Statistical Methods in Financial Risk Management

Transcription

1 Statistical Methods in Financial Risk Management Lecture 1: Mapping Risks to Risk Factors Alexander J. McNeil Maxwell Institute of Mathematical Sciences Heriot-Watt University Edinburgh 2nd Workshop on Risk Analysis in Economics and Finance Guanajuato Mexico May 2013 McNeil Statistical Methods in Financial Risk Management 1 / 32

2 QRM Q U A N T I TAT I V E R I S K M A N A G E M E N T Concepts Techniques Tools Alexander J. McNeil Rüdiger Frey Paul Embrechts P R I N C E T O N S E R I E S I N F I N A N C E McNeil Statistical Methods in Financial Risk Management 2 / 32

3 Overview 1 From Balance Sheets to Loss Distributions McNeil Statistical Methods in Financial Risk Management 3 / 32

4 1 From Balance Sheets to Loss Distributions McNeil Statistical Methods in Financial Risk Management 4 / 32

5 A Bank Balance Sheet Assets Bank ABC (31st December 2010) Liabilities Cash 10M Customer deposits 80M (and central bank balance) Securities 50M Bonds issued - bonds - senior bond issues 25M - stocks - subordinated bond issues 15M - derivatives Short-term borrowing 30M Loans and mortgages 100M Reserves (for losses on loans) 20M - corporates - retail and smaller clients Debt (sum of above) 170M - government Other assets 20M - property - investments in companies Equity 30M Short-term lending 20M Total 200M Total 200M McNeil Statistical Methods in Financial Risk Management 5 / 32

6 Balance Sheet of an Insurer Insurer XYZ (31st December 2010) Assets Liabilities Investments Reserves for policies written 80M - bonds 50M (technical provisions) - stocks 5M Bonds issued 10M - property 5M Investments for unit-linked 30M Debt (sum of above) 90M contracts Other assets 10M - property Equity 10M Total 100M Total 100M McNeil Statistical Methods in Financial Risk Management 6 / 32

7 Solvency of a Financial Institution The balance sheet equation asserts that value of assets = value of liabilities = debt + equity. A company is solvent at a particular time if the equity is nonnegative; otherwise it is insolvent. Insolvency should be distinguished from default which occurs if a firm misses a payment to its debtholders or other creditors. An otherwise solvent company can default because of liquidity problems. (Banks typically borrow short and lend long and are susceptible to increased costs of short-term refinancing as we found out in 2008.) An insolvent company is however very likely to experience liquidity problems and default. McNeil Statistical Methods in Financial Risk Management 7 / 32

8 Capital All notions of capital embody the idea of a loss-bearing buffer that ensures that the financial institution remains solvent. 1 Equity capital can be read from the balance sheet according to accounting equation and is a measure of the value of the company to the shareholders. Usually broken down into positions for shareholder capital retained earnings and other items. 2 Regulatory capital is capital an institution should have according to regulatory rules (Basel II/III for banks and Solvency II For European insurers). Rules specify amount of capital necessary to continue operating taking into account size and riskiness of positions. Usually specify the quality of the capital and hence the form it should take on the balance sheet (Tier 1 Tier 2). 3 Economic capital is an internal estimate of the amount of capital that a financial institution needs in order to control the probability of becoming insolvent typically over a one-year horizon. McNeil Statistical Methods in Financial Risk Management 8 / 32

9 Economic Capital: European Actuarial Approach Consider an insurer with current equity capital given by V t = A t B t. To ensure solvency in 1 year s time with high probability α company may require extra capital x 0 determined by x 0 = inf{x : P(V t+1 + x(1 + i) 0) = α} where i is one-year risk-free interest rate. If x 0 is negative company is well capitalized and money could be taken out; this can be thought of as an additional liability position of size x 0 not matched by an additional asset position so that equity capital is reduced. which shows that x 0 = inf{x : P( V t+1 x(1 + i)) = α} = inf{x : P(V t V t+1 /(1 + i) x + V t ) = α} V t + x 0 = q α (V t V t+1 /(1 + i)) where q α denotes a quantile. McNeil Statistical Methods in Financial Risk Management 9 / 32

