Contents. 3 FIXED-INCOME SECURITIES VALUATION Introduction Bond Valuation Introduction... 32

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1 Contents 1 YIELD CURVE CONSTRUCTION Introduction Market Conventions Business day Conventions Compounding Methods Spot interest rate Fundamental Interest Rate Curve Forward Yield Curve Interpolation Method for yield curve construction Introduction Yield Curve Fitting Linear Interpolation INCREMENTAL RISK CHARGE (IRC) Credit Metric model Multiperiod Model Multi-Period Credit Rating Simulation Position Valuation in multiperiod model FIXED-INCOME SECURITIES VALUATION Introduction Bond Valuation Introduction

2 3.2.2 Bond Valuation Without Credit Risk Bond Valuation With credit risk Calibration of Survival Probability The CDS-Cash Basis CDS Valuation How CDS works Pricing CDS Marking To Market A CDS Position Calibration of Survival Probability SHOCK OF SPREAD Shock on Spread Curves Description of the data Methodologies: calibration of spread shock Grouping in full grade Additive Spread Shock on Sovereign Spread Curves Multiplicative Spread Shock on Corporate Spread Curves Transform from Full Grades to Notches by Interpolation Interpolation of Term Structure Profit and Loss Calculation RESULTS AND DISCUSSION Application: Bond Valuation Risky Coupon Bond Results of Bond Valuation Results of the calibration of survival probability Application: CDS valuation Application: Shock of spread

3 6 CONCLUSION AND FUTURE WORK Conclusion Future Works

4 List of Figures 3.1 Mechanics of a default swap premium leg The protection leg following a credit event Histogram for issuers: BEAZHOMUSA for SEN 1year spread Histogram for issuers: BEAZHOMUSA for SEN 3 year spread Histogram for issuers: ALCALSTPA for SEN 1 year spread Histogram for issuers: ALCALSTPA for SEN 3 year spread P&L Calculation in Single Period Simulation

5 ABSTRACT As every financial products, fixed income assets price (such as bonds and CDS) varies gradually and continuously in market information. However, two bonds which have the same cash-flow, the same maturity could be traded at different prices. The reason is that fixed income products admit not only market risks but also credit risk. The bond price and credit default swap (CDS) traded in markets depend on the credit quality of issuers. In practice, credit quality is represented by credit rating. The credit rating is provided by rating agencies such as S&Q, Moodys... A change of the issuers rating could provide an important jump in the issuers spread curve. This jump of spread results a possible significant loss on bond and CDS position. In the risk management point of view, identifying and measuring impact of rating migration on fixed income portfolios are crucially important. The object is to modelise the jumps in the issuer s spread curve when migration happens. In the point of view of risk manager, this modelisation will be calibrated into market data in order to capture the possible loss induced by degradation. The thesis consists of six chapters. In chapter one, the construction of the yield curve is presented. Chapter two introduce Incremental Risk Charge. Some simple fixed-income securities such as bonds and CDS in a complete market satisfying the usual conditions are considered in chapter three. Valuation of these products is based on the Absence of Arbitrage opportunity argument. Next, the calibration of shock of spread is presented in chapter four. Practical applications of pricing, shock of spread and profit and loss (PnL) calculations on market data set are shown in chapter five. The provided results cover all necessary steps: formulas to compute price, procedure to construct and Matlab implementation. Finally, conclusions and future works are given in chapter six. 6

6 Chapter 1 YIELD CURVE CONSTRUCTION 1.1 Introduction The term structure of interest rates is defined as the relationship between the yield-tomaturity on a zero coupon bond and the bonds maturity. There is a clear relationship between bond prices, spot rates and forward rates, given any one of these sets (observed on the market), it is possible to calculate the other two. In this section, we present a selection of linear interpolation and flat extrapolation that are used in financial markets for construction of curves such as forward curves, basis curves, and most importantly, yield curves. Continuously and annual compounded rates are used relative to the case where we price all instruments consistently (at any time points) or discrete time. 1.2 Market Conventions Definition 1.1. Day count conventions are used to compute the holding period of investments such as bonds, swaps, loans etc and the interest accrued in turn. When the holding period ends in between two coupon payment dates, the interest accrued needs to be calculated from the previous coupon payment date or the date on which the bond is acquired whichever is later to the last day of holding. The interest accrued can be calculated in terms of the holding period, the interest rate, and the 7

7 number of days in an year as: Interest accrued = (Principal)(Interest rate)(factor). Where: the factor is calculated using one of the available conventions. Example A bond of face value 100 pays a coupon of 7% every year. The last coupon was paid on 31st Dec, 2010 and the bond was purchased on 1st Jan, As of 23rd Mar, 2011 the interest accrued on the bond using Actual/360 convention is: Holding period in days (Actual) = 81; Number of days in an year = 360. It impplies interest accrued = 100( )0.07 = There are no standardized day count calculations because the conventions differ in their ease of use, market settings and the nature of time period under consideration. The following some commonly cases of day-count conventions are mentioned /360 Method We consider day-count conventions(time to maturity) as follows: months and years are assumed 30 days long and 360 days long, respectively. The year fraction between date D 1 and D 2 of all conventions of this class calculate as: max(30 d 1, 0) + min(d 2, 30) + 360(y 2 y 1 ) + 30(m 2 m 1 1) 360 Where: d 1, m 1, y 1 and d 2, m 2, y 2 denotes days, months, years of date D 1 and D 2 respectively. This convention is listed on standard loan constant charts and is now typically used to determine mortgage payments. It is easy to calculate by hand compared with the actual days between two dates. 30/360 The number of days considered in a month is 30 in the numerator and a year is assumed to be 360 long in the denominator. This day-count convention used for corporate bonds, municipal bonds and agency issues. 30E/360 The date adjustment rules are as the following: If d 1 is 31, then change d 1 to 30. If d 2 is 31, then change d 2 to 30. It is also called Eurobond basis. 8

8 30E/360 ISDA The date adjustment rules are as the following: if d 1 is the last day of the month, then change d 1 to 30. If d 2 is the last day of February but not the termination date or d 2 is 31, then change d 2 to Actual methods The conventions of this class calculate the number of days between two dates, the period first day is included and the last day is excluded. Actual/360 A year is assumed to be 360 days long. The actual number of days between the two dates, D 1 = (d 1, m 1, y 1 ) included and D2 = (d 2, m 2, y 2 ) excluded are denoted D 2 D 1. The corresponding year fraction is D2 D In this case, notice that there are 31 days in January, March, May, July, August, October, December and 28 days in February, 30 days in April, June, September, November. This convention is used in money markets for short-term lending of currencies, including the US dollar and Euro, commercial paper, T-bills, and is applied in ESCB monetary policy operations, with Repurchase agreements. Spreads and rates on Actual/360 transactions are typically lower, e.g., 9 basis points than 30/360 method because monthly loan payments are the same for both methods. Because the borrower is paying interest for 5 or 6 additional days a year as compared to the 30/360 day count convention, this leaves the loan balance 1-2% higher than a 30/ year loan with the same payment. Actual/365 The convention is similar to the one above except for the fact that the number of days in a year is assumed to be 365 long. The corresponding year fraction is D2 D For example, in a period from February 1, 2005 to April 1, 2005, the factor is considered to be 59 days divided by 365. This count convention 9

9 is used for US treasury notes, US Treasury bonds. Actual/365L The computation of the factor requires three dates: the coupon start date (D 1 ), the accrual factor date (D 2 ) and the coupon end date (D 3 ). This convention requires a set of rules in order to determine the days in the year (denominator). It was originally designed for Euro-Sterling floating rate notes. It is used only to compute the accrual factor of a coupon. The formula is D 2 D 1 Denominator If F requency = 1 (Annual Coupons): If February 29 days is in the range from D 1 (exclusive) to D 3 (inclusive), then denominator is 366 days, else the denominator is 365 days. If F requency 1: If D 3 is in a leap year, the denominator is 366 days, else it is 365 days. Actual/365A The accrual factor is D 2 D 1 denominator where Denominator is 366 if 29 February is between D 1 (exclusive) to D 2 (inclusive) and 365 otherwise. Actual/Actual The numerator states the actual number of days in the holding period but the denominator states the number of days in a year is calculated as, actual number of days in the relevant coupon period multiplied by the coupon frequency (annual, semi-annual etc.). This count convention is used for US Treasury bonds. There are at least three different interpretations of actual/actual. For example, the actual/actual ICMA convention is used for euro-denominated bonds, the US treasury. A second method is known as the Actual/Actual AFB method. The third approach is the ISDA method. The numerator be equal to the actual number of days from the last coupon payment date or period end date (inclusive) to the current value date or period end date (exclusive) in all 10

