ECONOMICS AND FINANCE OF PENSIONS Lecture 12

Transcription

1 ECONOMICS AND FINANCE OF PENSIONS Lecture 12 MODELLING CREDIT RISK AND MORTALITY RISK Dr David McCarthy

2 Lecture outline Examine two of the major risks faced by DB pension plans & describe how these risks are modeled Sponsor covenant risk Mortality risk (These are just two of many types of risks)

3 Sponsor credit risk Systematically underfunded DB pension plans depend on the ability of the sponsor to make future contributions for their security So we need to Assess the credit risk of the sponsor Build a stochastic model of sponsor credit risk Although the theoretical grounds for this are well-established, in practice this is very new

4 Assessing sponsor credit risk Qualitative techniques can be used to assess the covenant risk of the sponsor Business outlook Financial metrics Credit rating Independent business review Your core reading is a little sceptical about Merton-type credit risk models, but they are actually widely used and very effective at estimating default probabilities But they are very difficult to calibrate in practice, which limits their usefulness e.g. for pricing corporate bonds and pension debt (!)

5 Types of credit risk models Structural models Try to model the firm s structure (assets, liabilities) explicitly (based on models by Robert Merton) Reduced-form models Assume default is an exogenous event Similar to mortality/transition models used by actuaries in other contexts

6 Basic Merton credit risk model This was devised by Robert Merton in 1974 Assumes that we are in the setting of the standard Black-Scholes model, so: Agents are price takers No transactions costs Unlimited short selling and no indivisibility of assets Borrowing and lending occurs at the same riskless rate r, which is constant Assume that the underlying assets of the corporation, A, follow a process given by a GBM: da A dt db t

7 Merton (1974) The company issues two securities: debt (with a face value of D, which matures at time T and which pays zero coupons) and equity, which is the residual claim Therefore, we can write out the payoff of debt and equity at time T as follows: B min( A, D) D max( D A,0) T T T E max( A D,0) T T We can therefore derive the no-arbitrage price of these two securities at other times by using risk-neutral pricing

8 Value of debt and equity Therefore, the value of debt and equity at time t is: BS B Dexp( r( T t)) P ( A, D,, r, T t) t BS E C ( A, D,, r, T t) t The bond price is therefore: Increasing in V Increasing in D Decreasing in r Decreasing in time-to-maturity t Decreasing in volatility (increasing volatility of assets increases the value of shareholders claims) t

9 Yield spreads We can use the Merton model to estimate yield spreads on corporate debt, by writing the yield on debt as: log( Bt ) log( D) y( t, T) t T As part of your QCS I want you to write out yield spreads on corporate debt for a given set of parameters

10 Merton-implied credit spreads Asset volatility = 0.2; interest rate = 0.05; D = 100

11 Generalising Merton s model Subordinated debt can easily be handled in the Merton model If you have senior debt, junior debt and equity, and the senior debt gets paid first, you can write: You can then write out the payments to each of the debt holders in terms of a portfolio of options on the underlying firm assets

12 Subordinated debt In the case of subordinated debt we therefore have the payoffs to senior and junior debt and equity as follows: For your QCS, I would like you to write a spreadsheet which calculates the implicit spreads over the risk-free rate for senior and junior debt

13 Debt issues expiring at different times We cannot use the Merton model to price debt issues maturing at different times as a portfolio of zero-coupon bonds This is because if a company pays debt by reducing its own assets, it reduces its assets and liabilities, and therefore changes the value of outstanding equity and debt If equity holders decide to pay the debt out of their own pockets, then we need to decide under which conditions this is actually optimal (rather than for them to declare bankruptcy) We therefore need to build a tree and value the debt from the back, exactly in dynamic programming, to produce a value of debt at the front

14 Corporate debt and pensions An unfunded pension liability is a debt on the balance sheet of the employer Contributions are payments that need to be made to service this debt, so the analogy with risky corporate debt is exact However, the ranking of pensions in the wind-up order is not clear Pensions are unsecured debt at wind-up, but the pension regulator can hold up a corporate transaction if the pension fund trustees are not satisfied Therefore, in principle and probably in practice, pension contributions are fairly high on the pecking order

15 Extensions of the Merton model The process governing the pension fund assets can be made consistent with the process governing firm default Risky interest rates Correlation with the stock market You can also make the asset process governing firm default subject to jumps Collin-Dufresne and Goldstein (2001) examine the case of dynamic corporate financial structure, by assuming that corporations try to maintain stable leverage ratios They show how this affects the structure of corporate yield spreads Anthony Neuberger and I used a jump-diffusion extension of CDG in our work for the PPF (2010)

16 Calibration of Merton models In principle, the Merton model is relatively simple to calibrate We assume that the value of assets A(0) and the value of the volatility of assets, sigma, are unknown We can estimate the value of the stock, given the values of the assets and sigma We can also estimate the volatility of the stock (using Ito s lemma) We therefore have two equations in two unknowns which we can solve jointly If leverage varies a lot over the period, this can be difficult, and there are numerical schemes to deal with this problem

17 Drawbacks of Merton-type models The actuarial core reading says that Merton-type models are difficult to use because they are expensive to implement and because pension funds may not have access to all the data that is required The second only applies if the sponsor is not listed The main problem with Merton models is that they are pretty good at measuring default, but not so good at measuring spreads (i.e. they are hard to calibrate) In fact, Huang and Huang (2003) claim that only about 1/3 of the spread on corporate bonds is due to default, the remainder is due to some other unidentified factor What do you think it might be?