10 Risk Measures Applied to Loss Distributions The sum V t + x 0 gives the solvency capital requirement (SCR) namely the available capital corrected by the amount x 0 and is a quantile of the distribution of L t+1 = V t V t+1 /(1 + i). For short horizons or simplicity we often neglect discounting to consider L t+1 = V t V t+1 the decrease in net asset value. More generally capital is computed by applying risk measures like q α to the loss distribution of L t+1. Alternative risk measures like expected shortfall e α (L) = (1 α) 1 1 α q u(l)du may be favoured. Sometimes the mean loss is subtracted from risk measure as expected losses are often reserved for on balance sheet. There is a rapidly growing mathematical finance literature on appropriate risk measures starting with [Artzner et al. 1999]. McNeil Statistical Methods in Financial Risk Management 10 / 32

11 1 From Balance Sheets to Loss Distributions McNeil Statistical Methods in Financial Risk Management 11 / 32

12 The Problem of Value All kinds of difficulties surround the issue of value. Do we mean book value or fair or market-consistent value? How is market-consistent value determined in the absence of liquid markets? The international fair-value accounting standard (IFRS 7) describes a 3-level fair value hierarchy: mark-to-market valuation at level 1; mark-to-model with fully observed market inputs at level 2; mark-to-model with unobserved inputs at level 3. Examples: level 1 stocks and shares; level 2 European call option; level 3 bespoke CDO tranche annuity portfolio. And asset/liability valuation in the future requires the forecasting of a complete financial market consisting of equities currencies yield curves credit spreads volatilities etc. McNeil Statistical Methods in Financial Risk Management 12 / 32

13 Mapping to Risk Factors at time t Suppose we denote the risk factors at time t by the vector Z t = (Z t1... Z td ). Examples of risk factors include equity prices exchange rates interest rates for different maturities credit spreads volatilities etc. In most cases risk factors are market observables at time t. Consider a portfolio of assets liabilities or both. The value of the portfolio at time t is given in general by the formula V t = f (t Z t ) (1) where f is the portfolio mapping. It contains information about the portfolio composition and incorporates the valuation models that may be needed to value the more complex (derivative) assets/liabilities with respect to the underlying risk factors Z t. McNeil Statistical Methods in Financial Risk Management 13 / 32

14 Mapping to Risk Factors at time t + 1 The value of an portfolio at time t + 1 will be V t+1 = f (t + 1 Z t+1 ) where we have assumed that the mapping function stays the same and there is no portfolio rebalancing or strategic asset reallocation. Modelling stochastic changes to portfolio composition is difficult. Another possibility is to have a set of deterministic rebalancing rules depending on Z t+1. From prespective of time t the future value V t+1 is stochastic (depending on random vector Z t+1 ) whereas V t is deterministic (can be computed based on time t information). McNeil Statistical Methods in Financial Risk Management 14 / 32

15 Analytical Models or Monte Carlo? There are three broad approaches to determining the distribution of V t+1 (or equivalently of the loss L t+1 = (V t+1 V t )). Analytical: by assuming simple stochastic models (e.g. models where Z t+1 given Z t has a normal or elliptical distribution) and simple approximate forms for f (e.g. linear) we can get closed form models for V t+1 given V t. Historical simulation: we can work with the empirical distribution of past risk factor increments: Z t Z t 1 Z t 1 Z t We can resample values from this distribution. Full Monte Carlo or economic scenario generation: set up a multivariate stochastic process which projects the risk factors into the future giving scenarios Z s for the economy at future times s > t. McNeil Statistical Methods in Financial Risk Management 15 / 32

16 1 From Balance Sheets to Loss Distributions McNeil Statistical Methods in Financial Risk Management 16 / 32