10 three cases, but vary from the denominator. Actual/Actual AFB The formula is similar to above convention but the denominator is equal to 365 (if the calculation period does not contain 29 February) or 366 (if 29 February falls within the Calculation Period or Compounding Period). The following is the calculations of period or compounding period in a term of more than one year. The number of complete years shall be counted back from the last day of the Calculation Period or Compounding Period. If the last day of the relevant period is 28 days February, the full year should be counted back to the previous 28 February. If 29 February exists, 29 February should be used. This number shall be increased by the fraction for the relevant period calculated. Actual/Actual ISDA This convention accounts for days in the period based on the portion in a leap year and the portion in a non-leap year.to compute the number of days, the period first day is included and the last day is excluded. In general, coupon payments will vary from period to period because of the different number of days in the periods. The formula applies to both regular and irregular coupon periods. The accrual factor is (Days in a non-leap year / 365) + (Days in a leap year / 366). How to a leap or non-leap year: If it is not evenly divisible by 4 (a whole number with no remainder), like 1997, it is not a leap year. If a year is divisible by 4, but not 100, like 2012, it is a leap year. If a year is divisible by both 4 and 100, like 2000, check if the year is divisible by 400. If a year is divisible by 100, but not 400, like 1900, 11

11 then it is NOT a leap year. If a year is divisible by both, then it is a leap year. So 2000 was indeed a leap year. Example Start date is December 30, 2010 and end date is January 2, The factor is = Start date is December 30, 2011 and end date is November 2, The accrual factor is = Start date is December 30, 2010 and end date is January 02, The factor is = Actual/Actual ICMA The computation of the factor requires three dates: the coupon start date (D 1 ), the accrual factor date (D 2 ) and the coupon end date (D 3 ). The factor is 1 F req ( D2 D1 D 3 D 1 ). Where: Freq is the number of coupons per year, D 2 D 1 is the numbers of days between D 1 and D 2. For regular coupon periods: Factor = 1 F req. For irregular coupon periods, the period has to be divided into one or more stub coupon periods (also called notional periods) to match the normal frequency of payment dates. The interest in each such period (or partial period) is then computed, and then the amounts are summed over the number of stub-coupon periods. This method ensures that all coupon payments are always for the same amount and all days in a coupon period are valued equally. This is the convention used for US Treasury bonds and notes, among other securities. Example: Notional: 10,000 Euro and fixed rate: 10%. Consider three dates: the coupon start date (D 1 = Nov 01, 2003), the accrual factor date (D 2 = Dec 31, 2003) and the coupon end date (D 3 = May 01, 2004) The actual numbers of days between D 1 and D 2 and between D 2 and D 3 is 61 days, 121 days, respectively. 12

12 ISDA Method: (10,000)(10%)( ) = Euro. ICMA Method: (10,000)(10%)( 182 ) = Euro. 182(2) AFB Method: (10,000)(10%)( ) = Euro. Actual/364 Each month is treated normally and the year is assumed to be 364 days. For example, in a period from February 1, 2005 to April 1, 2005, the Factor is considered to be 59 days divided by /1 This is used for inflation instruments and divides the overall 4 year period distributing the additional day across all 4 years i.e. giving days to each year. NL/365 The accrual factor is Numerator 365 where Numerator is D 2 D 1 1 if 29 February is between D 1 (exclusive) to D 2 (inclusive) and D 2 D 1 otherwise. Business days/252 The accrual factor is Businessdays 252 where the numerator is the number of business days (in a given calendar) from and including the start date (inclusive) to the end date (exclusive). This day count is used in particular in the Brazilian market. 1.3 Business day Conventions Business day conventions are a common practice to adjust non-business days into business days. The adjustment is done with respect to a specific calendar. (a) Following The following good business day is the adjusted date. For examples, start date is on Aug 18, 2011, period 1 month, end date is on sep 19,2011. (b) Preceding 13

13 The preceding good business day is the adjusted date. This convention is often linked to loans. This convention means that a translation of the amount that should be paid on or before a specific date. For example, start date is on Aug 18, 2011, period 1 month, end date is on sep 16, (c) Modified Following The following good business day is the adjusted date. Otherwise, the day is not in the next calendar month then the adjusted date is the preceding good business day. This is the most used convention for interest rate derivatives. For example, start date is on Jun 30, 2011, period 1 month and the end date is on Jul 29, As the rule, Aug 01 is in the next calendar month with respect to 30-Jul. (d) Modified following bimonthly The following good business day is the adjusted date. Otherwise, that day does not cross the mid-month (15th) or end of a month then the adjusted date is the preceding good business day. For example, start date 15-Sep- 2011, period 1 month: end date: 14-Oct The following rule would lead to Oct 17 which crosses the mid-month. (e) End of month The start date of a period is on the final business day of a particular calendar month, the end date is on the final business day of the end month (not necessarily the corresponding date in the end month). Example Start date is on Feb 28, 2011, 1-month period, the end date is on Mar 31, Start date is on Apr 29, 2011, 1-month period, the end date is May 31, Apr 30, 2011 is a Saturday, so Apr 29 is the last business day of the month. Start date is on Feb 28, 2012, 1-month period, the end date is on Mar 28, is a leap year and the 28th is not the last business day of the 14

14 month. 1.4 Compounding Methods 1. Continuously Compounded rates To value all instruments consistently (at any time points) within a single valuation framework, we need a risk-free yield curve( a continuous zero curve). It is more common for the market practitioner to think and work in terms of continuously compounded interest rates. The time 0 continuously compounded risk free rate for maturity T, denoted r T. The price of the zero-coupon bond can be calculated from its yield by B(0, T ) = exp( r T T ). Hence, the continuously compounded yield on this bond is r T = T 1 lnb(0, T ). It is the redemption amount earned at time t from an investment now at time 0 of 1 unit of currency in zero coupon bonds. In general, the continuously-compounded spot interest rate prevailing at time t for the maturity T and is the constant rate at which an investment of B(t, T) units of currency at time t accrues continuously to yield a unit amount of currency at maturity T. An alternative is simple compounding and defined as follows 2. The simply-compounded spot interest rate Definition 1.2. The simply-compounded spot interest rate prevailing at time 0 for the maturity T is denoted by L(0, T) and is the constant rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from B(0, T) units of currency at time 0, when accruing occurs proportionally to the investment time. A bond price can be expressed in terms of L as B(0, T ) = 1 (1+L(0,T )) T. It follows the LIBOR rates L(0, T ) = 1 B(0,T ). LIBOR rates are typically linked to zero- B(0,T )T coupon-bond prices by the Actual/360 day-count convention for computing time to maturity. 15

15 3. Annual Compounding rate A further compounding method that is considered is annual compounding. It is defined as follows: Definition 1.3. The annually compounded spot interest rate prevailing at time 0 for the maturity T is denoted by Y(0, T) and is the constant rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from B(0, T) units of currency at time 0, when reinvesting the obtained amounts once a year. Bond prices can be expressed in terms of annually-compounded rates as B(0, T ) = 1 1. It follows the annually-compounded rate Y (0, T ) = 1. A (1+Y (0,T )) T B(0,T ) T 1 year fraction (time to maturity) that can be associated to annual compounding is for example ACT Spot interest rate The most obvious fact is that B(0, T ) is decreasing in T. By contradiction method, suppose B(0, T 1 ) < B(0, T 2 ) for some T 1 < T 2. Then the arbitrageur will buy a zero coupon bond for time T 1, and sell one for time T 2, for an immediate income of B(0, T 2 ) B(0, T 1 ) > 0. In fact, such bonds rarely trade in the market. In so-called normal markets, yield curves are upwardly sloping, with longer term interest rates being higher than short term. The yield curve with downward sloping is called inverted. one or more turning points is called mixed. It is often stated that such mixed yield curves are signs of market illiquidity or instability. One can observe a consistent mixed shape over long periods in a stable market with reasonable liquidity of time. The arbitrage argument for a decreasing B(0, T ) function does not hold true with a real inflation linked curve because the actual size of the cash payments that will occur are unknown (as they are determined by the evolution of a price index, which is 16

