2. Consider T, T, T to be points on the term structure of interest rates and...

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2. Consider T₁, T₂, T₁ to be points on the term structure of interest rates and let r = r(t, T.) be the corresponding continuously compounded zero-coupon rates. You observe the following continuously compounded yield curve: **** T 1 234567699 8 r(t, T.) B₂ 1 4.40% 4.70% 1 4.85% 1 4.95% 1 -0.05 1 0.00 0.05 0.10 0.15 0.20 0.25 4.85% 4.75% 1 4.70% 1 4.60% 1 4.50% 1 10 4.40% 1 where 3₁ and 3₂ are the factor loadings in the interest rate model dr. = B₁,dp + B₁2dpa. (1) (a) How (in terms of the term structure shape) can the factors ₁ and 2 be interpreted? Find factor durations D₁ and D₂ for a 10-year 6% coupon bond with annual coupon payments. (b) Suppose that you want to hedge your holding of the coupon bonds from (a) against changes in the two factors. Suppose that through the repo market you can at no cost take a position in a 1-year (PS) and 10-year (PI) zero-coupon bonds. Find the weights ks and kr, in your hedging portfolio. (c) Suppose that the next day, due to a central bank announcement, the interest rates are as follows: T 1 2 3 B₂2 -0.20 -0.15 -0.10 4 5 6 7 8 9 10 r(t,T.) 4.20% 4.60% 4.90% 5.15% 5.35% 5.50% 5.60% 5.65% 5.70% 5.75% 1. You observe the following swap rates (0,7) (a) Find the discount factors for each maturity. Do they imply any arbitrage opportunity in the market? (b) Find the corresponding semiannually compounded interest rates (72) (c) What is the forward price for a 6% coupon bond with semiannual payments with delivery in 2 years and 2 year remaining maturity (at the delivery) P/(0,2,4)? 3 7207) 0.5 6.10% 1.0 6.50% 1.5 6.85% 2.0 12.5 2. Consider T₁. T.-T to be points on the term structure of interest rates and letr, 7) be the corresponding continuously compounded zero-coupon rates. You observe the following continuously compounded yield curve 4 3.0 3.5 4.0 TFCT) A 1 4.40% 2 5 6 7 6.08 % 7.08% 7.17% 7.20% 7.31% 10 -0.20 4.70% 1 -0.15 4.85% 1 4.95% 4.85% 4.75% 1 4,70 % 3 4.60% 1 4.50% 4.40% -0.10 -0.05 0,00 0.06 0.10 0.15 0.20 0.25 where, and are the factor loadings in the interest rate model dripudi+Bades B (a) How (in terms of the term structure shape) can the factors, and be interpreted? Find factor durations D, and D, for a 10-year 6% coupon bond with annual coupon payments 7967 4 4.2006 24.60% 3 4.3% 4 5.35% $5.35% • 5.50% 5.60% 85.65% 95.70% 10 5.75% (b) Suppose that you want to hedge your holding of the coupon bonds from (s) against changes in the two facton opose that through the repo market you can at no cost take a position in a 1-year (P) and 30-year (P) zero-coupon bonds. Find the weights ky and k in your hedging portfolio (c) Suppose that the next day, due to a central bank announcement, the interest rates are as follows: Page 11 How much has changed the value of the total portfolio that you found in 0? How much would the value of the initial unhedged portfolio (.e. the 10-year coupon bond from (a)) change without the hedge? Has the hedge worked? 2. Consider T₁, T₂, T₁ to be points on the term structure of interest rates and let r = r(t, T.) be the corresponding continuously compounded zero-coupon rates. You observe the following continuously compounded yield curve: **** T 1 234567699 8 r(t, T.) B₂ 1 4.40% 4.70% 1 4.85% 1 4.95% 1 -0.05 1 0.00 0.05 0.10 0.15 0.20 0.25 4.85% 4.75% 1 4.70% 1 4.60% 1 4.50% 1 10 4.40% 1 where 3₁ and 3₂ are the factor loadings in the interest rate model dr. = B₁,dp + B₁2dpa. (1) (a) How (in terms of the term structure shape) can the factors ₁ and 2 be interpreted? Find factor durations D₁ and D₂ for a 10-year 6% coupon bond with annual coupon payments. (b) Suppose that you want to hedge your holding of the coupon bonds from (a) against changes in the two factors. Suppose that through the repo market you can at no cost take a position in a 1-year (PS) and 10-year (PI) zero-coupon bonds. Find the weights ks and kr, in your hedging portfolio. (c) Suppose that the next day, due to a central bank announcement, the interest rates are as follows: T 1 2 3 B₂2 -0.20 -0.15 -0.10 4 5 6 7 8 9 10 r(t,T.) 4.20% 4.60% 4.90% 5.15% 5.35% 5.50% 5.60% 5.65% 5.70% 5.75% 1. You observe the following swap rates (0,7) (a) Find the discount factors for each maturity. Do they imply any arbitrage opportunity in the market? (b) Find the corresponding semiannually compounded interest rates (72) (c) What is the forward price for a 6% coupon bond with semiannual payments with delivery in 2 years and 2 year remaining maturity (at the delivery) P/(0,2,4)? 3 7207) 0.5 6.10% 1.0 6.50% 1.5 6.85% 2.0 12.5 2. Consider T₁. T.-T to be points on the term structure of interest rates and letr, 7) be the corresponding continuously compounded zero-coupon rates. You observe the following continuously compounded yield curve 4 3.0 3.5 4.0 TFCT) A 1 4.40% 2 5 6 7 6.08 % 7.08% 7.17% 7.20% 7.31% 10 -0.20 4.70% 1 -0.15 4.85% 1 4.95% 4.85% 4.75% 1 4,70 % 3 4.60% 1 4.50% 4.40% -0.10 -0.05 0,00 0.06 0.10 0.15 0.20 0.25 where, and are the factor loadings in the interest rate model dripudi+Bades B (a) How (in terms of the term structure shape) can the factors, and be interpreted? Find factor durations D, and D, for a 10-year 6% coupon bond with annual coupon payments 7967 4 4.2006 24.60% 3 4.3% 4 5.35% $5.35% • 5.50% 5.60% 85.65% 95.70% 10 5.75% (b) Suppose that you want to hedge your holding of the coupon bonds from (s) against changes in the two facton opose that through the repo market you can at no cost take a position in a 1-year (P) and 30-year (P) zero-coupon bonds. Find the weights ky and k in your hedging portfolio (c) Suppose that the next day, due to a central bank announcement, the interest rates are as follows: Page 11 How much has changed the value of the total portfolio that you found in 0? How much would the value of the initial unhedged portfolio (.e. the 10-year coupon bond from (a)) change without the hedge? Has the hedge worked?