17 Assumptions The current time is t and the value of a given asset portfolio is V t which is known or can be computed with appropriate valuation models. We are interested in value changes or losses over a relatively short time period [t t + 1] for example one day two weeks or month. Scaling will be applied to derive capital requirements for longer periods. We assume there is no change to the composition of the portfolio over the time period. The future value V t+1 is modelled as a random variable. We want to determine the distribution of the loss distibution of L t+1 = (V t+1 V t ). McNeil Statistical Methods in Financial Risk Management 17 / 32

18 Risk Factor Changes We introduce notation for the risk factor changes over the time horizon X t+1 = Z t+1 Z t. Typically historical risk factor data are available in the form of a time series X t n... X t 1 X t and these can be used to help model the behaviour of X t+1. We have L t+1 = (V t+1 V t ) = (f (t + 1 Z t+1 ) f (t Z t )) = (f (t + 1 Z t + X t+1 ) f (t Z t )) (2) McNeil Statistical Methods in Financial Risk Management 18 / 32

19 Portfolio of Stocks Consider d stocks; let α i denote number of shares in stock i at time t and let S ti denote price. The risk factors: following standard convention we take logarithmic prices as risk factors Z ti = log S ti 1 i d. Why? The risk factor changes: in this case these are X t+1i = log S t+1i log S ti which correspond to the so-called log-returns of the stock. The mapping formula (1) is V t = d α i S ti = i=1 d α i e Z ti. i=1 McNeil Statistical Methods in Financial Risk Management 19 / 32

20 Portfolio of Stocks II The loss is ( d ) d L t+1 = α i e Z t+1i α i e Z ti i=1 i=1 d ( = V t ω ti e X t+1i 1 ) i=1 where ω ti = α i S ti /V t is relative weight of stock i at time t. For short time intervals (small values of X t+1i ) this can be approximated by a first-order (linear) approximation. McNeil Statistical Methods in Financial Risk Management 20 / 32

21 Portfolio of Zero-Coupon Bonds Consider a portfolio of d default-free zero-coupon bonds with maturities T i and prices p(t T i ) 1 i d. (Assume p(t T ) = 1.) By λ i we denote the number of bonds with maturity T i in the portfolio. In a detailed analysis of the change in value of the bond portfolio one takes all yields y(t T i ) 1 i d as risk factors. The mapping at time t is given by V t = d λ i p(t T i ) = i=1 d λ i exp( (T i t)y(t T i )). i=1 Recall yield given by y(t T ) = (1/(T t)) ln p(t T ). McNeil Statistical Methods in Financial Risk Management 21 / 32

22 Portfolio of Zero-Coupon Bonds II The risk factor changes are the changes in yields X t+1i = y(t + 1 T i ) y(t T i )1 i d. Because yields for different maturities are highly correlated we would typically use factor models to reduce dimension. To model defaultable bonds we would add risk factors for default risk and spread risk. Consider d defaultable bonds issued by different companies. Let Y ti be a default indicator taking value 1 if company defaults in [0 t] and 0 otherwise. The mapping at time t could take form V t = d (1 Y ti )λ i exp( (T i t)(y(t T i ) + s i (t T i ))) i=1 where s i (t T i ) denotes spread for company i s bond. McNeil Statistical Methods in Financial Risk Management 22 / 32

23 Nelson-Siegel Model In this model [Nelson and Siegel 1987] the yield curve is modelled by three factors y(t T ) = F t1 + k 1 (T t η t )F t2 + k 2 (T t η t )F t3 + ignored error The functions k 1 and k 2 are given by k 1 (s η) = 1 exp( ηs) k 2 (s η) = k 1 (s η) exp( ηs). ηs lim T y(t T ) = F t1 so the first factor is usually interpreted as a long-term level factor. F t2 is interpreted as a slope factor because the difference in short-term and long-term yields satisfies lim T t y(t T ) lim T y(t T ) = F t2. F t3 has an interpretation as a curvature factor. McNeil Statistical Methods in Financial Risk Management 23 / 32