16 unknown). Thus, for a real curve the B(0, T ) function is not necessarily decreasing. Spot Interest Rate calculations: The rate earned on a zero-coupon bond is also called the spot interest rate. Spot rate has a positive, upward slope. A coupon bond may be written as a linear combination of zero-coupon bonds. An implied zero-coupon interest rate structure can therefore be derived from the yields on coupon bonds. If B(0, T i ), i = 1, 2,..., n and T n = T denotes the actual prices of zero coupon bond with different maturities and known $ 1 nominal values, the price of a coupon bond with $ 1 nominal value and coupon C is derived: P (0, T ) = B(0, T 1 )C + B(0, T 2 )C B(0, T )(1 + C) (1.1) Conversely, the coupon bond prices P (0, T i ) with the coupon payment C are known, the implied zero-coupon term structure can be derived through an iterative process as follows: P (0, T 1 ) = B(0, T 1 )(1 + C). It implies B(0, T 1 ) = P (0,T1) 1+C and so on for the prices B(0, T i ), i = 2,..., n B(0, T ) = P (0, T ) B(0, T n 1)C... B(0, T 1 )C 1 + C In practices, an error term ɛ is added to the equation (1.1) because it presents the effects of liquidity, taxes, and other factors. Generally, the price is estimated using cross-sectional regression against all the other bonds in the market, as follows. Denote P (0, T i ) coupon bonds with the corresponding coupon C i. P (0, T i ) = B(0, T 1 )C i + B(0, T 2 )C i B(0, T )(1 + C i ) + ɛ i Where: C i the coupon on the i th bond. T i the maturity of the i th bond. The regressor parameters are the coupons paid on each coupon-payment date, and the coefficients are the prices of the zero coupon bonds. 17

17 1.6 Fundamental Interest Rate Curve The zero-coupon curve at a given date t can be obtained from the market data of interest rates. The graph of this curve is defined as follows: Definition 1.4. The zero-coupon curve ( or yield curve or term structure of interest rates) at time t is the graph of the function L(t, T ) t < T t + 1 (in years) T Y (t, T ) T > t + 1 (in years) r(t, T ) T > t (in years) It is a plot at time t of simply-compounded interest rates for all maturities T up to one year and of annually compounded rates for maturities T larger than one year or rates with different compounding conventions. Another curve is called Zero-bond curve and defined as follows: Definition 1.5. The zero-bond curve at time t is the graph of the function T B(t, T ), T > t which, because of the positiveness of interest rates, is a T-decreasing function starting from B(t, t) = 1. Such a curve is also referred to as term structure of discount factors. 1.7 Forward Yield Curve If we can borrow at a known rate now at time 0 to date T 1, and borrow from T 1 to T 2 at a rate known and fixed at 0, then equivalently we can borrow at a known rate at 0 until T 2. Clearly the no arbitrage equation is B(0, T 1 )B(T 1, T 2 ) = B(0, T 2 ) (1.2) Where: B(T 1, T 2 ) is the forward discount factor for the period from T 1 to T 2, to ensure no arbitrage, this value is at time 0 with the information available at that time. It is calculated in terms of the forward rate governing the period from T 1 to T 2, denoted f(t 1, T 2 ) satisfies: B(T 1, T 2 ) = exp( f(t 1, T 2 )(T 2 T 1 )) 18

18 Where the forward rate is defined: f(t 1, T 2 ) = lnb(0, T 2) lnb(0, T 1 ) T 2 T 1 = r T 2 T 2 r T1 T 1 T 2 T 1 (1.3) Taking the limit of above equation, we get the instantaneous forward rate for a tenor of t (denoted f(t)). f(t) = d lnb(0, t) dt = d dt r tt (1.4) It implies that f(t) = r t +r tt. So the forward rate will lie above the normal yield curve and below the inverted yield curve. By integrating, r t t = t It implies that B(0, t) = exp( t f(s)ds) Also 0 0 f(s)ds r i t i r i 1 t i 1 T i T i 1 = 1 T i T i 1 Ti T i 1 f(s)ds (1.5) Above equation means that the average of the instantaneous forward rate over any of intervals [t i 1, t i ] is equal to the discrete forward rate for that interval. By simple calculations, we get r t t = r i 1 T i 1 + Ti T i 1 f(s)ds (1.6) t [T i 1, T i ]. This is a crucial interpolation formula: given the forward function, it is easy to find the risk free function. 1.8 Interpolation Method for yield curve construction Introduction In general, the main idea of the interpolation problem is as follows: Given some data y as a function of x, so x 1, x 2,..., x n and y 1, y 2,..., y n known. An interpolation method aims to construct a continuous function y(x) satisfying y(x i ) = y i. In our context, the 19

19 y values might be the risk free rates, forward rates,and so on. The x values might be time T 1, T 2,..., T n. There are available interpolation functions. The choices of these functions depends on the nature of the problem, one imposes additional requirements such as continuity, differentiability, twice differentibility, boundary conditions, and so on Yield Curve Fitting In spot yield curve section, an approach to estimate term structure is mentioned. The approach assumes that the cash flows of the bond without default is payable on specific discrete dates, to each of which a set of unrelated discount factors are applied. These discount factors are be considered as regression coefficients; the bond cash flows be the independent variables and the bond price at each payment date is the dependent variable. This type of regression produces a discrete discount function, not a continuous one. McCulloch (1971) proposes a more practical approach, using polynomial to produce a continuous and linear function, so the ordinary least squares regression technique can be employed. James Langetieg and Wilson Smoot (1981) extended McCulloch method that fits cubic splines to zero-coupon rates instead of the discount function and uses nonlinear methods of estimation. The term structure can be extracted from the complete set of discount factors. These discount factors can themselves be derived from the price of default-free bonds trading in the market using the bootstrapping technique. However, the drawbacks are that it is unlikely that the complete set of bonds in the market will pay cash flows at precise six-months intervals from today to thirty years from now or longer, which is necessary for the bootstrapping derivation to work. Hence, cash flows should adjusted to receive at irregular intervals or, in the case of longer maturities, not at all. Another issue is that bootstrapping calculates discount factors for terms that are multiples of six months, but in reality, non-standard periods may be involved, particularly in pricing derivative instruments. A third drawback is that bonds market prices often reflect investor considerations such as the liquidity of the bond, the tax treatment of the cash 20

20 flows, the bid-offer spread and the bonds trade continuously. To handle above considerations, smoothing techniques are used in the derivation of the discount function. We consider the empirical method which merely try to find a close representation of the term structure at any point in time, given some observed interest rate data. It is required that the derived yield curve should be smooth, not oversmoothing, as this might cause the elimination of valuable market pricing information. All inputs to the yield curve may or may not replicated exactly (priced back exactly after the construction of that curve). We need to consider the problem of error minimization when there is a large set of inputs. We prefer an approach with fewer inputs and hence a perfect replication is feasible. Even the curve perfectly is required to replicate the price of the input instruments, the yield curve is not constructed uniquely; need to select an interpolation method with which to build the curve. In this section, a common smoothing technique, linear interpolation are presented. Suppose we have the rates r 1, r 2,, r n at the nodes T 1, T 2,, T n and need to determine the rate r(t) where t may not be equal one of the T i. For any t between T 1 and T n, linear interpolation are used. For any t less than T 1 or beyond T n, a flat extrapolation is used. That is, the value of r(t) will be that rate found at the nearer of T 1 or T n Linear Interpolation This method interpolates missing yield curve caused by associated gaps in the set of observed bond prices from actual yields. Typically, the methods can be formulated as implicitly linear interpolation on the spot rate function, or forward rate function. Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not differentiable at the point. In words, the error is proportional to the square of the distance between the data points. However, this method is very stable, is trivial to implement before the more sophisticated methods. 21

21 1. Linear on Rates(spot and forward rate) Suppose T 1 < T 2 <... < T n and r i function value at these time points. We want to calculate spot rate from 0 to t (r t ) at every t. Suppose r t is linear on each interval [T i 1, T i ]. This allows to calculate r t at every t [T i 1, T i ] by the following formula: r t = t T i 1 r i + T i t r i 1 (1.7) T i T i 1 T i T i 1 Using equation (1.5), we get forward rate formula f(t) = 2t T i 1 r i + T i 2t r i 1 (1.8) T i T i 1 T i T i 1 f is undefined at t since the function r(t)t is not differentiable at t. For equation (1.7), when t approaches t i, the term r i 1 reaches zero. However, the equation (1.8), lim t T + i f(t) lim t T f(t). It means that the forward jumps at nodes. i Moreover, this interpolation does not prevent negative forward rates. For example, given points T i 1 = 1 < T i = 2 years, and the corresponding spot rate r i 1 = 8% and r i = 5%, for t = 1.84 year, forward rate f(t) = is calculated. 2. Linear on the log of rates. For T i 1 t T i, the interpolation formula is It implies the rate formula ln(r t ) = t T i 1 ln(r i ) + T i t ln(r i 1 ) T i T i 1 T i T i 1 r t = r t T i 1 T i T i 1 This method prevents negative interest rate. i r T i t T i T i 1 i 1 By equation (1.4), we get forward rate formula f(t) = r t (1 + However, the forward also jumps at each node. t T i T i 1 ln( r i r i 1 )) 22

