Challenges of Managing Interest Rate Risk: Part 7—How to get the Most Insight out of Company's Assets and Liabilities
By Dariush Akhtari
Risk Management, April 2025
In this final part of the series, I will evaluate two different novel techniques for approximating change in value of a portfolio due to the change in the interest rates that do not rely on the ALM metrics discussed so far. These approaches use deterministic runs created to understand the range where ALM metrics could be relied on as discussed in Part 2. I will then argue that one of these techniques could be more superior to the use of key rates for managing one’s portfolio when they are used in approximating value change for a change in interest rate. This technique is not meant to replace the use of key rates for asset allocation, yet it can be a reasonable addition for investment and risk management professionals.
Introduction
So far, we have used ALM metrics in approximating the value of a portfolio under various rate environments. In Part 2, I indicated projecting and evaluating discounted value of cash flows of the portfolio as a way to identify the inflection points of the portfolio. This made me wonder if I could use these runs further in assessing what the value of the portfolio would be in a different rate environment even if the rate changes were not parallel. I identified two techniques, one using the portfolio values under various rate shifts and the other using the produced cash flows under different shifted rates. Below, I will elaborate on the two techniques and will compare them to the results of the outcome when using key rates.
A Different Approach: Technique 1 (Using Portfolio Values)
In Part 4, I had already identified two techniques to convert the change in the spot curve to only one rate to be applied to the duration and convexity to calculate the impact of rate change to the portfolio value. One technique was using a yield by calculating an internal rate of return (IRR) of the discounted value of the original cash flows using the current rate and comparing it to the yield calculated as the IRR using the same cash flows and the portfolio value. The difference between these two IRRs would be the “yield change” that would be applied to the duration and convexity. The other technique was to use a weighted average of the change in each tenor’s rate change where weights were the impact of the discounted value of the tenor’s cash flows to the portfolio value.
In order to assess a fair comparison with the key-rate method, I decided to only use 10 shifts of 20 bps each from -60 bps to +140 bps so that I would use the same number of runs as I had used in the calculation of the five key rates. This meant that I would have 11 discounted values in each of the environments that I had tested. For each of these environments, I ran a cubic spline to the 11 calculated values, which would allow me to calculate my expected value of the portfolio given the change in the yield or the weighted average of the rate change. Table 1 in the Appendix shows these values. Additionally, I only compared them with the five key rate cases using the cumulative technique using 2-bp shifts to forwards.
Comparison
The discounted value approach seemed to produce more accurate results as compared to the use of yield in the A cases but materially worse in the B and C cases. However, neither of these were consistently as good as the approximations using key rates. Note that this technique uses one “equivalent” rate for a yield curve change that we had previously identified that would not produce a decent approximation when comparing the key rate results versus using duration and convexity. Interestingly, this approach produces materially similar results to the use of duration and convexity if one compares the results in Table 1 (see Appendix) to the ones in Figure A3.1 in Part 5. As a result, I suggest that any technique that tries to reduce the change in interest-rate curve to one rate change should not be used.
A Different Approach: Technique 2 (Using Cash Flows)
So far, we have used ALM metrics in approximating the value of a portfolio under various rate environments. However, one can note that the portfolio value is simply the discounted value of the cash flows of the portfolio under the then-rate environment. This made me think of directly approximating cash flows of the portfolio given a rate shift. For this reason, I captured cash flows of the portfolio under equal shifts of 20bps from -40 bps to +160bps. This resulted in 10 additional runs similar to using the five key rates. This resulted in 11 sets of cash flows (including base run), which I could use to approximate cash flow for any tenor (years 1 to 30 in my example). Within each rate environment, to calculate the cash flow for any tenor given the shift in the rate in that tenor, I could interpolate (linearly, quadratic, cubic, or using cubic spline) using that tenor’s 11 cash flows using the change in the rate in that tenor. Once I had approximated my 30 years of cash flows, I simply would discount them using that environment’s spot rates to produce the portfolio’s value.
Since I had 11 values for each tenor I could use a number of interpolation techniques. The first was using cubic spline. I discarded this technique since the fitted curve would not necessarily go through every value. As such, I decide to use piecewise splines. This meant that for a cubic spline, I would use the closest four rate shifts to the rate change I was interested in. Similarly, for a quadratic spline, I would use the closest three rate shifts to the rate change I was interested in. And for linear interpolation, I would use the closest two points enveloping the rate change. This allowed me to calculate expected cash flow for any tenor given the rate change in that tenor.
Next, I needed to decide whether the rate change would be the change in the spot or the forward rate in that tenor. I decided to test both and compare the results.
Upon comparison of the results using various interpolation techniques, I noticed that there was not a materially different winner among them. In fact, linear interpolation produced materially the same result as quadratic or cubic splines. Next, was comparing the results using linearly interpolated cash flows using spot rate change versus forward rate change and further comparing these to the ones using five key rates.
Forward vs. Spot
As it can be seen in Table 2 in the Appendix, once again, in general, the use of change in forwards produced more accurate results. Interestingly, using forward change produced 0% error in all of the B cases. However, the error percentage in C cases was quite large. Yet, one needs to also note that the denominator for the C cases for my example is small since C cases had materially close-discounted value to the base case and any small change would produce a large percentage error.
Forward vs. Five Key Rates
Since the use of forward rate produced materially more accurate results, I compared the results to the approximations I had produced using five key rates. As it can be seen in Table 2, in most of the A cases, this technique produces more accurate results than one would get using five key rates. As indicated earlier, the B cases produce an approximation that had 0% error. However, the results for C cases were materially off using percent change, but on a value base, they were reasonable approximations. So, when there was a material change in value, this technique even outperformed using five key rates.
Conclusion
When one needs to quantify the impact of a rates change to a portfolio without running actuarial models, which could be cumbersome and time-consuming, one needs to rely on approximation techniques. Early on, duration and convexity were the tool. As these metrics were only useful when interest rate curve moved in parallel, the use of key rates became prevalent in some parts of the industry. In the prior parts of this article series, I highlighted that there were a couple of issues with how the industry calculated and used key rates. First was the use of gross up-factor to ensure the sum of key rates matched the duration. Note that due to the interaction of key rates with one another, their sum does not match the duration and convexity (see Part 3). The second issue was the calculation shift to the spot rates was used as opposed to a shift to the forward rates. I introduced the use of cumulative key rates (see Part 3) to ensure the additivity of key rates to the duration and further capture the cross-key rates’ impact to eliminate the need for materially additional runs to capture the cross-key rate impact. I also highlighted that shifts to forward rates need to be used in their calculation (see Part 6). Additionally, to avoid using convexity or key rate convexity, some used larger shifts to the rate in the calculation of duration or key-rate duration, which I further showed that smaller shift size combined with key-rate convexity would materially produce a more accurate approximation (see Part 6).
While value impact quantification due to rate change is important, for risk management, knowing how the cash flows would change would be a more useful tool. Use of key rates could only provide approximations to the portfolio value change, but it would not provide any information as to the change in cash flow (needed for liquidity analysis). Using the same number of runs to calculate key rates, I have introduced a novel technique that produces reasonably similar value change approximation with the added benefit of providing the expected cash flow change in a new rate environment. This technique is not to replace the use of key rates for managing or acquiring assets backing the liabilities, but to augment it. One can additionally evaluate potential cash flow needs under a certain interest rate environment and compare it to the expected cash flows from the asset portfolio to evaluate liquidity needs under such a rate environment.
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.