University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives FNCE 4040 – Derivatives Chapter 4 Interest Rates
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Goals • Discuss the types of rates needed for Derivative Pricing • Continuous Compounding • Yield Curves • Risk • Forward Rate Agreements (FRA)
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Types of Rates • For the purpose of this class there are three types of interest rates that are relevant – LIBOR – Risk-free rates – Interest Rates on collateral • Important but out of scope rates include: – Treasuries – Overnight Interest Rate Swaps (OIS)
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives LIBOR
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives LIBOR • London Interbank Offered Rate – This is the rate of interest at which a bank is prepared to borrow from another bank. – It is compiled for a variety of maturities ranging from Overnight to 1 year – It exists on all 5 currencies – CHF, EUR, GBP, JPY and USD – It is compiled once a day ICE Benchmark Administration (IBA)
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives LIBOR Process • Once a day major banks submit the answer to the following question “At what rate could you borrow funds, were you to do so by asking and then accepting inter-bank offers in a reasonable market size just prior to 11am London time?”
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Uses of LIBOR • LIBOR rates are used for – Interest Rate Futures • This is a futures contract whose price is derived by the interest paid on 3-Month LIBOR – Interest Rate Swaps • These are derivative instruments that “swap” LIBOR for fixed interest rates generally for three or six month period. The maturity of these tends to be 3 to 50 years – Mortgages • Some Adjustable Rate Mortgages are linked to LIBOR rates – Benchmark rate for short-term borrowing in the market. – There has been a scandal surrounding LIBOR for the past few years. If interested see the appendix.
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives RISK FREE RATE
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives The Risk-Free Rate • Derivatives pricing originally depended upon a “risk - free” rate – The risk-free rate traditionally used by derivatives practitioners was LIBOR – Treasuries are an alternative but were considered to be artificially low for a number of reasons • Treasury bills and bonds must be purchased by financial institutions to satisfy a variety of regulatory requirements. Increases demand and decreases yield • The amount of capital a bank has to have in order to support an investment in treasury bills and bonds is lower • Treasuries have a favorable tax treatment
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives The Risk-Free Rate • In this course we will generally assume that risk-free rates exist and they will be given to you. • We will assume that LIBOR is the risk-free rate • We will give you rates.
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Collateral Based Discounting • Derivatives pricing theory has moved to Collateral Based Discounting – The yield curve relevant for discounting depends on the collateral agreement – Every derivatives contract might have a different yield curve • When we discuss pricing we will work through at least one collateralized example
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives THE YIELD CURVE
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Yield Curve • When pricing derivatives we will need a yield curve. • For our purposes a yield curve will consist of – Yields to specified maturities, – A methodology for interpolating missing yields, – A methodology for calculating forward rates (rates that are for borrowing/lending starting in the future)
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Theories of the Term Structure • Liquidity Preference Theory: forward rates higher than expected future zero rates • Market Segmentation: short, medium and long rates determined independently of each other • Expectations Theory : forward rates equal expected future zero rates – The Derivatives market uses this theory.
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives CONTINUOUSLY COMPOUNDED ZERO RATES
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Continuous Compounding • The compounding frequency used for an interest rate is the unit of measurement • All else being equal, a more frequent compounding frequency results in a higher value of the investment at maturity • In this class interest rates will be quoted as continuously compounded zero rates – Except when we are discussing a specific instrument or market – for example LIBOR or swap rates.
