NO.PZ2023091802000081
问题如下:
A portfolio manager is
analyzing the impact of yield changes on two portfolios: portfolio ASD and
portfolio BTE. Portfolio ASD has two zero-coupon bonds and portfolio BTE has
only one zero-coupon bond. 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 of the decrease in the values of the two portfolios due to the combined effects of duration and convexity?
选项:
Portfolio ASD Portfolio BTE
A.USD 102,000 USD 65,000
B.USD 110,000 USD 70,000
C.USD 118,000 USD 74,000
D.USD 127,000 USD 79,000
解释:
Step 1 - Calculate the values of the two portfolios
before increases in yield:
Portfolio ASD
PA = 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
PB = 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
DA = weighted-average
durations of the two zero-coupon bonds
= DA*WA + DB*WB
= 3*(740,818.22/1,147,387.88) + 9*(406,569.66/1,147,387.88) = 5.13
Portfolio BTE
DB = duration of portfolio BTE
= 6.00 (duration is approximately same as maturity for a
zero-coupon bond).
Step 3 – Note the convexities given for
the two portfolios (no need to calculate):
CA = 34.51; and CB
= 36.00
Step 4 - Estimate the changes in
portfolio values due to the yield change (y) and the
effects of duration and convexity:
Change in bond value = ΔP = -P*D*Δy + ½*P*C*(Δy)2
Portfolio ASD
ΔPA = -PA*DA*Δy + ½*PA*CA*(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
ΔPB
= -PB*DB*Δy + ½*PB*CB*(Δy)2
=
-618,783.39*6.00*0.02 + 0.5*618,783.39*36*(0.02)2
=
-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.
题里给的是macaulay duration 计算中用的是effective duration?
不应该用mac d/(1+Y)来计算effective duration吗?怎么直接就用mac d了呢?