CTSC 445 Final Exam Summary sset and Liability Management Unit 5 - Interest Rate Risk (References Only) Dollar Value of a Basis Point (DV0): Given by the absolute change in the price of a bond for a basis point (0.0%) change in the yield. Forward Rates: ( + s k ) k ( + f k ) = ( + s k+ ) k+ Duration: D m = (y ) D = t>0 t t(+y ) t = t>0 t t(+y ) t (modified) = ( + y )D m (regular) D p = k i= w id i, w i = nii (portfolio) D F W = i= s i si=s i D F W = t= t te t s t (Fisher-Weil; alt. below) D Q = t>0 t t( + s t ) t (quasi-modified) D m,t = s t st=s t D m,t = { t t ( + s t ) t t t e ts t D e m = (y y) (y + y) 2 y D m, = (Ã ) t s t = k+ t Convexity: C = (partial; alt. below) discrete case continuous case (effective) (key rate), where s t t < t t k+ t s tk + s tñ C F W = t 0 t(t+)t(+y ) t 2 i= 2 s 2 i si=s i t t k t k+ t s tk+ t k < t < t k+ t > tñ C F W = t= t2 t e t s t C e m = (y y) 2 +(y + y) ( y) 2 Change estimation: (standard) (Fisher-Weil; alt. below) (effective) (y + y) (y ) D m y +C ( y)2 2 ( change) (y + y) (y ) (y ) D F W s + 2 C F W ( s) 2 (% change) k= D m,k s k (% change) New = ( + D m,k s k ) (new value of ) Remarks: The Macaulay duration of a zero-coupon bond is equal to its maturity. 2 Unit 6 - Immunization Target Date Immunization: Let V k (y) be the value of a portfolio of securities at time k (measured in years) for a given ytm y (assume annual effective rate). In the target date immunization scenario, we want to match the target date of the portfolio with the duration of the portfolio since for any ŷ. Single Liability Immunization: V D (ŷ) V D (y ) t>0 t( + y ) t = L( + y ) k, t>0 t t( + y ) t = kl k ( + y ) k If the assets are are symmetric about the time of the liability, put half of the PV in the first asset and half in the second asset Multiple Liability Immunization: (Redington s Basic Conditions; RBCs) Let S(y) = (y) L(y). Then the immunization conditions are:. S(y) = (y) L(y), (i) S(y ) = 0 [match PV] 2. S (y ) = 0 [match duration] 3. S (y ) > 0 [dispersion / convexity condition] Immunization Strategies: Bracketing Strategy: If we have liability cash flows t L < t L 2 <... < t L n and asset cash flows at t < t L and t + > t L n then if RBC (i) + (ii) is satisfied then so is (iii). If
M 2 Strategy: Let M 2 = t>0 w t (t D ) 2, w t = t(+y ) t, then if M 2 M L 2 and RBC (i) + (ii) hold we have RBC (iii) holding. Consider the probability measure P (T = t) = te r t t. Then for a portfolio, we have D = E[T ], C = E[T 2 ], M 2 = V ar[t ] = C D 2 Generalized Redington Theory: If N t = t L t, let the current surplus be denoted by S = t>0 N tp (0, t) and a shocked surplus (caused by interest rate changes) be denoted by Ŝ = t>0 N t ˆP (0, t) Let g(t) = ˆP (0,t) P (0,t) and n t = N t P (0, t) which will imply that Ŝ S = t>0 n t g(t) If t>0 n t = 0, t>0 tn t = 0, {n k } k>0 undergoes a +,, + sequence, and g(t) is convex, then Ŝ S 0. 3 Unit 8 - Interest Rate Models General risk-neutral equation: For a payoff of V T at time T, the value at time 0 is V 0 = E [V T e T r(t)dt] 0 The mean return of a stock is used in assessing the probabilities associated with threshold default models, whereas the risk neutral rate is used in pricing (in the Black Scholes and Merton models) Properties of the Continuous Time Models: The Rendleman-Barter (lognormal) does not capture mean-reversion, but disallows negative interest rates; it is an equilibrium model The Vasicek model captures mean-reversion but does allows negative interest rates; it is an equilibrium model The Cox-Ingersoll-Ross model is an improvement on the Vasicek model since it captures mean reversion while disallowing negative interest rates; it is an equilibrium model Monte-Carlo Simulation: Monte Carlo is a method to estimate E[X] for a statistic X using the estimator n i= x i The exact steps are:. We simulate a set of discount factors {v, v 2,..., v N } 2. Find the simulated price N i= c tv t 3. Repeat steps and 2, n times where n is large; at the end of the process, we have n simulated prices c 0,.., c n 0 4. The estimated price of the security is given by n i= ci 0 For simulation of the discount factors, write v t = e t 0 r(s)ds e (r0+r+...