22 3. Linear on discount factors The interpolation formula is It implies a rate formula: B(0, t) = t T i 1 B(0, T i ) + T i t B(0, T i 1 ) T i T i 1 T i T i 1 r t = 1 t ln[ t T i 1 exp( r i T i ) + T i t exp( r i 1 T i 1 )] T i T i 1 T i T i 1 Again, the forward jumps at each node. 4. Linear on the log of discount factors This method corresponds to piecewise constant forward curves. This method is very stable and trivial to implement. Raw interpolation has constant instantaneous forward rates on every interval T i 1 < t < T i. From equation (1.6), we have f(t) = riti ri 1Ti 1 T i T i 1. Then, from equation (1.7) we have, r t t = r i 1 T i 1 + (t T i 1 ) r it i r i 1 T i 1 T i T i 1 By simplifying, we get the interpolation formula on that interval r t t = t T i 1 r i T i + T i t r i 1 T i 1 T i T i 1 T i T i 1 5. Linear interpolation on the log of Probability Assume a constant hazard rate between T i 1 and T i. Hence, t [T i 1, T i ], Q(τ > t) = exp( λ i 1 (t T i 1 ))Q(τ > T i 1 ) 1 (ln(q(τ > t)) ln(q(τ > T i 1 )) t T i 1 1 = λ i 1 = (ln(q(τ > T i )) ln(q(τ > T i 1 )) T i T i 1 It implies that function ln(q(τ > t)) linear on [T i 1, T i ]. For any t [T 1, T n ], we have.. Q(τ > t) = Q(τ > T i 1 )( Q(τ > T i) Q(τ > T i 1 ) ) t Ti 1 T i T i 1 23

23 Chapter 2 INCREMENTAL RISK CHARGE (IRC) The drawback with Value at Risk measures is that they are completely based on historical data. The worst case generated by VaR measure can not be worse than what has happened before. In fact, the traditional Value at Risk Model could underestimate risk as well as do not capture default and migration risks for portfolio. The final guidelines for computing capital for the incremental risk charge (IRC), released in 2009 by the Basel Committee on Banking Supervision, considered as a complement to the traditional value at Risk Model. Following the Basel Committees requirement in determining the Incremental Risk Capital Charge, the 99.9% value-at-risk estimate over a one-year capital horizon is required to ensure the default and migration risks of credit products, taking into account two guided standards. The first standard is that the liquidity horizons applicable to individual trading positions or sets of positions. In practice, however, a majority of banks have chosen to use a liquidity horizon of one year for all positions. The second standard is the assumption of a constant level of risk of the portfolio. It means that banks re-balance trading positions to maintain initial level of risk at the end of liquidity horizon. All positions whose credit characteristics have improved or deteriorated are replaced by positions with the equivalent credit characteristics at the start of the liquidity horizon. Correlation structure among obligors such as issuer correlations, auto-correlation is also considered in case of calculations of IRC. 24

24 The approach uses credit rating transition matrix, given the initial credit rating in order to response to the increasing amount of exposure to credit risk related products in banks trading book. The migration of credit rating are considered as function of the underlying asset value of a firm. Asset values are random and simulated by some distribution with underlying issuer correlations. If the simulated asset value falls below a threshold, calculated from the transition matrix, the issuer changes its credit rating. The transition matrix is combined with the simulated asset values and credit ratings of the underlying issuers in the portfolio are created. Depending on the changes in credit ratings, the total portfolio value is measured. The simulation is repeated many times to get a distribution of portfolio values after which a Value at Risk at the confidence level 99.9 percent can be determined. This is idea of the Incremental Risk Charge. 2.1 Credit Metric model Changes in value caused not only by possible default events, but also by rating migration(downgrades or upgrades) in credit quality. CreditMetrics allows to assess changes in debt value exposed to in a portfolio caused by changes in obligor credit quality, including the default risk. It is a good starting point for an Incremental Risk Charge model. It is considered as a framework to apply to the valuation and risk of nontradable assets such as loans and privately placed bonds. Creditmetrics allows to address a full text of a portfolio such as corporate bond, treasury bond, market-driven instruments, financial letters of credits, derivatives (swap,future,...) and so on. This involves estimating a probability distribution of credit losses by carrying out a Monte Carlo simulation of the credit rating changes. Consider the BBB-rated issuer. Let R = AV i t 1,t 2 denotes the asset return. Φ(x) denotes the cumulative standard normal distribution function evaluated at x. The notations of thresholds are simplified Z CRt. Then the thresholds are calculated recursively by the following formulas. P (BBB D) = P (R < Z BBB D ) = Φ(Z BBB D ) 25

25 P (BBB CCC) = P (Z BBB D < R < Z BBB CCC ) = Φ(Z BBB CCC ) Φ(Z BBB D ) P (BBB B) = P (Z BBB CCC/C ) < R < Z BBB B ) = Φ(Z BBB B ) Φ(Z BBB CCC/C ) P (BBB BB) = P (Z BBB B < R < Z BBB BB ) = Φ(Z BBB BB ) Φ(BBB BBB) P (BBB BBB) = P (Z BBB BB < R < Z BBB BBB ) = Φ(Z BBB BBB ) Φ(Z BBB BB ) P (BBB A) = P (Z BBB BBB < R < Z BBB A ) = Φ(Z BBB A ) Φ(Z BBB BBB ) P (BBB AA) = P (Z BBB A < R < Z BBB AA ) = Φ(Z BBB AA ) Φ(Z BBB A ) Example The probability of default is approximately percent. From above formula, P (R < Z BBB D ) = Φ(Z BBB D ). Hence, Z BBB D = Φ 1 (P (R < Z BBB D )) Φ 1 ( ) 4.08 This means that an BBB-rated issuer defaults if the simulated change in asset value is less than standard deviations. The generation of the credit rating CR t+k,i at the end of considered risk horizon k, using the initial credit rating CR t,i (and the simulated issuer asset return ( x i ) underlying the credit position i), is expressed as follows: D x i < Z CRt D ICR t+k,i = CCC B BB BBB A AA AAA Z CRt D < x i < Z CRt CCC Z CRt CCC < x i < Z CRt B Z CRt B < x i < Z CRt BB Z CRt BB < x i < Z CRt BBB Z CRt BBB < x i < Z CRt A Z CRt A < x i < Z CRt AA Z CRt AA < x i < Z CRt AAA x i the simulated issuer asset return and be the standard normally distributed random variable. 26

26 2.2 Multiperiod Model In comparison with credit metric model, the disadvantage of credit metric model is that it only captures 1-year single horizon for all positions without rebalancing. Therefore, a single-period model is extended into a 4-periods model over risk horizon of a year to capture two criteria from the guidelines of the Basel Committee Multi-Period Credit Rating Simulation A four-period simulation model is incorporated in The Incremental Risk Model [Stel, Yannick V.D., 2009]. Each period is three months so all four periods add up to the total capital horizon of one year. The four-periods model could be seen as four individual simulations according to the methods used in CreditMetrics Model. Note that the one-year transition probabilities have to transferred into the 3-months transition probabilities. At the end of each single period, a position may be re-balanced if the liquidity horizon of that position is reached or if a default occurs. By the constant risk assumption, the position is replaced by a new hypothetical position with characteristics equal to the initial position when re-balancing occurs. The total value of each position at the end of the capital horizon (one year from simulation date) will be the value of the bond at the end of the year added with re-balancing results from earlier periods. ICR t+k,i is used to define as above system to determine CR t+k,i as following rule: D x t+k,i < Z ICRt+k,i D CR t+k,i = CCC B BB BBB A AA AAA Z ICRt+k,i D < x t+k,i < Z ICRt+k,i CCC Z ICRt+k,i CCC < x t+k,i < Z ICRt+k,i B Z ICRt+k,i B < x t+k,i < Z ICRt+k,i BB Z ICRt+k,i BB < x t+k,i < Z ICRt+k,i BBB Z ICRt+k,i BBB < x t+k,i < Z ICRt+k,i A Z ICRt+k,i A < x t+k,i < Z ICRt+k,i AA Z ICRt+k,i AA < x t+k,i 27

