School of Education, Culture and Communication Division of Applied Mathematics

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1 School of Education, Culture and Communication Division of Applied Mathematics MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS Bridging the Gap: An Analysis into the relationship between Duration and Key Rate Duration by Andile Dumisani Ndiweni Masterarbete i matematik / tillämpad matematik DIVISION OF APPLIED MATHEMATICS MÄLARDALEN UNIVERSITY SE VÄSTERÅS, SWEDEN

2 School of Education, Culture and Communication Division of Applied Mathematics Master thesis in mathematics / applied mathematics Date: Project name: Bridging the Gap: An Analysis into the relationship between Duration and Key Rate Duration Author: Andile Dumisani Ndiweni Version: 14th June 2016 Supervisor(s): Jan Röman and Ying Ni Reviewer: Milica Ran cić Examiner: Professor Anatoliy Malyarenko Comprising: 30 ECTS credits

3 Abstract As financial instruments continue to increase in their complexity, the ways in which risk is measured needs to adapt accordingly. In this paper the aim is to communicate the importance of using Key Rate Duration as a risk measure as opposed to the more popular Duration when measuring risk of fixed income instruments as well as the relationship between the two. The paper will give insights into the deficiencies of the Duration measure with particular emphasis on when interest shifts on the term structure are done in a non-parallel manner. It will also show how when shifts are done in a non-parallel manner Key Rate Duration is the better measure. Finally this paper will show, using market data relating to Swedish interest rates, that individual nodes have varying risk buckets and these can be substantial thereby creating potential risk compliance issues. Keywords: Duration, Key Rate Duration, Term Structure, Interpolation, Bootstrapping

4 Acknowledgements I convey my special thanks to the Swedish Institute (SI) for awarding me a full scholarship to study the masters program Financial Engineering at Mälardalen University. To my supervisors Jan Röman and Ying Ni for their guidance during the writing of this thesis paper.

5 Contents 1 Introduction Background Trading floor perspective Risk management perspective Literature Review Duration Key Rate Duration Problem Formulation Fixed Income Instruments Bonds Valuation of Bonds Forward Rate Agreements Valuation of FRAs Interest Rate Swaps Valuation of Swaps Chapter Summary Curve Construction Term structure estimation methods Mathematics of Yield Curves Interpolation Linear Interpolation Smoothing of curve Chapter Summary Duration Modified duration Convexity Use of Duration to hedge interest rate exposure Duration shortcomings Using Yield to Maturity for Discounting Non-infinitesimal shift Flat yield curve and parallel shifts

6 4.5 Chapter Summary Key Rate Duration Key Rate Convexities Key Rate Durations and Value at Risk Analysis Limitations of KRD Choice of Key Rates Shape of Key rates Loss of efficiency Chapter Summary Implementation Curve construction Data set Data set Calculation of risk FRA calculation Swaps Calculation Chapter Summary Conclusion 51 8 Notes on fulfilment of Thesis objectives 52 Bibliography 54 A Calculation of Rates 57 A.0.1 Cash Deposits A.0.2 FRAs A.0.3 Swaps B Matlab code 60 C Proof of Theorem

7 List of Figures 1.1 The floor trading perspective Risk manager view with shift at node t i Risk manager view with shift at node t i Risk manager view with shift at node t i Risk manager view Diagram depicting bond relationship Key time periods in FRA contracts Diagram depicting plain vanilla swap relationship Duration error when shift is large Comparison of two bonds and duration calculation Key Rate Duration Shift illustration Key Rate Duration and Duration Shift relationship Key Rate Duration Shift on spot curve Key Rate Duration Shift effect on forward curve Discount rate curve Zero rate curve Forward rate curve Discount rate curve Zero rate curve Forward rate curve FRA risk calculation using data set FRA risk calculation using data set Swap risk calculation for data set Swap risk calculation for data set

8 List of Tables 6.1 Table showing interest data as on 09 July Table showing interest data as on 06 April Table showing risk buckets per node for data set Table showing risk buckets per node for data set Repriced swap price Table showing risk buckets per node for data set Table showing risk buckets per node for data set

9 Chapter 1 Introduction In the financial services industry it is of paramount importance to ensure that risk relating their products, be calculated both accurately and provide relevant information to for decision making. The research to be undertaken is to communicate the differing methodologies between the Risk Department and the Trading Floor for calculating risk relating to interest rate movements on fixed income instruments and "Bridging the Gap" between the two. 1.1 Background To better understand the purpose of the research the differing methodologies need to be explained using a Forward Rate Agreement (FRA) for illustration. The first will be the view of Trading floor perspective The Trading Floor calculates risk by shifting quotes of the fixed instrument as shown in Figure 1.1, Figure 1.1: The floor trading perspective 7

10 and this is done as a way of hedging their trades [19]. The value at risk would then be calculated as Risk = notional amount time change in interest rate. The calculation of interest rate risk by the trading floor is very similar to Duration. This is due to both calculations being based on shifting the interest only of the specific instrument Risk management perspective On the other side of the coin, the Risk Department shifts the zero coupon curve (primarily done to aggregate the risk in multiple currencies) in specific nodes as shown in Figures The shifting done at these curve-nodes is similar to Key Rate Duration. Figure 1.2: Risk manager view with shift at node t i The first shift on the rates is at t i but the instrument is shifted to an interest rate change determined by interpolation methods [8]. Figure 1.2 would show interest shift risk associated with that node. Next is to shift the node at t i 1 in this case only one part of the instrument is affected the shift. Figure 1.3: Risk manager view with shift at node t i 1 Finally a shift is done at the node t i+1 When all the affected nodes have been shifted the result is as in Figure 5. The relationship between these two calculations is Trading floor risk = m Risk department risk, calculated at each node t i, i=1 and is similar to the relationship of Key Rate Duration(KRD) and Duration [12]. 8

