Tài chính doanh nghiệp - Topic 12: Measuring and managing interest rate risk with financial forwards, futures and swaps l. gattis

1Topic #12Measuring and Managing Interest Rate Risk with Financial Forwards, Futures and SwapsL. GattisA Course in Financial Risk Management2What Do Financial Risk Managers Do? (IMM)Identify risks that affects the viability of your firmMarket Risks: Equity, Interest Rates, Currency, CommodityCredit Risks: Bond and counterparty defaultMeasure exposure to identified risksPositions, VaR/CaR/EaR, Stress TestsDuration, Convexity, DV01Mitigate risksLayoff, Accept, Mitigate, Hold Capital3Interest Rate RiskMarket risk is the exposure to market price and interest rate changesInterest rate risk (IRR) is the exposure to interest rates changesBond Price Exposure: long bond positions are exposed to rising interest rates (short positions / falling rates)General Rate exposure: All rates falling/risingSpread exposure: spread between asset classes change (e.g., long corporate bonds and short Treasury’s when corporate spreads widen relative to treasury’s)Yield curve exposure: exposure to changes in the shape of the yield curveBorrowing rates: exposed to rising interest rates when need new funding or existing loans/bonds matures ; “Refunding Risk”Investment rates: exposed to lower rates on planned investments or when existing investments mature “Re-investment Risk”Interest rate risk management (IRRM) is the identification, measurement and mitigation of interest rate risk4Learning ObjectivesStudents can understand and can recall how DV01, Duration, Convexity, VaR, and Stress Tests are used to measure interest rate riskhow interest rate forwards, futures, and swaps are used to manage interest rate risk5Refunding Risk and FRAs 120 Days 211 Days rquarterly=1.5% rquarterly=2%Borrow 100m +100m -101.5m -102mConsider the problem of a borrower who wishes to hedge against increases in the cost of borrowing. We consider a firm expecting to borrow $100m for 91 days, beginning in 120 days from today, in June. The loan will be repaid in September on the loan repayment date. In this example, we suppose that the effective quarterly interest rate at that time can be either 1.5% of 2%, and that the implied June 91-day forward rate is 1.8%.A FRA is a contract that guarantees a borrowing or lending rate on a given notional principal amount. Suppose the FRA rate is the implied forward rate of 1.8%.Mcd.Example 7-11JuneSep6Forward Rate AgreementsFRAs are over-the-counter contracts that make a payment based on the difference between a forward rate and an actual ratesCan be settled at maturity (in arrears) or the initiation of the borrowing or lending transactionFRA settlement in arrears: (ractual- rFRA) x notional principalAlthough the FRA does not involve actual borrowing or lending, it can be combined with loans to “lock into” a fixed borrowing cost. The all-in-cost of the loan plus FRA payments are certain.7Forward Rate AgreementsAll-in-cost of funding using spot market loan and FRA payment in arrears 100*(.015-.018)8Eurodollar futuresThe Eurodollar Futures Contract is a very liquid and actively traded FRA. The standard terms areNotional Amount $1 MillionTerm 3 MonthsDelivery Dates Quarterly for 10 years (plus two serial months)Reference Rate 3-Month LIBOR (British Bankers Assoc. Index)Price Quote 100 – (Annualized 3-month Rate)Delivery Cash Settled - Thus when using E$, need (1/(1+r)) less contracts This is called taking tailed position in the contractSettlement Long: (rfutures – ractual)x1M/4 (rates are annualized here) Short: (ractual–lrfutures)x1M/4 also equal to F0t-(100-ractual)x100x25 ($25 is the DV01 of the contract, value of one annualized basis point)Payment Payment due at beginning (June Contract is based on the Jun-Sep 3-mo rate paid in june (beginning)9IRR of Bond PositionsBondholders are exposed to the devaluation of bond prices if the bond’s yield (YTM) rises.Duration is a measure of the interest rate risk of bonds, however there are several duration measures that are usedMacaulay DurationModified DurationEffective DurationEmpirical Duration10Macaulay DurationMacaulay duration is a measure of the “average” maturity of the stream of payments associated with a bond. Specifically, it is a weighted-average of the length of time until the bond’s remaining payments are made, with the weights equal to the present value of each cash flow relative to the price of the bond. Macaulay Duration is equal to:The duration of a portfolio of bonds is equal to the weighted average