Valuation in a world of CVA, DVA, and FVA a tutorial on debt securities and interest rate derivatives

www.ebook3000.com 10511_9789813222748_tp.indd 4/7/17 4:15 PM b2530   International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank www.ebook3000.com b2530_FM.indd 01-Sep-16 11:03:06 AM 10511_9789813222748_tp.indd 4/7/17 4:15 PM Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library VALUATION  IN  A  WORLD  OF  CVA,  DVA,  A ND  FVA A Tutorial on Debt Securities and Interest Rate Derivatives Copyright © 2018 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN 978-981-3222-74-8 ISBN 978-981-3224-16-2 (pbk) Desk Editor: Shreya Gopi Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore www.ebook3000.com Shreya - 10511 - Valuation in a World of CVA, DVA, and FVA.indd 06-07-17 3:42:46 PM July 6, 2017 14:3 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-fm page v Contents Introduction ix About the Author xvii Chapter I An Introduction to Bond Valuation Using a Binomial Tree I.1 I.2 I.3 I.4 I.5 Valuation of a Default-Risk-Free Bond Using a Binomial Tree with Backward Induction Pathwise Valuation of a Default-Risk-Free Bond Using a Binomial Tree Recommendations for Readers Study Questions Answers to the Study Questions Chapter II Valuing Traditional Fixed-Rate Corporate Bonds II.1 II.2 II.3 II.4 The CVA and DVA on a Newly Issued 3.50% Fixed-Rate Corporate Bond The CVA and DVA on a Seasoned 3.50% Fixed-Rate Corporate Bond The Impact of Volatility on Bond Valuation via Credit Risk Duration and Convexity of a Traditional Fixed-Rate Bond v 11 11 13 19 23 28 31 July 6, 2017 14:3 vi Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-fm page vi Valuation in a World of CVA, DVA, and FVA II.5 Study Questions II.6 Answers to the Study Questions Endnotes 39 40 44 Chapter III Valuing Floating-Rate Notes and Interest Rate Caps and Floors 47 III.1 III.2 III.3 III.4 CVA and Discount Margin on a Straight Floater A Capped Floating-Rate Note A Standalone Interest Rate Cap Effective Duration and Convexity of a Floating-Rate Note III.5 The Impact of Volatility on the Capped Floater III.6 Study Questions III.7 Answers to the Study Questions Endnotes 48 54 56 61 63 65 67 71 Chapter IV Valuing Fixed-Income Bonds Having Embedded Call and Put Options IV.1 IV.2 IV.3 Valuing an Embedded Call Option Calculating the Option-Adjusted Spread (OAS) Effective Duration and Convexity of a Callable Bond IV.4 The Impact of a Change in Volatility on the Callable Bond IV.5 Study Questions IV.6 Answers to the Study Questions Endnote 73 78 80 83 86 88 92 Chapter V Valuing Interest Rate Swaps with CVA and DVA V.1 V.2 V.3 V.4 A 3% Fixed-Rate Interest Rate Swap The Effects of Collateralization An Off-Market, Seasoned 4.25% Fixed-Rate Interest Rate Swap Valuing the 4.25% Fixed-Rate Interest Rate Swap as a Combination of Bonds www.ebook3000.com 73 93 94 103 106 111 July 6, 2017 14:3 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-fm page vii vii Contents V.5 Valuing the 4.25% Fixed-Rate Interest Rate Swap as a Cap-Floor Combination V.6 Effective Duration and Convexity of an Interest Rate Swap V.7 Study Questions V.8 Answers to the Study Questions Endnotes 114 119 127 127 136 Chapter VI Valuing an Interest Rate Swap Portfolio with CVA, DVA, and FVA Valuing a 3.75%, 5-Year, Pay-Fixed Interest Rate Swap with CVA and DVA VI.2 Valuing the Combination of the Pay-Fixed Swap and the Hedge Swap VI.3 Swap Portfolio Valuation Including FVA — First Method VI.4 Swap Portfolio Valuation Including FVA — Second Method VI.5 Study Questions VI.6 Answers to the Study Questions Endnotes 137 VI.1 138 142 145 150 155 155 161 Chapter VII Structured Notes VII.1 An Inverse (Bull) Floater VII.2 A Bear Floater VII.3 Study Questions VII.4 Answers to the Study Questions Endnote 163 163 172 178 179 182 Chapter VIII Summary 183 References 189 Appendix: The Forward Rate Binomial Tree Model Endnotes to the Appendix 193 206 b2530   