+rt )t and then simulate the sample path {r,..., r n } It is generally used when a pricing problem is too difficult to solve analytically Discrete Binomial Trees and Embedded Options: Using backwards recursion, the general formula is: V (t, n) = q(t, n) V (t +, n + ) + [ q(t, n)] V (t +, n) + i(t, n) There are two approaches to pricing bonds with embedded options: () price the bond directly (2) price the components (option-free and option) We always start with V (T, k) = F for k =,..., n For a callable bond (option part) the option payoff at node (t, n) is E(t, n) = max(0, B(t, n) K) and V (t, n) = max(e(t, n), H(t, n)) where H(t, n) depends on the previous V (t+, n+) and V (t+, n) results and B(t, n) is the price of the option-free component For a putable bond, the algorithm is the same except now E(t, n) = max(0, K B(t, n)) Interest Rate Caps and Floors: Let L be the notional amount of the loan Caps are used to protect the borrower of a loan from increases in the interest rate. It is formed by a series of caplets. t time t, the payoff from a caplet is L(i t K) + if settled in arrears L(i t K) + if settled in advance Floors are used to protect the lender of a loan from decreases in the interest rate. It is formed by a series of floorlets. t time t, the payoff from a floorlet is L(K i t ) + if settled in arrears L(K i t ) + if settled in advance Black-Derman-Toy Model: In this model, q(t, n) = 2, and the interest node relationship is given as i(t, n + ) = i(t, n)e 2σ(t) or equivalently i(t, n) = i(t, 0)e 2σ(t) n 2
To calibrate with s ts and σ ts we use: r 00 = s Solve r t0 with t ( + s t+ ) t+ = (t, n) + i(t, 0)e 2kσ(t) k=0 This model is an arbitrage-free model Option djusted Spread: Reasons for the spread: Compared to option-free bonds, bonds with embedded options come with repayment/reinvestment risk. Using the calibrated model if we compute the price of such a bond, we will have the theoretical price, this may differ from the actual market price. The OS is a fixed/flat spread over the rates of the calibrated free that gives the theoretical price is equal to market price. Prepayment/reinvestment risk for a callable bond can be defined as the risk that the principal with be repaid before maturity, and that the proceeds will have to be invested at a lower interest rate. OS is the rate such that the binomial interest rate lattice shifted by the OS equates the new theoretical price with the market price (uniform shift) The OS of an option free bond is 0 Here are the steps to compute V + /V :. Given the security s market price, find the OS. 2. Shift the spot-rate curve by a small quantity y. 3. Compute a binomial interest-rate lattice based on the shifted curve obtained in Step 2. 4. Shift the binomial interest-rate lattice obtained in Step 2 by the OS. 5. Compute V + /V based on the lattice obtained in Step 4. The V + /V values are used in the calculation of effective duration and convexity through the formulas: Dm e = V V + 2V 0 y, Ce m = V + 2V 0 + V V 0 ( y) 2 4 Unit 9 - Value-at-Risk (VaR) Standard Definition of VaR: The formal definition for VaR is implicitly defined throughif we have a non-negative surplus and matched duration, then the portfolio of assets and liabilities will have V D (ŷ) V D (y ), D = D L where y is the current ytm and ŷ is a shift in the ytm, then the realized rate of return can never fall below its initial yield. P (L n > V ar α,n ) = F Ln (V ar α,n ) = α where L n is the loss random variable. It is also equivalent to V ar α,n = inf{l R F Ln (l) α} = inf{l R P (L n > l) α} for general distributions (i.e. discrete, continuous, and mixed) lternatively, V ar can be interpreted as the change in portfolio value V = V n V 0 = L n since V ar α,n is such that P (L n V ar α,n ) = α = P ( V V ar α,n ) = α Remark that VaR is, in general, never sub-additive Conditional Tail Expectation: This is the average loss that can occur if loss exceeds V ar α,n. For a loss distribution L n and confidence α this is CT E α,n = E[L n L n V ar α,n ] all l w/ L V ar = α,n l P r(l n = l) all l w/ L V ar α,n P r(l n = l) In general CTE is sub-additive for continuous distributions and not sub-additive for discrete distributions lternate Definition (One Factor): We can re-write V ar as V ar α,n = V 0 (σ z α n nµ ) = V 0 (σ n z α µ n ) where z α = Φ (α) and Φ(α) = P (N (0, ) α) If µ = 0 then nv ar α, = V ar α,n lternate Definition (Two Factor): We can re-write V ar as V ar α,n = V 0 (σ V z α µ V ) where the two factor representation is V = V n V 0 = V 0 (w ( + R ) + w 2 ( + R 2 )) V 0 and R V = V V 0 N (µ V, σv 2 ) with µ V = w µ + w 2 µ 2, σv 2 = w2 σ 2 + w2σ 2 2 2 + 2ρw w 2 σ σ 2 3
