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15
Forward Rates
The forward rate is the future zero rate implied by today’s term structure of interest rates
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
c 0.050.5 c 0.0581.0 c 0.0641.5 e e e 2 2 2 c 0.0682.0 100 e 100 2 to get c=6.87 (w iths.a. compoundin g)
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
9
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
Rm Rc m ln1 m Rc / m Rm m e 1
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
5
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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
13
The Bootstrap Method continued
To calculate the 1.5 year rate we solve
4
5
5.0
5.3
6.2
6.5
17
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
Formula for Forward Rates
Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded. The forward rate for the period between times T1 and T2 is
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
3
Continuous Compounding
(Page 83)
In the limit as we compound more and more frequently we obtain continuously compounded interest rates $100 grows to $100eRT when invested at a continuously compounded rate R for time T $100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is R
8
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 3e y 0.5 3e y 1.0 3e y 1.5 103e y 2.0 98.39 to get y = 0.0676 or 6.76% with cont. comp.
(100 100 P)m c A
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
11
Sample Data (Table 4.3, page 86)
Bond Time to Annual Bond
Principal
(dollars)
Maturity
(years)
Coupon
(dollars)
Price
(dollars)
100
100 100 100 100
0.25
0.50 1.00 1.50 2.00
0
0 0 8 12
97.5
94.9 90.0 96.0 101.6
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
3e 0.050.5 3e 0.0581.0 3e 0.0641.5 103e 0.0682.0 98.39
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
4
Conversion Formulas
(Page 83)
Define Rc : continuously compounded rate Rm: same rate with compounding m times per year
Interest Rates
Chapter 4
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
1
Types of Rates
Treasury rates LIBOR rates Repo rates
14
Zero Curve Calculated from the Data (Figure 4.1, page 88)
12
Zero Rate (%)
11
10.681 10.469
10
10.808
10.536
10.127
Maturity (yrs)
9 0 0.5 1 1.5 2 2.5
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
10
Par Yield continued
In general if m is the number of coupon payments per year, P is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date
12
The Bootstrap Method
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 10.256% with quarterly compounding This is 10.127% with continuous compounding Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010
6
Example (Table 4.2, page 85)
Maturity Zero Rate (years) (% cont. comp.) 0.5 1.0 1.5 2.0 5.0 5.8 6.4 6.8
