# Lecture 2 Bond pricing. Hedging the interest rate risk

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1 Lecture 2 Bond pricing. Hedging the interest rate risk IMQF, Spring Semester 2011/2012 Module: Derivatives and Fixed Income Securities Course: Fixed Income Securities Lecturer: Miloš Bo ović

2 Lecture outline Theoretical bond pricing Risks related to fixed income securities Hedging the interest rate risk 1

3 Theoretical bond pricing 2

4 No arbitrage pricing: an example Suppose we have 5 riskless zero-coupon bonds: Series Maturity (yrs.) Price A1 A2 A3 A4 A

5 No arbitrage pricing: an example What is the price of a 5-year bonds with N = and annual coupon rate of 5%? Year CF

6 No arbitrage pricing: an example Strategy for the issuer: Let the price be P Buy 5 of each riskless bonds and another 100 from A5 series. Cash flows are: Yr CF buy CF issue CF net ( ) P 500 P

7 No arbitrage pricing: an example Conclusion: Under no arbitrage it has to be P =

8 No arbitrage pricing: general formula Write the formula in terms of discount factors/rates Generalize to get: 7

9 Yield to maturity IRR of cash flow generated by the bond 8

10 Basic bond valuation Step 1: Determine the cash flow Step 2: Determine the discount rates Step 3: Find the PV of discounted cash flows 9

11 Basic bond valuation Step 1 Cash flow Possible problems: Prepayments, call or put provisions Variable-rate bonds Convertible bonds 10

12 Basic bond valuation Step 2 Discount rates Zero-coupon yield curve (Lecture 1) Step 3 Present value P = T t t t= 1 (1 + rt ) CF 11

13 The price 12 In general: = + = T t t t t r CF P 1 ) (1 ( ) = + = + = T t T t r r CF r CF P ) (1 ( ) ( ) ( ) T T t T T t r N r r c r N r c P = = = ) (1 1 For annuities: If the discount rate is fixed:

14 Price vs. time Ceteris paribus, the price: Decreases with time for premium bonds Increases with time for discount bonds Stays constant with time for par bonds 13

15 Risks related to fixed income securities 14

16 The origins of risk Changes in discount rates Equivalently: changes in YTM Defaults on payments Uncertainty in the payment dynamics External (e.g. macroeconomic) factors 15

17 Factors that influence the interest rate risk Time to maturity Coupon rate Embedded options 16

18 Example All bonds have the same yield None of them has an embedded option Which one has the highest/lowest interest rate risk? Series coupon maturity (yrs.) ¼ 6 ½ 4 ¾ 8 ½

19 Default The failure to promptly pay interest or principal when due Any borrower whose liability is past due for more than 90 days is technically in default 18

20 Types of default The inability to pay The unwillingness to pay Also known as strategic default 19

21 A distinction Default is any failure by a person or business to meet its liabilities. Insolvency means that the debtor is unable to pay its debt. The term is usually used for business entities in financial industry. Bankruptcy is a legal proceeding involving a person or business in default. It involves imposing a court supervision over one s financial affairs. 20

22 Credit ratings Credit ratings are qualitative measures of payment ability for certain types of debt They are assessed by credit rating agencies using many risk factors, including systemic ones Ratings are typically assigned to: Sovereigns Banks Corporations 21

23 Credit ratings In the S&P/Fitch rating system, AAA is the best rating. After that comes AA, A, BBB, BB, B, and CCC The corresponding Moody s ratings are Aaa, Aa, A, Baa, Ba, B, and Caa Bonds with ratings of BBB (or Baa) and above are considered to be investment grade Some banks have their own internal rating systems for large borrowers 22

24 Credit scores Credit scores are quantitative measures of payment ability They are assessed directly by lenders using only factors specific for the borrowers Scores are typically assigned to retail clients 23

25 Altman s Z-score Credit risk factors for manufacturing companies: X 1 = Working Capital/Total Assets X 2 = Retained Earnings/Total Assets X 3 = EBIT/Total Assets X 4 = Market Value of Equity/Book Value of Liabilities X 5 = Sales/Total Assets 24

26 Altman s Z-score Z = 1.2X X X X X 5 Z>3.0 default is unlikely 2.7<Z<3.0 we should be on alert 1.8<Z<2.7 there is a moderate chance of default Z<1.8 there is a high chance of default 25

27 Estimating default probabilities Alternatives: Use historical data Use bond prices or asset swaps Use CDS spreads Use Merton s model 26

28 Historical data Historical data provided by rating agencies can be used to estimate the probability of default 27

29 Cumulative average default rates % ( , Moody s) Time (years) Aaa Aa A Baa Ba B Caa

30 Interpretation The table shows the probability of default for companies starting with a particular credit rating A company with an initial credit rating of Baa has a probability of 0.170% of defaulting by the end of the first year, 0.478% by the end of the second year, and so on 29

31 Do default probabilities increase with time? For a company that starts with a good credit rating default probabilities tend to increase with time For a company that starts with a poor credit rating default probabilities tend to decrease with time 30

32 Rating transition matrix (% probability, Moody s, ) Initial Rating at year end Rating Aaa Aa A Baa Ba B Caa Ca C Default Aaa Aa A Baa Ba B Caa Ca C Default

33 Other types of risk Embedded options Liquidity risk Currency risk Inflation (PP) risk Rare event risk Uncertainty (in the Knight sense) 32