International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank www.ebook3000.com b2530_FM.indd 01-Sep-16 11:03:06 AM July 6, 2017 14:3 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-fm Introduction The financial crisis of 2007–09 fundamentally changed the valuation of financial derivatives Counterparty credit risk became central Before September 2008, the thought of a major investment bank going into bankruptcy was unthinkable Post-Lehman, that risk is a critical element in the valuation process Bank funding costs rose dramatically during the crisis A proxy for bank funding and credit risk is the LIBOR-OIS spread (LIBOR is the London Interbank Offered Rate and OIS is the Overnight Indexed Swap rate) That spread was 8–10 basis points before the crisis, peaked at 358 basis points at the time of the Lehman default, and has since stabilized but still remains above the pre-crisis level In addition to recognizing the impact of credit risk and funding costs to banks, regulatory authorities since the crisis have imposed new rules on capital reserves and margin accounts This has led to a series of valuation adjustments to derivatives and debt securities, collectively known as the “XVA” These include CVA (credit valuation adjustment), DVA (debit, or debt, valuation adjustment), FVA (funding valuation adjustment), KVA (capital valuation adjustment), LVA (liquidity valuation adjustment), TVA (taxation valuation adjustment), and MVA (margin valuation adjustment) A problem, however, is that the models used in practice to calculate the XVA are very mathematical, and sometimes dauntingly so ix page ix July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app Appendix The Forward Rate Binomial Tree Model This book uses a one-factor binomial forward rate tree model for the term structure of interest rates to illustrate the impact of CVA, DVA, and FVA on the valuation of debt securities and derivatives The one factor is the short-term interest rate That means that all interest rate volatility is realized through changes in the shortterm rate To keep the model simple, this is assumed to be the 1-year rate and the underlying benchmark bonds are assumed to make annual coupon payments In particular, the forward rate trees are based on the Kalotay-Williams-Fabozzi (KWF) model.1 Buetow and Sochacki (2001) provide a detailed review of this and other one-factor term structure models, including the Ho-Lee, Hull-While, Black-Karasinski, and Black-Derman-Toy models The KWF model is chosen for this introduction to valuing risky bonds and derivatives because it is familiar to many finance professionals in that it has been used in the CFA R (Chartered Financial Analyst) examination readings since 2000 Exhibit A-1 displays the primary binomial tree that is used The initial 1-year rate on Date is 1.0000% At the end of the first year on Date 1, the 1-year rate for the second year will be either 3.6326% or 2.4350% An important feature to this model is that the odds of the rate going up or down are assumed to be 50-50 for all nodes in the tree Below each forward rate in parenthesis is the probability of attaining that particular node in the tree On Date at the end of the second year, the possible 1-year forward rates are 5.1111%, 3.4261%, 193 page 193 July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app Valuation in a World of CVA, DVA, and FVA 194 Exhibit A-1: Binomial Forward Rate Tree for 20% Volatility Date Date Date Date Date 8.0842% (0.0625) 6.5184% (0.1250) 5.1111% (0.2500) 3.6326% (0.5000) 1.0000% (1.0000) 5.4190% (0.2500) 4.3694% (0.37500) 3.6324% (0.3750) 3.4261% (0.5000) 2.4350% (0.5000) 2.9289% (0.3750) 2.2966% (0.2500) 2.4349% (0.2500) 1.9633% (0.1250) 1.6322% (0.0625) Expected Forward Rate 3.0338% 3.5650% 3.7971% 3.9329% and 2.2966% with probabilities of 0.25, 0.50, and 0.25, respectively This is a recombinant tree in that the 3.4261% outcome could have been obtained from the 1-year rate initially having gone up or down [Note that “up” and “down” refer to the relative movement at each node without regard to the actual change in the interest rate.] This forward rate tree is derived from an underlying sequence of arbitrarily chosen annual coupon payment, risk-free benchmark government bonds “Risk-free” here is with respect to default risk but not inflation risk—presumably the government issuer can and will print money to assure that there is no default to the bondholder The coupon rates and prices for these bonds are given in Exhibit A-2 www.ebook3000.com page 194 July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app Appendix page 195 195 Exhibit A-2: Underlying Benchmark Coupon Rates, Prices, and Yields to Maturity Date Coupon Rate Price Yield to Maturity 1.00% 2.00% 2.50% 2.80% 3.00% 100 100 100 100 100 1.00% 2.00% 2.50% 2.80% 3.00% This is the par curve for the benchmark bonds out to five years in that each bond is priced at par value so that the coupon rate equals the yield to maturity Also, there is no accrued interest because the time to maturity is an integer The next step is to bootstrap the discount factors corresponding to each date A discount factor is the present value of one unit of currency received at a future date The Date-1 discount factor, denoted DF1 , is simply 1/1.010000 = 0.990099 because the 1-year rate is 1.0000% The bootstrapping technique requires an initial zerocoupon bond to launch the procedure Typically, this comes from the money market, for instance, the yield on a short-term Treasury bill Here the 1-year annual payment bond implicitly provides the starting point The Date-2 discount factor is the solution for DF2 in this equation: 100 = (2 ∗ 0.990099) + (102 ∗ DF2 ), DF2 = 0.960978 Note that the future cash flows on the 2-year, 2% bond are and 102 and the price is 100 Similarly, the Date-3 discount factor is the solution for DF3 , whereby the results from the previous steps are used as inputs — that is the hallmark of bootstrapping The cash flows on the 3-year, 2.5% bond are 2.5, 2.5, and 102.5 and the price is again 100 100 = (2.5 ∗ 0.990099) + (2.5 ∗ 0.960978) + (102.5 ∗ DF3 ), DF3 = 0.928023 July 6, 2017 14:13 196 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app page 196 Valuation in a World of CVA, DVA, and FVA To generalize, the discount factor for the nth date (DFn ) is: DFn = − CRn ∗ n−1 j=1 DFj + CRn (A1) CRn is the coupon rate for the n-year bond on the par curve The implied spot (or zero-coupon) rates that correspond to the discount factors are easily calculated The “0× n” spot rate, meaning a rate that starts on Date and ends on Date n and is denoted Spot0,n , is calculated using this formula: Spot0,n = DFn 1/n −1 (A2) For example, the 3-year (“0 × 3”) implied spot rate is 2.5212% Spot0,3 = 0.928023 1/3 − = 0.025212 The implied spot rates are used to validate the accuracy of the valuations produced using the binomial tree model and to calculate spreads over the benchmark rates The implied forward, or projected, rates are also calculated from the series of discount factors For the forward rate between Date n−1 and Date n, denoted Forwardn−1,n , this formula is used: Forwardn−1,n = DFn−1 −1 DFn (A3) As an example, the 1-year rate, two years forward, is 3.5512% It is the 1-year rate between Dates and It can be called the “2 × 3” forward rate; in practice, this rate sometimes is designated the “2y1y” rate Forward2,3 = 0.960978 − = 0.035512 0.928023 These forward rates are the baseline around which the binomial tree is built.2 The series of discount rates, spot rates, and forward rates are shown in Exhibit A-3 www.ebook3000.com July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app Appendix page 197 197 Exhibit A-3: Discount Factors, Spot Rates, and Forward Rates Time Frame Discount Factor Spot Rate 0×1 0×2 0×3 0×4 0×5 0.990099 0.960978 0.928023 0.894344 0.860968 1.0000% 2.0101% 2.5212% 2.8310% 3.0392% Time Frame Forward Rate 0×1 1×2 2×3 3×4 4×5 1.0000% 3.0303% 