawback of the model ‘The main drawback of this model fs that it allows the short rate of interest to take negative values, which is not always realistic (although negative inleroet rales have been seen in some counities in recent years). (0) Alternative model ‘The Cox-ngersoltRoss (CIR) model avoids this problem by incorporating the curcent interest rate inthe volatility term, ‘The SDE for this modet Is ert) = 9(0- re) ot +a, ‘Subject CT8, April 2010, Question 6 (i) Two-state model for credit ratings ‘See Core Reading paragraphs 54 and 55. (i) Jarrow-Lando-Turmbul model See Core Resting paragraphs 57 and 58 Subject C78, October 2010, Question 3 ‘See Core Reading paragraphs 35 t0 38 Subject C78, October 2010, Question 4 (ia) Three examples of a credit event ‘Seo Core Reading paragraph 44, @IFE: 2079 Examinations Page 65