27 for k = 3, 6, 9, 12 and i = 1,..., n, where CR t+k 3,i if CR t+k 3,i D and k 3 q i h ICR t+k,i = CR t,i if CR t+k 3,i = D CR t,i if k 3 = q i h for h = 1, 2, 3, Position Valuation in multiperiod model The four-periods model can be considered as four individual simulations in Credit Metric model. A 1-year probability transition matrix is exchanged into a 3-months probability transition matrix. At the ending of each period, a credit product may rebalance if a credit event (liquidity horizon is reached or default) happen, where this position is replaced by new position with the same characteristics as initial position. The initial risk of a position is defined by class, size, credit rating and issuers correlation. The total value of a position at a 1-year risk horizon from the simulation day will be the sum of the value of bond at risk horizon and rebalancing results the previous period. We evaluate each position at the end of the capital horizon (1 year). For simplicity, only corporate bond portfolios will be assessed in the model estimations. In the case of credit rating is not default, to obtain the values of the bonds at the end of each single-period in the model for each possible credit rating (except default), the remaining cash flows of each position i are discounted at the forward zero credit curve of the relevant credit rating: V CR t+k,i t+k,i = M i m=k CF m,i (1 + f CRt+k,i k m,t ) k for k = 3, 6, 9.12 months and i = 1,..., n positions and CR t+k,i D, D =default. Where: CF m,i is cash flow (coupon payment or notional payment) CR t+k,i is the simulated credit rating at time t + k (the end of each single-period in the model) 28

28 M i is the maturity of the bond in months ( m is in months as well) f CRt+k,i k m,t the forward zero credit yield of rating CR t+k,i observed at time t, referring to the period between k months (the point for which the value of the position is calculated) and m months from now (the time at which the cash flow occurs) Here, Forward zero credit rate f k m,0 is derived by adding a credit spread for credit rating of issuer to the forward zero risk free rate of issuer between time point k and m (in years). The forward zero risk-free rates are calculated from the current zero yield curve. No market risk is considered, so the forward rate between two time points is a deterministic function of the interest rates at these time points. To calculate the forward zero risk-free rate, consider two alternate ways to invest money from t = 0 to t = m. Either buy a zero coupon bond at time 0 maturing at time m with interest rate r m, or buy a zero coupon bond at time 0 maturing at time k with interest rate r k and reinvest a new zero coupon bond at time k with maturity at time m to the forward rate f k m,0. Since the forward rate is known at time 0, both alternatives are risk-free and the two investments should be equivalent. Mathematically, (1 + r k ) k (1 + f k m,0 ) (m k) = (1 + r m ) m where: r k and r m are zero coupon rate between 0 and k, between 0 and m, respectively. The model can handle different yield curves for different positions in the portfolio. Here is the valuation expression to account for possible rebalancing results at the end of each single period in the valuation of position i at the end of the capital horizon. V t+12,i = V CRt+12,i t+12,i for i = 1,.., n and R t+k,i = V CR t+k,i t+k,i + (1 + f CRt,i 3 12,t ) 3 4 Rt+3,i +(1 + f CRt,i 6 12,t ) 1 2 Rt+6,i +(1 + f CRt,i 9 12,t ) 1 4 Rt+9,i V ICR t+k+3,i t+k,i for k = 3, 6, 9 The value of position i at the end of the capital horizon is the sum of value of the position at the end of the last period (t + 12) month, given its simulated credit rating 29

29 at that moment and the rebalancing results of the previous single-periods (R t+k,i for k = 3, 6, 9) which are carried to the end of the capital horizon at a forward zero credit yield. The rebalancing result R t+k,i is equal to the value of the position at the end of each single-period (V CRt+12,i t+12,i ) given the simulated credit rating at that moment (CR t+12,i ) minus the value of the position at the same point in time(v ICR t+k+3,i t+k,i ) given the credit rating at the start of the next period ICR t+k+3,i More explanation, ICR t+k+3,i = CR t+k,i CR t,i CR t,i rebalancing not occur CR t+k,i = D liquidity happen By constant risk, the position is sold at the end of each period and that the same position with the input credit rating of the next period is bought again. Rebalancing occurs when issuers default or liquidity horizon of the positon is reached. R t+k,i = 0, position value is not changed if rebalancing not occur. R t+k,i = V CR t+k,i t+k,i V CRt,i t+k,i if rebalancing does occur. The difference displays exactly what the losses or gains of rebalancing are at that moment. Because these gains or losses do not occur at the end of the capital horizon (1 year), they are defined as the future cash flow discounted by applying the relevant forward zero credit yield (or more specifically, the relevant forward accrual rate). (1 + f CRt,i k 12,t ) 12 k 12 k = 3, 9, 12. The difference between the initial porfolio value and the sum of all these total position value changes captures the difference in value of the total portfolio. This allows each data points simulated and simulations are made for each time interval in order to create a distribution of changes in the total portfolio value. In the case that an issuer defaults, default value is calculated as follows: V D t+k,i = RV CR t+k 3 t+k,i where the recovery rate R β(0.550, 0.709) The main problem here is that the recovery rate is highly uncertain or volatile in practice need to be estimated. The uncertainty is achieved by utilizing a distribution. 30

30 Chapter 3 FIXED-INCOME SECURITIES VALUATION 3.1 Introduction A large number of new fixed-income instruments have been developed and introduced into the financial market, including put bonds, zero-coupon convertibles, interest rate futures and options, credit default swaps, and swaptions, and so on. The total value of the fixed income assets is about two thirds of the market value of all outstanding securities. Hence, it is important to study about fixed-income valuation. Firstly, the chapter provides a literature review of the studies on default-free bonds, then explore defaultable bonds. Next, the pricing formula of credit default swap based on risk-neutral pricing theory is presented. The following sections presents a more formal analysis, then explain the traditional approach to pricing for instruments, making certain assumptions to keep the analysis simple. Suppose that we have a riskless (bank account) S 0 which plays the role as the market s numeraire. The riskless asset S 0 admits a deterministic return r(.) which could depend on time so that its dynamic can write as the following: 31

31 { t ds 0 (t) S 0 (t) = r(t)dt or S 0(t) = exp 0 r(s)ds where r(t) is the instantaneous short-term rate at time t. } with S 0 (0) = 1. Before going to the detail of the valuation, recall that the Absence of Arbitrage opportunity and the numeraire allows us to bring all future cash-flows back to the present. For example, an amount C t at time t is equivalent to: C 0 = C { t } t S 0 (t) = exp r(s)ds C t = D(0, t)c t 0 { where D(0, t) = exp } t 0 r(s)ds in known as discount factor. 3.2 Bond Valuation Introduction This section consider pricing of defaultable bonds in the context of sovereign bonds and corporate bonds. The valuation of bond is more complicated relative to the case when the possibility of default or rating migration of the issuer is taken into. The payoffs of defaultable bonds depend on the occurrence of credit event. Therefore, modeling the default probability (credit risk) is the key issue for pricing the defaultable bonds. The difficulties comes from the followings: 1. The time of default of the issuer is random and unknown so that the probability of default need to be estimated. 2. The recovery rate (denoted by R and supposed to be constant) is unknown. Standard recovery schemes such as recovery of treasury and recovery of nominal are presented. In order to evaluate fixed income securities taken into account credit risk of the issuer, the survival probability of each issuer need to be calibrated. This curve will be transformed into a spread curve, called zero coupon cash. This zero coupon cash spread is different from the spread curve deducted from CDS market and the actuarial rate 32

32 curve. The downgrade of rating could be represented by a jump on the zero coupon cash spread. Therefore, this curve spread also allows to calibrate the shock of spread in case of changes of credit rating. 1. Market data: sovereign and coupon Bonds, consisting The cash flow of 19,178 positions in bond portfolio, including coupon payment and nominal at specific payment date of each position. The cash flow of 42,060 positions in CDS portfolio, including spread premium at specific payment date and principal notional at specific payment date at default. Zero coupon interest rate curve defined by issuer. The evaluation of instruments relies on spread, hence on rate curves. For each issuer, it has to transform an actuarial rate curve into a survival probability curve. The rate curve is defined by a basket of the most liquid obligations thanks to bootstrap method. When bootstrapping a bond curve, the rate curve is defined by a basket of the most liquid obligations because liquidity reduces the transaction cost and prices for liquid instruments change first. The approach of evaluation relies on this rate curve by supposing that the actuarial rate curve represent all information used to evaluate the instruments of the issuers. Therefore, it is a good indicator to evaluate the survival probability of the issuer. 2. Data Treatment(Calibration) Since cash flows of coupon bonds can be seen as a linear combination of elementary cash flows of zero coupon bond, only zero coupon bonds is considered. Denote 0 = T 0 < T 1 <... < T n = T the maturities of zero coupon bond. B(t, T i ) and B(t, T i ) the price of default risk-free and risky zero coupon bond. The tenors supposed are discrete such as 3 months, 6 months, 1 year, 2 years, 5 years, 7 years, 10 years, 15 years, 20 years. An flat extrapolation is considered for tenors more than 20 years. 33