11 Figure 1.4: Risk manager view with shift at node t i+1 Figure 1.5: Risk manager view Duration = m KRD at t i. i=1 where m is the number of key rates to be used. These two different methods have implications regarding whether the trades are within the prescribed risk limits. This would affect compliance on one hand as well as lead to blind spots regarding unstated risk from the different nodes. As this is an applied science course the aim is not to deliver a theoretical output but a usable product, hence the aim is to deliver a better general bootstrap for determining the risk. 1.2 Literature Review Duration Originally defined in 1938 by Professor Macaulay, Duration [14] is the mean length of time that would pass before the present value would be returned by a stream of known fixed payments. Duration is given by [10] where D = t i is the time when cash flow occurs, c i is the cashflow, n i=1 ( ci e yt ) i t i. (1.1) B 9

12 B is the bond price and y is the yield to maturity. Though initially used as a replacement for maturity it was Hicks who first demonstrated the risky-proxying property of Duration [14]. This led to it s wide usage as a risk-proxy but it would be Hopewell and Kaufman [13] that would bring to light inconsistencies regarding the use of Duration as a risk measure. They noted that the relationship between the maturity and interest rate sensitivity of bonds does not hold for bonds selling at a discount. This led to the following key limitations of duration as a risk-proxy: when a 100 basis point change occurs, Duration as measure of risk is useless, when shifts on the yield curve are non-parallel, Duration becomes a poor risk measure, due to it s use of yield to maturity as opposed to forward interest rates Duration requires a flat term structure. Despite the shortcomings of Duration it is widely used, for the purposes of this study the non-parallel shift deficiency will be the main defiency to be researched Key Rate Duration Given the weakness of Duration in dealing with non-parallel shifts Key Rate Duration model was [11] proposed as an alternative. It s premise is the shifts in the term structure as a discrete vector representing the changes in the key spot rates of various maturities. Interest changes at other maturities are derived from these values via linear interpolation, it is important to note that only linear interpolation can be used as other interpolation methods require more data points. 1.3 Problem Formulation Currently Traders use Duration as their main form of risk proxy when dealing with Fixed Income instruments and related portfolios. Though this measure is attractive to them as they can use it for the purposes of hedging it falls short at identifying risk at key rates. As these rates can at times move independent of other rates it leaves them exposed to unforeseen risk. The centre of interest for this thesis document is the utilisation of Key Rate Duration as a method for calculating of risk in Fixed Income instruments. The type of instruments to be used in this study are Deposits, Forward Rate Agreements and Interest Rate Swaps. The study will utilise bootstrapping techniques to obtain the term structure curve and then linear interpolation to determine the risk amounts in the individual key rates. The results of this research show that when calculating interest movement dependent risk in Fixed Income instruments Key Rate Duration provides a solution with better insights. It gives the Trader(viewed as primary user of study) both the total risk during the entire period of the instrument and the risk due to interest rate movements that occur at only one rate. This 10

13 will result in the Trader gaining an understanding of how different key rates carrying varying risk and hence require differing strategies. In Chapter 2 the study will detail the instruments to be used for building the term structure curve that will also be analysed to show how key rate shifts operate. In Chapter 3 the focus will be on the mathematics of how to build a term structure and detail the bootstrapping techniques used and the importance of curve smoothness in obtaining accurate results. Chapter 4 gives insight into Duration as a risk proxy and its utilisation as a tool for hedging. The will also be a discussion on how it has certain deficiencies and how these compromise the results that one obtains. Chapter 5 deals with how Key Rate Duration is utilised to give good insights on the amount of risk at key rates. Chapter 6 involves the building of a term structure curves using bootstrapping techniques discussed in Chapter 3. The next is to calculate risk based on Duration and then Key Rate Duration. From these relationships between the two will shown and also that amount of risk in certain key rates can at times significantly higher than that which is calculated from Duration. 11

14 Chapter 2 Fixed Income Instruments This chapter deals with the main types of Fixed Income Instruments and how they are valued. These are also the key instruments used in the construction of an interest rate curve and also be used to show the risk implications of Duration and Key Rate Duration. 2.1 Bonds A bond is the most commonly known fixed income security. The general theory is that an Issuer of the bond sells it to an Investor who is lending money to the issuer in return for interest payments and the repayment of the money borrowed [7]. Figure 2.1: Diagram depicting bond relationship Valuation of Bonds The theoretical price of a bond is the PV of the interest payments and the principal and calculated as follows where C is the coupon payment, PV = n i=1 C p(0,t i ) + M p(0,t n ). 12

15 p(0,t i ) is the discounting factor at the period T i, M is the principal amount and n is number of payment periods. The different types of bonds are Callable bonds, Putable bonds, Convertible bonds, Zero-coupon bonds and Foreign currency bonds. 2.2 Forward Rate Agreements A Forward Rate Agreement, is an over-the-counter contract that a reference rate (normally LIBOR rate) will be exchanged for a specified interest rate during a specified future period of time. This instrument is used to both hedge interest rate exposure and speculate on interest rates. To better understand an FRA and it s valuation familiarisation with the following terms, needed [19] Notional (N)- the principal underlying the contract, Trade date - the date on which the contract is dealt, Spot date - the date in with the contract must be delivered, Settlement date (T 1 )- the date on which the contract period begins, Fixing date - the date on which the reference rate is determined, Maturity date (T 2 )- the date on which the contract expires, FRA rate (r FRA )- the rate at which the FRA is traded, Reference rate (r Re f )- the rate which is agreed upon to be used to reference FRA contract on and Settlement sum - the difference between FRA rate and reference rate as a percentage of the notional and is paid on the settlement date. Figure 2.2 shows these dates as they would be on a timeline [19]. 13

