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Chapter 8

Chapter 8. Bond Models and Interest Rate Options. §1. Interest rates and Forward Rates. Bond: Face value $1,000 (in USA) Coupon bond: pay interest periodically. (every six months or every year) Zero-coupon bond: pay no interest before maturity, but the price is discounted.

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Chapter 8

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  1. Chapter 8 Bond Models and Interest Rate Options

  2. §1. Interest rates and Forward Rates Bond: Face value $1,000 (in USA) Coupon bond: pay interest periodically. (every six months or every year) Zero-coupon bond: pay no interest before maturity, but the price is discounted. Coupon rate: periodic interest payment (annualized) as % of face value. Current yield: annual payment as % of the price.

  3. Coupon rate Example: • Face value (Principal): $1,000 • Maturity: 10 years • Interest rate: 5% • Coupon bond: • Interest paid every 6 months of $25, or every year of $50 • If the current price is $950, then the current yield is 50/950 = 0.0526 = 5.26% • If the current price is $1,050, then the current yield is 50/1050 = 0.0476 = 4.76%

  4. Yield-to-maturity: % of return (annualized) if the bond is hold to maturity. For the coupon bond, the yield of maturity R is the solution of the following equation P = current price, F = face value I = interest payment, n = # of interest payments

  5. Back to the previous Example: • If the current price is $950, then the yield-to-maturity is 5.66% > 5.26%. • If the current price is $1,050, then the yield-to-maturity is 4.38% < 4.76%. • For the coupon bond: • If price < face value, then • Coupon rate < current yield < yield-to-maturity. • If price > face value, then • Coupon rate > current yield > yield-to-maturity.

  6. For the zero-coupon bond (ZCB): Suppose the interest is compounded continuously and the interest rate is r, then P = price, F = face value, T = maturity In the previous example, we thus have In which case, the annual yield is 5%. For simplicity, we will mainly deal with ZCB with interest compounded continuously.

  7. Suppose the ZCB has face value $1 and matures at time T, if the price is P(t) at time t, then the yield-to-maturity (or yield) R(t) is given by Yield curveY(t): Y(t) is the annualized interest rate for a bond maturing in t years. Let P(0,t) be the price of ZCB with face value $1 maturing in t years, then

  8. Forward Interest Rate: rate of interest implied by current interest rates for periods of time beginning in the future. Example: Suppose the 1-year interest rate is 8% and the 2-year interest rate is 8.5%. The forward rate for the 2nd year, denoted by f(1,2), is the interest rate for the 2nd year that, when combined with 8% for year one, gives 8.5% for the 2-year rate. Thus So f(1,2) = 0.09.

  9. In general, if r1 is the T1-year rate and r2 is the T2-year rate of interest with T1 < T2, then the forward interest rate between T1 and T2 is given by Let Y(t) be the yield curve, then

  10. As T2→ t, we get This is called the instantaneous forward rate at time t. New Notation: Let f(0,t) denote the instantaneous forward rate at time t. This can be regarded as the interest rate at time t as seen from today.

  11. Thus the forward rate can be calculated from the yield curve. On the other hand, Thus Y(t) is the average forward rate between now and time t.

  12. §2. Zero-Coupon Bonds Let P(t) be the price at time t of a ZCB with face value $1 and maturing at time T. If the interest rate is constant, r, then If the interest rate is variable, r(t), then Thus

  13. Forward Rates and ZCBs Let P(0,T) be the price at time t = 0 of a ZCB with face vale $1 and maturing at time T, then So

  14. Suppose P(t,T) be the price at time t of a ZCB with face vale $1 and maturing at time T, then So Meaning of f(t,T): The forward rate at time T as seen at time t.

  15. §3. Swaps Suppose two companies SSI and FCTC both wish to borrow $10 million. The interest rates available are as follows Floating Rate Fixed Rate 6-month Treasury rate SSI 8% 6-month Treasury rate + 1% 10% FCTC SSI wants to borrow at a floating rate and FCTC whishes to borrow at a fixed rate.

  16. How to determine these rates? Strategy: SSI borrows at fixed rate 8% FCTC borrows at floating rate T + 1% Then they swap, that is, SSI pays a floating rate to FCTC FCTC pays a fixed rate to SSI.

  17. Fair Price: Friendly solution. SSI pays at T to FCTC FCTC pays at 8% to SSI. Then the actual rate are as follows SSI: Pay 8% + T, Receive 8%, Net T FCTC: Pay T +1% + 8%, Receive T, Net 9% What do you think about this solution?

  18. Fair Price: Arithmetic average solution. Total saving is: (T + 10%) – (8% + T + 1%) = 1% SSI pays at T – 0.5% to FCTC FCTC pays at 8% to SSI. Then the actual rate are as follows SSI: Pay 8% + T – 0.5%, Receive 8%, Net T – 0.5% FCTC: Pay T +1% + 8%, Receive T – 0.5%, Net 9.5%

  19. Fair Price: Geometric average solution. SSI pays at TR to FCTC FCTC pays at 10%R to SSI. How to determine R? TR + 10%R = T + 9% Thus R = (T+0.09)/(T+0.10) R is the discount rate Any other solutions?

  20. 8.5% – T SSI FCTC T + 1% 8% BANK Simple Way of Payment in arithmetic average: Let SSI send (or receive) one payment at X to FCTC to resolve the matter. Net payment of SSI is X + 8% = T – 0.5% Thus X = T – 8.5%. 1.5% 8% T + 1%

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