3.5512% 3.7658% 3.8766% Before describing the process by which the binomial tree is derived, notice that for each date the implied forward rates in Exhibit A-3 are lower than the expected forward rates in Exhibit A-1 The 1-year implied forward rate for the timespan between Date and Date is 3.8766%, whereas the expected rate given the five possible outcomes and their probabilities is 3.9329% (8.0842% ∗ 0.0625) + (5.4190% ∗ 0.2500) + (3.6324% ∗ 0.3750) + (2.4349% ∗ 0.2500) + (1.6322% ∗ 0.0625) = 3.9329% This difference arises because the KWF model assumes a log-normal distribution for the forward rate This assumption alleviates the problem of negative nominal rates present in some term structure models (e.g., Ho-Lee and Hull-White) that use a normal distribution for the rates Log-normality means that the percentage change in rates is normally distributed Therefore, the rates in the tree for each date are not distributed symmetrically around the forward rate When volatilty is increased, the rates at the top of the tree go up more than the rates at the bottom go down July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app Valuation in a World of CVA, DVA, and FVA 198 The log-normality assumption is particularly important when interest rates are at historic lows While negative real rates can and occur, it is not common (but not impossible as we have seen recently in some Asian and Eurozone markets) for nominal rates to go below zero, especially on uncollateralized interbank borrowing and lending rates such as LIBOR In any case, it is possible to adjust the model to allow rates to become negative but that goes beyond this introduction to valuation The binomial forward rate tree is designed to branch out around the implied forward rate for each date This is illustrated in Exhibit A-4; the dotted line is the forward curve For example, the Date-1 rates of 3.6326% and 2.4350% are above and below the “1 × 2” implied forward rate of 3.0303% How much above and below depends critically on the assumed annual volatility This tree assumes a constant annual standard deviation of 20% for each year The KWF model is among the class known as arbitrage-free term structure models because it is calibrated to assure that the valuation obtained for each benchmark bond matches the market price, which is assumed to be par value.3 Using the log-normal distribution for Exhibit A-4: The Binomial Tree Branches Out around the Forward Curve Forward Rates 8% 7% 6% 5% Forward Curve 4% 3% 2% 1% www.ebook3000.com Dates page 198 July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in Appendix b2856-app page 199 199 the forward rate, the initial trials for Date-1 rates moving up and down are given by: 3.0303% ∗ exp − 0.04 + 0.20 = 3.0303% ∗ 1.197217 = 3.6279% 3.0303% ∗ exp − 0.04 − 0.20 = 3.0303% ∗ 0.802519 = 2.4319% The general expression for this pattern, whereby σ is the assumed annual standard deviation for the forward rate and T is the relevant time period, is: Forward Rate ∗ exp − √ σ2 ∗ T ±σ∗ T (A4) In this simplified example, σ is 0.20 and T is These initial rates of 3.6279% and 2.4319% are then tested and calibrated using the backward induction method of valuation to eliminate any arbitrage opportunity This is illustrated in the upper panel to Exhibit A-5 Notice that the scheduled cash flows on the 2-year benchmark bond (per 100 of par value) are placed directly across from the node on the binomial tree The first coupon payment of on Date is across from the Date-0 rate of 1.0000% The final coupon payment and principal redemption for a total of 102 are across from the two possible Date-1 forward rates If the 1-year rate goes up to 3.6279% from 1.0000%, the value of the bond on Date is 98.4291 (= 102/1.036279) If the rate goes “down” to 2.4319%, the bond value is 99.5784 (= 102/1.024319) The next step is critical to understanding how backward induction works The key assumption is that the value of the bond on Date is the