.. slideconf:: :slide_classes: appear ============================================================================== Futures on Interest Rates ============================================================================== Common Interest Rates ============================================================================== - Treasury rates. - U.S. Treasury rates are rates earned by on bills, notes and bonds. .. raw:: - London Interbank Offer Rate (LIBOR). - Short-term borrowing rate between banks. - Published by British Bankers Association (BBA) on 10 currencies and 15 borrowing periods. Based on survey of AA-rated banks. .. raw:: - Fed Funds Rate. - The overnight lending rate between banks which keep deposits at the Federal Reserve. Day Counts ============================================================================== Different day count conventions exist for determining the period over which interest bearing instruments accrue interest. .. raw:: - Actual/Actual (U.S. Treasury bonds and notes). .. raw:: - 30/360 (Corporate agency, municipal bonds). .. raw:: - Actual/360 (T-bills, commercial paper). T-bill Quotations ============================================================================== T-bill prices are quoted using a *discount rate*. .. raw:: - The discount rate is the *annualized* interest earned as a percentage of final face value. .. raw:: The *cash price* of a T-bill is defined as .. math:: \begin{align} P & = 100 - \frac{n}{360} Q. \end{align} .. raw:: - :math:`\smash{Q}` is the quoted price of the bond (the discount rate). .. raw:: - :math:`\smash{n}` is the number of days remaining in the life of the bond. Treasury Bond Quotations ============================================================================== Treasury bonds are quoted in dollars and thirty-seconds of a dollar. .. raw:: - For example, on Mar 5, 2015, the 30-year semi-annual Treasury bond with 11\% coupon rate maturing July 10, 2038 was quoted as 95-16, or \$95.50. .. raw:: - The quotations do not include accrued interest. .. raw:: - Accrued interest is the prorated amount of the next coupon that must be paid to the seller, since they will not receive the coupon at a later date. .. raw:: - The quoted price is the *clean price*. .. raw:: - The *dirty price* includes accrued interest. Accrued Interest ============================================================================== Treasury bond coupon rates are always quoted in *annual* terms. .. raw:: - An 8\% coupon on a semiannual bond means that a 4 dollar coupon is paid every 6 months (assuming \$100 face value). .. raw:: - In the example above, the 11\% coupon rate means \$5.50 is paid every 6 months, on Jan 10 and Jul 10. .. raw:: - If you buy the bond on Mar 5: - 54 days have elapsed since Jan 10 - There are a total of 181 days between Jan 10 and Jul 10. - Accrued interest is :math:`\smash{\frac{54}{181}\$5.50 = \$1.64}`. .. raw:: The dirty price of the bond is: :math:`\smash{\$95.50 + \$1.64 = \$97.14}`. Treasury Futures ============================================================================== Treasury futures typically provide a menu of instruments that can be delivered. .. raw:: - Treasury bond futures: any bond with 15 to 25 years to maturity. .. raw:: - Ultra T-bond futures: any bond with maturity over 25 years. .. raw:: - 10-year Treasury note futures: any bond/note with 6.5 to 10 years to maturity. .. raw:: - 5/2-year Treasury note futures: a note with about 5/2 years remaining and original maturity less than 5.25 years. .. raw:: - Conversion factors compensate recipient for differentials in delivered instruments. Conversion Factors ============================================================================== The party with the short futures position chooses the instrument to deliver. .. raw:: - The cash price paid by the recipient (long position) is .. math:: \begin{align} \text{Most recent settlement price} \times \text{Conversion factor} + \text{Accrued interest}. \end{align} .. raw:: - The short party determines the *cheapest-to-deliver* bond using data on conversion factors and settlement prices. Treasury Futures Quotations ============================================================================== Each futures is for delivery of \$100,000 face value of bonds. .. raw:: - Treasury bond and Ultra T-bond futures: quoted in thirty-seconds of a dollar per \$100 face. - Identical to spot market. .. raw:: - 10-year Treasury note futures: quoted in half of a thirty-second. .. raw:: - 5/2-year Treasury note futures: quoted in quarter of a thirty-second. Treasury Futures Quotations ============================================================================== .. image:: InterestFutures/bondFuturesTable.png :width: 8in :align: center Treasury Futures Price ============================================================================== The value of a Treasury futures is somewhat ambiguous because the cheapest-to-deliver bond and delivery date aren't precise. .. raw:: - Assuming a