16 2.2.1 Valuation of FRAs Figure 2.2: Key time periods in FRA contracts The valuation of the FRA is then derived as which rate you receive your payout in and discounted using the riskfree rate at T 2 which will be r 2 [10]: 1. r FRA = r Re f where the value is 0, 2. if payout is r FRA then V FRA = N(r FRA r Re f )(T 2 T 1 )e r 2T 2 and 3. if payout is r Re f then V FRA = N(r Re f r FRA )(T 2 T 1 )e r 2T Interest Rate Swaps A swap is an over-the-counter agreement between two counterparts to exchange periodic interest rate payments where it s dollar amount is based on some pre-determined principal. The agreement gives certain dates when cash flows are to be paid and the way in which they are to be calculated [15]. The best type of swap to illustrate the concept with is a plain vanilla swap as it operates of the basic definition of a swap. Swaps are extremely liquid due to the huge size of the swaps market. Figure 2.3: Diagram depicting plain vanilla swap relationship Valuation of Swaps In interest rate swaps no principle payments are made, the most basic illustration of the valuation is [7] 14

17 from view of a floating-rate payer as a long position in a fixed rate bond and a short position on a floating rate bond V swap = B f ixed B f loating, from view of a fixed rate payer as a long position in a floating rate bond and a short position on a fixed rate bond. V swap = B f loating B f ixed. To then get a value for the swap it, necessary to determine the fixed leg and floating leg. Floating leg To understand the valuation of the floating leg the following terms need to be explained discount factors P(T i 1,T i ) - this allows for valuing of money received in T i at T i 1 where T i 1 T i and 0 is present with the following relationship ( ) P(0,Ti ) P(T i 1,T i ) =, (2.1) P(0,T i 1 ) forward rates f(t i 1,T i ) - are rates of interest implied by current zero for periods of time in the future. It s relationship with discount factors is shown below with δ i being the day count fraction f (T i 1,T i ) = 1 ( ) 1 δ i P(T i 1,T i ) 1, As floating-rate payments are set at the beginning of the payment period and paid at the end of the period. As these payments are on the interest only for that period they are similar to zero coupon bonds. The value of the swap can therefore be seen as a sum of zero coupon bonds with payments calculated using forward rates, leading to the following relationship The first thing do to simplify PV f loating = PV f loating = n f loating i=1 n f loating δ i f (T i 1,T i )P(0,T i ), i=1 ( δ i δ i 1 P(T i 1,T i ) 1)P(0,T i which leads to PV = n f loating i=1 P(0,T i 1 ) P(0,T i ) = P(0,T 0 ) P(0,T n f loating ), and the resulting final relationship is PV f loating = 1 P(0,T n f loating ). ), 15

18 Fixed leg The pricing of the fixed leg is simpler as it is the discounting of the swap rate = C as shown PV f ixed = n f ixed δ i CP(0,T i ). i=1 It is important to note that the frequencies of the fixed and floating payments need not be similar hence n f ixed and n f loating [19]. 2.4 Chapter Summary In this chapter the discussion is on the key fixed asset instruments and their valuation. The first was bonds which are the simplest and most commonly known instrument and also the factors that affect their pricing. The next were Forward Rate Agreements and Swaps and their valuation. These instruments are of importance when constructing a yield curve as will be discussed in the next chapter. 16

19 Chapter 3 Curve Construction One of the key purposes of a term structure is to allow the consistent valuing of instruments under a singular valuation framework. The framework to be used is the continuous zero curve as it is standard format for all option pricing formulae [7]. The accuracy of the model implementation are highly dependent on material found in this chapter. 3.1 Term structure estimation methods There exist two term structure estimation methodologies [8] 1. Theoretical term structure methods typically based on an explicit structure for a variable known as the short rate of interest, whose value depends on a set of parameters that might be determined using statistical analysis of market variables with Vasicek [20] being a notable example. 2. Empirical methods, unlike the theoretical methods, are independent of any model or theory of the term structure. The empirical methods utilise observed interest rate data and then uses these data points to try and find a close representation of the term structure at any point in time. For the purposes of this research the focus will be on the empirical methods only. 3.2 Mathematics of Yield Curves The term structure of interest rates is defined as the relationship between the yield to maturity on a zero coupon bond and the bond s maturity. From the outset, continuously-compounded rates will be used as most derivatives are modelled in continuous time [1]. To develop the zero curves certain assumptions need to be made Market trades continuously over the time horizon, Market is frictionless, 17

20 Market is competitive every trader, Market is efficient, Market is complete, No arbitrage and All trades are rational. These assumptions though important in the development of the zero curve do not occur in reality. The first step is to give the zero yield in terms of the bond price. This relationship is given by where t T t and t is current time, T is maturity time, P(t,T ) price of the bond and P(t,T ) = exp( (T t)r(t,t )), (3.1) r(t,t ) yield at time t of a bond that matures at time T. This forms the basis of the mathematics of a zero yield curve. The forward price of a bond is provided that t < T 1 < T 2 is given by ( ) P(t,T2 ) P(t,T 1,T 2 ) =, (3.2) P(t,T 1 ) at the same time an implied forward rate, as seen at time t for the period T 1 to T 2 symbolised as f (t,t 1,T 2 ) is defined by P(t,T 1,T 2 ) = exp( (T 2 T 1 ) f (t,t 1,T 2 )), Using Equation (3.1) and substituting it into Equation (3.2) the following is obtained exp( (T 2 t)r(t,t 2 )) = exp( (T 1 t)r(t,t 1 )) exp( (T 2 T 1 ) f (t,t 1,T 2 )), (3.3) Rearranging Equation (3.3) yields a result that shows how the forward rate is determined ( ) (T2 t)r(t,t 2 ) (T 1 t)r(t,t 1 ) f (t,t 1,T 2 ) =, (3.4) (T 2 T 1 ) In Equation (3.4), the period forward rate is defined. However, the instantaneous forward rate is of much greater importance in the theory of the term structure. The instantaneous 18