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Continuous Compounding • A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T • Continuous compounding means that an investment is instantaneously reinvested. • In practical terms this means – $100 grows to $100 × 𝑓 𝑆 𝑑 𝑈 when invested at a continuously compounded rate 𝑆 𝐷 to time 𝑈 – Conversely, $100 paid at time 𝑈 has a present value of $100 × 𝑓 −𝑆 𝑑 𝑈 , when the continuously compounded discount rate to time 𝑈 is 𝑆 𝑑
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Practice Remember: PV = 𝐺𝑊 ∗ 𝑓 −𝑆 𝑑 𝑈 Continuously Maturity Present Future Compounded (years) Value Value Zero Rate 1 4.0000% 961 1,000 2 3.0000% 2,000 2,124 1.5 6.0000% -6,398 -7,000 3 1.5000% 4,000 4,184
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Other Interest Rates • The quoting convention for quoted interest rates involves a daycount convention. • Through this one can compute the interest owed. • There are two examples we will use in class – ACT/360 – The interest owed is 𝑠𝑏𝑢𝑓 × 𝐵𝑑𝑢𝑣𝑏𝑚 𝐸𝑏𝑧𝑡 𝑗𝑜 𝑞𝑓𝑠𝑗𝑝𝑒 360 – ACT/365 – The interest owed is 𝑠𝑏𝑢𝑓 × 𝐵𝑑𝑢𝑣𝑏𝑚 𝐸𝑏𝑧𝑡 𝑗𝑜 𝑞𝑓𝑠𝑗𝑝𝑒 365
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Practice 𝐽𝑜𝑢𝑓𝑠𝑓𝑡𝑢 = 𝑂𝑝𝑢𝑗𝑝𝑜𝑏𝑚 × 𝑠𝑏𝑢𝑓 × 𝐵𝑑𝑢𝑣𝑏𝑚 𝐸𝑏𝑧𝑡 𝑗𝑜 𝑞𝑓𝑠𝑗𝑝𝑒 𝐸𝑏𝑧𝑑𝑝𝑣𝑜𝑢 Start End Daycount Rate Notional Interest (days) (days) Basis 0 365 3.00% 360 1,000,000 30,417 0 365 4.00% 365 1,000,000 40,000 182 365 3.00% 360 1,000,000 15,250 180 290 2.00% 365 1,000,000 6,027
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Conversion • We will often have rates given in a particular form and have to convert to another. • We can do this by computing the investment return from the given rate and using this to compute the unknown rate, or equating PVs: 𝑒𝑏𝑧𝑡 𝑓 𝑠 𝑑𝑑 ∗𝑒𝑏𝑧𝑡/365 = 1 + 𝑠 360 𝐵𝐷𝑈/360 1 𝑄𝑊 = 𝑓 −𝑠 𝑑𝑑 ∗𝑒𝑏𝑧𝑡/365 = 𝐵𝐷𝑈/360 𝑒𝑏𝑧𝑡 1 + 𝑠 360
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Practice 𝑒𝑏𝑧𝑡 𝑒𝑏𝑧𝑡 365 = 𝑓 𝑠 𝑑𝑑 ∗𝑒𝑏𝑧𝑡/365 1 + 𝑠 360 = 1 + 𝑠 𝐵𝐷𝑈/360 𝐵𝐷𝑈/365 ACT/360 ACT/365 C. comp. Start End Rate Rate Rate 3.00% 3.0417% 2.9963% 0 365 4.00% 4.0556% 3.9755% 0 365 3.00% 3.0417% 3.0187% 182 365
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives INTERPOLATION BETWEEN RATES
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Interpolation • When interpolating between rates we will linearly interpolate continuously compounded zero rates. • The advantages of doing this are: – It is easy to explain and implement – It has great risk properties • Sophisticated spline techniques are common in the market.
University of Colorado at Boulder – Leeds School of Business – FNCE-4040 Derivatives Linear Interpolation • If we know continuously compounded zero rates 𝑨 1 and 𝑨 2 for two times 𝑢 1 and 𝑢 2 then for time 𝑢 between 𝑢 1 and 𝑢 2 we define 1 + 𝑠 2 − 𝑠 1 𝑠 𝑢 = 𝑠 𝑢 − 𝑢 1 𝑢 2 − 𝑢 1 𝑠 2 𝑠 1 𝑢 𝑢 1 𝑢 2
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