Challenges of Managing Interest Rate Risk: Part 5—How to get the Most Insight out of a Company's Assets and Liabilities
By Dariush Akhtari
Risk Management, December 2024
Editor’s note: The author extends heartfelt gratitude to Michel Perrin for his unwavering dedication, investing countless hours in thoughtful review, insightful discussions, and meticulous analysis of the various approaches. His expertise, patience, and collaborative spirit were invaluable to this endeavor, and his contributions significantly enhanced the quality and depth of this work.
This part of this series of articles will introduce various approaches under two different methods used to combat the non-additivity of key rates and will compare them under a number of rate environments. In the next part, I will highlight another misstep in the calculation of key rates and how to resolve it.
Defining Methods and Rate Environments
In Part 2 of this series, I used a flat spot curve to evaluate how each approach in the calculation of key rates will perform in approximating value change. In this article, I will use a term structure for the spot rates based on actual spot rates during the one-and-a-half years starting in the fourth quarter of 2019 to the first quarter of 2021. Below, two methods are defined under which four approaches are compared. Additionally, I added key rates 15 and 25 to work with seven key rates instead of five.
Method 1 – Centered: Centered metrics use the combination of up and down shifts in the calculation of KRkDV01 and KRkCV01 (i.e., metrics). This means that for the metrics at a point, both rate increases and decreases from that point are used.
Method 2 – Single-sided: Single-sided metrics use a one-sided shift, i.e., only up or down shifts are used for the metrics for the up or down metrics calculation, respectively. Since for KRkCV01 calculation three points are required, I added +/- half the shift in at the desired direction for the one-sided method. As a result, single-sided metrics require even more runs.
Approach 1 – Scaled key rate: In this approach, two scales are created, namely the up-shift scale and the down-shift scale. The up/down-shift scales are created by taking the ratio of the change in value had the entire spot rates been shifted up/down to the sum of the changes to the value created using up/down-shift key rates. These scales are applied to the change in value due to the up-shift and down-shift in the key rates accordingly. The key-rate DV01s (KRkDV01s) and key-rate convexities (KRkCV01s) are then calculated using these scaled values. It should be noted that the sum of KRkDV01s would equal DV01 and, similarly, the sum of KRkCV01 would equal CV01. The scales are applied to capture the cross-key rate impacts.
Approach 2 – Cumulative key rate: In this approach, the shifts for the calculation of key rates are accumulated so that the last key-rate shift equals the parallel shift to the entire spot rates. The key rate’s impact for each key rate is the difference between the calculated impact using cumulated shift less that generated from its immediately prior cumulated key rate change. In this method, by design, the sum of key-rate metrics match those created by the parallel shift to the entire spot rates. As indicated in Part 3, the interaction between all earlier key rates with the current key rate is captured in the current key rate (see Appendix 1).
Approach 3 – Unscaled key rate: In this approach, key rates are not scaled, but all the interactions between key rates are additionally captured through cross-key rates (four separate combinations for each x and y key rates, i.e., up/down-shift in key rate x with up/down-shift in key rate y). This approach requires many more runs. However, all the interactions are captured independently and used in the approximation of value change due to a change in the spot rates.
Approach 4 – Parallel: In this approach, simply DV01 and CV01 are used in the approximation of value change due to a change in the spot rates. Obviously, this approach requires the least number of runs.
I will leverage the 2-GIC portfolio example used in Part 2 of this article series, along with the six actual spot rates from 4Q19 to 1Q21 (see Appendix 2). I chose these rates not only for the fact that they are based on rates that have been experienced, but also because during these six quarters there were periods of large and nonparallel rate shifts, including twists. I generated additional spot rates by shifting these rates up to create environments where the portfolio of GICs would be in various moneyness to evaluate the effectiveness of various approaches in the approximation of the value change. The valuation date is set to be 1Q20, i.e., the starting spot rate is as of 1Q20. Three different spot rate sets were created by adding a flat spread (3%, 0%, and 5%) to the starting spot rates. These can be viewed as a spread earned over the risk-free rate (i.e., the expected earned rate) or simply a different interest rate environment. All the metrics are calculated from the valuation date based on the used rates (i.e., spot at 1Q20 + the spread). For the calculation of the actual value in a different environment, the portfolio is valued assuming the rates instantaneously changed to the then rate. The expected values use various metrics calculated using the four aforementioned approaches. I performed 42 comparisons consisting of two methods, three spread shifts (3%, 0%, and 5%) and seven rate environments.
- “A” cases refer to a 3% addition to each spot rate: This was done to force at least one of the two GICs in the portfolio to get in or out of the inflection point (depending on the change in rate the approximation is made for), thus producing an environment where convexity metrics are volatile and potentially using ALM metrics would not provide accurate approximations.
- “B” cases refer to no addition to each spot rate: This would result in the products in the portfolio to reasonably be in-the-money (spot rates were below the guarantee rates) resulting in losses. Any rate change will directly impact the profit.