21 forward rate for time T, as seen at time t, is denoted by f (t,t ) and is the continuously compounded rate defined by where f (t,t ) = lim h 0 f (t,t,t + h) = r(t,t ) + (T t)r T (t,t ), (3.5) r T (t,t ) = r(t,t ), T A re-arrangement of Equation (3.1) gives the following Differentiating Equation (3.6) with respect to T gives ln(p(t,t )) = (T t)r(t,t ), (3.6) P T (t,t ) P(t,T ) = y(t,t ) + (T t)y T (t,t ), (3.7) Finally, by direct comparison of Equation (3.4) and Equation (3.7), ( ) PT (t,t ) f (t,t ) =, P(t,T ) Equation (3.4) implies f (t,t) = y(t,t) and noting that P(t,t) = 1, Equation (3.7) can be used to obtain P T (t,t ) r(t,t) = lim = lim P T (t,t ). T t P(t,T ) T t Though defined in terms of the zero yields, the zero curve can be defined in terms of the instantaneous forward rates [1]. In this case, the zero curve is defined by the instantaneous forward rates f (t) for t T t. The zero yield in terms of the instantaneous forward rate is obtained by integrating Equation (3.5) and yields T r(t,t )T = r(t,t)t + t f (t,u)du. This gives an overview of the essential mathematical theory of zero curves. In the following sections, the relevance of this theory to the interpolation of zero curves is shown and a look into smoothest forward-rate interpolation. 3.3 Interpolation To construct zero curves from market data, it s assumed that the n data values are {(T 1,y 1 ),(T 2,y 2 ),...,(T n,y n )}, where 0 T 1 < T 2 <... < T n < T are the times to maturity of n 1 zero coupon bonds. 19

22 Mathematical theory of zero curves discussed earlier assumes that the value of r(t, T ) for 0 T < t is known. In reality, the current zero curve is not defined by this infinite set of values implied by the complete market assumption but, rather, by a set of discrete data values each value comprising a time to maturity and a zero rate [2]. To use the mathematics of zero curves derived earlier, the discrete set of values should be extended to an infinite set. To achieve this define the current zero curve by a combination of the set of discrete data values and a method for interpolating those values. Given the need for interpolation, suppose a time variable t is given such that t i 1 < t < t i. The desire is to attain the estimated value of r(t) given that the rates for r(t i 1 ) and r(t i ) are known, this is where interpolation comes in. There are various interpolation methods with varying levels of accuracy and difficulty, it is important to note the following when comparing them [9]: The positivity and continuity of the forward rates to avoid arbitrage, The localisation of interpolation, which relates to if an input is changed does the interpolation function only change in close proximity or the change is more widespread, The stability of forward rates; this is observed by changing an input in the interpolation function and noting the maximum change on the forward curve, The delta risk assigned to hedging instruments that have maturites close to the given tenors or is it more widespread and The effect the number of instruments has on it; if it is small does it give an exact empirical curve and on large set does the algorithm give a small degree of error Linear Interpolation The most common form of interpolation is linear and the methods formulated are thereby also implicitly linear. There are numerous ways that linear interpolation can be applied these are Linear on rates To determine r(t) given the following conditions t i 1 < t < t i the interpolation formula is ( ) ( ) t ti 1 ti t r(t) = r i + r i 1, t i t i 1 t i t i 1 Using the relationship that f (t) = dt d (r(t))t the following is obtained for forward rates f (t) = d (( t 2 ) ( tt i 1 tti t 2 ) r i + )r i 1, dt t i t i 1 t i t i 1 differentiation with respect to t yields ( ) ( ) 2t ti 1 ti 2t f (t) = r i + r i 1. t i t i 1 t i t i 1 20

23 Linear on discount factors As similar approach to that done on the rates can be taken on the discount factors then ( ) ( ) t ti 1 ti t D(t) = D i + D i 1, t i t i 1 t i t i 1 Given the relationship D(t) = exp( r(t)t) the following calculation can be done r(t) = 1 (( ) ( ) t t ln ti 1 ti t D i + )D i 1, t i t i 1 t i t i 1 This allows for the use of the relationship f (t) = dt d (r(t))t to obtain forward rates by firstly ( 1 t f (t) = i t i 1 D i t 1 ) i t i 1 D i 1 t t i 1 t i t i 1 D i + t, i t t i t i 1 D i 1 ( ) D i 1 D i f (t) =. (t t i 1 )D i + (t i t i 1 )D i 1 as can observed the forward rate is not continuous. Linear on the log of rates In this instance the interpolation formula for the conditions t i 1 < t < t i is ( ) ( ) t ti 1 ti t ln(r(t)) = ln(r i ) + ln(r i 1 ), t i t i 1 t i t i 1 which gives the following rate formula r(t) = ( r t t i 1 t i t i 1 i r t i t t i t i 1 i 1 The key failing of this interpolation method especially in the current interest rate environment is that it does not allow for negative interest rates. To determine the forward rate first define differentiation leads to ( t ti 1 ln(r(t)t) = t i t i 1 ) ln(r i ) + ( ti t ) t i t i 1 ( ( ) 1 1 r(t)t f (t) = ri ln + 1 ), t i t i 1 r i 1 t, ) ln(r i 1 ) + ln(t), 21

