为什么普通资产组合和barbell的duration计算方式不一样

问题如下：

A portfolio manager is analyzing the impact of yield changes on two portfolios: portfolio ASD andportfolio BTE. Portfolio ASD has two zero-coupon bonds and portfolio BTE has only one zero-couponbond. Additional information on the portfolio is provided in the table below:

To assess the potential effect of a parallel shift in the yield curve on portfolio values, the manager runs a scenario in which yields increase by 200 bps across all points of the yield curve. In addition, the manager estimates a convexity of 34.51 for portfolio ASD and 36.00 for portfolio BTE. Assuming continuous compounding, which of the following are the best estimates to the decrease in the values of the two portfolios due to the effects of duration and convexity?

选项：

A.USD 102,000丨USD 65,000

B.USD 110,000丨USD 70,000

C.USD 118,000丨USD 74,000

D.USD 127,000丨USD 79,000

解释：

Step 1 - Calculate the values of the two portfolios before increases in yield:

Portfolio ASD

P_A = Value before yield increase: 1,000,000*exp(-0.1*3)+1,000,000*exp(-0.1*9)

= USD 740,818.22 + USD 406,569.66 = USD 1,147,387.88

Portfolio BTE

P_B = Value before yield increase: 1,000,000*exp(-0.08*6) = 618,783.39

Step 2 - Calculate the duration of the two portfolios before increases in yield:

Portfolio ASD

D_A = weighted-average durations of the two zero-coupon bonds

= D_A*W_A + D_B*W_B = 3*(740,818.22/1,147,387.88) + 9*(406,569.66) = 5.13

Portfolio BTE

D_B = duration of portfolio BTE = 6.00 (same as maturity, zero-coupon bond).

Step 3 – Note the convexities given for the two portfolios (no need to calculate):

C_A = 34.51; C_B = 36.00

Step 4 - Estimate the changes in portfolio values due to the yield change (Δy) and the effects of duration and convexity:Change in bond value = D_P = -P*D*Δy + ½*P*C*(Δy)²

Thus,

Portfolio ASD

ΔP_A = -P_A*D_A*Δy + ½*P_A*C_A*(Δy)²

= -1,147,387.88*5.13*0.02 + 0.5*1,147,387.88*34.51*(0.02)2

= -117,722.00 + 7,919.27 = USD -109,802.73

Portfolio BTE

ΔP_B = -P_B*D_B*Δy + ½*P_B*C_B*(Δy)²

= -618,783.39*6.00*0.02 + 0.5*618,783.39*36*0.0004

= -74,254.00 + 4,455.24 = USD -69,798.76

A is incorrect. The change in value for both portfolios are wrongly computed as the parameter 0.5 is left out in the convexity formula.

C is incorrect. The changes in value for both portfolios do not consider the effect of convexity.

D is incorrect. Changes in value for both portfolios are wrongly computed by inserting a negative sign (rather than a positive) in the convexity part of the formula.