34 Hedging the interest rate risk 33

35 Measuring the interest rate risk Direct assessment full valuation: Recalculate portfolio value for every imaginable change in the TS. Scenario analysis: Parallel shifts Non-parallel shifts 34

36 Full valuation Q: Which scenarios to consider? Banks are usually required to simulate possible changes in their portfolio values. Typical changes to be considered are 200 bp up and down uniformly. 35

37 Full valuation alternatives Consider only the most likely scenarios These are known as stylized scenarios Consider historical scenarios Determine the risk factors that account for the most of the variation in changes Usually done via PCA 36

38 Full valuation problems Can be very computationally intensive The more diversified portfolio the worse Even without securities with complex payoff structure (e.g. derivatives) Simpler methods are needed for quick valuation 37

39 Simpler methods Use simple measures based on first- or second-order Taylor approximation: Duration Convexity Key limitations: Assume small and parallel changes in TS Neglect stochastic nature of volatility 38

40 Hedging principles Basic principle: Reduce the dimensionality of the problem as much as possible. First step: duration hedging Consider only one risk factor parallel shifts Assume a flat yield curve Assume only small changes in the risk factor Beyond duration Relax the assumption of small interest rate changes Relax the assumption of a flat yield curve Relax the assumption of parallel shifts 39

41 Hedging principles Use a proxy for the term structure: YTM It is an average of the whole term structure If the term structure is flat, it is the term structure We will study the sensitivity of the price of the bond to changes in yield: Change in TS means change in yield 40

42 Duration No-arbitrage price of a bond that provides cash flow c i at time t i is given by P = n! i=1 CF i ( 1+ y) t i where y is its yield to maturity. 41

43 Duration Duration is defined as the weighted sum of cash flow maturities: D = n! i=1 w i t i where weights are relative discounted cash flows: It is easy to check that w i = 1 CF i P (1+ y) t i n! w i = 1 i=1 42

44 Modified duration The first-order approximation of the price change is then: where!p P " 1 P #P #y!y = \$M!y M =! 1 P "P "y = 1 P n t i CF # i = D (1+ y) t i+1 1+ y i=1 is called the modified duration. 43

45 Example Consider a 10-year bond paying a 6% annual coupon The bond trades at par Find its duration and modified duration Suppose that the yield increases by 1%: Calculate the approximate change in price of the bond using the modified duration Compare it to the exact value 44

46 Duration hedging Principle: immunize the value of a bond portfolio with respect to changes in yield Hedging instrument may be Bond Swap Future Option Assume all instruments having same yield 45

47 Duration hedging limitations Duration hedging is Very simple Built on very restrictive assumptions Assumption 1: Small changes in yield The value of the portfolio could be approximated by its first order Taylor expansion OK when changes in yield are small, not OK otherwise This is why the hedge portfolio should be re-adjusted reasonably often 46

48 Duration hedging limitations Assumption 2: Yields are the same initially In other words, the interest rate risk is simply considered as a risk on the general level of interest rates Assumption 3: Yields are the same always In other words, we have assumed that the yield curve is affected only by a parallel shift 47

49 Convexity The second-order approximation of the bond price change is given by where is called the convexity.!p P " #M!y C (!y ) 2 C = 1 P! 2 P!y 2 = 1 P n " i=1 t i (t i +1)c i (1+ y) t i+2 48

50 Example (continued) Consider a 10-year bond paying a 6% annual coupon The bond trades at par Find its convexity Suppose that the yield increases by 1%: Calculate the approximate change in price of the bond using the modified duration and convexity Compare it to the exact value 49

51 Duration-convexity hedging Principle: immunize the value of a bond portfolio with respect to changes in yield Needs two hedging instrument because we want to hedge one risk factor up to the second order Assume all instruments having same yield 50

52 Example Portfolio at date t P = \$ MDur = 6.76 Conv = Hedging instrument 1 H 1 = \$ MDur 1 = Conv 1 = Hedging instrument 2: H 2 = \$ MDur 2 = Conv 2 = Assume yields are approximately the same 51

53 Portfolios Duration and convexity can be defined similarly for portfolios of bonds and other interest-rate dependent securities The duration of a portfolio is the weighted average of the durations of the components of the portfolio. Similarly for convexity. 52

54 What duration and convexity measure Duration measures the effect of a small parallel shift in the yield curve Duration plus convexity measure the effect of a larger parallel shift in the yield curve However, they do not measure the effect of non-parallel shifts 53

55 Accounting for changes in shape of the term structure Bad news: not only the yield curve is not flat, but also it changes shape! Aforementioned methods do not allow to account for such deformations We need to use additional risk factors We also need to regroup different risk factors to reduce the dimensionality of the problem 54

56 Accounting for changes in shape of the term structure To properly account for the changes in the yield curve, one has to get back to pure discount rates Another simplification: use continuous compounding The challenge is that we are now facing many risk factors Reduce the dimensionality of the problem by using parametric approach 55

57 Example: Nelson-Siegel Mechanics of the model: changes in beta parameters imply changes in discount rates, which in turn imply changes in prices One may easily compute the sensitivity of discount rates with respect to each parameter beta Very consistent with factor analysis of interest rates in the sense that they can be regarded as level, slope and curvature factors, respectively 56

58 Example: Nelson-Siegel Hedging principle: immunize the value of a bond portfolio wrt changes in parameters of the model Needs 3 hedging instruments because we want to hedge 3 risk factors 57

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