24 finally giving the forward rates as ( ( ) ) t ri f (t) = r(t) ln + 1. t i t i 1 r i 1 ( ) t As can be observed if t i t i 1 ln ri r i 1 < 1 the forward rates can become negative leading to arbitrage opportunities. Raw Interpolation Linear on log of discount factors The following method is very stable and is usually a starting point for interpolation investigations. It is used to ensure no errors occur in fancier methods by comparing the raw method with the more sophisticated method[9]. The first step in this method is to define the forward discount factor which is given by, Z(t) = exp( t 0 f (s)ds), (3.8) The discrete forward rate for the interval [t i 1,t i ] is equal to the average of the instantaneous forward rate over any the given intervals, this will leads to r i t i r i 1 t i 1 t i t i 1 = which is an important interpolation formula. 1 t i t i 1 ti t i 1 f (s)ds, ti 1 r(t)t = r i t i + f (s)ds, t [t i 1,t i ], t i Given the relationship in equation this formula can be re-written as ( ) ri t i r i 1 t i 1 r(t)t = r i t i + (t i t i 1 ), t i t i 1 by making t i t i 1 a common denominator the following relationship is obtained ( ) ( ) t ti 1 ti t i 1 r(t)t = r i t i + r i 1 t i 1. t i t i 1 t i t i 1 This method guarantees that all instantaneous forwards are positive which is key to avoiding arbitrage, because every instantaneous forward is equal to the discrete forward for the parent interval. As can be seen r(t)t is a log on the discount factors. Linear interpolation methods have continuity difficulties associated with them. Thus, they should not be used for anything other than basic yield curve construction. 22

25 Polynomial Interpolation Polynomial interpolation is the process of approximating complicated curves (yield curve being one). It smears out sharp edges to deliver a continuous curve. If we have n + 1 discrete data points (t i,r(t i )), to construct an interpolation polynomial to fit the data to a polynomial of degree n through all the points [19] this can be simplified as p(t) = a n t n + a n 1 t n a 2 t 2 + a 1 t + a 0. p(t) = n i=0 To solve for multiple points p(t i ) = r(t i ) a system of linear equations can be utilised giving t0 n t0 n 1 t n t 0 1 a n r(t 0 ) t1 n t1 n 1 t1 n 2... t 1 1 a n = r(t 1 ).. tn n tn n 1 tn n 2... t n 1 a 0 r(t n ) Alternatively using Lagrange polynomials the following general equation can be obtained Cubic Spline Interpolation p(t) = n i=0 0 j n j i a i t i, t t j r(t i ). t i t j A polynomial spline is a function which is piecewise in each interval a polynomial, with the coefficients arranged to ensure at least that the spline coincides with the input data (and so is continuous). It is preferred to polynomial interpolation as it gives similar results without having any instability. In a cubic spline the aim is to find the coefficients (a i,b i,c i,d i ) for 1 i n 1. Given these coefficients, the function value at any term t will be [6] r(t) = a i + b i (t t i ) + c i (t t i ) 2 + d i (t t i ) 3, It is important to note that the equation is three times differentiable as shown. r (t) = b i + 2c i (t t i ) + 3d i (t t i ) 2, r (t) = 2c i + 6d i (t t i ), r (t) = 6d i. The common constraints when using splines are: 1. the interpolating function needs to meet the given data points, 23

26 2. the entire interpolating function is continuous and 3. the entire function is differentiable, this is important as when the forward function f (t) = dt d (r(t))t is continuous the following is obtained: f (t) = d dt (a i + b i (t t i ) + c i (t t i ) 2 + d i (t t i ) 3 )t, differentiation with respect to t yields f (t) = a i + b i (2t t i ) + c i (t t i )(3t t i ) + d i (t t i ) 2 (4t t i ). This method of using splines is very good at producing smooth curves both forward and zero and this shall be explored in the following section. 3.4 Smoothing of curve The smooth interpolation of interest rates is of keen interest for risk managers. Though smooth interpolation gives an aesthetically appealing curve, there is little published research on numerical benefits. There exist an intuitive belief that smooth interpolation gives more accurate results [2]. What was proposed by them is the following, they defined the smoothest forward rate curve on an interval (0,T ) as one that minimises the function Z = x t ( f (u)) 2 du, (3.9) which is a common mathematical expression used in defining smoothness. Using this definition it can be shown that a cubic spline fitted to the discount function produces the smoothest possible discount function according to this definition of smoothness. The maximum smoothness criterion is useless unless it is combined with observable points on the yield curve. If the observable points take the form of m given zero-coupon bond prices Equation (3.9) will be minimised subject to Equation (3.8) acting as a constraint. In this case let f (t) = f (t;a 1,a 2,...,a n ) be the forward rate curve as a known function of parameters a 1,a 2,...,a n. Introducing the Lagrange multipliers λ i for i = 1,...,m, the minimisation problem can be written as min(z + The solution then becomes [ x ( f (u)) 2 du + a i t m i=1 m i=1 λ i x[p(t i ) P i ]), [ x ] ] λ i x exp( f (u)du) + P i. t i 24

27 To then determine the maximum smoothness term structure within all possible functional forms a theorem from Vasicek [20] is utilised. Theorem 1. The term structure f (t), 0 t T of forward rates that satisfies the maximum smoothness criterion while fitting observed points is a fourth-order spline with the cubic term absent and given by, f (t) = c i t 4 + b i t + a i, for t i 1 < t t i, for i = 1,2,..,m + 1, where 0 = t 0 < t i <... < t m < t m+1 = T. The coefficients a i,b i,c i for i = 1,2,...,m + 1 such that the following equations are satisfied: c i t 4 i + b i t i + a i = c i+1 t 4 i + b i+1 t i + a i+1, for i = 1,2,...,m, 4c i ti 3 + b i = 4c i+1 ti 3 + b i+1 for i = 1,2,...,m, 1 5 c i(ti 5 ti 1 5 ) + 1 ( ) 2 b i(ti 2 ti 1) 2 Pi + a i (t i t i 1 ) = ln, i = 1,2,...,m, c m+1 = 0. The proof to this theorem can be found in Appendix C. P i Chapter Summary In this chapter the discussion was around the process in which forward and zero coupon curves are constructed. A look at the mathematics in which the zero and forward curves a based leading to how this is implemented using interpolation methods. To finish of the chapter a discussion how best to improve accuracy and smoothness in a yield curve detailed. 25

