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5.1 Interest Rate Markets Chapter 5 5.2 Types of Rates • Treasury rates • LIBOR rates • Repo rates 5.3 Zero Rates 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 Example 5.4 Bond Pricing • To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate • In our example, the theoretical price of a two-year bond providing a 6% coupon semiannually is 5.5 Bond Yield • The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond • Suppose that the market price of the bond in our example equals its theoretical price of 98.39 • The bond yield is given by solving to get y=0.0676 or 6.76%. 5.6 Par Yield • The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value. • In our example we solve The Bootstrap Method 5.7 How do we calculate the zero rates? Bond Time to Annual Bond Principal Maturity Coupon Price (dollars) (years) (dollars) (dollars) 100 0.25 0 97.5 100 0.50 0 94.9 100 1.00 0 90.0 100 1.50 8 96.0 100 2.00 12 101.6 5.8 Zero Curve Calculated from the Data Zero Rate (%) 10.681 10.808 10.469 10.536 10.127 Maturity (yrs) 5.9 Day Count Conventions in the US Day count conventions matter so that you can correctly calculate the interest payments over a period of time. When you buy or sell interest-bearing securities, you need to adjust the price for accrued interest. 5.10 Forward Rates The forward rate is the future zero rate implied by today’s term structure of interest rates 5.11 Calculation of Forward Rates Zero Rate for Forward Rate an n -year Investment for n th Year Year (n ) (% per annum) (% per annum) 1 10.0 2 10.5 ? 3 10.8 ? 4 11.0 ? 5 11.1 ? 5.12 Forward Rates (Comment on Textbook) • For continuously compounded interest rates: • For discrete compounding (such as money market rates): – where L (S) denotes longer-term (shorter-term) rate, TL (TS ) denotes the number of days in the longer-term (shorter-term) period, and TF denotes the number of days in the forward period – You will use the discrete compounding method when you calculate forward rates for money market instruments like LIBOR term deposits 5.13 Forward Rate Agreement • A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period • An FRA is equivalent to an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate • An FRA can be valued by assuming that the forward interest rate is certain to be realized 5.14 Theories of the Term Structure • Expectations Theory: forward rates equal expected future zero rates • Market Segmentation: short, medium and long rates determined independently of each other • Liquidity Preference Theory: forward rates higher than expected future zero rates 5.15 Treasury Bill Futures (Chicago Mercantile Exchange) • The Contract –Physical delivery of $1 million in T-bills maturing in 90, 91, or 92 days from the first contract delivery date –Tick size = $25 per basis point on a discounted basis • Quotation and Pricing in the Cash Market –Cash T-bills are discount securities –The annualized discount yield, DY, on a T-bill maturing in n days with the current cash price, Y DY = [(360/n)]x(100 – Y) The cash (or invoice) price for a T-bill with n days to maturity (per $100) is: Y = 100 – DY*(n/360) where DY is quoted as 5.0%, for example 5.16 US Treasury Bills: Futures Pricing • T-Bill Futures quoted price, Z: Z = (100 – corresponding Treasury bill quoted price ) Z = 100 - (360/n)*(100 - Y) • The futures cash invoice price = 10,000[100 - (n/360)*(100 - Z)] – the futures price is a function of the spot cash price of Treasury Bills. 5.17 Eurodollar Futures Contracts • Among the most liquid and actively traded futures contracts in the world • Eurodollar: a dollar deposit in a U.S. or foreign bank outside the U.S. • The contract: – Underlying: 90-day hypothetical Eurodollar CD – Contract Size: $1 mm face value – Minimum Price Move: (Tick): $12.50 • One basis point is worth $25 • One tick, worth $12.50, thus is equal to one-half of basis point – Contract Months: March, June, Sep, Dec, serial months, spot month – Maturities: Available through 10 years 5.18 Eurodollar Futures Quotation • The settlement price when a contract matures is F = 100 - R where R is the 90-day LIBOR – R is annualized relative to a 360-day money market year – R is expressed as a percentage to two decimals • For example: R=8.25% => F = 91.75 • Because Eurodollars CDs are bilateral, the quotes are bank- specific – The CME surveys several London banks each day to determine their quoted LIBOR rates – The two highest and the two lowest quotes are discarded – R is the average of the remaining survey quotes 5.19 Eurodollar Futures Pricing • In general, Eurodollar Futures are priced prior to expectation based on risk-neutral expectations model of interest rates. Thus, the The quoted Eurodollar futures price gives us a type of forward rate. Ft,T = 100 – ET[RT] where RT is 90-day rate that will be offered on 90-day Eurodollar deposit at time T • For short maturities the Eurodollar futures interest rate can be assumed to be the same as the corresponding forward interest rates. For longer terms, a convexity adjustment is often made. 5.20 T-Bonds futures (CBOT) • Quotation and Pricing in the Cash Market Cash price = Quoted price + Accrued Interest • T-Bond contract allows delivery of any bond with at least 15 years to maturity or first call. It is based on 8%, 15 year bond. $100,000 nominal face value – Party with the short position chooses bond – About 30 deliverable bonds – Beginning with March 2000 contracts based on 6% • Value of a tick: Minimum price change is 1/32nd or $31.25 per contract • Conversion Factors: – Quoted price is based on the assumption of a 6% hypothetical coupon standard – Approximate price, in decimal form, at which the bond would (as of the first delivery day of the month) yield 6% to maturity (rounded to whole quarters) or to first call if callable. 5.21 T-bonds & T-notes futures: Cheapest-to-deliver • The party with the short position will choose which of the available deliverable bonds is the “cheapest -to-deliver” – bond for which the following is the least: Quoted spot price - (Quoted futures price x Conversion factor) • Factors determining the cheapest-to-deliver – Yields > 6%, conversion factors favor delivery of low-coupon, long-maturity bonds – Yields < 6%, conversion factors favor delivery of high-coupon, short-maturity bonds – Upward (downward) sloping yield curve favors delivery of long (short) maturity bonds 5.22 U.S. Treasury Bonds Futures Pricing • Treasury Bonds: cash price = quoted price + accrued interest • Treasury Bond Futures (per $1,000): Full Invoice price = (quoted futures price)×(conversion fct.) + accrued int. • Quoted Futures Price – Over the life of the futures contract, the bond that is the “cheapest” to deliver changes – The quoted futures price is based on the bond that is expected to be the cheapest to deliver – Imbedded options in the contract 5.23 Duration • Duration of an instrument which provides cash flow c i at time t i is • where B is the bond price and y is the continuously compounded bond yield • This leads to 5.24 Duration Continued • When the yield y is expressed with compounding m times per year • The expression is referred to as the “modified duration” 5.25 Duration Matching • This involves hedging against interest rate risk by matching the durations of assets and liabilities • It provides protection against small parallel shifts in the zero curve