33 Linear interpolation is applied for tenors between 1 and 20 years. 3. Valuation: The valuation of instruments in case of credit rests on the definition of payment at default. In contrast to the case of corporate, in which all holders of bond with the same seniority are deal with the same liquidation procedure, the sovereign default is synonymous with restructuring: reduction of nominal, of coupon, lengthening the maturity, transformation of instruments. Therefore, the recovery is difficult to precise because it depends on the future negotiation. 4. Recoveries: A default brings to cash stream(principal and coupon payments) in fraction of face value. It is called recovery of nominal. A default brings to restructure (in fraction of treasury). Recovery of nominal: The holder of instrument receives a fraction R of the nominal at credit event. The nominal policy is simpler to implement but it admits certain drawback. For example, A long term bond s price written on a very risky issuer could be more expensive than the same bond written on the risk-free issuer. Another example, zero coupon bond with zero rate 5%, time to maturity 20 years and nominal of $ 1. Price of zero coupon bond without default is exp( 0.05(20)) = 0.38 while the price of zero coupon when default happen is R N = = 0.4. Recovery of Treasury: All cash flow (including face value and coupons) after credit event (default or migration) are adjusted to the fraction R which corresponds to the valuation of risk-free interest rate. It is more convenient in practice since the exposure at default is, in general, random and defined by revaluation of assets and liabilities of the firm after credit event. 5. Exposure at Default (EAD) Assumption Nominal Policy: the exposure to default is fixed, i.e, EAD = N. 34

34 Treasury policy: The exposure is the present value of all future cash-flows discounted at risk-free rate, i.e, for t < s < T, EAD(s) = B(s, T i )C i + NB(s, T ) T i s Bond Valuation Without Credit Risk Uncertainty of economy is modeled in a filtered probability space (Ω, F = (F t ) t 0, P) satisfying usual conditions. Denote Q denote the neutral risk probability which is equivalent to P. The price on a zero-coupon bond at t for $ 1 receiving at T from a risk-free issuer is B(t, T ) = E Q [1D(t, T ) F t ] The right hand side of the equation is the discount factor at time t. The equation is satisfied the Absence of Arbitrage Opportunity argument because of the following reasons: Consider a scenario in which a market participant can either invest D(t,T) units of cash in a money market account, for a return of $1 at time T, or buy a risk-free zero-coupon bond that has a maturity value of $1 at time T. The bond and the money market are both risk-free and have identical payouts at time T, and generate the cash flow at time T. The bond must have a value equal to the initial investment in the money market account D(t,T) because of constant interest rate. The equation is satisfied. Equivalently, by contradiction, suppose the bond price B(t, T ) is less than D(t,T). In this case, the investor borrows D(t,T) at the money market rate, then use B(t, T ) to buy zero coupon bond. At maturity the bond pays $1, which the investors use to repay the loan. But they still have profit D(t, T ) B(t, T ). Similarly, if the bond is priced higher than D(t,T). In this case, investors could sell the bond and invest D(t,T) in the money market account. At maturity, the investor pay $ 1 to the bond holder while he gets the same $ 1 on the money market account. And investor would still get the profit at t, B(t, T ) D(t, T ). Therefore, arbitrage opportunities 35

35 would present. For more general, the price of a risk-free coupon bond with nominal N giving coupon C i at T i (t = T 0 < T 1 <... < T n = T ) is n P (t, T ) = E Q [ D(t, T i )C i + ND(t, T ) F t ] i=1 n = C i B(t, T i ) + N.B(t, T ) i= Bond Valuation With credit risk It is natural to consider zero coupon bond in the case credit risk. The credit risk is the risk that the quality of the issuer or the possibility of bankruptcy for the issuer have a significant impact on the rate of return of a security. A lower-quality issuer will sell at a lower price and thus offer a higher yield than a similar security issued by a higher quality issuer. To avoid unnecessary complications at this point, recovery rate R is assumed nonrandom and recovery R on the coupon payment date that immediately follows a default is received by the coupon bond holder. Risky zero-coupon bond Let us denote τ the time of default or rating migration of the issuer, then the price of a risky zero coupon bond of 1$ face value at time t with maturity T is defined B(t, T ) = E Q t [ND(t, T )1 {τ>t } + R.EAD(τ).D(t, τ).1 {τ T } F t ] (3.1) Suppose that τ is independent of riskless rate diffusion and that EAD(τ) is approximated by its value at the middle of period, the risky zero coupon bond s price could be written as: B(t, T ) = NB(t, T )Q(τ > T ) + R.EAD( T i )B(t, T i )Q(τ T ) (3.2) Next, the pricing of coupon bonds is presented. Risky coupon bond The value of a risky bond with nominal N maturing at time T paying coupons C i can 36

36 be partitioned into two components: valuation of payments without credit risk, and valuation of payments with credit risk. The first is equal to the value of the bond independent of the recovery value while the latter will vary with the recovery form: the recovery of nominal and the recovery of treasury. The price of risky bonds is characterised by the issuer s default probability, the exposure at default (EAD) and the recovery rate, R, in case of default. n P (t, T ) = E Q [ D(t, T i )(C i 1 {τ>ti } + R.EAD(τ)1 {Ti 1<τ T i }) i=1 +ND(t, T )1 {τ>t } F t ]. Suppose that τ is independent of risk-free rate diffusion and that EAD(τ) is approximated by its value at the middle of period, the risky bond s price could be written as n P (t, T ) = B(t, T i )[C i Q(τ > T i ) i=1 +R.EAD( T i )Q(T i 1 < τ T i )] + NB(t, T )Q(τ > T ) (3.3) Where Q(τ > T i ) is the probability of survival Calibration of Survival Probability 1. Data includes CDS market data (CDS spreads) and bond market data (bond spreads) Using CDS market data for corporate issuers: CDS spread is more relevant for credit risk and CDS spread takes into account hedging cost. Using bond market data for sovereign issuers: CDS index spread does not totally reflect the credibility of a country. 2. Recovery rate: market recoveries Senior corporate and sovereign obligors: 40% Subordonate corporate obligors and Greece: 20%. 37

37 3. Methodologies: Recall the discussion about the calibration of the zero coupon cash spread which is equivalent to the probability of default in introduction. This discussion suggests a calibration from risk-free yield curve and the yield curve of the issuer. Recovery of treasury The equation (3.1) can be written B(t, T ) = NB(t, T )Q(τ > T ) + RNB(t, T )(1 Q(τ > T )) (3.4) Where: B(t, τ)b(τ, T ) = B(t, T ). The equation (3.4) allows to calculate directly the probability of survival from the price of zero coupon bonds (risk-free and risky) as follows. The implied survival probability is Q(τ > T ) = Recovery of nominal: The equation (3.2) is written: B(t, T ) = NB(t, T )Q(τ > T ) + RN B(t,T ) B(t,T ) R 1 R n B(t, T i )(Q(T i 1 < τ T i ) (3.5) i=1 From the Equation (3.5), we get the recurrence equations B(t, T k ) = [B(t, T k ) RB(t, T k )]Q(τ > T k ) [RΣ k i=1 (B(t, T i 1 ) B(t, T i ))Q(τ > T i 1 )] RB(t, T 0 ) The survival probability curve is calibrated by bootstrap method. It reduces to solving a linear system. Bootstrap: the linear system AQ = B. Where: Q = (Q(τ > T i )), i = 1,...n calibrated vector of survival probabilities. B(t, T 1) RB(t, T 1) R[B(t, T 2) B(t, T 1)] B(t, T 2) RB(t, T 2). 0 0 A =... R[B(t, T 3) B(t, T 2)] B(t, T n 1) RB(t, T n 1) 0 R[B(t, T 2) B(t, T 1)] R[B(t, T 3) B(t, T 2)]... R[B(t, T n) B(t, T n 1)] B(t, T n) RB(t, T n) B(t, T 1) RB(t, T 1).1 B =. B(t, T ) RB(t, T 1).1 The vector of survival probabilities: Q = A 1 B where A 1 defined. 38

