5.2 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 (Table 5.1, page 95)
5.3 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
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. aka YTM Suppose that the market price of the bond in our example equals its theoretical price of The bond yield is given by solving to get y= or 6.76%.
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
Par Yield continued In general if m is the number of coupon payments per year, d is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date
5.4 Determining Treasury Zero Rates Treasury zero rates can be calculated from the prices of instruments that trade. One way to do this is the bootstrap method. To see how this works, consider the following example:
Sample Data for Determining the Zero Curve (Table 5.2, page 97) BondTime toAnnualBond PrincipalMaturityCouponPrice (dollars)(years)(dollars) Half the stated coupon is paid every 6 months. An amount 2.5 can be earned on 97.5 during 3 months. The 3-month rate is 4 times 2.5/97.5 or % with quarterly compounding This is % with continuous compounding
The Bootstrapping the Zero Curve Similarly the 6 month and 1 year rates are % and % with continuous compounding BondTime toAnnualBond PrincipalMaturityCouponPrice (dollars)(years)(dollars)
The Bootstrap Method continued To calculate the 1.5 year rate we solve to get R = or % BondTime toAnnualBond PrincipalMaturityCouponPrice (dollars)(years)(dollars) Similarly the two-year rate is %
Zero Curve Calculated from the Data (Figure 5.1, page 98) Zero Rate (%) Maturity (yrs)
5.5 Forward Rates The forward rate is the future zero rate implied by today’s term structure of interest rates
Calculation of Forward Rates Table 5.4, page 98 Zero Rate forForward Rate an n -year Investmentfor n th Year Year ( n )(% per annum)
Formula for Forward Rates Suppose that the zero rates for maturities T 1 and T 2 are R 1 and R 2 with both rates continuously compounded. The forward rate for the period between times T 1 and T 2 is
Instantaneous Forward Rate The instantaneous forward rate for a maturity T is the forward rate that applies for a very short time period starting at T. It is where R is the T-year rate
Upward vs. Downward Sloping Yield Curve For an upward sloping yield curve: Fwd Rate > Zero Rate > Par Yield For a downward sloping yield curve Par Yield > Zero Rate > Fwd Rate
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
Forward Rate Agreement continued (Page 100) An FRA is equivalent to an agreement where interest at a predetermined rate, R K 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 We can make arbitrage arguments
Theories of Term Structure Yield Maturity Liquidity Preference Upward bias over expectations Fails to explain downward sloping yield curve Market Segmentation Preferred Habitat Explains both downward and upward sloping yield curves D5D5 D 10 D15D15 S5S5 S 10 S15S15 Expectations D5D5 D 10 D15D15 S 10 S15S15 S5S5
Day Count Conventions in the U.S. (Pages ) Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (in period) 30/360 Actual/360 The interest earned between two dates is:
5.9 Treasury Bond Price Quotes in the U.S Cash price = Quoted price + Accrued Interest since last coupon date
Treasury Bill Quote in the U.S. If Y is the cash price of a Treasury bill that has n days to maturity the quoted price is This is referred to as the discount rate
5.10Treasury Bond Futures Page 104 Cash price received by party with short position = Quoted futures price × Conversion factor + Accrued interest Each contract is for the delivery of $100,000 face value of bonds. Suppose the quoted futures price is 90-00, the conversion factor is and the accrued interest at the time of delivery is $3 per $100 of face value: 90× = $127.2 per $100 of face value or $127,200
Conversion Factor The conversion factor for a bond is approximately equal to the value of the bond on the assumption that the yield curve is flat at 6% with semiannual compounding
CBOT T-Bonds & T-Notes Factors that affect the futures price: –Delivery can be made any time during the delivery month –Any of a range of eligible bonds can be delivered –The wild card play
If Z is the quoted price of a Eurodollar futures contract, the value of one contract is 10,000[ (100-Z)] A change of one basis point or 0.01 in a Eurodollar futures quote corresponds to a contract price change of $ Eurodollar Futures (Page 110)
Eurodollar Futures continued A Eurodollar futures contract is settled in cash When it expires (on the third Wednesday of the delivery month) Z is set equal to 100 minus the 90 day Eurodollar interest rate (actual/360) and all contracts are closed out
Forward Rates and Eurodollar Futures (Page 111) Eurodollar futures contracts last out to 10 years For Eurodollar futures we cannot assume that the forward rate equals the futures rate
Forward Rates and Eurodollar Futures continued
Duration of a bond that provides cash flow c i at time t i is where B is its price and y is its yield (continuously compounded) This leads to Duration
Duration Continued When the yield y is expressed with compounding m times per year The expression is referred to as the “modified duration”
Convexity The convexity of a bond is defined as
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
5.14 Duration-based hedging strategies
Hedging in Interest Rate Futures A mortgage lender who has agreed to loan money in the future at prices set today can hedge by selling those mortgages forward. It may be difficult to find a counterparty in the forward who wants the precise mix of risk, maturity, and size. It’s likely to be easier and cheaper to use interest rate futures contracts however.
Duration Hedging As an alternative to hedging with futures or forwards, one can hedge by matching the interest rate risk of assets with the interest rate risk of liabilities. Duration is the key to measuring interest rate risk.
Duration measures the combined effect of maturity, coupon rate, and YTM on bond’s price sensitivity –Measure of the bond’s effective maturity –Measure of the average life of the security –Weighted average maturity of the bond’s cash flows Duration Hedging
Calculating Duration Calculate the duration of a three-year bond that pays a semi-annual coupon of $40, has a $1,000 par value when the YTM is 8% semiannually.
Calculating Duration Duration is expressed in units of time; usually years.
Duration The key to bond portfolio management Properties: –Longer maturity, longer duration –Duration increases at a decreasing rate –Higher coupon, shorter duration –Higher yield, shorter duration Zero coupon bond: duration = maturity
5.25 On August 1 a portfolio manager has a bond portfolio worth $14 million. The duration of the portfolio in October will be 7.1 years. The December Treasury bond futures price is currently and the cheapest-to-deliver bond will have a duration 0f 8.8 years at maturity How should the manager immunize the portfolio against changes in interest rates over the next two months?
5.25 The treasurer should short Treasury bond futures contract. If bond prices go down, this futures position will provide offsetting gains. The number of contracts that should be shorted is