present value of the expected value on Date 1, assuming at the probability of the rate going up is 0.50 and probability of rate going down is also 0.50 Those 50-50 odds make the calculations easy — that is why the discrete version of the KWF binomial model is so useful for pedagogy The expected value of the bond on Date is [2 + (0.50 ∗ 98.4291 + 0.50 ∗ 99.5784)], the scheduled coupon payment plus the probability-weighted average of the two possible prices Discounting that amount back to Date using the 1-year July 6, 2017 14:13 200 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app Valuation in a World of CVA, DVA, and FVA Exhibit A-5: Calibrating the Forward Rates on the Binomial Tree for Date Upper Panel: The Initial Test Date Date 98.4291 3.6279% 100.0037 1.0000% 102 99.5784 2.4319% Calculations: Date 102 = 98.4291 1.036279 102 102 = 99.5784 1.024319 [2 + (0.50*98.4291 + 0.50*99.9784)] = 100.0037 1.010000 Lower Panel: The Final Calibration Date 100.0000 1.0000% Date Date 98.4246 3.6326% 102 99.5753 2.4350% Calculations: 102 = 98.4246 1.036326 102 102 = 99.5753 1.024350 [2 + (0.50*98.4246 + 0.50*99.9753)] = 100.0000 1.010000 rate of 1.0000% gives 100.0037 [2 + (0.50 ∗ 98.4291 + 0.50 ∗ 99.5784)] = 100.0037 1.010000 The problem is that this price is too high because the 2-year bond needs to be priced at par value (specifically, 100.0000) to meet the no-arbitrage condition www.ebook3000.com page 200 July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app Appendix page 201 201 These two trial forward rates for Date need to be increased a bit to get the present value down to 100.0000 But, importantly, they need to be raised proportionately to preserve constant volatility Given log-normality and the assumed annual standard deviation of 20%, the ratio of the up and down rates emanating from each node in the tree is 1.491825 The general relationship is: exp − σ exp − σ ∗T 2 ∗T +σ∗ −σ∗ √ √ exp − σ T = T exp − σ ∗T 2 ∗T = exp(2 ∗ σ ∗ ∗ exp(σ ∗ √ T) √ ∗ exp(−σ ∗ T ) √ T) (A5) For σ = 0.20 and T = 1, exp(0.40) = 1.491825 The calibration process entails raising the rates by a small amount until the price rounds off to 100.0000 while keeping the proportionality factor This final result is shown in the lower panel of Exhibit A-5 — the two possible rates for the 1-year bond on Date turn out to be 3.6326% and 2.4350% [Note that all calculations are done on a spreadsheet to preserve precision and the rounded values are shown for consistency in presentation.] Now the model builder moves out to Date This is displayed in Exhibit A-6 The three initial forward rates in upper panel are 5.2978%, 3.5512%, and 2.3804% The middle rate is the implied forward rate between Dates and given in Exhibit A-3 The first and third rates are set to preserve constant 20% volatility: 3.5512% ∗ 1.491825 = 5.2978% and 3.5512%/1.491825 = 2.3804% Working through the tree using backward induction produces a price on the 3-year, 2.50% annual coupon payment bond of 99.8814, which is below the no-arbitrage target price of 100.0000 Lowering the initial rates proportionately by trial and error (or using Solver in Excel) eventually leads to a price that rounds to 100.0000 Those rates, shown in the lower panel of Exhibit A-6, are 5.1111%, 3.4261%, and 2.2966%.a a Because the rates are rounded to four digits, the ratio between adjoining rates is not always exactly equal to 1.491825 For instance, 3.4261%/2.2966% = 1.491814 July 6, 2017 14:13 202 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app page 202 Valuation in a World of CVA, DVA, and FVA Exhibit A-6: Calibrating the Forward Rates on the Binomial Tree for Date Upper Panel: The Initial Test Date Date 99.8814 1.0000% Date Date 97.3430 5.2978% 102.5 97.1354 3.6326% 2.5 2.5 98.9849 3.5512% 99.6250 2.4350% 102.5 2.5 100.1168 2.3804% 102.5 Lower Panel: The Final Calibration Date Date Date 97.5160 5.1111% 97.2766 3.6326% 100.0000 1.0000% 2.5 99.7234 2.4350% Date 102.5 2.5 99.1046 