28 Chapter 4 Duration It is important to compute the change in value of fixed income instruments due to changes in interest rates that they are based on. The key to measuring interest-rate risk is the accuracy of the estimate of the value of position after an adverse rate change. In this chapter the discussion will be around Duration and convexity [7] this is the measure that will be compared to Key Rate Duration the model. The Duration of a bond, as its name implies, is a measure of how long on average the holder of the bond has to wait before receiving cash payments. A zero-coupon bond that lasts n years has a duration of n years. However, a coupon-bearing bond lasting n years has a duration of less than n years, because the holder receives some of the cash payments prior to year n [10]. Suppose there exists a bond that provides the holder with cash flows c i at time t i where 1 i n. The bond yield y(continuously compounded) and bond price B are related by B = The Duration of the bond, D, is defined as D = n i=1 n i=1 c i exp( yt i ), (4.1) t i ( ci exp( yt i ) B When a small change δ in the yield is considered, it is approximately true that When differentiated it becomes B = y ), B = db y, (4.2) dy n i=1 c i t i exp( yt i ), As can be observed the relationship between yield and bond price is negative which is analogous to them being inversely proportional to each other. With the use of Equation (4.1) the 26

29 relationship for duration and bond price is this can then be written as B = BD y, B B = D y. (4.3) Equation (4.3) is an approximate relationship between percentage changes in a bond price and changes in its yield. It is easy to use and is the reason why duration is a popular measure. 4.1 Modified duration The preceding analysis is based on the assumption that y is expressed with continuous compounding. If y is expressed with annual compounding, it can be shown that the approximate relationship in Equation (4.3) becomes B = BD y 1 + y, More generally, if y is expressed with a compounding frequency of m times per year, then D = D 1 + y, m sometimes referred to as the bond s modified duration. It allows the duration relationship to be simplified to B = BD y. 4.2 Convexity The duration relationship applies only to small changes in yields. This is illustrated in Figure 4.1 ([7]), which shows the relationship between the percentage change in value and change in yield for two bond portfolios having the same duration. In Figure 4.2 ([7]) gradients of the two curves are the same at the tangent line. This means that both bond portfolios change in value by the same percentage for small yield changes and is consistent with Equation (4.2). For large yield changes, the portfolios behave differently. Bond A has more curvature in its relationship with yields than Bond B. A factor known as convexity measures this curvature and can be used to improve the relationship. A measure of convexity is the second derivative of a bond with respect to the yield as shown C = 1 d 2 n B B dy 2 = i=1 ( ci t 2 i exp( yt i) B ), 27

30 Figure 4.1: Duration error when shift is large Figure 4.2: Comparison of two bonds and duration calculation 28

31 To then compute a change in bond price the following relationship exists, B = db dy y + 1 d 2 B 2 dy 2 y2, this then gives us the final argument where the answer is similar to Duration but adjusted for the curvature of the bond yield relationship B B = D y C( y)2. Despite it offering an improved answer when compared to duration, convexity is still not a good estimator for non-parallel shifts. 4.3 Use of Duration to hedge interest rate exposure One of the uses of Duration is as a way to hedge against interest rate exposure. To observe this benefit, consider at time 0 a default-free zero coupon bond with a payout at time T. If interest rate is taken to be constant and continuously compounding, the result is the following where, B is the price of the bond, B = exp( rt ), r is the constant interest rate per unit time and T is the time in the future when the bond matures. For a coupon paying bond where B = n i=1 CF i is the ith cash flow at time t i and n is the total number of cash flows. CF i exp( rt i ), This leads to defining Duration which is the relationship B y B = D, With this relationship attained, the assumption of a constant interest rate r is then replaced with a forward interest rate curve and denoted by f (t). This replacement gives [ T ] B = exp f (t)dt, 0 29

32 For a coupon paying bond B = n i=1 [ T CF i exp 0 ] f (t)dt, In this context Duration is given by [ D = n i=1 T icf i exp ] T i 0 f (t)dt. B If the price of the bond is differentiated with respect to a parallel shift in the forward curve, the result as r approaches 0 is n [ B Ti ] y = T i CF i exp f (t)dt = B, i=1 0 Substituting B y B = D. This equation shows the benefit and downfall of Duration as a measure of interest rate exposure. This shows that for any forward interest rate curve as long as the shift is small and parallel Duration can be utilised. 4.4 Duration shortcomings Duration though used widely as being a proxy of interest rate sensitivity and the usefulness in immunising bond portfolios has serious shortcomings Using Yield to Maturity for Discounting The first key shortcoming is that for a set of certain payments, CF(t), and the present value function at time t, P(t), the definition of duration is this measure is then generally changed into D = n i=1 t icf(t i )P(t i ) n i=1 CF(t i)p(t i ), D = tcf(t)exp( yt) CF(t)exp( yt). (4.4) or a discrete period discounted equivalent. These two definitions are equivalent if and only if the discount rate implicit in the present value function is the constant y [6]. Theorem 2 [14] was developed to show that when the two definitions differ only the first definition can be used. 30