date and a particular bond, the price is: .. math:: \begin{align} F_0 & = (S_0 - I) e^{rT} \end{align} .. raw:: - :math:`\smash{I}` is the present value of coupons for the remainder of the bond's life. .. raw:: - :math:`\smash{r}` is the risk-free interest rate. .. raw:: - :math:`\smash{T}` is the life of the bond in years. Treasury Futures Price Example ============================================================================== Suppose the following characteristics of cheapest-to-deliver bond: - Quoted bond price is \$115. - 12\% coupon rate and bond is semi-annual. - 270 days until maturity. - Conversion factor of 1.6000. - Risk-free rate is 10\% per annum. - 60 since last coupon. - 122 days until next coupon. - Futures contract expires 35 days before bond maturity. Treasury Futures Price Example ============================================================================== .. image:: InterestFutures/bondFuturesExample.png :width: 8in :align: center Treasury Futures Price Example ============================================================================== Cash price of bond includes accrued interest: .. math:: 115 + \frac{60}{60+122} \times 6 = 116.978. .. raw:: Present value of \$6 coupon received in 122 days (0.3342 years): .. math:: \smash{6e^{-0.1 \times 0.3342} = 5.803.} .. raw:: Life of futures contract is 270 days (0.7397 years), so the cash price is: .. math:: \smash{F_0 = (116.978 - 5.803) e^{0.1 \times 0.7397} = 119.711.} Treasury Futures Price Example ============================================================================== At expiry, there are 148 days of accrued interest, so quoted price is: .. math:: \begin{align} 119.771 - 6 & \times \frac{148}{148+35} = 114.859 \\ \Rightarrow \frac{114.859}{1.6000} & = 71.79. \end{align} Eurodollar Futures ============================================================================== - A Eurodollar is a dollar deposited in a bank outside of the U.S. .. raw:: - 3-month Eurodollar futures are most popular interest rate futures traded at CME Group. - They are futures contracts on 3-month LIBOR to be paid on \$1m principle at the expiry date. .. raw:: - Eurodollar futures have maturities in the four nearest months and then Mar/Jun/Sep/Dec for up to 10 years. Eurodollar Futures Quotes ============================================================================== A Eurodollar futures quote is 100 minus the futures interest rate: .. math:: \begin{align} Q & = 100 - R. \end{align} .. raw:: - The rate is an APR and is expressed in precent. .. raw:: Since the contract is for 3 months (0.25 years), the contract price is defined as: .. math:: \begin{align} P & = 10,000 \times (100 - 0.25 R) = 10,000 \times (100 - 0.25 (100 - Q)). \end{align} .. raw:: - The contract price is the difference between the principle (\$1m) and the interest paid on the principle. .. raw:: - A 0.01\% change in the futures rate or futures quote causes a \$25 change in paid interest or contract price. Eurodollar Futures Table ============================================================================== .. image:: InterestFutures/eurodollarFuturesTable.png :width: 8in :align: center Eurodollar Futures Example ============================================================================== In the previous table: .. raw:: - The June 2013 futures price on May 13, 2013 is .. math:: \begin{align} 10,000 \times (100 - 0.25(100-97.725)) & = \$999,312.5. \end{align} .. raw:: - The June 2013 futures price on Jun 17, 2013 (expiry) is .. math:: \begin{align} 10,000 \times (100 - 0.25(100-97.615)) & = \$999,037.5. \end{align} .. raw:: - The price fell by 11 basis points or :math:`\smash{\$25*11 = \$275}`. .. raw:: - The buyer benefits if rate (price) falls (rises). Hedging with Eurodollar Futures ============================================================================== Suppose it is currently April 27 and you plan to lend \$1 million for 3 months on May 24 at the 3-month LIBOR rate. .. raw:: - You can hedge yourself by purchasing a Eurodollar futures contract with May 15 expiry. .. raw:: - If the current quoted price of the contract is :math:`\smash{Q_0 = 98.80}`, the implied LIBOR rate is :math:`\smash{1.2\%}` per year or :math:`\smash{1.2/4 = 0.3\% = 0.003}` for the 3-month period May 14 - Aug 14. .. raw:: - The cash price is :math:`\smash{P_0 = \$10,000(100-0.25*(100-Q)) = \$997,000}`. Hedging with Eurodollar Futures ============================================================================== - If the price rises to :math:`\smash{P_1 = \$998,000}` on May 14 when you close out the contract, you gain \$1000 on the futures. .. raw:: - However, the implied 3-month LIBOR will be the value :math:`\smash{R_1}` such that .. math:: \smash{\$998,000 = \$10,000*(100-0.25 R_1)} or :math:`\smash{0.8\% = 0.008}` per year or :math:`\smash{0.8/4 = 0.2\% = 0.002}` for the 3-month period May 14 - Aug 14. .. raw:: - As a lender, you lose :math:`\smash{\$1,000,000 (0.003-0.002) = \$1,000}`, which balances the gain on the futures.