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  1. 03-9&-10Another Financial Engineering Metaphor: Creating Stockholder Value and Managing Enterprise Risk by Synthetic Balance-Sheet Surgery Edward J. Kane Boston College Edward J. Kane, BC, 03-9&10

  2. Fully Integrated Process of FSF Risk Management Risk Analysis Risk Adjustment IRR & Credit Risk Appraisal Originating & structuring deals Info. Services Edward J. Kane, BC, 03-9&10

  3. Financial Engineering Can Simplify the Calculation of Duration-Based Measures of the IRR in any Instrument or Position. • Begin by stripping coupon bond into two parts whose prices sum to the price of the bond: • A single-payment or zero-coupon principal repayment of F to be made at maturity n. Its PDV is =PDVF. • Coupons can be analyzed as an n-year annuity. Value is the product of the annual coupon C and the PDV of an n-period unit annuity: Edward J. Kane, BC, 03-9&10

  4. The duration of a n-period bond DB can be found as the PDV-weighted sum of the durations of the bond’s two components: Query: What makes this a “financial engineering approach” to calculating DB? ANS. It relies on the concept of synthetic replication and the Law of One Price. Edward J. Kane, BC, 03-9&10

  5. DF is trivially equal to n. • Dn is the same as the PV-weighted duration of the n-year “unit annuity” because C can be factored out of each payment. Dn = (4.a) Edward J. Kane, BC, 03-9&10

  6. Brute Force Approach to First Duration Exercise: Edward J. Kane, BC, 03-9&10

  7. It is much easier to use the financial-engineering method to solve our first duration exercise (C=$60, F=$1,000, R=.10, n=10) than to work the problem by the formula that defines duration conceptually: PVF = 385.54 PVn = 368.67 = = 600[1-.3855] PVB = $754.21 Edward J. Kane, BC, 03-9&10

  8. Financial-Engineering Formulas That Might Be Used to Calculate Duration on Exams Edward J. Kane, BC, 03-9&10

  9. PRACTICAL USES OF D* AND pvbp In practice, asset-liability managers no longer need to make tedious calculations. Portfolio valuations and duration/convexity calculations are routinely produced by computer software. • The link between decision-support technology and information management and contracting at FSFs is becoming stronger day by day. • Regulations requiring lazy managers to adopt a benchmark IRR information system help to protect FSF stockholders from managerial incentive (i.e., ethical) conflict. Edward J. Kane, BC, 03-9&10

  10. Because the durations of instruments and portfolios vary as interest rates change, “static” control strategies are inadequate. • It is a disastrous mistake to suppose that the duration of an institution’s net worth is a constant that managers may set once a quarter or so and then forget about. [• Reminder: IRR management may be likened to a householder’s trimming the hedges on the boundaries of its yard. The managerial implications of this metaphor turn on the need to expect “rain” (=adversity or good luck) and to plan to respond to the amount of rain that falls. This trimming must be planned to occur from time to time, and to occur more frequently the more it rains.] Edward J. Kane, BC, 03-9&10

  11. Regular readjustment of hedges resembles "trimming." Continual realignment of the durations of the two sides of an FSF’s balance sheet is called “dynamic hedging.” • Dynamic hedging seeks to offset changes in durations of A and L as R changes and options are exercised. Contrast with mindless once-and-for-all or passive “static” strategies for coping with interest-rate change. • Difference lies between totally controlling and partially and temporarily managing interest-rate risk exposure. • Dynamic Hedging is indispensable because an institution cannot afford not to “sell” hedges to its customers. Edward J. Kane, BC, 03-9&10

  12. Both Voluntary vs. Involuntary Portfolio Mismatching Occur • Customers routinely hold valuable imbedded options that let them retime loan and deposit contracts: e.g., for early loan repayment, for refinancing, for early deposit withdrawal, and for activating credit lines; for taking down annuities or policy loans in life insurance contracts. The values and durations of these options change with interest rates. • Customers pay handsomely for optionality. Compensation paid for optionality is a profitable part of almost every deal a bank writes. • Customers often fail to appreciate the cost of these options. • FSFs often make options troublesome to exercise. Edward J. Kane, BC, 03-9&10