18 Moody s KMV approach To cope with this problem, Moody s KMV implement a model of the corporation based on the Merton model (with some input from Vasicek and Kealhofer giving KMV) The model has 5 classes of debt They estimate something called the distance to default, which in Merton s model is defined as: E(log( V )) log( D) T This is just the amount x such that: N( x) Pr(Company defaults at time t) T

19 Distance to default Moody s provide a mapping from distance to default defined slightly differently from above to expected default frequency (EDF) based on their historical database which maps DD s to EDF s This mapping is proprietary, and, they claim, quite accurate Other practical problems which need to be surmounted include: Defining the default point (KMV use 0.5* long-term liabilities + shortterm liabilities) Measuring the liabilities Accurately specifying the capital structure

20 Empirical evidence on credit spreads How do unfunded pension liabilities affect credit spreads? If not at all, then it is hard to argue that they affect default probabilities in any way (or at least hard to argue that the market values firms accurately) Cardinale (2007) estimates an equation relating the credit spread on a corporations bonds to various pension and other firm characteristics He finds a significant, asymmetric effect, in that pension fund underfunding affects credit spreads, while overfunding is less significant Pension leverage is twice as expensive as other leverage

21 Structural models Advantages: Allow simultaneous valuation of a firm s securities and determination of default probabilities Permit the determination of optimal capital structure (in the pensions context this means pension funding and investment policy) Risk premia arise endogenously so don t need to be externally specified Disadvantages Struggle to simultaneously predict default rates and observed spreads on corporate debt with plausible parameter values (Cannot replicate existing firm balance sheets easily) Seems that credit risk only explains about 1/3 of the spread on good quality corporate bonds, more for poorer quality bonds (Huang and Huang, 2003)

22 References for structural models Leland (2008), on WebCT Has a bias towards structural models (so beware!) But summarises most developments in structural models well (take his new model with a pinch of salt though)

23 Reduced form models These are typically credit ratings-based models Use historical default probabilities of different credit ratings and a transition matrix of changes in credit rating to estimate long-term survival probabilities Difficult to estimate risk premia Have to use generator matrices and try to calibrate to bond prices But stable ratings might still have varying default probabilities (e.g. credit crisis) Correlations have to be exogenously imposed (unlike structural models) (Jump-diffusion models are to some extent a compromise between structural and reduced-form models because of the introduction of the exogenous jump to default)

24 References for reduced-form models Lando (2004) is an excellent introduction Perraudin et al (2007) shows how a ratings-based credit derivative model might be calibrated (on WebCT)

25 Demographic models When insurance first began to be written, actuaries were unsure of the correct level of mortality, let alone the idea the mortality may be improving over time So they invented the with profits system, whereby profits mainly from mortality would be distributed to survivors to compensate them for incorrect past levels of charges The CMI collects mortality information about insured lives and pensioners and the ONS collects information about national mortality BUT little is actually known about how to model future uncertainty in life expectancy

26 Important facts about longevity risk There is little reason for it to be correlated with asset price risk, so it is a safe working assumption that it is largely uncorrelated this means that the asset pricing implications particularly for pricing pension liabilities are less than they would be if it were correlated, but longevity is still important because fund payoffs are often non-linear functions of assets and liabilities Longevity changes only slightly from year to year But over the recent past, most of the changes in the UK (and other developed economies) has been in a similar direction, resulting in a large aggregate change There is little consensus over what a likely range of uncertainty is (and we have little data to help us)

27 UK mortality This graph shows the log of the crude death rate by age and year for the UK (civilian) population Relative improvements at young and older ages Senescence Selection humps during wars

28 A limit to life expectancy? Even though our own bodies may age and decay, life retains the ability to renew itself (our germ cells do not age) There does not appear to be a natural limit to human life expectancy

29 Life expectancy in the UK has been improving Healthy life expectancy has been improving at roughly the same rate as total LE

30 Regional / lifestyle discrepancies in mortality in the UK Large regional / lifestyle / social class-related discrepancies in life expectancy and life expectancy improvement in the UK

31 Model risk and stochastic mortality Model risk is particularly important in stochastic mortality modelling All the models rely on data, and relying exclusively on the data is implicitly assuming that future mortality shocks will be like past mortality shocks The last fifty years have been relatively stable from a mortality point of view Therefore many commentators believe that the true level of mortality uncertainty illustrated by standard mortality models is significantly understated This is even before the effects of parameter uncertainty are taken into account