- “C” cases refer to a 5% addition to each spot rate. This would result in the products in the portfolio to be currently out-of-the-money (forced rates to be above the sum of guarantee and pricing spread) and the profit to be reasonably fixed at the expected spread (i.e., cash flows are reasonably fixed) unless rates drop.
The seven comparison scenarios are:
- Compares 1Q20 to 2Q20
- Compares 1Q20 to 3Q20
- Compares 1Q20 to 4Q20
- Compares 1Q20 to 1Q21
- Compares 1Q20 to 4Q19
- Compares 1Q20 to 1Q20+100bps
- Compares 1Q20 to 1Q20+50bps
I developed tables capturing actual and approximated change in values in addition to their magnitude and absolute percentage errors. I performed these analyses using both centered and single-sided methods to evaluate which will produce better estimates. In addition, I used 0.5-, 1-, 2-, 10-, and 50-bp shifts to evaluate whether smaller or larger shifts produce more accurate approximations.
Comparison of Various Methods to Approximate Change in Value
Centered versus one sided: Comparing the tables in Appendix 3, it seems that a centered measure for the metrics provided a better approximation to value change from instantaneous rate changes, which comes as a surprise. For example, the A1 Unscaled approach for centered produced approximations that are materially more accurate than those produced by single-sided using any shift size. However, one would have expected that incorporating the exact direction of the change in a key rate and using a metric calculated using changes from that direction would have produced a more accurate approximation. I believe that the reason for this is that if in the particular direction of rate change an inflection point occurs, the calculated metrics at that level gets applied to the entire rate change for the approximated value, whereas in a centered approach, the metrics are averaged and the impact is muted. For this reason, I recommend using centered calculated metrics for the approximation of value change. Additionally, fewer runs are required for centered versus single-sided, in addition to producing more accurate results, at least for this example.
Comparison of Various Approaches to Approximate Change in Value
Since the centered method produced more accurate approximations, from this point on comparisons of various approaches will only use the centered tables. Additionally, when comparing results from various approaches, it seems that in the majority of the cases, the cumulative approach produces more accurate approximations than all other approaches. To appreciate this, I will compare the cumulative approach with every other approach.
Cumulative versus Scaled: Comparing the tables in Appendices 3, one can easily note that in the scaled cases, the approximation is extremely inaccurate in the A cases, when convexity is volatile and similar to the cumulative in B and C cases. The inaccuracy in the A cases may be a result of the use of the scaling factors since when rates are close to the inflection point the scales used could be materially away from 100%. In addition, in the cumulative approach, a smaller shift produces more accurate results in general consistent with “true” capturing of first and second order moments.
Cumulative versus Parallel: As seen in the tables in Appendix 3, other than in cases A3, A4 and A5, when both approaches did not produce accurate approximation, the cumulative approach dramatically outperforms the parallel approach. This comparison reflects the importance of the use of key rates versus only using DV01 and CV01. Segmenting the spot curve into smaller periods where rates seem to move closer to level combined with applying metrics for these smaller periods (i.e., using key rates) is expected to produce more accurate results.
Cumulative versus Unscaled: The expectation here is that the unscaled approach would outperform the cumulative approach since the interactions between the key rates are explicitly captured accurately through additional runs that have reflected cross-key rates as opposed to in the cumulative approach where these interactions have been implicitly captured. Considering only small shift sizes of 0.5 bp and 1 bp, the cumulative approach outperforms the unscaled approach. Note that when cumulative does not produce accurate results, nor does unscaled, and in other cases, they both are materially as accurate. This result is unintuitive since one would expect that the more exact approach (explicitly capturing of cross-key-rate interactions) would produce more accurate results all the time. In addition, it should further be noted that the unscaled approach requires substantially more runs to capture the key-rate interactions (98 runs versus 14 runs using seven key rates—in general “n” times more runs when “n” key rates are calculated).
I examined this phenomenon to understand why the unscaled does not always outperform the cumulative approach. This led me to the realization that when calculating key rates, forward rates need to be shifted not the spots. I will explore this in the next part.
Appendix 1
Below, I will show how the cumulative key-rate’s convexities reasonably capture the combination of key-rate convexities plus the convexities from the cross-key rates using the Unscaled approach. Table A1.1 is the calculated values using 1-bp shift for the A cases. Since the convexity is generally a small number, all the shown numbers are multiplied by 1000; thus, the difference is 1000 times bigger than it appears. From Table A1.1, one can see how the Cumulative key rates capture the cross-key rates reasonably well. In general, all the cross-key rates between x and y (x<y) are added to the y period in the Cumulative key rates.
Statements of fact and opinions expressed herein are those of the individual authors and are not necessarily those of the Society of Actuaries, the newsletter editors, or the respective authors’ employers.
Dariush Akhtari, FSA, FCIA, MAAA, is chief actuary at Converge RE. He can be contacted at dakhtari@converge-re.com.