3.4261% 102.5 2.5 100.1988 2.2966% 102.5 The next step is to get trial rates for Date 3, taking the calibrated Date-1 and Date-2 rates (rounded to four decimals) as inputs The general pattern is to use the implied forward rate for the middle rate when the number of nodes for the date is odd, for instance, three nodes for Date and five nodes for Date Then the other trial rates are set using the proportionality factor to preserve constant volatility www.ebook3000.com July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in Appendix b2856-app page 203 203 When the number of nodes is even, for instance, two nodes for Date and four nodes for Date 3, the implied forward rate is adjusted up or down, as demonstrated above for the Date-1 calibration Exhibit A-7 shows the initial test and final calibration trees for Date The initial rates are 6.7258%, 4.5084%, 3.0221% and 2.0258% These spread out around the “3 × 4” forward rate of 3.7658%, which is shown in Exhibit A-3 The middle two rates are calculated as for Date 1: 0.04 + 0.20 0.04 − 0.20 3.7658% ∗ exp − 3.7658% ∗ exp − = 3.7658% ∗ 1.197217 = 4.5084% = 3.7658% ∗ 0.802519 = 3.0221% The outer two rates follow from these: 4.5084% ∗1.491825 = 6.7258% and 3.0221%/1.491825 = 2.0258%, using the proportionality factor for 20% volatility The trial rates produce a value of 99.8944 for the 2.80%, 4-year benchmark bond After lowering them a bit — and keeping them proportional — the final rates of 6.5184%, 4.3694%, 2.9289%, and 1.9633% result in a value of 100.0000 for the bond The initial test rates for Date start with the “4 × 5” forward rate of 3.8766% These are shown in Exhibit A-8 The rates above and below preserve proportionality: 3.8766% ∗ 1.491825 = 5.7831%, 5.7831% ∗ 1.491825 = 8.6274%, 3.8766%/1.491825 = 2.5985%, and 2.5985%/1.491825 = 1.7419% [Note: the spreadsheet behind these numbers uses the full precision proportionality factor, exp(0.40)] The value for the 3%, 5-year benchmark bond turns out to be 99.7795 Trial-and-error search leads to Date-4 rates of 8.0842%, 5.4190%, 3.6324%, 2.4349%, and 1.6322% These value the benchmark bond at par value This “artisanal” approach to model-building, in particular, bootstrapping the forward curve and calibrating the rates in the tree year by year, is intended to reinforce the idea of no arbitrage In practice a computer model can be built to get the tree instananeously and without rounding the rates to four digits In any case, once the tree is developed, it can be used in applications other than valuation For example, the sensitivity of the total return facing a buy-and-hold July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app page 204 Valuation in a World of CVA, DVA, and FVA 204 Exhibit A-7: Calibrating the Forward Rates on the Binomial Tree for Date Upper Panel: The Initial Test Date Date Date Date 96.3216 6.7258% 95.2739 5.1111% 2.8 98.3653 4.5084% 2.8 98.5001 3.4261% 2.8 99.9940 2.4350% 2.8 96.1927 3.6326% 99.8944 1.0000% 100.7576 2.2966% 99.7844 3.0221% Date 102.8 102.8 102.8 2.8 100.7589 2.0258% 102.8 Lower Panel: The Final Calibration Date Date Date 95.4254 5.1111% 96.3175 3.6326% 100.0000 1.0000% 2.8 100.0826 2.4350% 98.6072 3.4261% 2.8 100.8320 2.2966% Date Date 96.5092 6.5184% 102.8 2.8 98.4963 4.3694% 2.8 99.7844 2.9289% 102.8 2.8 100.8206 1.9633% www.ebook3000.com 102.8 102.8 July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app Appendix page 205 205 Exhibit A-8: Calibrating the Forward Rates on the Binomial Tree for Date Upper Panel: The Initial Test Date Date Date Date 93.0302 6.5184% 93.2601 5.1111% 95.2095 3.6326% 99.7795 1.0000% 100.3452 2.4350% 98.0761 3.4261% 97.0232 4.3694% Date Date 94.8195 8.6274% 103 97.3690 5.7831% 103 99.1562 3.8766% 99.8492 2.9289% 101.5011 2.2966% 100.3913 2.5985% 101.8150 1.9633% 101.2366 1.7419% 103 103 103 Lower Panel: The Final Calibration Date Date Date Date Date 95.2961 8.0842% 93.4118 6.5184% 93.5715 5.1111% 94.4681 3.6326% 100.0000 1.0000% 100.5319 