Delta Normal Method: For a portfolio with multiple factors, we have through a first order Taylor expansion, dv i= V f i df i = where i = V/ f i df i = i= i= f i i df i f i = f i i R i i= Let S t, B t be the equity and debt values and of a firm at time t respectively; these are modeled as stochastic processes Denote V t = S t + B t where V t is the firm s value ssume that no dividends are paid and a payment B is paid at time T from the firm issuing a bond t time T we have We can then compute V ar(dv ) = σv 2 = i i ) i=(f 2 V ar(r i )+2 f i f j i j Cov(R i, R j ) B T = min(v T, B) = B max(0, B V T ) i j and so V T is the payoff of a call option S T of strike B, B units of a T year ZCB and assuming that µ V, we can approximate VaR as V ar α,n σ V z α For the special case of options, dv = ds = S 0 ds S 0 = S 0 R S where is the delta of the option. Thus we can use the approximation V ar(dv ) = S0 σ S = σ V = V ar α, = σ V z α S T = max(0, V T B) This is because at time T, if V T < B, the whole firm liquidates its assets to debtholders since it has defaulted and missed a payment In the former case, since shareholders are paid last, they get nothing Thus default occurs when V T < B Merton s Model: Merton s model assumes V t behaves as Brownian motion and implies 5 Unit 0 - Credit Risk Remark that in computing probabilities, we tend to use the Black-Scholes formula that involves µ V (Merton s model), but in pricing, we use the formula that involves the risk-free rate r (options pricing) Types of models: Static v. Dynamic: static models are for credit risk management while dynamic models are for pricing risky securities Structural and Threshold v. Reduced-form: Threshold models are when default occurs when a selected random process falls under a threshold; reduced form models are when the time to default is modeled as a non-negative random variable whose distribution depends on a set of economic variables Challenges of Credit Risk Management: Lack of public information and data; interpreted as-is Skewed loss distributions; problems of frequent small profits and occasional large losses Dependence modeling; defaults tend to happen simultaneously and this impacts the credit loss distribution Structural Models of Default: where B t N(0, t). dv t = µ V V t dt + σ V V t db t = V t = V 0 e (µ V σ V /2) 2 +σb t This implies that V t is lognormally distributed and compute quantities like P (default) = P (V T B) = P (ln V T ln B) ( = P N (0, ) ln B ln V 0 ( µ V σv 2 /2) ) T σ V T Going back to the first point of this section, let r be the risk-free rate. If a security has a payoff of h(v T ) at time T, then its price is E Q (e rt h(v T )) where this expectation is done under the risk-neutral measure. This is equivalent to V t = V 0 e (r σ2 V /2)t+σ V B t which is the Black-Scholes framework under r Threshold Models: Used to model default in the case of a portfolio of securities issued by a large number of obligors 4
This is a generalization of Merton s model where firm i defaults if V T,i < B i In a general threshold model, firm i defaults if its associated critical random variable X i falls below some threshold d i Threshold Model Notation: Let d ij be the critical threshold of firm i at rating j (e.g. credit rating) Let D = [d ij ] m n R m n where X i < d i implies default Let S i be the state of firm i with S i {0,,..., n} and S i = j d ij < X i d i(j+) with d i,0 =, d i(n+) = S i = 0 is true iff there is default Let Y i = χ Xi(T )<d i, the default indicator variable for X i We denote the marginal cdf of X i through the following equivalent forms: p i = P (X i d i ) = F Xi (d i ) = F i (d i ) = P (Y i = ) M = m i= Y i is the number of obligors who have defaulted at time T L = m i= δ ie i Y i is the overall loss of the portfolio where e i is the exposure of firm i and δ i is the fraction of money that is lost from default 2. If U = U 2 then F (u, u 2 ) = P ( u 2 U u ) 3. If U = U 2 then F (u, u 2 ) = P (U min(u, u 2 )) These results are similar if U, U 2 N (0, ) and U = U 2 in the second case; this gives us some copulas:. C ind (u, u 2 ) = u u 2 2. C neg (u, u 2 ) = max(u + u 2, 0) 3. C pos (u, u 2 ) = min(u, u 2 ) Generalization is easily done for more than two variables with similar dependence structure This can be seen in the Gauss copula of the form C Σ (u,..., u m ) = Φ Σ (φ (u ),..., φ (u m )) Note that C(u, u 2 ) = u + u 2 is not a copula pplications of Copulas: They are mainly useful in calculating binary results for firms which are of the form P (d j < X < d j2, d Bj < X B < d Bj2 ) which is usually calculated by drawing the encompassing region and re-writing the expression in terms of additions and subtractions of cdfs The default correlation is given as ρ(y i, Y j ) = E(Y iy j ) p i p j ( p i p 2 i )( p j p 2 j ) Intro to Copulas: copula is a joint distribution of uniform random variables such that C(F X (u ), F X2 (u 2 )) = F X,X 2 (u, u 2 ) which implies that C(u, u 2 ) = F X,X 2 (F X (u ), F X 2 (u 2 )) It has the property that C(u, ) = C(, u) = u C(u, 0) = C(0, u) = 0 C is increasing in u and u 2 Special Copulas: Suppose that U, U 2 Unif(0, ). If U U 2 then F (u, u 2 ) = F U (u )F U2 (u 2 ) 5