Q6 cant figure out about the yield curve factor model…
Ans: A parallel shift of the yield curve would result in a loss across each key rate duration given a sensitivity of 1. For example, a 100 basis point (bp) parallel shift would generate an approximately 4.7% loss in value. A flattening of the yield curve in the long end would result in a loss given a sensitivity of -1. For example, a 100 bp decline in the 30 year key rate duration would result in a loss of approximately 2.9% (-100*-1*-8.7*.333). There is no impact from curvature, since the curve did not “twist”.
I dun understand how to derive (-100*-1*-8.7*.333) anybdy can help?
Exhibit 3 – Factor Movements per One Standard Deviation Shift and
Portfolio Key Rate Durations
Year
5
10
30
Parallel
1
1
1
Steepness
1
0.5
-1
Curvature
0.5
0
1
Key Rate Durations
1.8
3.6
8.7
Assuming rates change as described by Akron and based on Exhibit 3, the impact on the portfolio as outlined in Module 6 would be most likely be a loss in value from changes in: level and a gain from changes in steepness. level and a loss from changes in steepness. steepness and a gain from changes in curvature.
I think that is a poorly worded explanation. What probably they are trying to say is that when there is a parallel shift of 100 bp, the portfolio value comes down by 4.7% (which is 14.1 *.333) and the impact of the 30 year duration portfolio is 2.9% (8.7% *.333).There is an implicit assumption that the portfolio is equally distributed among the three durations.
For example if the portfolio was distributed as 25% in 5 year duration, 25% in 10 year duration and 50% in 30 year duration, then the impact of 100 bps change on the total portfolio would be (.25*1.8+.25*3.6+.5*8.7 = .45+.9+4.35 which is a total decline of 5.7%. Intuitively also it makes sense right, if you are loading up on securities with the longest duration, the decline will be much more. Now reverse it and change the numbers to 50% in 5 year duration, 25% in 10 year duration and 25% in 30 year duration. The decline will be (.5*1.8 + .25*3.6+.25*8.7= 3.975%)
Thx man. I see I understand your point in the parallel shift but I don’t understand the change in steepness point.
For example, a 100 bp decline in the 30 year key rate duration would result in a loss of approximately 2.9%(-100*-1*-8.7*.333)
Its like why do i pick the long end when the curve flattens? coz i guess if we pick the short end it will result in a gain? coz the table is 1, 0.5,-1. Is it?
The formula is ( -Change in level - Change in steepness - Change in curvature). So a -1 sensitivity corresponds to a opposite movement. For example if the interest drops by 100 bps, the portfolio gains (since bond prices go up), but since the sensitivity of steepness is -1, there will be a opposite movement, in this case the portfolio will lose value which is 8.7 *-1*.33 = 2.9% drop.
The definition of ‘steepness’ is when there are changes in two parts of the curve and the changes are not uniform, the short and the long. When the curve flattens, there is change only in the long end of the curve, where the interest has decreased, which should have given a gain, but in this case the sensitivity is -ve, hence the portfolio value decreases.
I’m confuse abt the steepness part. Why (A)-loss in value from changes in level and a gain from changes in steepness is incorrect
When rate rises evenly, the parallel shift of 100 bps (rising) will reduce portfolio value by 4.7% (refer bsbharath’s 1st comment), thus loss of value due to change in level.
For steepness part, under 5-year rate has 1 as sensitvity. So upward rising of 100bps will result upward shift of 100pbs under 1 year rate and downward shift for 30 year-rate with -1 as sensitvitiy. This result a loss under 1 year-rate and gain under 30 year rate. From here, if I were to follow CFAI texbk pg144-145, it seems that the fornula will change to ( -Change in level + Change in steepness - Change in curvature) . Am I missing out anything?
Module 6: “Consider a portfolio of zero-coupon bonds that mature at different times in the future. Changes in interest rates are not always parallel across maturities, so let’s analyze what happens as rates change across the yield curve. Let’s assume that the portfolio has sensitivities to factors as provided in Exhibit 3. The portfolio has equal weightings in each key rate duration and an effective duration of 4.7. I would like you to assess the impact on the return of the portfolio if rates rise evenly across the curve and also when the curve flattens but does not twist.”
This question/answer makes no sense to me. I am hoping someone can chime in.
