.. slideconf:: :slide_classes: appear ============================================================================== Forward Rate Agreements ============================================================================== Zero Rates ============================================================================== A *zero rate* for maturity :math:`\smash{T}` is the annual rate of interest earned on a :math:`\smash{T}` period investment. .. raw:: - There is a single payment at the end of the investment (no intermediate payments). .. raw:: - It is sometimes called the *spot* rate. .. raw:: - It is equivalent to the yield on a zero-coupon bond. Zero Coupon Bond ============================================================================== Suppose that the price of a 3-year zero-coupon bond is 97-05. - The continuously-compounded 3-year zero rate is .. math:: \begin{align} 97.15625 e^{r_3 3} & = 100 \\ \Rightarrow r_3 & = \frac{1}{3} \left(\log(100) - \log(97.15625)\right) \\ & = 0.00961656. \end{align} Zero Curve ============================================================================== .. image:: InterestFutures/zeroRateTable.png :width: 8in :align: center Zero Curve ============================================================================== .. image:: InterestFutures/zeroRatePlot.png :width: 8in :align: center Forward Rates ============================================================================== Forward rates are future spot rates implied by current spot rates. Consider two investments: .. raw:: - Investment 1: At :math:`\smash{t_0}` invest \$100 in a :math:`\smash{t_2}` year zero coupon bond, earning :math:`\smash{r_2}` (continuously compounded). .. raw:: - Investment 2: At :math:`\smash{t_0}` invest \$100 in a :math:`\smash{t_1}` year zero coupon bond (where :math:`\smash{t_2 > t_1}`), earning :math:`\smash{r_1}`, and at :math:`\smash{t_1}` roll the proceeds into a :math:`\smash{t_2-t_1}` year zero coupon bond. .. raw:: - The forward rate is the spot rate that would have to prevail at :math:`\smash{t_1}` for the investments to be equal. .. math:: \begin{gather} 100e^{r_1\times t_1}e^{r_f \times (t_2-t_1)} = 100e^{r_2 \times t_2} \\ \Rightarrow r_1 t_1 + r_f (t_2-t_1) = r_2 t_2 \\ \Rightarrow r_f = \frac{r_2 t_2 - r_1 t_1}{t_2-t_1}. \end{gather} Forward Rate Example ============================================================================== .. image:: InterestFutures/forwardRateTable.png :width: 8in :align: center Forward Rates: Alternative ============================================================================== Rearranging the forward rate formula: .. math:: r_f = r_2 + (r_2-r_1) \frac{t_1}{t_2-t_1}. .. raw:: - If the zero-curve is upward sloping between :math:`\smash{t_1}` and :math:`\smash{t_2}`, :math:`\smash{r_f > r_2 > r_1}`. .. raw:: - If the zero-curve is downward sloping between :math:`\smash{t_1}` and :math:`\smash{t_2}`, :math:`\smash{r_f < r_2 < r_1}`. Notes on Forward Rates ============================================================================== - Spot rates are *currently* available interest rates for investments of different maturities. .. raw:: - Forward rates are *implied future* spot rates for a single maturity. .. raw:: - Forward rates are not the same as future spot rates. - They are guaranteed rates for the future period, obtained by investing in current spot rates of differing maturities. Forward Rate Agreements ============================================================================== A Forward Rate Agreement (FRA) is a contract to fix an interest rate for borrowing/lending on a specific principal amount for a specific period of time. .. raw:: - The contract is over the counter. .. raw:: - The benchmark interest rate is typically LIBOR. .. raw:: - If LIBOR is below the contracted rate at maturity, the borrower pays the interest differential on the principal to the lender and vice versa. .. raw:: - The principal and time period are variables. .. raw:: - Interest is due at end of period, but present value typically paid at beginning. FRA Example ============================================================================== A company enters into an FRA to receive 4\% on \$100m for a 3-month period starting in 3 years. .. raw:: - If LIBOR is 4.5\% in 3 years, cash flow to the lender at 3.25 years is: .. math:: \smash{100,000,000 \times (0.04 - 0.045) \times 0.25 = -\$125,000.} .. raw:: - The present value is paid at year 3: .. math:: - \frac{125,000}{1+0.045 \times 0.25} = -123,609. .. raw:: - Note that for FRAs, interest rates are typically not quoted with continuous compounding, but with compounding frequency equal to term of agreement. FRAs vs Eurodollar Futures ============================================================================== FRAs and Eurodollar futures are similar. .. raw:: - Both fix an interest rate for a future period of time, tied to LIBOR. .. raw:: - Eurodollar futures are exchange traded, \$1m principal and 3-month LIBOR. .. raw:: - FRAs are over the counter, variable principal, variable term. .. raw:: - Eurodollar futures commit to pay the difference of principal and interest payment at beginning of period. .. raw:: - FRAs commit to pay interest differential on principal at end of period. FRA Notation ============================================================================== Suppose an FRA is set so that company :math:`\smash{X}` lends to :math:`\smash{Y}` between :math:`\smash{T_1}` and :math:`\smash{T_2}`. .. raw:: - :math:`\smash{R_K}`: Fixed interest rate set in FRA. .. raw:: .. raw:: - :math:`\smash{R_F}`: Forward LIBOR rate (determined today) between :math:`\smash{T_1}` and :math:`\smash{T_2}`. .. raw:: - :math:`\smash{R_M}`: Actual LIBOR observed at :math:`\smash{T_1}`. .. raw:: - :math:`\smash{L}`: Principal. FRA Cash Flows ============================================================================== At :math:`\smash{T_2}` the cash flow from :math:`\smash{Y}` to :math:`\smash{X}` (possibly negative): .. math:: \smash{L(R_K-R_M)(T_2-T_1)} .. raw:: If settled at time :math:`\smash{T_1}`, the present value is: .. math:: \frac{L(R_K-R_M)(T_2-T_1)}{1+R_M(T_2-T_1)} FRA Valuation ============================================================================== An FRA is worth zero if :math:`\smash{R_K = R_F}`. .. raw:: - At inception, :math:`\smash{R_K}` is set equal to :math:`\smash{R_F}` and the FRA has zero value. .. raw:: - As time elapses, :math:`\smash{R_F}` fluctuates and the value of the FRA changes. .. raw:: - Prior to :math:`\smash{T_1}`, the value of the FRA is computed by substituting :math:`\smash{R_F}` for :math:`\smash{R_M}` and computing the present value: .. math:: \smash{L(R_K-R_F)(T_2-T_1)e^{-R_2T_2}.} .. raw:: - :math:`\smash{R_2}` is the continuously compounded risk-free rate. FRA Valuation Example ============================================================================== It is currently Jan 1, 2010. .. raw:: - On Jan 1, 2009, a company entered into an FRA to receive 5.8\% (semi-annual compounding) and pay LIBOR on \$100m between July 1, 2011 and Dec 31, 2011. .. raw:: - The current LIBOR forward rate for July 1 - Dec 31, 2011 is 5\% (semi-annual compounding). .. raw:: - The 2-year risk-free interest rate (continuous compounding) is 4\%. .. raw:: - The value of the FRA on Jan 1, 2010 is: .. math:: \smash{100,000,000 \times(0.058 - 0.050) \times 0.5 \times e^{-0.04\times 2} = \$369,200.}