  13. The better proprietary FSF models of IRR “internally” adjust “interest spreads” to price the optionality that imbedded puts and calls conveys to its customers. • Maturity of a passbook is not really zero. Maturity is entirely at the customer’s option. The speed of customer deposit runoff caused by rising market interest is not fixed. It depends on how quickly an FSF resets its yields when and as market interest rates rise. • Correspondingly, on the other side of an FSF balance sheet, deposit flow is also an endogenous variable in the system. The SPEED of loan runoffs rises when interest rates fall. Edward J. Kane, BC, 03-9&10

  14. OPTIONALITY AS A SOURCE OF CONVEXITY • FSF managers and regulators know that DA and DL change with interest rates, not just because the PDV of given positions change, but because interest-sensitive customers can alter these positions by prepayments, deposit flows, and loan requests that activate implicit or explicit credit lines. • Long-run survival in a repeat business forces FSFs to accept customer-initiated variation in the timing of funds flows. Edward J. Kane, BC, 03-9&10

  15. Optionality is increasingly recognized as creating by itself a negative “asset convexity” and positive “liability convexity.” • Increases in R lower values of “borrower-callable” assets but leave values of “depositor-puttable” liabilities (as approximate “par floaters”) more or less unchanged. • Decreases in R do what to same values? Edward J. Kane, BC, 03-9&10

  16. WE CAN ONLY APPROXIMATE PERCENTAGE BOND-PRICE CHANGES WITH DURATION Edward J. Kane, BC, 03-9&10

  17. More on Convexity Setting an appropriately weighted gap = 0 serves to “immunize” NW, but only for infinitesimally tiny changes in r. “Infinitesimal” means “negligible in size.” The criterion that defines an infinitesimally “immunized” state is that the first derivative = zero . • If the second derivative, , is positive or negative, a “convexity” in interest sensitivity is said to exist. The larger the absolute value of convexity, the more the (N,rA) curve departs from a straight line. • Even higher derivatives may be dealt with by software.] [ Edward J. Kane, BC, 03-9&10

  18. Why is Convexity Helpful? 1. Strategically: Using it helps to immunize Net worth against large movements in r. • Substantively: In two positions, the algebraically more convex position becomes shorter faster for a given rise in r and becomes longer faster for a given fall in r. Query: For what movement in interest rates are liability positions more convex than asset positions and vice versa? Edward J. Kane, BC, 03-9&10

  19. Importance of Convexity Grows with D of Position: Comparing Two Money-Market Funds Please calculate how much a 2% fall in yield will raise the value of a “dollarsworth” of each fund: dP-D*dr+(.5)(convexity)(dr)2 Edward J. Kane, BC, 03-9&10

  20. Self-StudyA Last Complication in Calculating D For a real-world bank, RA and RL would actually be weighted-average rates of return. The response of rates of return on individual assets and liabilities to movements in a reference “market” rate R need not be uniform either in sign or in magnitude. Assuming a uniform response keeps calculations simple. Edward J. Kane, BC, 03-9&10

  21. Swap: A particular kind of forward contract between two counterparties Interest-Rate Swap: An agreement to exchange the coupon interest flows from two different hypothetical or “notional” instruments over a series of future settlement dates. Each coupon-swap agreement creates synthetically a financial instrument that could not otherwise be traded in financial markets. Aim is: • to reduce borrowing costs • to hedge interest-rate risk, or • to speculate (i.e., “gamble”) on the future course of interest rates. Edward J. Kane, BC, 03-9&10

  22. The maturity of a swap iscalled its “tenor.” • The usage traces to tenor’s Latin meaning as a “course of continuous or uninterrupted process.” • Two nonfinancial meanings of tenor: Besides its additional meaning as a “voice quality,” tenor can also mean the “subject of a metaphor.” Edward J. Kane, BC, 03-9&10

