清华大学：《期权、期货衍生产品》（英文版） Chapter5 InterestRatesFutures

5. Interest rates Futures Summary Asset Prices Depend on the Interest Rates Spot Rates and Forward rates How to determine the price of the interest Rate Futures How to Hedge interest rate risk Charles cao

1 Charles Cao 5. Interest Rates Futures ◼ Summary ◼ Asset Prices Depend on the Interest Rates ◼ Spot Rates and Forward Rates ◼ How to Determine the Price of the Interest Rate Futures ◼ How to Hedge Interest Rate Risk

Spot Rates Interest rates on zero-coupon bonds a The n-year spot rate is the interest rate on an n-year investment of the zero coupon bond No intermediate payment(coupon) Both the principal and interest will be paid on the last day Charles cao 2

2 Charles Cao Spot Rates ◼ Interest rates on zero-coupon bonds ◼ The n-year spot rate is the interest rate on an n-year investment of the zero￾coupon bond ◼ No intermediate payment (coupon) ◼ Both the principal and interest will be paid on the last day

Forward rates For a l-year zero-coupon bond, the spot rate is10% For a 2-year zero-coupon bond the spot rate is12% What is the interest rate for the second year? a Forward Rate: interest rates implied by current spot rates for periods of time in the future Charles cao 3

3 Charles Cao Forward Rates ◼ For a 1-year zero-coupon bond, the spot rate is 10% ◼ For a 2-year zero-coupon bond, the spot rate is 12% ◼ What is the interest rate for the second year? ◼ Forward Rate: Interest rates implied by current spot rates for periods of time in the future

Forward Rates: EXample Maturity Spot Total Forward (year) Rate(%) Returns Rate (%) n1=10.0100e 0.100米1 110.52 12345 2=10.5100105*2 12337/2=1.0 3=10.8100108313826/3=1.4 r4=110100e 0.110*4 15527f4=116 =11.1 100e 0.111*5 174.195=115 Charles cao

Forward Rates: Example Maturity (year) Spot Rate (%) Total Returns Forward Rate (%) 11 5 11 6 11 4 11 0 5 4 3 2 f . f . f . f . = = = = 5 4 3 2 1 11 1 11 0 10 8 10 5 10 0 5 4 3 2 1 r . r . r . r . r . = = = = = 100 110 52 0 100 1 e . . =  100 155 27 0 110 4 e . . =  100 123 37 0 105 2 e . . =  100 138 26 0 108 3 e . . =  100 174 19 0 111 5 e . . =  Charles Cao 4

The relationship between the spot and Forward Rates: Example 1 ■1 year zero,1=10% .2-year zero, r ,=10.5% a in each year, the rate is 10.5% Infer the forward rate, f, the interest rate from the beginning to the end of the second year Charles cao

5 Charles Cao The Relationship Between the Spot and Forward Rates: Example 1 ◼ 1-year zero, r1 = 10% ◼ 2-year zero, r2 = 10.5% ◼ in each year, the rate is 10.5% ◼ Infer the forward rate, f2 , the interest rate from the beginning to the end of the second year

The relationship between the spot and Forward Rates: Example 1 1=10% f2=? 10.5% 10.5% Charles cao

6 Charles Cao The Relationship Between the Spot and Forward Rates: Example 1 r1 = 10% f2 = ? r2 = 10.5% r2 =10.5%

The relationship between the spot and Forward Rates: Example 1 100e*/2 2 100e 1*1+f2*1_。72*2 3 米1 1 2 P*2-n1 f2=0.105*2-0.10=0.1=11% Charles cao

7 Charles Cao The Relationship Between the Spot and Forward Rates: Example 1 f . . . % f r r r f r e e e e e r f r r f r 0 105 2 0 10 0 11 11 2 1 1 2 100 100 2 2 2 1 1 2 2 1 1 2 1 1 2 1 2 2 1 2 2 =  − = = =  −  +  =  = =  +     

The relationship between the spot and Forward Rates: Example 2 2-year zero, /2 =10.5% 3-year zero, B3=10.8% Infer the forward rate, f, the interest rate from the beginning to the end of the third year Charles cao

8 Charles Cao The Relationship Between the Spot and Forward Rates: Example 2 ◼ 2-year zero, r2 = 10.5% ◼ 3-year zero, r3 = 10.8% ◼ Infer the forward rate, f3 , the interest rate from the beginning to the end of the third year

The relationship between the spot and Forward Rates: Example 2 72=10.5 2=105% f3 3=10.8%73=108%73=10.8 Charles cao

9 Charles Cao The Relationship Between the Spot and Forward Rates: Example 2 r3 = 10.8% r r3 = 10.8% 3 = 10.8% r2 = 10.5% f r2 = 10.5% 3 = ?

The relationship between the spot and Forward Rates: Example 2 100e72*2e/31=100已3*3 e2*2+/3*1 =e3*3 P*2+∫3米 3 J∫3 33 3-2*2 ∫3=0.108*3-0.105*2=0.114=11.4% Charles cao

10 Charles Cao The Relationship Between the Spot and Forward Rates: Example 2 f . . . . % f r r r f r e e e e e r f r r f r 0 108 3 0 105 2 0 114 11 4 3 2 2 1 3 100 100 3 3 3 2 2 3 3 2 1 3 2 1 3 2 3 3 2 3 3 =  −  = = =  −   +  =  = =  +     