2.4350% 98.3006 3.4261% 97.2963 4.3694% Date 103 97.7053 5.4190% 103 99.3898 3.6324% 100.0406 2.9289% 101.6592 2.2966% 100.5517 2.4349% 101.9472 1.9633% 101.3458 1.6322% 103 103 103 July 6, 2017 14:13 206 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app Valuation in a World of CVA, DVA, and FVA investor in the 5-year benchmark bond to coupon reinvestment risk can be analyzed along each possible path For example, suppose that the 1-year rate tracks the topmost path in the tree so that coupon payments can be reinvested each year at 3.6326%, 5.1111%, 6.5184%, and 8.0842%, respectively The 5-year total return is 117.0891 per 100 of par value ((((((3 ∗ 1.036326) + 3) ∗ 1.05111) + 3) ∗ 1.065184) + 3) ∗ 1.080842 + 103 = 117.0891 The 5-year holding-period rate of return is 3.2056% along this particular path, found as the solution for “rate” in this expression: 100 = 117.0891 , (1 + rate)5 rate = 0.032056 This is 20.56 basis points higher than the 3.00% yield to maturity on the 5-year benchmark bond However, the probability of realizing this path is only 0.0625, as shown in Exhibit A-1 The range of possible total returns that are consistent with the model and the assumption of 20% volatility can calculated in the same manner In summary, the KWF model used in this tutorial is a one-factor model of the term structure of interest rates Movement in the shortterm rate, here the 1-year rate, is only factor driving volatility in interest rates The model assumes constant volatility in that the up and down rates branching out from each node maintain the same ratio The model also assumes that the forward rates follow a lognormal distribution That assumption means that the expected forward rates are above the implied forward rates Finally, the KWF model assumes no arbitrage in that it is calibrated to price correctly the underlying benchmark bonds Endnotes to the Appendix The original source for the KWF model is a 1993 article by Kalotay, Williams, and Fabozzi in the Financial Analysts Journal It is described and used in various Fabozzi textbooks, including Fixed Income Analysis and Fixed Income Markets, Analysis, and Strategies, as well as in a 2001 practitioner-oriented textbook by Finnerty and Emery Another useful www.ebook3000.com page 206 July 6, 2017 14:13 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-app Appendix page 207 207 resource is a 2007 book by Miller, which describes the version of the model that is used by Bloomberg Equations (A1), (A2), and (A3) are simplified because of the annual coupon payment assumption These are the general formulas whereby Aj and An are the day-count fractions for the jth and nth period: DFn = Spotn = Forwardn−1,n = − CRn ∗ n−1 j=1 DFj ∗ Aj + CRn ∗ An DFn 1/n −1 ∗ An DFn−1 −1 ∗ DFn An For example, if the underlying bonds make semiannual payments and the day-count convention is actual/actual, Aj might be 181/365 and An 184/365 In practice, daily discount factors are needed to value the multitude of debt securities and derivatives that are on a financial institution’s balance sheet, thereby requiring interpolation between rates and prices on observed bonds Tuckman and Serrat (2012) provide a thorough discussion of term structure models, the differences between equilibrium and arbitrage-free models, and the multi-factor models that are used in practice, including the LIBOR Market Model ... 14:4 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-ch01 Valuation in a World of CVA, DVA, and FVA 12 Exhibit I-7: Pathwise Valuation of a 5-Year, 1.50%, Annual Payment Bond Path Date... 14:4 Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-ch01 Valuation in a World of CVA, DVA, and FVA All calculations in this book are done on an Excel spreadsheet and the rounded values... Valuation in a World of CVA, DVA, and FVA - 9in x 6in b2856-ch01 Valuation in a World of CVA, DVA, and FVA coupon and principal payments to the right of the nodes in another color, and the calculated