38 3.2.5 The CDS-Cash Basis 1. The CDS-Cash Basis The CDS-cash basis defined as the difference between CDS premium and cash spread. In practice, a CDS-cash basis may be nonzero because of the following fundamental factors. Cheapest-to-deliver (CTD) feature of CDS contracts: The values of deliverable obligrations can differ at the settlement time of the CDS contract. For example, a restructuring of the reference entity s debt affect on the market value of bonds and loans differently. This means that the protection buyer is long a CTD option. The buyer may hand over the cheapest one among the full range of deliverables to the protection seller. In contrast, the buyer of an asset swap will receive the post-default value of the specific fixed-rate bond or underlying loan after the default of the reference entity. Default-contingent exposure in asset swaps: Unlike the CDS, the asset swap such as the interest rate swap embedded in the asset swap continues to risk exposure after the reference entity defaults. Accrued premiums in CDS contracts: This factor may create a negative CDS-cash basis. The fact is that the protection seller in a CDS receives the accrued premiums in the event of default by the reference entity. The asset swap buyer does not have that benefit, they must be compensated by a higher asset swap spread. Funding risk in asset swaps: This factor contribute to a negative CDS-cash basis. The participants in a CDS contract face no such uncertainties while the asset swap buyers may face. The asset swap buyers may fund for buying the underlying bond through a short-term loan. Liquidity: This factor may potentially affect on the CDS-cash basis. The CDS market may be less liquid than the cash market because of the large amount of bonds outstanding on the CDS market. That lead to push CDS premium above the corresponding cash spread as the protection buyers have 39

39 to compensate for the greater illiquidity the sellers face. As the result, the CDS-cash basis is positive in this case. Counterparty credit risk: It is a potential factor in the pricing credit default swap. 2. Calibration of the cash-cds Basis Zero spread derived from CDS market and the cash market should be to add a base to reflect the differences in liquidity between two instruments and reflect the fact that the basket of calibration take only the most liquid respectively. CDS cash basis is calibrated from minimizing the value of difference between the theoretical price of the instrument of maturity T, for a zero spread s(t) with a recovery rate R and a base cash CDS b at time t and the market price of the bond according to the parameter b of all search base as [b min, b max ]. 3.3 CDS Valuation The credit default swap has revolutionized the trading of credit risk. Over the past five years it has become popular used, accounting for 72.5% of a total outstanding market notional currently estimated to be around $2.3 trillion. The default swap market is truly global, with contracts linked to the credit risk of a wide array of US, European and Asian corporate names as well as to a number of sovereigns How CDS works The series of payments of the default swap spread (a constant quarterly fee as a percentage of its notional value, is called CDS spread or premium) made to the protection seller until maturity or to the time of the credit event, whichever occurs first, including the payment of premium accrued from the previous premium payment date until 40

40 the time of the credit event, known as the premium leg. This size of these premium payments is calculated from a quoted default swap spread which is paid on the face value of the protection. Figure 3.1: Mechanics of a default swap premium leg If a credit event does occur before the maturity date of the contract, there is a payment by the protection seller, known as the protection leg. This payment equals the difference between par and the price of the cheapest to deliver asset of the reference entity on the face value of the protection and compensates the protection buyer for the loss. It can be made in cash or physically settled format. If there is no default event before maturity, the protection seller pays nothing. Figure 3.2: The protection leg following a credit event Example How CDS Works: Example Two parties enter into a 5-year CDS on Mar 20, 2012 with 100 million dollars notional principal and quarterly payments of 22.5 basic points. 1. In the case of no credit event. The buyer receives no payoff and pay (100, 000, 000) = 225, 000 dollars each quarter until Mar 20,

41 2. In case of occurrence of credit event. Physical settlement contract: the buyer has the right to sell bonds at face value. Cash settlement contracts: The auction indicates that the bond is worth 35 million dollars per 100 million dollars of face value after 3 months of default. The cash payoff would be = 65 million dollars Pricing CDS 1. Premium Leg Given a CDS contract with nominal N, receiving a spread S at T 1 < T 2 <... < T n = T, and providing a protection at default, then the value of premium leg is n P V (P remium) = E Q [ S i D(t, T i )1 {τ>ti }] = i=1 n S i B(t, T i )Q(τ > T i ) i=1 A future premium due at T i will only be received if the reference entity has not defaulted before T i. 2. Accured Premium Typically, the issue of accrued premium is addressed by adding half an accrual period to the premium leg, which amounts assuming that, should a default occur, it will on average take place midway through the period. n AccuredP remium = B(t, T i )Q(T i 1 < τ T i ) S i 2 3. Protection Leg i=1 Within the default intensities approach, this timing problem can be solved on condition that the credit event may happen on each small time interval [s,s+ds]. The steps are described below. Calculate the probability of surviving to some future time s(denote Q(τ > s)). 42

42 Compute the probability of a credit event in the next small time increment ds which is given by λ(s)ds. At this point an amount (1 R) is paid, and we discount this back to today t at the risk-free rate B(t, s). We then consider the probability of this happening at all times from s = t to the maturity date T. As a result, the present value of the protection leg at time t in continuous time is given. P V (protection) = N(1 R)E Q [D(t, τ)1 {τ T }] = N(1 R) T t B(t, s)dq(τ s) (3.6) dq(τ s) the probability of default between s and s + ds knowing that there was no default before s. In a model with a current hazard rate λ, this amount is written dq(τ s) = Q(τ > s)λ(s)ds. In the event of finding default in the middle of the period, the equation (3.6) is approximated to P V (protection) = N(1 R) Marking To Market A CDS Position N B(t, T i )Q(T i 1 < τ T i ) Marking a CDS position to market is the act of determining todays value of a CDS agreement that was entered into at some time in the past. The value of a CDS contract i=1 to a protection buyer. V (t, T ) = n S i B(t, T i )Q(τ > T i ) i=1 n + B(t, T i )Q(T i 1 < τ T i ) S i 2 i=1 n N(1 R) B(t, T i )Q(T i 1 < τ T i ) (3.7) i=1 43

43 3.3.4 Calibration of Survival Probability By equating the premium leg and protection leg (including the accrued premium), we get Q(τ > T ) = n 1 i=1 B(t, T i )(1 R Si 2 )(Q(τ > T i 1) Q(τ > T i )) B(t, T )S n + B(t, T n )(1 R Sn 2 ) n 1 i=1 B(t, T i)q(τ > T i )S i + B(t, T n )(1 R Sn 2 )Q(τ > T n 1) B(t, T )S n + B(t, T n )(1 R Sn 2 ) Using the formula, find Q(τ > T i ),i = 1,...n. starting with a 1-period CDS contract it is simple to work out Q(τ > T 1 ) = B(t, T 1 )(1 R S1 2 ) B(t, T )S 1 + B(t, T 1 )(1 R S1 2 ) for a 2-period CDS contract, knowing Q(τ > T 1 ) it is simple to work out Q(τ > T 2 ) = B(t, T 1 )(1 R S1 2 )(Q(τ > T 0) Q(τ > T 1 )) B(t, T )S 2 + B(t, T 2 )(1 R S2 2 ) B(t, T 1 )Q(τ > T 1 )S 1 + B(t, T 2 )(1 R S2 2 )Q(τ > T 1) B(t, T )S 2 + B(t, T 2 )(1 R S2 2 ) continue to work out Q(τ > T 3 ), Q(τ > T 4 ),... 44

44 Chapter 4 SHOCK OF SPREAD 4.1 Shock on Spread Curves Description of the data Data from Bond Market Curves are zero coupon spread from the zero coupon rate calculated by the method of valuation of bonds as follows: The zero Coupon spread can be introduced as a discount factor between prices of risky zero coupon bond and the price non risky zero-coupon bond. B(t, T ) B(t, T ) = 1 ((1 + S(T )) T where: zero coupon risk-free rate is the actuarial rate of German state in 10 years for the Europe region or US rate for outside Europe region. These zero coupon rates are the rates in the bond market that can restore market prices. Zero coupon spread curves are derived from the database of industry sectors. Data from CDS Market Zero spreads are extracted from the survival probability from CDS curves. Spread curves are from industry sectors. Ratings Data Ratings are from S & P. They come from Credit Pro, including scale AAA,AA+,AA,..., 45