24 A Derivative Portfolio Consider portfolio consisting of one standard European call on a non-dividend paying stock S with maturity T and exercise price K. The Black-Scholes value of this asset at time t is C BS (t S t r σ) where C BS (t S; r σ) = SΦ(d 1 ) Ke r(t t) Φ(d 2 ) Φ is standard normal df r represents risk-free interest rate σ the volatility of underlying stock and where d 1 = log(s/k ) + (r + σ2 /2)(T t) σ T t and d 2 = d 1 σ T t. While in BS model it is assumed that interest rates and volatilities are constant in reality they tend to fluctuate over time; they should be added to our set of risk factors. McNeil Statistical Methods in Financial Risk Management 24 / 32

25 Deriving Loss Distribution The risk factors: Z t = (log S t r t σ t ). The risk factor changes: X t = (log(s t /S t 1 ) r t r t 1 σ t σ t 1 ). The mapping: V t = C BS (t S t ; r t σ t ) The loss could be calculated from (2). For derivative positions it is quite common to calculate linearized loss ( ) 3 L t+1 = f t (t Z t ) t + f zi (t Z t )X t+1i where f t f zi denote partial derivatives. t is the length of the time interval expressed in years since Black-Scholes parameters relate to units of one year. i=1 McNeil Statistical Methods in Financial Risk Management 25 / 32

26 The Greeks From Balance Sheets to Loss Distributions It is more common to write the linearized loss as ( ) L t+1 = Ct BS t + CS BS S tx t+11 + Cr BS X t+12 + Cσ BS X t+13 in terms of the derivatives of the BS formula. C BS S Cσ BS Cr BS Ct BS is known as the delta of the option. is the vega. is the rho. is the theta. Sometimes delta-gamma (second-order) approximations are used. McNeil Statistical Methods in Financial Risk Management 26 / 32

27 1 From Balance Sheets to Loss Distributions McNeil Statistical Methods in Financial Risk Management 27 / 32

28 Uses of Risk Measures Risk measures are used for a variety of purposes: We assume that risk capital is calculated by applying a risk measure to a loss distribution. Our interpretation. Risk measure gives amount of capital that needs to be held to back a position with loss L t+1 so that the position is acceptable to an (internal/external) regulator. Most commonly used risk measures (like VaR) are based on the loss distribution of L = L t+1. We denote the loss distribution function by F L so that P(L x) = F L (x). McNeil Statistical Methods in Financial Risk Management 28 / 32

29 VaR and Expected Shortfall Let 0 < α < 1. We use Value at Risk is defined as VaR α = q α (F L ) = F L (α) (3) where we use the notation q α (F L ) or q α (L) for a quantile of the distribution of L and F L for the (generalized) inverse of F L. Provided E( L ) < expected shortfall is defined as ES α = 1 1 α 1 α q u (F L )du. (4) McNeil Statistical Methods in Financial Risk Management 29 / 32

30 Losses and Profits Loss Distribution probability density Mean loss = % VaR = % ES = 3.3 5% probability McNeil Statistical Methods in Financial Risk Management 30 / 32

31 Expected Shortfall From Balance Sheets to Loss Distributions For continuous loss distributions there is a more intuitive way of understanding expected shortfall - it is the expected loss given that the VaR is exceeded. For any α (0 1) we have ES α = E(L L VaR α ). For a discontinuous loss distribution we have the more complicated expression ES α = E(L L > VaR α ) + VaR α 1 α P(L > VaR α ) 1 α [Acerbi and Tasche 2002].. McNeil Statistical Methods in Financial Risk Management 31 / 32

32 References For Further Reading For Further Reading Acerbi C. and Tasche D. (2002). On the coherence of expected shortfall. J. Banking Finance 26: Artzner P. Delbaen F. Eber J. and Heath D. (1999). Coherent measures of risk. Math. Finance 9: Nelson C. and Siegel A. (1987). Parsimonious modeling of yield curves. Journal of Business 60: McNeil Statistical Methods in Financial Risk Management 32 / 32