32 Lee-Carter model Developed by demographers at the University of California in 1992 Has become the standard stochastic mortality model Says that log( m ) a b k x, t x x t The log of the central death rate of someone aged x in year t Average mortality curve at each age Sensitivity of mortality rate at each age to index k Single parameter controlling random changes in mortality over time k t is assumed to follow some stochastic process, so, in the original paper: k k t 1 t t 1

33 Lee-Carter model Cannot allow for complicated mortality changes across age and time such as the cohort effect because its simple form assumes that deviations from average mortality at each age respond in exactly the same way to the overall mortality index change Furthermore, it s a one-factor model so changes in mortality across the entire age distribution are perfectly correlated Haberman and Renshaw extend the Lee-Carter model to allow for cohort effects in their 2006 paper But do not cope with the correlation problem Hard to estimate (& obtain standard errors of parameters)

34 Cohort effect is significant Allowance for cohort effect affecting the mortality of the generation born in Source: Haberman and Renshaw (2006)

35 Lee-Carter estimates of k(t) This is from work I did with David Miles LHS shows the value of k(t) RHS is an estimate of the standard error of the estimates Mortality becomes much more stable post-war; using data from this period only may ignore some risk

36 Narrow LC confidence intervals Figure shows the 95% confidence interval of the life expectancy of a man aged 65 in ten years, as a function of the standard error of the k(t) parameter in the Lee-Carter model What is an appropriate number? (LC estimates are typically low)

37 Other stochastic mortality models Cairns-Blake-Dowd (2006) model Model q(x,t) as a logistic function of two random factors: q( t, x) exp( A ( t 1) A ( t 1)( t x)) 1 exp( A ( t 1) A ( t 1)( t x)) First factor affects mortality equally at each age Second factor has a greater effect at older ages Their model assumes a particular functional form for mortality over the age distribution Doesn t allow for cohort effects

38 Cairns-Blake-Dowd (2006) However, their model is amenable to fairly simply statistical estimation, and furthermore allows the market price of mortality risk to be estimated (using the observed prices of traded mortality bonds) For instance, they assume that (but in principle you could do anything you like): A μ A Cε t 1 t t 1 and obtain the following estimates: ˆ ˆˆ μˆ ; V CC'

39 Cairns-Blake-Dowd (2006) They don t provide confidence intervals for life expectancy, but they do provide them for the survival function In subsequent work they look at this question again

40 Life Metrics ( JP Morgan has done a lot of work trying to develop a market in instruments which can be used to hedge mortality We will talk more about them in the next class on how to manage risk Today we will talk about the part of this work which has been to develop: standard indices which can be used to measure mortality, and models which can be used to project it reliably, including stochastic models

41 Cairns et al (2008) Backtests various mortality models Including Lee-Carter, Haberman-Renshaw, Cairns- Blake-Dowd, and extensions Paper is a useful summary of all these mortality models It also implements them over various time horizons using past data to see how well they would have performed

42 Models used by Cairns et al (2008) M1 = Lee-Carter M2B = Haberman-Renshaw M3B = Currie (simplified Haberman-Renshaw) M5 = Cairns-Blake-Dowd M6 = Cairns-Blake-Dowd plus cohort effect (I) M7 = Cairns-Blake-Dowd plus cohort effect (II) For model formulae please see appendix of paper, on WebCT

43 95% prediction interval without parameter uncertainty 95% prediction interval with parameter uncertainty Rolling 10-year window forecasts Projection based on 10 years of data with rolling end date Actual 2006 value Most models need parameter uncertainty to produce confidence bands which contain realised value

44 Rolling horizon predictions Most models need parameter uncertainty to produce confidence bands which contain realised value Lee-Carter with parameter uncertainty seems to perform best, but none is great

45 Olivier-Smith and others There is a natural analogy between interest rates and mortality rates in that the survival curve can be thought of as a discounting function: t t 0 0 s S(0, t x) 1 exp( ( x, s) ds) exp( r ds) B(0, t) Survival curve Force of mortality Instantaneous short rate Price of a zerocoupon bond So, if, at a given age x, you had a complete set of 0-coupon mortality bonds, you could in theory estimate a process for the force of mortality which was consistent with no arbitrage

46 Complications The price of a zero-coupon mortality bond may not be observable in the market, and certainly cannot be arbitraged, rendering the exercise questionable The present value of such a bond would need to include the effect of interest as well as mortality (not trivial because you don t know when the mortality bond is going to pay) You need to estimate a process for the whole surface of the force of mortality, because presumably the force of mortality at age x is not too unrelated to that at age x+1 Cairns (2007) and Olivier and Smith (2004) have done work along these lines

47 Other stochastic mortality models The CMI has recently embarked on a great deal of work examining the different mortality models Their results are published in a series of working papers, which I can heartily recommend p30_cover.xml They also fit P-spline models, which are based on fitting a series of spline functions to the age-time mortality surface, and obtaining distributions of likely outcomes (note: P-spline sample paths are not sample paths)

48 Demographic state variables Mortality is just one of the demographic factors affecting pension funds Others are new entrants; early leavers; early & ill-health retirements etc These are usually of less financial significance (why?) And in fact, if the exit payment on the other decrements equals share of fund, don t even have to modelled at all! (why)