  23. More Swaps Terminology: • Notional Value = assumed face amount P used by contract in translating contract interest rates into cash flows; • Selling on market or at-the-market =both halves have equal value: no premium or discount; • Selling off-market = instances where obligations imposed by swap are not equally valuable at current interest rates. E.g., the PF and RF values might be substantial, when PV and RV are not. [• Rates on Reference Securities.] Edward J. Kane, BC, 03-9&10

  24. Typically, the notional instruments on which swaps are based have an exact or rough cash-market counterpart. The different notional instruments have the same maturity and differ as to whether the contract interest rate on the instrument is fixed (Rt) or floating ( ). • The party that is long the fixed-rate obligation is usually of higher credit standing than the short. This side Pays Fixed, Receives Variable = PFRV or RVPF. • RxPz Notation mimics the order of a balance sheet. • Cash Settlement: only the net cash-flow difference is paid on any settlement date. Why is this efficient? Edward J. Kane, BC, 03-9&10

  25. Let P be the “notional principal” that is used to calculate the interim cash flows. P need not be precisely the same as the principal on the underlying cash instruments. Rather in an OTC market it is a negotiated variable that can be used to equalize the initial market value of the two sides of the swap. • At each settlement date (typically every six months), the floating rate is ordinarily reset, although the settlement and reset dates are in principle contracting variables. The reset makes the payment due at the next reset date knowable in advance = “nonstochastic.” • The check written for the difference at each settlement date is called the difference check. The payline equals the absolute value of ( - R) P. • At some dates, the payoffs go from the fixed side to the floating side; at other dates, funds flow the other way. Edward J. Kane, BC, 03-9&10

  26. Financial-Engineering Insight: Swaps are synthetic substitutes for financial intermediation which is an ancient financial-engineering substitute for direct finance. • Opportunity Costs of contracting for a swap parallel the three benchmarks we identified in comparing the costs of direct and indirect finance. 1) Expenses of shopping for best deal 2) Expenses of Due Diligence 3) Contracting and Enforcement Expense Edward J. Kane, BC, 03-9&10

  27. Intuitive Perspective • FSF managers may use synthetic transactions to hedge or transfer unwanted business or financial risks. The trick is to strip and sell off cash-flow outcomes managers do not want to retain: as entailed by the surgery metaphor. • Accommodating customers’ interest in shedding or acquiring particular risks is the central idea in making a market in derivatives and in writing insurance policies. Edward J. Kane, BC, 03-9&10

  28. As an incremental balance sheet, every interest-rate swap has two parts: a “pay half” and a “receive half.” Each side accepts an obligation and receives a claim to something valuable: -- PFRV: Pay Fixed Rate, Receive Variable Rate -- PVRF: Pay Variable Rate, Receive Fixed Rate [Concept of a “half-swap”: “Pay half”  an on-balance-sheet liability; “Receive half “ an on-balance-sheet asset] Edward J. Kane, BC, 03-9&10

  29. What is the PDV of owning both sides of an interest-rate swap? ANS. Net flows constitute a string of zero payments. • Why not describe the swap as paying and receiving “fixed” and “floating?” ANS. The acronyms for both sides would be identical: RFPF and RFPF. • If value of RFPV side is zero, why must the value of the RFPV side also be zero?ANS. Abstracting from dealer market-making “vigorish,” the value of each side equals minus the value of the other. Edward J. Kane, BC, 03-9&10

  30. Standard RFPV Swaps: Example July 1, 1998 initiate fixed-for-floating interest rate swap with notional principal of $1,000,000 and tenor of 2 years January 1, 1999 RECEIVE 5% per annum fixed rate PAY OUT July 1, 1998 six-month LIBOR rate (4.5%) July 1, 1999 RECEIVE 5% per annum fixed rate PAY OUT six-month LIBOR rate set January 1, 1999 January 1, 2000 RECEIVE 5% per annum fixed rate PAY OUT six-month LIBOR rate set July 1, 1999 July 1, 2000 RECEIVE 5% per annum fixed rate PAY OUT six-month LIBOR rate set January 1, 2000 Edward J. Kane, BC, 03-9&10