of the duration of the bonds in the portfolio.11Macaulay Duration (cont.)Example 7.63-year annual coupon (6.95485%) bond Price = Par = $1Macaulay Duration:12Modified DurationDuration allows us to estimate the effect on a bond or bond portfolio’s value of a parallel shift in the yield curve.The modified duration is Macaulay’s duration divided by one plus the yield to maturity. It can be shown that this is equal to minus the slope of the price-yield function divided by the price.Duration is thus a measure of interest rate risk for bonds. Specifically, it’s the % change in price for a 100 bps change in the YTM.13Duration (cont.)Example 7.4 & 7.5 3-year zero-coupon bond with maturity value of $100,000Bond price at YTM of 7.00%: = 100000/1.07^3=81,629.79Macaulay duration: 3Modified Duration: 3/1.07=2.804 Duration Estimate of Bond price at YTM of 7.1%81,629.79-2.804*.001*81,629.79=81,400.90Bond Price using the new YTM? 100000/1.071^3=81,401.35Difference of $.45 from duration estimate The duration estimate of price always underestimates the value of the bond because the price-yield relationship is convex and duration is a linear estimate!Graphical Interpretation of Convexity PriceYieldP(y)Slope of the tangent line is the -Dmod/P. For small changes in yield duration predicts price changes well. For larger changes in yield a correction for convexity (curvature) must be made.7.0%P0=81,629.79acb7.1%P1=81,400.90P1=81,401.35Diff=$0.45ConvexityBond price is a convex function of yield. This means that as the yield changes, the bond price will be above the price that would be predicted using the duration measure. On top of the price change estimated using duration, the price will change by the additional amount (based on convexity):The more convex a bond’s price, the more valuable the bond is, given a specific coupon rate. Another way of looking at this is the higher the convexity, the lower the yield for the bond.Total % change in bond price after yield change is given byBonds that are not callable are said to have “positive convexity”. Callable bonds (including Mortgage Backed Securities) have negative convexity.16Double-click for excel functionsGraphical Interpretation of Convexity PriceYieldP(y)7.0%P0=81,629.79acb7.1%P1=81,400.90P1=81,401.35Diff=$0.45Convexity Adjustment+.5*.0012*10.48*81,629.79=$.4318Duration Hedging and Duration$Suppose you bet that Kmart will recover and its bond prices will rise as long as the general level of interest rates are unchanged?Hedge IRR by shorting another bond with same duration dollars (dur$=Bond Price x Modified Duration)Suppose we buy a Kmart bond with time to maturity t1, price B1, and Macaulay duration D1How many (N) of another bond with time to maturity t2, price B2, and Macaulay duration D2 do we need to short to eliminate sensitivity to interest rate changes? The hedge ratio:Example 7.8We own a 7-year 6% annual coupon bond yielding 7%Want to match its duration by shorting a 10-year, 8% bond yielding 7.5%You can verify that B1=$94.611, B2=$103.432, D1=5.882, and D2=7.297 19Double-click for excel functions20Kmart Example (Duration Hedging)Long 10 of kmart bonds with 1000 FaceValue = 10x946.11=$9,461.10Short 7.41 of hedged bond with 1000 FaceValue = -7.408x103,432=$7,662.09Initial Portfolio Value = $9,461.10-$7,662.09=$1799.01If rates rise 50 basis points (both bond prices fall)Kmart Value Change = -5.50x.005x9,461.10 = -260.05Hedge Bonds Change= -6.79x.005x-7,662.09.45 = 260.05Total Portfolio Value Change = 0Portfolio Duration (aka equity dur.) is the market value weighted averagePortfolio Duration = 5.5*wkmart+6.79whedge=5.5x5.26+6.79x(-4.26)=0However, if these bonds have unequal convexity dollars, the price changes would not perfectly offset because of the convexity adjustmentNumbers may differ due to rounding, see worksheet on next slides21Kmart Example (Duration and Convexity Hedging)Suppose the convexity of the kmart and hedge bond are 30 and 20 respectively.If rates rise 50 basis points (both bond prices fall)Kmart Value Change =-260.05+½x30x.0052x9,461.1 =-256.50Hedge Bonds Change=260.05+½x20x.0052x-7,662.09 = 258.13Total Portfolio Value Change = $1.63In this case, you are not completely hedged because you are long more convexity dollars than you are long. (but this is good)Portfolio Convexity = 30*wkmart+20whedgePortfolio Convexity = 30x5.26+20x(-4.26)=+72.59Portfolio Value Change = +½x(72.59)x.0052x 1799.01 =+1.6322Kmart Example (Duration and Convexity Hedging)Positive Convexity Makes Investors Smile!23Kmart Example – Spread RiskLong 