33 Theorem 2. For infinitesimal shifts in the yield curve, the percentage change in value of any asset with fixed payments is proportional to Macaulay s duration measure as defined in Equation (4.4) if and only if the entire yield curve undergoes a uniform additive displacement, dy(t) = dy for all t. Proof. An assets value is V = CF(t) exp( y(t)t), thus dv V = tcf(t)exp( y(t)t)dy(t) = Ddy, V if dy(t) = dy and proves sufficiency. To prove necessity, its important to recall that the duration of a pure discount bond is its maturity. For two pure discount bonds of maturities t 1 and t 2, ) ( dp(t1 ) P(t 1 ) ( dp(t2 ) P(t 2 ) which equals ( t1 t2 ) only if dr(t 1 ) = dr(t 2 ). ) = t 1dr(t 1 ) t 2 dr(t 2 ) Non-infinitesimal shift Duration also noted as having key errors as shown in Figure 4.2 this is also illustrated [14] in Theorem 3. Theorem 3. A portfolio of positive payments with duration T and current value V equal to the present value of $1 in T years, will, after a non-infinitesimal shift of size δ in the entire yield curve r(t), be worth more than the new present value of $1 in T years. Proof. To prove this theorem first consider only separate portfolios of two separate payments at times t 1 and t 2 with t 1 < T < t 2. The general case follows immediately by induction. Consider two pure discount unit bonds of maturities t 1 and t 2. The prices of these bonds before and after the interest rate shift are denoted by V i = exp( r(t i )t i ) and V i = exp( r (t i )t i ) where r (t)t = r(t) + δ. Consider a portfolio holding n 1,n 2 > 0 of these bonds with current value V and duration D or and since n 1 V 1 + n 2 V 2 = V exp( r(t )T ), n 1 V 1 V t 1 + n 2V 2 V t 2 = T, n i V i = V (t j T ) (t j t i ), V i = V i exp(δt i ), (4.5) 31

34 the post shift value of the portfolio is V = n 1 V 1 + n 2 V 2 = n 1 exp( δt 1 ) + n 2 exp( δt 2 ), = V t 2 t 1 [(t 2 T )exp( δt 1 ) + (T t 1 )exp( δt 2 )], after substituting for n i V i. The post shift ratio of the value of this portfolio to that of the pure discount bond of maturity T is with V Q = exp( r (T )T ) = V exp(δt), V Evaluating this expression using Equation (4.5) gives Q = [(t 2 T )exp( δt 1 ) + (T t 1 )exp( δt 2 )] t 2 t 1, dq dδ = (t 2 t)(t t 1 ) [exp(δ(t t 1 )) exp(δ(t t 2 ))] >0 0< as δ >0 0< t 2 t, 1 d 2 Q dδ 2 = (t 2 t)(t t 1 ) [(T t 1 )exp(δ(t t 1 )) + (t 2 T )exp(δ(t t 2 ))] > 0. t 2 t 1 (4.6) From Equation (4.6) its observed that this ratio reaches a unique minimum at a zero shift where it has the value one. For any non zero shift, the portfolio value will be greater than that of the pure discount bond Flat yield curve and parallel shifts The weakness of Duration when dealing with non-parallel shifts and requiring a flat yield curve were brought to light by Theorem 4 [14]. Theorem 4. Yields to maturity on all assets with known fixed payments can change by same amount if and only if the yield curve flat (yields to maturity on pure discount bonds of all maturities are the same) and makes parallel shifts[6]. Proof. Let y(t) denote the yield to maturity on a pure discount bond of maturity t. Assuming that all yields do make an identical change on a sloped yield curve. Consider two maturities, t 1 and t 2 with differing yields. Without loss of generality take y(t 1 ) < y(t 2 ). Now consider an asset with payments at times t 1 and t 2 and no other payments, its yield to V = 2 i=1 CF(t i )exp( y(t i )t i )) = 2 i=1 CF(t i )exp( yt i )), (4.7) satisfies y(t 1 ) < y < y(t 2 ) since the exponential function is monotonic. If an infinitesimal shift of the assumed type occurs, then dy = dy(t 1 ) = dy(t 2 ). This interest rate shift causes a value 32

35 change of dv = 2 i=1 CF(t i )t i exp( r(t i )t i )dr(t i ) = Now multiplying V in Equation (4.7) and adding dv dy 2 i=1 CF(t i )exp( yt i ))dy. (4.8) from Equation (4.8) t 1 V dv dy = (t 1 t 2 )CF(t 2 )exp( y(t 2 )t 2 ) = (t 1 t 2 )CF(t 2 )exp( yt 2 ). This however can only be true if y(t 2 ) = y resulting in a contradiction. Thus yields to maturity cannot make identifiable shifts on all of an arbitrary set of assets if the yield curve is not flat. It is this case that this study takes primary interest in as the deficiencies relating to Theorems 2 and 3 are dealt with to some extent with the measure convexity. 4.5 Chapter Summary Duration is a widely used measure for risk which has good properties in terms of ease of calculation and use for large portfolios. It can also be a good tool for hedging risk. It s main downside is errors that occur due to parallel shifts and non-infinitesimal shifts. In the next chapter an alternative measure that deals with it s shortcoming relating to non-parallel shifts is discussed. 33

