A review on article "Yield curves and interest rate sensitivity - the devil is in the tail"
The original article "Yield curves and interest rate sensitivity - the devil is in the tail" is available at https://www.dhirubhai.net/pulse/yield-curves-en-rentegevoeligheid-het-venijn-zit-hem-van-dijk/
The described Alternative method looks interesting to me. It reminds me the old methodology, when all assets and liability cash flows were mapped/distributed linearly to/between chosen 9 liquid term points/buckets (i.e. in such case 35Y swap cash flow would indeed be split between 30Y and 40Y liquid term points only). Such method, of course, would not provide the accurate results, because the assets and liabilities cash flows in reality have the payment dates, which do not coincide with the chosen 9 liquid term points.
The Alternative method, described in article, involves the cash flows' sensitivity measurement towards 100 swap terms, and only afterwards those sensitivities are allocated to the chosen 9 liquid term points.
In my practice, I didn't meet such Alternative method implemented in LDI engines, however there may be indeed many methods proposed of how to get the reliable LDI results.
I also read Hagan / West article, which was mentioned in the article (https://web.math.ku.dk/~rolf/HaganWest.pdf), and agree that interpolation method (linear, splines, monotone convex.) should be chosen together with YC valuation principle (e.g. bootstrap, UFR, etc.) - and together they provide the final estimated zero coupon YC .
In the same time, indeed, there is a wide variety of interpolation methods with own pros and cons. A summary table with comparison is provided in the mentioned Hagan / West article on page 79. According to this article the monotone convex (unameliorated version) method is the optimal one (which is probably available in SimCorp Dimension). In the Hagan/West article I didn't notice B-spline method (only quadratic and cubic splines). However, I assume the B-spline results should be comparable to natural cubic spline results, which are less optimal than Monotone convex (unameliorated version) method (according to Hagan/West article).
Also, interesting statement is made regarding the mentioned 35Y swap: you mentioned that such swap should obviously contribute into 50Y liquid term point sensitivity. In general, I met 2 approaches in LDI and YC shocks theory: "Approach 1)" - when the final estimated zero coupon curve is shocked, and "Approach 2)" - when swap par rates are shocked and then a new (shocked) YC is re-estimated (based on shocked swap par rates). Both approaches have arguments to exist. In case of 1) approach the 35 Y swap would not be sensitive to 50Y liquid term point shock, because 40Y liquid term point already exists/is chosen (whereas the Key Rate Duration (KRD) related shocks at liquid term points are expected to be independent). Therefore, according to 1) approach, for example, the 35Y zero coupon swap would contribute only to 30Y and 40Y liquid term points. In case 35Y swap would have non-zero coupons - they would also contribute to earlier liquid term points (i.e. <= 40Y). This approach sounds logical to me: in case we have 35Y zero coupon asset bond then, taking into account that KRD shocks are expected to be independent for our chosen 9 liquid term points shock structure, I would not expect such 35 Y zero coupon bond to be sensitive to 50Y liquid term point (as the result it would not expect to contribute to 50Y liquid term point KRD).
For "Approach 1)" often the linear interpolation is used in order to construct the shock structure (with independent shocks for each of 9 liquid term points). However, it doesn't mean that YC itself (to which such linear shock structure is applied) becomes linearly interpolated: the linear shock structure is applied to non-linear estimated zero coupon YC rates. As the result, we will get the non-linear shocked estimated zero coupon curve.
The advantages of "Approach 1)" :
1) it keeps KRD shocks (applied to liquid term points) independent (as it should be according to KRD definition)
2) a sum of KRDs stays close to dollar duration (or modified duration) as it should be according to KRD purpose & definitions: a sum of linear shocks result in the total parallel shock/shift, which is used in dollar/modified duration definition
If "Approach 2)" is chosen: in such case the swap par rates are shocked before YC zero coupon rates get estimated, what means that shocks, applied to liquid term points e.g. 2Y and 5Y, potentially can also have impact on 35Y zero coupon bond. Besides, if we add B-spline interpolation method - then 50Y point sensitivity could also get affected. For example, "Approach 2)" often is used for UFR curve, because UFR transformation changes the market swap curve - consequently, the results in case of "Approach 1)" and "Approach 2)" might differ significantly for UFR curve case.
The advantages of "Approach 2)":
1) provides more realistic results in case of UFR curve (which differs from the "market swap curve" - based hedge)
2) apply shocks directly to swap par rates instead of final estimated zero coupon curve (whereas swap par rates are indeed the initial raw market data to be affected by shocks)
For "Approach 2" we still might apply linearly interpolated shock structure (based on 9 liquid term points) to 17 key swap par rates, whereas the re-estimated YC itself doesn't have to use linear interpolation.
To conclude:
1) There are indeed many interpolation methods. Probably Monotone convex (unameliorated version) method is even more optimal than proposed Bezier spline interpolation method (according to Hagan/West article)
2) Linear interpolation for shock structure (i.e. of independent shocks across chosen 9 liquid term points) looks like a reasonable choice to me, because
A) YC itself might still use the non-linear interpolation method for estimation of zero coupon rates
B) linearly interpolated shock structure is aggregated to the resulting total parallel shift/shock, which is necessary in order to ensure that 9 Key Rate durations would sum up to the total dollar / modified duration (according to definition of KRD and dollar/modified duration), whereas dollar/modified duration is based on the parallel shift/shock by definition
C) it ensures that shocks, applied to 9 liquid term points, stay independent (according to classical definition of KRD): for example, in case we have 30Y, 40Y, 50Y liquid term points among 9 liquid term points -then 35Y zero coupon asset bond should not be expected to contribute to KRD of 50Y liquid term point
D) linear shock structure should not underestimate 50Y KRD, because the linear shock structure for 50Y liquid term point would look like: [0bp shock at 40Y; linear interpolation;1bp shock at 50Y; 1bp constant shock for all YC points after 50Y], providing the independency of 50Y shock with other shocks, applied locally to the rest of 8 liquid term points
3) Linearly interpolated shock structure may be applied to the final estimated YC zero coupon rates (Approach 1) or to the swap par rates "Approach 2" before YC re-estimation. However, "Approach 2" usually gets considered for UFR case (which is not market curve based hedge)
4) In case the dependent shocks are expected to be used across 9 liquid term points: for example, 35 zero coupon asset bond is expected to be sensitive to 50Y liquid term point (taking into account that 40Y liquid term point exists) - then the correlation matrix would be required and, as the result, it would rather result into a specific stress test scenario than the independent classical KRDs for standard LDI process
5) It should be possible to reconstruct "Approach 1)" and "Approach 2" in Dimension
Also, a nice article from JPMorgan regarding linearly interpolated shock structure to be used for Key Rate Durations in LDI hedging process: https://www.pionline.com/assets/docs/CO7299731.PDF
Best regards,
Sergii Oliinyk
?? SimCorp Dimension Business Analyst & Project Manager