10 of kmart bonds with 1000 FaceValue = 10x946.11=$9,461.10Short 7.41 of hedged bond with 1000 FaceValue = -7.408x103,432=$7,662.09Initial Portfolio Value = $9,461.10-$7,662.09=$1799.01If Treasury yields rise 50 basis points, but Kmart yields only rise 40 bps (both bond prices fall, but Kmart outperforms)Kmart Value Change = -5.50x.004x9,461.10 = -208.14Hedge Bonds Change= -6.79x.005x-7,662.09.45 = 260.05Total Portfolio Value Change = 51.91You Win! Kmart spreads “narrowed” to Treasuries and your portfolio gains (You were rate “delta neutral”, but exposed to Kmart spread “widening”) 24Other Duration MeasuresMacaulay Duration: Modified DurationEffective DurationComputed by completely revaluing the bond when rates are increased by 100 bps and then rates are decreased by 100 bps. Effective duration is the average percentage price change. More accurate than modified duration because it takes into account the convexity of the price yield function.Empirical DurationA measure of duration calculated by "backing into" the duration value using changes in observed market prices resulting from changes in prevailing rate. (e.g. look historically the value change of a bond, the last time rates fell by 100 bps, takes into account other factors such as credit risk)Many firms report the portfolio DV01 which is change in portfolio value for a one basis point change (which is not useful if there is high convexity!)DV01 (dollar value of one basis point)=.0001*D*V0A.k.a.; BPV (basis point value)25Mortgage FinanceMortgage backed securities (MBS) have negative convexity because and their values are always less than the duration prediction becauseAs rates fall, homeowners refinance and prepay their mortgage, returning capital to the lender (at par) which must be reinvested at the new lower rate mortgages (rarely priced above par for this reason---- limiting appreciation)As rate rise, homeowners hold on to their mortgages longer, delaying the return of capital to lenders just when the lender would like to reinvest it at a higher rate (increasing losses)PYPYNormal Bonds“Positive Convexity”MBS“Negative Convexity”10026Mortgage FinanceNegative Convexity Makes Investors Sad!27Mortgage FinanceOne way to hedge MBS convexity is to issue callable debt which also has negative convexity. Selling negatively convex bonds reduces equity negative convexityAlternatively, banks buy options that make payments if rates change Interest rate caps and floorsOptions on swaps (Swaptions) 28Orange County, Duration and VaR(Stulz Box 9.2)The Orange County Treasurer, Bob Citron, lost about $1.6B in an investment pool, nearly bankrupting one of America’s most affluent counties.The investment pool was long $20B of assets and financed with 12.5B of reverse repo borrowing: portfolio value was $7.5B. (leverage was 2.67:1)The pool had a duration of 7.4 when rates rose 261 bps in 1994Loss Estimate = -7.4x.0261x7.5B=-$1.45 (close to reported loss)29Orange County, Duration and VaR(Stulz Box 9.2)If Orange County Officials were aware of the size of Citrons bets, it could have been avoided. (Identification/Measurement)Stress Test (+/- 50 bps change)-D x .005 x V=-7.4 x .005 x 7.5B = -278M-D x -.005 x V=-7.4 x -.005 x 7.5B = +278MO.C. Duration VaRSigma of monthly rate changes (1984-93) = 4.2% / monthAnnual Sigma = 4.2% x Sqrt(12) = 16.63%The 5-year note in Dec 93 had a yield of 5.22%The Sigma of the rate = 16.63% x 5.22% = 87 basis pointsO.C. VaR(1yr, 5%) = 7.4 x 1.65 x .0087 x 7.5B = $80030Non-Parallel Rate Changes(Yield Curve Reshaping Risk)Duration and convexity risk measures assume all interest rates fall and rise by an equal amount (parallel shift in interest rates)Changes in the yield curve could significantly affect portfolioExample: long 30 yr bond, short 10 year bondeven if compute optimal hedge ratio, you are exposed a yield curve steepening from the 10 to 30 year sectorsYou could create stress tests to measure expose to yield curve re-shapingExample: Exposure to 50 bps steepening or flattening31Treasury bond/note futuresWSJ listings for T-bond and T-note futuresSuppose you need to short or buy bonds to manage duration. An alternative to trading bonds is to short or long a bond futures contracts. Futures market provide more liquidity and involve less cash (only margins are required)32Treasury bond/note futures (cont.)Long T-note futures position is an obligation to buy a 6% bond with maturity between 6.5 and 10 years to maturityThe short party is able to choose from various maturities and coupons: the “cheapest-to-deliver” bond