"Consider a portfolio of zero-coupon bonds that mature at different times in the future. Changes in interest rates are not always parallel across maturities, so let’s analyze what happens as rates change across the yield curve. Let’s assume that the portfolio has sensitivities to factors as provided in Exhibit 3. The portfolio has equal weightings in each key rate duration and an effective duration of 4.7. I would like you to assess the impact on the return of the portfolio if rates rise evenly across the curve and also when the curve flattens but does not twist. ”
Year 5 10 30
Parallel 1 1 1
Steepness 1 0.5 -1
Curvature 0.5 0 1
Key Rate Duration 1.8 3.6 8.7
Assuming rates change as described by Akron and based on Exhibit 3, the impact on the portfolio as outlined in Module 6 would be most likely be a loss in value from changes in:
level and a loss from changes in steepness.
steepness and a gain from changes in curvature.
level and a gain from changes in steepness.
Answer is A: A parallel shift of the yield curve would result in a loss across each key rate duration given a sensitivity of 1. For example, a 100 basis point (bp) parallel shift would generate an approximately 4.7% loss in value. A flattening of the yield curve in the long end would result in a loss given a sensitivity of -1. For example, a 100 bp decline in the 30 year key rate duration would result in a loss of approximately 2.9% (-100*-1*-8.7*.333). There is no impact from curvature, since the curve did not “twist”.
Wiley refers to “steepness” as “twists” and “curvature” as “butterfly shifts”. Is this wrong? I can’t find the reference in the curriculum. Also, it appears the answer key is just running a parellel shift in the entire KRD, and not incorporating the flattener. Can anyone understand this?
Hi guys! I also spent some time on looking this question and answer. I think that key in looking only 30y sensitivity is in the wording in text : “and also when curve flattens”. The first part is easy and explained up in comments, if interest rate increase, portfolio will decrease in value. But if the yield curve flattent that means that LT interest rates decreased by more than ST rates (or in CFAI explanation, ST change is 0 and LT decrease by 100 bp). Similar exercise (although, much more simpler) is presented in 6.4 section of Term structure and interest rate dynamics (Example 11, question 2). If somebody have other view on this thing please share!
Gain or loss resulting from parallels shift = (1/3)*1%*1.8 + (1/3)*1%*3.6 + (1/3)*1%*8.7 = -4.7% Gain or loss resulting from steepness = (1/3)*1%*1.8 + (1/3)*0.5%*3.6 + (1/3)*(-1%)*8.7 = -1.7% And there is no impact from curvature, since the curve did not “twist”. From above, we can judge that most losses come from parallels shift and the rest of losses come from steepness. In total, the impact on the portfolio would be most likely be a loss in value from changes in level (-4.7%) and a loss from changes in steepness (-1.7%). Answer B coincides with it.
Gain or loss resulting from steepness = (1/3)*1%*1.8 + (1/3)*0.5%*3.6 + (1/3)*(-1%)*8.7 = -1.7% This suggests that with flattening, shorter-maturity bonds gained and a longer-maturity bond lost, which makes no sense, should be the exact opposite.
But if the yield curve flattent that means that LT interest rates decreased by more than ST rates (or in CFAI explanation, ST change is 0 and LT decrease by 100 bp).
And why would it result in a loss?
Also, sensitivity to steepness of -1 should mean that sensitivity to flattening is -(-1) or 1, am I wrong? The answer still doesn’t make sense to me
I realize that I am late to the party (as always), but the key is to look at this as a taylor expansion.
Total change in value = change in value from change in level + change in value from change in steepness (which is the first derivative of the level) + change in value from change in curvature (which is the second derivative of level) + higher order terms (but who asks about those)
A flattening of the curve (100 bp decline in long term rates, no change in short term rates) as in the CFAI example leads to loss from change in steepness (as indicated by the sensitivity of -1), but that does not mean that the portfolio will actually lose value. You are only looking at the steepness term. Staying within the CFAI example the total change of the portfolio value would be:
Total change in value = change from decline in long term rate = change in value from change in level (= 1/3*100bp*1*8.7 = 2.9%) + change in value from change in steepness (= 1/3*100bp*-1*8.7 = -2.9%) + change in value from change in curvature (this term is zero, as stated in the example) + higher order terms (these are neglected here) = 2.9% + -2.9% = 0
So while in the CFAI example there will be a decline in portfolio value due to a change in steepness (-2.9%) there will also be an increase in value due to the falling long-term interest rate (2.9%) to leave the portfolio value unchanged.