  31. Hypothetical Cash Flows from Plain-Vanilla RFPV Interest Rate Swap for Firm ANotional Principal = $1,000,000, Tenor = 2 years Why the lag in the pay floating rate? Reduce uncertainty of PV How does the “1/2” figure in? (semiannual settlement) Edward J. Kane, BC, 03-9&10

  32. Standard Swaps with Market Maker* Pays LIBOR + Pays LIBOR + Firm A Bank Firm B • Bank acts as a market-maker, providing “immediacy” by simultaneously borrowing and lending at LIBOR, and borrowing at a fixed rate but lending at this fixed rate plus about 3 basis points. • In exchange for the spread, the bank performs services. It absorbs default risk of the counterparties and may temporarily warehouse either side of the swap. Receives 4.985% Receives 5.015% * Arrows point to receiver and away from payer Edward J. Kane, BC, 03-9&10

  33. Indicative Pricing Schedule Banks Charge RFPV and RVPF Counterparties Note: TN stands for “Treasury Note” Edward J. Kane, BC, 03-9&10

  34. Performance Risk in Swaps: Role of swapbrokers and dealers: • Most swaps are arranged by a third party called a dealer that accepts the nonperformance risk by taking other sides of each counterparty’s position and undertakes the search for a matching counterparty. • Dealerstake a spread which they load into the fixed-rate side of each swap (about 3 b.p.). • Vs.Brokers, who perform their search function only for a commission & take no risk. Edward J. Kane, BC, 03-9&10

  35. Issuer Perspectives on the Pay Halves: The issuer of an FR bond is obligated to pay a fixed stream of coupons; the VR issuer accepts a PV obligation. • How does a fixed-rate (FR) bond of maturity n differ from a VR bond of the same maturity? ANS: By the “difference” being swapped. • A swap transaction exchanges the value of the physical coupons issuers owe on a FR and a VR Bond. The FR coupon sizes the PF obligation FR coupon obligation + RFPV=VR obligation VR coupon obligation + RVPF=FR obligation Edward J. Kane, BC, 03-9&10

  36. Bondowner Perspectives on the Receive Halves: Bond owners could equally well swap coupons: FR claim + PFRV=RV claim VR claim + PVRF=RF claim [FR coupon is a Pay-Fixed obligation of the Issuer. What is a RV coupon?] Edward J. Kane, BC, 03-9&10

  37. -- This exercise shows that Interest Swaps may be interpreted as a synthetic “missing link” between FR and VR instruments: completes these two markets. --Any VR bond can be decomposed logically into the sum of a FR bond and an appropriate swap. -- The parallel decomposition of an FR bond is the sum of a VR bond and an appropriate swap. Edward J. Kane, BC, 03-9&10

  38. New Topic: Using swaps to manage IRR begins with knowing how to calculate the Duration of a Swap. • Think of a swap as establishing an incremental balance sheet • Duration of a Swap is the duration of the net value of long and short positions Edward J. Kane, BC, 03-9&10

  39. Our goal is to explain how to Use Swaps to Lessen IRR of market cap. This means you must learn to Calculate Duration at least for Default-Free Interest-Rate Swaps • Recall that --when weighted by proportionate present value-- durations of the two halves of anycontract add: Dswap w1DPF + w2DRV . • One weight will be negative. Why? It is an obligation, in an incremental balance sheet. • If swap is “on market,” w1=-w2, but net worth of position is zero, so we can’t divide by it to calculate relative weights. Traders arbitrarily set w’s equal to 1 in this case. Edward J. Kane, BC, 03-9&10