In exchange for the delivery the long pays the short the “invoice price.”Invoice price = (Futures price x conversion factor) + accrued interestPrice of the bond if it were to yield 6%PMT=5/2=2.5N=7x2=14Y=6/2=3%FV=100Compute PV =94.35PMT=7/2=3.5N=8x2=16Y=6/2=3%FV=100Compute PV =106.28Cheapest to deliverWe’ll ignore this33Double-click for excel functions34Bloomberg Ranking of CTD35Introduction to SwapsA swap is a contract calling for an exchange of payments, on one or more dates, determined by the difference in two prices or interest rates.Alternatively, a swap contract can simply be viewed as a forward contract with multiple delivery datesTypesInterest rate swapCommodity swapCurrency Swap36Interest Rate SwapsInterest rate swaps make payments based on the difference between two rates-- a fixed swap rate (specified in the contract) and a short-term floating rate (which are observed on future dates)One party pays the fixed swap rate and the other party pays the floating rate.The party that pays the fixed swap rate is call the “payer”The notional principle of the swap is the amount on which the interest payments are based.The life of the swap is the swap term or swap tenor.If swap payments are made at the end of the period (when interest is due), the swap is said to be settled in arrears.37An example of an interest rate swapXYZ Corp. has $200M of floating-rate debt at LIBOR, i.e., every year it pays that year’s current LIBOR.XYZ would prefer to have fixed-rate debt with 2 years to maturity.XYZ could enter a swap, in which they receive a floating rate and pay the fixed rate, which is 6.4823%.012Floating Debt - 1Yr LIBOR -1Yr LIBORReceive Floating +1Yr LIBOR +1Yr LIBORPay Fixed -6.4823% -6.4823%Debt + SWAP -6.4823% -6.4823%timeSWAP38The swap curveA set of swap rates at different maturities is called the swap curve.The swap spread is the difference between swap rates and Treasury-bond yields for comparable maturities. It a measure of the credit risk premium.Swap spread39Swaps and Duration ManagementA 10-year, pay-fixed, receive floating, quarterly payment swap is similar to shorting a 10-year coupon bond and buying a floating rate bond.The duration of the floating rate “leg” will be less than a quarter year The duration of the fixed rate “leg” will be the same as a fixed rate bondTherefore, swaps can be used to manage duration risk.If portfolio duration is long (positive) and you want it to be zeroShort Treasuries, Treasury Futures or other bond with the same duration dollars as portfolioEnter into pay fixed swap of similar maturity as Treasury aboveNote that swaps usually have no value at inception, so you cannot calculate duration dollars, for swaps you must calculate the duration dollars for both legs of the swapSwap advantagesLower basis risk if hedging a portfolio of corporate bondsNo cash exchangedLiquidity40ProblemsSuppose you own a 10-year, 7.5 modified duration, 105 convexity bond that is selling for $10,000. What is the estimated price of the bond if its yield falls .25% (25 basis points)? (compute using duration and convexity) Suppose you own fifty 3-year annual payment bonds that have a coupon rate of 15%, YTM of 10%, Macaulay duration of 2.64, par of $1,000, and price of $1,124.34?. You want to hedge your risk to interest rate changes by shorting a quantity of 10-year zero-coupon bonds that have a par of $1,000 and price of $450.00. How many 10-year zeros do you need to short to eliminate your duration exposure? What is the modified duration of a 3-year, $1,000 par, 25% annual coupon bond that is selling for $1,200? Suppose you are long a Kmart Bond that is selling for $100,000 and has a modified duration and convexity of 10 and 200, respectively. You are short a Walmart bond that is selling for $100,000 and has a modified duration and convexity of 15 and 400, respectively. What is your net gain or loss if the yield on both bonds increases by 50 basis points (.50%)?  Billy Bob is short the T-note futures contracts at futures price of 101.50. He has chosen to deliver a 5-year, $100,000 par, semiannual payment, 5% annual coupon rate bond that is selling for $102,250. What is the total profit (Invoice – Bond Price) for Billy to deliver? (Hint: the conversion factor is the calculated price (as a percentage) of the bond using a 6% annualized YTM)  41Exam FormulasEurodollar Payoff= [Futures price - (100 - rLIBOR)] x 100 x $25FRA settlement in arrears: (ractual - rFRA) x notional principalInvoice price = (Futures price x 6% YTM conversion factor) + accr. interest