45 CCC+, CCC,CC,C,D and NR. Sample The period of historical data for the calibration of spread shock is from July 1st, 2000 to June 21st, This period is long enough to calibrate spread shock effectively. Data Format sovereign: Data includes zero coupon spread curves from industry database and from market database zero coupon bond curves from industry database. 132 sovereign credit rating from Pro, including ratings from 44 countries in Euro region and ratings from 88 countries in other regions at specific days over the period. France Estonia Ukraine Finland Norway Latvia Montenegro Georgia Russia Liechtenstein Greece Romania Albania Sweden Belgium Portugal Macedonia Austria Denmark Germany Italy Switzerland Luxembourg Malta United Kingdom Ireland Netherlands Slovenia Croatia Lithuania Azerbaijan Spain Iceland Hungary Czech Republic Andorra Bosnia Herzegovina Turkey Cyprus Bulgaria Serbia Belarus Poland Table 4.1: List of Euro countries used in the calibration of sovereign spread 46

46 Corporate: Data includes spread curves of corporate products: senior, subordonnate, senior coverage from issuers in the industry database in terms of maturity 1, 3, 5,7,10 years. Corporate Ratings from Credit Prod are given in 13,863 issuers, including rating for 8,655 issuers in US region, 2,105 issuers from euro region and 3,103 issuers from other region. For a specific date, data can be used if spread and rating are available. Histogram Figure 4.1: Histogram for issuers: BEAZHOMUSA for SEN 1year spread Figure 4.2: Histogram for issuers: BEAZHOMUSA for SEN 3 year spread Figure 4.3: Histogram for issuers: ALCALSTPA for SEN 1 year spread 47

47 Figure 4.4: Histogram for issuers: ALCALSTPA for SEN 3 year spread Some histograms are plotted for two issuers. Following the histogram, spread is decreasing with respect to time. For example, 1 year spread is greater than 3 year spread for issuers: BEAZHOMUSA and ALCALSTPA. Another example, in practice, there are no difference between 30 year risky zero coupon rate of Italy and 30 year risk free zero coupon rate of Germany Methodologies: calibration of spread shock 1. The calibration is realised in full grade and shock of spread for notch grade is obtained by interpolation. 2. Construction of spread curve for each rating as the average of all issuers spread admitting the same rating,i.e., S i (t) = 1 #R i (t) k R i(t) S k (t) where R i (t) is the set of all issuers in the sample whose notation is i at time t. 3. Assume the multiplicative shock for corporates and additive shock for Sovereign Grouping in full grade A full grade rating class is set up to obtain a sufficient number per class to increase the representation support and to improve the quality of the estimate. The rating are considered seven full grades: AAA, AA, A,BBB,BB,B and CCC. For example, ratings such as AA+, AA and AA- are considered identical and are grouped in the AA rating as follows: 48

48 Grouping in full grade: Average Spread for the full grade: A = A + notch A notch A notch S A (t) = P (A + notch ) S (A + notch )(t) + P (A notch) S (Anotch )(t) + P (A notch ) S (A notch )(t) Where: P (A + notch ) + P (A notch ) + P (A notch) = 1 P (A notch ),P (A notch ),P (A notch) denote a proportion of number of issuers of each notch and number of issuers of full grade. Improvement of the quality of the estimators: Segmentation of the notch grade can greatly reduce the number of individuals per class but it does not reduce the dispersion within each category. That will lead to undermine the quality of the estimate. Grouping in full grade allows to increase the support time and number of individuals contributing to the estimates. That allows to increase the smoothness of the spread curve by rating class. Grouping in the full grade allows to improve the quality of the estimators. The curve obtained from the combination presents better regularity Additive Spread Shock on Sovereign Spread Curves The additive spread shock are calibrated from the historical market data in the following way ij (T ) = S i (T ) S j (T ) so that the spread of a sovereign issuer after migration from i to j is S j (T ) = S i (T ) + ij (T ) 49

49 4.1.5 Multiplicative Spread Shock on Corporate Spread Curves The multiplicative spread shock are calibrated from the historical market data in the following way ij (T ) = S i (T ) S j (T ) so that the spread of a corporate issuer after migration from i to j is S j (T ) = S i (T ) ij (T ) Transform from Full Grades to Notches by Interpolation A log-linear interpolation is realized for corporate because the spread curve has in the exponential form. In contrast, a linear interpolation is applied in case of sovereigns, for which an additive shock is applied. A notch is considered as a unit. The methodology is as follows: spread shock on sovereign: (s) 1 2 is between (s)1 1 and (s)1 3. (s)2 1 is between (s) 1 1 and (s)3 1, given in the following table. Spread shock at notch can be given (s) 1 1 (s) 1 2 (s) 1 3 (s) 2 1 (s) 2 2 (s) 2 3 (s) 3 1 (s) 3 2 (s) 3 3 from the following equation systems: (s) 1 2 = w (s)1 1 + (1 w) (s)1 3 (s) 2 2 = w (s)2 1 + (1 w) (s)2 3 (s) 3 2 = w (s)3 1 + (1 w) (s)3 3 (s) 2 1 = w (s)1 1 + (1 w) (s)3 1 (s) 2 3 = w (s)1 3 + (1 w) (s)3 3 In the case, the spread shocks at the four corners are given as the following table. (s) 2 3 will be calculated from the spread shock at the upper right corner (s)1 4 and the diagonal. Similarly, (s) 3 2 will be calculated from the spread shock at the lower left corner (s) 4 1 and the diagonal. 50

50 0 *. (s) 1 4 * 0 (s) 2 3. (s) * (s) 4 1. * 0 spread shocks on corporate: (s) 1 2 is between (s)1 1 and (s)1 3. (s)2 1 is between (s) 1 1 and (s)3 1, given in the following table. Spread shock at notch can (s) 1 1 (s) 1 2 (s) 1 3 (s) 2 1 (s) 2 2 (s) 2 3 (s) 3 1 (s) 3 2 (s) 3 3 be gotten in the following equation systems: (s) 1 2 = exp(wln( (s)1 1 ) + (1 w)ln( (s)1 3 )) (s) 2 2 = exp(wln( (s)2 1 ) + (1 w)ln( (s)2 3 )) (s) 3 2 = exp(wln( (s)3 1 ) + (1 w)ln( (s)3 3 )) (s) 2 1 = exp(wln( (s)1 1 ) + (1 w)ln( (s)3 1 )) (s) 2 3 = exp(wln( (s)1 3 ) + (1 w)ln( (s)3 3 )) In the case, the spread shocks at the four corners are given as the following table. (s) 2 3 will be calculated from the spread shock at the upper right corner (s)1 4 1 *. (s) 1 4 * 1 (s) 2 3. (s) * (s) 4 1. * 1 and the diagonal. Similarly, (s) 3 2 will be calculated from the spread shock at the lower left corner (s) 4 1 and the diagonal Interpolation of Term Structure Plots used for the construction of the term structure of shocks involve six maturities: 1 year, 3 years, 5 years, 7 years, 10 years and 20 years. These plots 51

51 correspond to the blocks for which the internal system offers a number satisfying calibration curves for the shock. The deformation method is based on a deformation of the zero spread. The method used to shock a new spread from the initial spread will be used to calculate the PnL. Interpolation: Shock calibration of corporate and sovereign issuers tenors are obtained by log linear interpolation and linear interpolation respectively between two shock tenors. The shock applied are interpolated from these shock pillars( for example, the 15 year is calculated from 10 year and 20 years) by log linear interpolation for corporate issuers and by linear interpolation for sovereign issuers. Extrapolation: For any spread less than 1 year or beyond 20 year, we use a flat extrapolation. The extrapolation of the term structure of shocks is constant for the lower 1-year tenor and beyond 20-years tenors. Spread shock with maturity less than 1 year will be subject the same shock that applies to plot 1 year, and similarly, the spread shock on the maturity beyond 20 years will be applied the shock 20 year. 1. Log linear Interpolation for Corporate Issuers Spread shock on maturity t between T 1 and T 2 can be obtained: ln( ij (t)) = ln( ij (T 1 )) T 2 t T 2 T 1 + ln( ij (T 2 )) t T 1 T 2 T 1 2. Linear Interpolation for sovereign issuers The impact of migration from rating i to j on maturity t between T 1 and T 2 can be calculated: ij (t) = ij (T 1 ) T2 t T 2 T 1 + ij (T 2 ) t T1 T 2 T 1 52

52 4.2 Profit and Loss Calculation Positive jump of an issuer s spread reduces market prices of instrument written on this issuer, hence produces a loss for policyholders. Figure 4.5: P&L Calculation in Single Period Simulation 53