Hi,
Referring to CFAI book as below under 6.4 Managing Yield Curve Risk, To make the discussion more concrete, consider a portfolio of 1-year, 5-year, and 10-year zero-coupon bonds with $100 value in each position; total portfolio value is therefore $300. Also consider the hypothetical set of factor movements shown in the following table: Year 1 5 10 Parallel 1 1 1 Steepness -1 0 1 Curvature 1 0 1 Because DL is by definition sensitivity to a parallel shift, the proportional change in the portfolio value per unit shift (the line for a parallel movement in the table) is 5.3333 = 16/[(300)(0.01)]. The sensitivity for steepness movement can be calculated as follows (see the line for steepness movement in the table). When the steepness makes an upward shift of 100 bps, it would result in a downward shift of 100 bps for the 1-year rate, resulting in a gain of $1, and an upward shift for the 10-year rate, resulting in a loss of $10. The change in value is therefore (1 – 10). Ds is the negative of the proportional change in price per unit change in this movement and in this case is 3.0 = –(1 – 10)/[(300)(0.01)]. Considering the line for curvature movement in the table, Dc = 3.6667 = (1 + 10)/[(300)(0.01)]. Thus, for our hypothetical bond portfolio, we can analyze the portfolio’s yield curve risk using Equation (22) ∆P/P=-5.33333∆XL - 3.0∆Xs -3.6667∆Xc
My question is pertaining to the Steepness portion bolded. Am I right to say that because of (-1) sensitivity for 1-year rate, thus we input -ve sign infront of the formula in order to obtain 3.0? If we switch the 1-year sensitivty and 10-year (Steepness) where 1-year sensitivity = (-1) & 10-year sensitivity = (1), does it mean that the formula change to -(-1+10/[(300)(0.01)] =-3 & Equation 22 will become
∆P/P=-5.33333∆XL + 3.0∆Xs -3.6667∆Xc?