  40. The duration DF of the Pay- or Receive-Fixed (RF) half of a just-agreed-upon interest-rate swap may be calculated as the duration of a stream of fixed-rate payments due on an n-period unit annuity (=Dn,1) • One can’t strictly “calculate” the duration of the pay-variable-rate (PV) half of this swap position because one cannot know how the “reference yield” will move. • By convention, traders assign an “implied duration” equal to the amount of time that is remaining until the next interest-rate “reset date.” The Receive-Variable half of a RVPF is a modified “par floater.” Value resets to par at each reset date. Edward J. Kane, BC, 03-9&10

  41. Query: As we approach each quarterly or semiannual reset date, what is the algebraic sign of the duration of the RFPV side of an interest rate swap? ANS. Duration (RF) = DF > 0 Duration (PV)  0 Duration of Net Position = DF - 0 > 0. Edward J. Kane, BC, 03-9&10

  42. What must be the duration of the reverse RVPF side of an on-market interest-rate swap at the same date? Duration must be 0 - DF < 0. • This negative duration is the “building block” used in constructing a swaps position to offset the IRR imbedded in a short-funded FSF’s market cap (S) by its book of customer-initiated core business. Edward J. Kane, BC, 03-9&10

  43. SWAP DURATION EXERCISE Suppose tenor of a RVPF swap is 5 years, notional amount B is $100 million, the FR bond yield index I is 10%, and settlement occurs once a year. a. At contracting date: DV = 1 year (= overly long reset date) DF= Dn,1= = 11 – 8.190 = 2.81 years b. Convention for on-market swaps sets weights at unity. Dswap = DV –DF = [1-2.81] = -1.81 years. Edward J. Kane, BC, 03-9&10

  44. If reset date were quarterly, DF would shorten to 2.42 and DV to .25. Dswap would be -2.17 years. • Effect of increasing n (i.e., increasing swap “tenor”) affects the DURATION of the fixed half only: • What would be DF in a 20-year swap with the same 10% yield? 11 - 7.5 years Edward J. Kane, BC, 03-9&10

  45. To explore how to use the Duration of Swaps in risk management, it is easier to employ “Modified Duration,” D*. D* = Reminder: The percentage price of any income stream responds to a tiny change in yield as follows: Edward J. Kane, BC, 03-9&10

  46. D* formulation expresses the approximate price sensitivity of a positionworth P to a specified basis-point change in the yield. • Setting R=.01%=.0001, the formula -PD*R gives the marginal “price value of a basis point:” pvbp. • Netting this value across both halves generates the marginal net interest-rate sensitivity of the swap itself. Edward J. Kane, BC, 03-9&10

  47. In an on-market swap, each side has the same value. • For R=.0001, we may write the pvbp of an on-market RVPF swap to a parallel change in market yields as pvbp of the two halves: • Dimensionality of pvbp lies in the currency in which the swap obligations are expressed: $, ¥, etc. Edward J. Kane, BC, 03-9&10

  48. As an example, suppose F is equal to $1 at contracting date and DPF=7.7 and R = .10.  .06¢ per b.p. Edward J. Kane, BC, 03-9&10

  49. How to Use the pvbp of a Swap to Manage Interest-Rate Risk: Sample Problem Suppose a bank’s market capitalization S is $220 million and the modified duration of S is: a. If R=10%, what is the pvbp of S? ANS. x one b.p. = ($220 mil.)(.0001) = (-4) (200 mil.) (.0001) = -$8 mil. (.01) = -$80,000 per b.p. at the margin (interpret) Edward J. Kane, BC, 03-9&10

  50. b. If the pvbp of an on-market RVPF swap with a notional value of $1 is .00060909, how large would the notional value of the swap F have to be to neutralize the interest-rate sensitivity of S? ANS. $80,000=F(.00060909) = F x per-dollar pvbp F = $131.34 mil. Doublecheck (looks at “hedge ratio” ): Relative variability of a “dollarsworth” of S and a dollarsworth of the swap is minus the ratio of modified durations of the two positions: F  .60 ($220 mil.) = $132 mil. = Edward J. Kane, BC, 03-9&10