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 + B₁2dp2. (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 B₂2 -0.20 -0.15 -0.10 3 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% 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 + B₁2dp2. (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 B₂2 -0.20 -0.15 -0.10 3 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%