36 Chapter 5 Key Rate Duration Though Duration is a good measure of the interest rate risk exposure of a bond or a portfolio, it s main assumption is that the spot yield curve shift is parallel. This leads to nonparallel shifts, factors such as steepness or curvature can lead to estimation errors [11]. Key Rate Duration introduced by [11] is a vector representing the price sensitivity of a fixed income instrument to each key rate change. The key rates normally relate to the main quoted instruments examples being O/N, 1 month and 4 year. The sum of the key rate durations is identical to the duration. D = m i=1 KRD i. where m is the longest maturity normally 30 years. The principal of Key Rate Duration is that a chosen key rate at t i is shifted by y(t i ) but the key rate after t i+1 and before t i 1 do not shift. Figure 5.1 ([11]) shows how it would look and in some instances has been termed triangulation. Figure 5.1: Key Rate Duration Shift illustration The values of the rates t i 1 < t < t i and t i < t < t i+1 affected can then be computed using interpolation methods giving a more accurate view on the impact of a shift on that key rate. To compute KRD we first define s(t,t i ) as the ith basic key rate shift of term t, with the level of shift being y(t), these rate shifts are 1 i m. As stated earlier the summation of the shifts gives a value similar to duration and can be shown in Figure 5.2 [11]. 34

37 Figure 5.2: Key Rate Duration and Duration Shift relationship The computation is split into three scenarios 1. i = 1 the first key rate normally overnight rate y(t 1 ) if t < t 1, s(t,t 1 ) = y(t i ) t 2 t t 2 t 1 if t 1 t t 2, 0 if t > t < i < m these are all the key rates between the first and last y(t i ) if t < t i 1, y(t i ) t 2 t i 1 t s(t,t i ) = i t i 1 if t i 1 t t i, y(t i ) t i+1 t t i+1 t i if t i t t i+1, 0 if t > t i i = m the last key rate normally 30 year rate 0 if t < t m 1, s(t,t m ) = y(t m ) t t m 1 t m t m 1 if t m 1 t t m, y(t i ) if t > t m. If all these individual non-parallel shifts are done at the key rates the end result is y(t) = s(t,t 1 ) + s(t,t i ) s(t,t m ). 35

38 Now if when the relationship in Equation (4.3) is used the following is obtained The final result obtained is the following 5.1 Key Rate Convexities B i B = KRD i y(t i ). m B B = KRD i y(t i ). i=1 When a shift occurs to the term structure that is non-infinitesimal, the Key Rate Duration framework extends to deal with the second-order nonlinear effects of the key rate shifts [18]. These are given as key rate convexities and defined as KRC i, j = KRC j,i = 1 2 P P y(t i ) y(t j ), for every pair of key rates. The set of key rate convexities can be represented by a symmetric matrix of dimension m KRC 1,1 KRC 1,2... KRC 1,m KRC 2,1 KRC 2,2... KRC 2,m KRC =.... KRC m,1 KRC m,2... KRC m,m The relationship between convexity and key rate convexities is given by CON = m i=1 m KRC i, j. j=1 5.2 Key Rate Durations and Value at Risk Analysis Value at risk is defined as the maximum loss in the portfolio value at a given level of confidence over a given horizon. Given a multivariate normal distribution for the key rate changes, the portfolio return is distributed normally under a linear approximation, with a mean equal to [18] and variance equal to µ R = M i=1 KRD i µ y(i), σ 2 R = M i=1 M KRD i KRD j cov[ y(i), y( j)], (5.1) j=1 36

39 where µ y(i) is the mean change in the ith key rate and cov[ y(i), y( j)] is the covariance between changes in the ith and the jth key rates. The Value at Risk of the portfolio at c percent confidence level is given as VaR c = V 0 (µ R z c σ R ), where z c is c percentile of a standard normal distribution. If the holding period of the VaR is very small, the expected return and express VaR simply as VaR c = V 0 z c σ R, (5.2) substituting Equation (5.1) into Equation (5.2) VaR c = V 0 z c M M KRD i KRD j cov[ y(i), y( j)]. (5.3) i=1 j=1 The VaR solution given in Equation 5.3 doesn t not apply when key rate changes are not normally distributed. 5.3 Limitations of KRD Choice of Key Rates The Key Rate Duration model does not give guidance into which rates are viewed as Key Rates this then leads to varying numbers of key rates. In his article Ho[11] recommends using as many as 11 key rates, also risk managers can also base it on the maturity structure of the portfolio. In Sweden and in this study the key rates to be observed are: {1d,2d,3d,1W,1M,2M,3M,6M,9M,1Y,2Y,3Y,4Y,5Y,7Y,10Y,12Y,15Y,20Y,25Y,30Y } Shape of Key rates The shape of Key shifted curved is a historically implausible shape of Figure 5.3 shows an example of effect on forward rates that the key rate shift has. In order to address this shortcoming, a natural choice is to focus on the forward rate curve instead of the zero-coupon curve. This method is called the Partial Derivative Approach. This approach uses the relationship Z(t) = 1 t t exp( f (s)ds), As in the section dealing with interpolation it leads to Z(t) = 1 t 0 t f (i 1,i). i=1 37

40 Figure 5.3: Key Rate Duration Shift on spot curve Figure 5.4: Key Rate Duration Shift effect on forward curve 38

41 This leads to forward rates being the simple averages of the corresponding forward rates and imply that the present value of a cash flow due at time t is ( ) CF t PV = exp( t. i=1 f (i 1,i)) This equation therefore states that the market price is affected by all forward rates preceding the maturity date. The partial duration is then defined as ( PD i = 1 ) P, P f (i 1,i) with Duration = PD i = KRD i. Though giving similar outputs the profiles of the key rate duration and partial duration are noticeable different Loss of efficiency Since each key rate change is assumed to be independent of the changes in the rest of key rates, the model deals with movements in the term structure whose probabilities may be too small to worry about. Yet historical volatilities of interest rates provide useful information about the behaviour of the different segments of the term structure, and the key model disregards this information. The use of the key rate model for interest rate risk management imposes too severe restrictions on portfolio construction that leads to increased costs and a loss of degrees of freedom [18]. 5.4 Chapter Summary Key rate duration offers a good alternative to duration especially when relating to non-parallel shifts in the term structure. Key rate convexity provides additional accuracy when dealing with non-infinitesimal shifts. 39