Fixed-income securities pricing and hedging rely on forward-rate curves. W e now consider the input market rates for the construction of the forward-rate curve in a generic derivative system. The input data needed to construct the curve are described below. The input data are typically obtained either from vendor sources o r from the desk in the case of a broker-dealer BD’.
The choice of which input rates are used depends to some extent on the algorithm chosen to construct the yield curve. The yield curve is used to construct the set of discount factors employed for discounting future cash flows2. The input rates used also depend on the currency, since different instruments are available in different currencies3. The rates include various types. For the US market, these are cash rates, ED futures, Libor swaps, and US Treasuries.
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The Close of Business: Rates of course vary during the course of the trading day.Official pricing to be put into the books and records will generally use rates from the close of business (COB). It is best if these COB rates are saved automatically to a database so that later analysis is facilitated.
’Future and Forward Jargon: Do not mix up the word “future” in a phrase like “future cash flow” (cash to be paid some time hence) with “a future” (a contract to buy/sell something at a fixed price at some date hence). Also, do not mix up the word “forward’
in a phrase like “forward time period” with “a forward”, which is “a future” modified by a small “convexity” correction, as we shall discuss later.
Yield Curves for Different Currencies: The main currencies for the largest swap markets and their notations are USD ($US), JPY (Japanese Yen), EUR (Euro), and GBP (Great Britain Pound) or STG (Sterling). Financial futures and swaps are available at different maturities for these currencies. The use of liquid instruments is in principle important for reliable pricing. For other currencies including emerging markets there is less choice, and there may only be a few instruments available to construct the yield curve, liquid or not.
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Cash Rates
Cash rates a r e short-term “money-market’’ rates’, usually involving the cash Libor interest rates4 paid f o r deposit times of various lengths o u t to 1 year f o r deposits in London, e.g. US Libor (USD “Eurodollar” deposits), JPY Libor (JPY
“Euroyen” deposits) etc.
Futures
For USD, t h e Eurodollar (ED) futures, based on 3-month ED deposits, are used ii.
ED futures are cash-settled (i.e. no delivery of a security as f o r treasury futures).
T h e price of a future Pf is between 0 and 100, a n d corresponds t o a future rate rf = 100 -
Pf .
The ED futures h a v e a notional of $1 MM a n d correspond t o a 3M forward time period5, so a lbp change in rates6 corresponds t o a price change7 $dPf = $25. This is the change in t h e interest for a $IMM deposit over!A year’, if rates were t o change b y 0.0001/yr. Using Excel spreadsheet notation’
we h a v e $dP, = $10 6
*
0.25yr*
0.0001 / yr = $25 per bp change in rate. T h eLibor and Hibor: The acronym Libor stands for “London Inter Bank Offer Rate”.
Libor rates in different currencies sometimes have different names (e.g. HIBOR for Hong Kong dollar).
Quantity and Time Notation: Abbreviations for quantity [MM = million; B = billion;
M, K = thousand]. Abbreviations for time [Y or yr = year; M = month; D = day].
Basis Points, Units and Some Advice: Small changes in rates are measured in “basis points” or bp. Numerically one bp = 0.0001. Actually there is an implicit time unit, since interest rates are usually quoted in amounts paid per year (e.g. r = 4%/yr) so the relevant
“one basis point” change in an annual rate r would be dr = Ibp/yr = O.OOOI/yr. Usually the symbol bp is used without the units. While it sounds trivial, dropping the units easily leads to confusion. The potential quant is highly advised to put all units in all equations, at least for himself/herse&
Sensitivities of Futures in other Currencies: The price changes for futures for 1 bp/yr change in rates for other currencies have other conventions, but they are all determined by the same sort of equation as presented in the text.
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Money Market Day-Count Conventions: The meaning of “one year” changes from currency to currency for these futures. For USD, there are considered to be 360 days/yr but for GBP there are 365 days/yr. These “money-market day count” conventions are not the same as the conventions for bonds. And you thought that it was stupid to have different units for the measurement of electricity!
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’Arithmetic Notation: Finance people often denote multiplication by a star
*
and powers by a caret A used in Excel spreadsheets, so 4*4*4 = 4”3 = 64.Forward Curves 15 value of the future increases if rates decrease. A given ED future will have its value determined at a definite IMM fixing or reset date”.
Libor Swaps
A Libor swap is an exchange of fixed payments determined by a “fixed rate” for floating payments based on US Libor values at various times in the future. A
“swap rate” is that fixed rate which makes the swap value zero, or indifferent to the choice of fixed or floating payments. USD Libor swap rates are available with high precision. They serve as constraints, since swaps priced by the swap model using the model yield curve have to agree with the input swap rates. Historically the swap rates are expressed as spreads to treasuries”. We look at swap pricing in Ch. 8. The USD swap rates based on Libor are now the most liquid instruments available in general, even more than treasuries, and the swap rates are now tending to become considered as the fundamental rates to which other rates are compared. Swap rates in other currencies (e.g. Euro, GBP, JPY) provide similar constraints for input to pricing deals in those currencies.
Treasuries
For the USD curve, these are US-government notes and bonds. For other currencies, the available government securities are used. US treasuiy securities are auctioned from time to time. The last-auctioned (“on-the-run”) securities are those that are the most liquid. For the Libor curve, treasuries are really only used for representing the swap rates as swap spreads to treasuries.
Treasuries are used directly to construct the Constant Maturity Treasury (CMT) curve that is used to price CMT deals. The CMT curve takes account of the various yields and results from a model-dependent interpolation’* to get a IMM Fixing/Reset dates and Settlement dates for Futures: The (IMM = International Monetary Market) “fixing” or “reset” dates, at which the values of the ED futures are determined, are the 3rd Mondays of March, June, September, and December in future years (abbreviated e.g. as MAR04 for March, 2004). Payment (settlement) is made 2 days later, a tradition started by snail mail. This is a minor but annoying complication, since the risk of an instrument depending on a Libor future is zero after that rate is set, so rate uncertainty or diffusion only goes up to Monday, but an extra 2 days of discounting for the cash flow should be included up to Wednesday. Conventions are different for different currencies. Finally, there have recently been introduced E D futures other than 3M (e.g. 1M).
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Bid, Offer, Mid Swap Spreads: There is a bid and offer side to the swap spreads. The bid might be 40 bp/yr and the offer 44 bp/yr. This means that a potential fixed rate payer is ready to pay 40 bp/yr and a potential fixed rate receiver wants to receive 44 bp/yr (above the corresponding treasury rate). The “mid” swap spread is the average (42 bp/yr).
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l 2 Off the Run Govies and C M T “Off-the-run” securities are. securities from previous
auctions. They may be included in the algorithm giving the CMT rates through interpolations. However, off-the-run securities are less liquid because many bonds have
nominal value for a treasury yield”’ at any maturity. Note that of course securities d o not have their original time to maturity since they were sold so the “10 year treasury” will be plotted at somewhat less than 10 yearsl3.