Challenges of Managing Interest Rate Risk: Part 2—How to get the Most Insight out of a Company's Assets and Liabilities

By Dariush Akhtari

Risk Management, May 2024

There appear to be inflection points in the value of an instrument when there is a shift in the interest rates. These inflection points could materially impact the approximation of value change using ALM metrics. In this part of the multipart series, I will provide:

• How to identify the inflection points;
• the range of rate changes where ALM metrics may be relied upon, i.e., the “validity” range, outside of which recalculation of the ALM metrics is required and the potential there would be a need for rebalancing the portfolio;
• a technique to improve on the approximation of value change beyond the validity range; and
• a technique to convert non-linear change on the yield curve to be used with duration and convexity.

In the next parts, I will introduce key rate metrics, their advantages, their limitations, the existing pitfalls of how they are calculated and used, and finally a technique to improve their use.

How to Deal with the Inflection Points

In the graph of an instrument’s value against parallel movement in the interest rates there are points where the convexity immediately prior and after such point is materially different in size. I denote these inflection points. To examine this, we will consider a portfolio of two GIC contracts with the following characteristics. One contract has 2% minimum guarantee rate and 1.5% expected profit while the other is a GIC with 3% minimum guarantee rate and 2% expected profit. Five hundred dollars is initially invested in each and they each have 10% per year partial surrender and full surrender at year 30. The graph of the contract for various flat spot curves is depicted in Figure 1. I use a flat spot curve for this illustration and use “metrics” to refer to ALM metrics, DV01 and CV01.

As expected from the product characteristics, there is a change in the convexity of the graph at 3.5% (2% minimum guarantee + 1.5% profit margin) as can be seen in Figure 1, and another at 5% (3% + 2%).  Comparing the coefficient of the x² for the fitted graph to present value of profits “PV(Profits)” below 3.5% versus the one fitted to 3.5% to 5% indicates that the convexity of the latter graph while having the same sign is roughly 40% of the former. This results in the possibility that the calculated metrics would not provide reasonable approximations for values that cross the points where metrics (in particular, convexity) change materially.

Figure 1
PV of Spreads and Fitted Curve to Pre and Post 3.5%

How to Identify Such Points

Since the convexity is the culprit, we need to calculate convexity at various rate shifts for a range where we expect rates to change and identify where convexity changes. This will allow us to be better equipped to trust our approximation or at least be able to perform some adjustments to reflect inaccuracy in using the metrics at face value. Calculating the convexity for the above simple examples using flat spot curve is an easy task. However, many insurance products need stochastic model runs with thousands of scenarios ran, and it is not feasible to perform multiple runs to identify value change for large rate changes.

The goal is to calculate DV01 and CV01 using 1-, 5-, or 10-bp shifts to the spot rates for a range of say -100 bps to +100 bps and then using the ratio of consecutive convexities to identify a large change. Note that with 1-bp shifts (recommended) this requires 200 sets of stochastic scenarios. However, calculating 200 sets of stochastic scenarios requires material runtime. One could replace the stochastic runs with the combination of deterministic runs using the valuation date’s spot rate and the option adjusted spread (OAS). OAS for an instrument is calculated as the flat spread over the spot curve that equates the present value (PV) of what is being measured (generally cash flows or profits) using a deterministic scenario (generally the spot or forward curve) to the one produced stochastically. We would use DV01 and CV01 using these deterministic runs to adjust the true DV01 and CV01 calculated by the stochastic runs for the expected change in rates where estimation is needed.

Simple Approach: Assume OAS does not Change Over +/-100 bps Shift Over the Spot Curve

Calculate the PV of interested measure (cash flows or profit) using spot curve with the desired shift level increments from -100 bps to +100 bps. Note that for the calculation of the PV, OAS would further be added to the spot curve in addition to any shift. Calculate DV01 and CV01 at each shifted rate and take the ratio of consecutive CV01s. When this ratio materially deviates from 100%, that is where an inflection point occurs, and it could highlight when approximating rate shifts that cross such points might not provide an accurate result. However, one can improve the approximation by bifurcating the rate change to the level that is from the current rate level to the level of rates where convexity changes the remaining portion. Then the approximation for the first part uses metrics at the valuation date with the amount of rate change in the first part and the second part of the approximation uses the calculated metrics at a point immediately after the convexity changes for the second portion with the remaining portion of rate change. An example outlined in Appendix 1 makes this clearer and will further highlight the approach and its effectiveness.

More Involved Approach: Assume OAS Varies for +/-100 bps Shift to the Spot Curve

Most companies likely produce values, for risk management purposes, using +/-100 bps shifts. Even if these runs are not performed, it is a good practice to add these to current calculated values. Using stochastic runs, OAS can be calculated for spot curve -100 bps, at the spot curve, and for spot curve +100 bps. Let’s call these OAS-, OAS, and OAS+. Calculate present values as in the previous simple approach except that instead of a flat OAS, for negative shifts use interpolated OAS derived from OAS- and OAS and for positive shifts use interpolated OAS derived from OAS and OAS+. 

Since spot rates do not move in parallel, one needs to convert the shift to the spot rates to one single shift to be used with the metrics. To calculate such shift in rates, the weighted average of change in the spot rate can be applied by using the contribution of the PV of each cash flow to the value, as illustrated in Appendix 2. This more involved approach is highly recommended to be added to all ALM work in addition to identifying at what rate shifts inflection points will occur for the portfolio.

Appendix 1

Let’s work with the example provided in section 1. The portfolio consists of two GICs each starting with $500, one with 3% minimum guarantee rate and 2% expected profit margin and the other with 2% minimum guarantee rate and 1.5% expected profit margin. Each GIC partially withdraws 10% of its account value per year and is fully surrendered after 30 years. Let’s assume the current spot rate is 3% and we are interested in approximating the value of this portfolio at 2.5%, 3.3% and 3.8%. Using the flat spot curve, the value of PV of profits (spreads and no expense) is calculated with 10bp shifts. While for simplicity I used 10bp shifts, I recommend using 1bp to 2bp shifts.

Table 1.1
A Simple GIC Example

Looking at the column “C Ratio” (the ratio of the then convexity to the one with one shift size lower) we notice a large change at 3.5%. So, this means that we may need to be cautious in our approximation of value at 3.8% since it is a point past 3.5%.  Initially we will use the below formula for getting an approximated value.

Approximate value at x% = PV at 3% + DV01 * shift + CV01 / 2 * shift², where shift in bps = 10000 * (x% - 3%)

Table 1.2
A Simple GIC Example

As expected, approximation at 3.8% produces a large error that we would have not been aware of had we blindly calculated these values. However, we know that there is a change of convexity at 3.5%.  So, to calculate the value at 3.8%, we would bifurcate the calculation into two parts. First is from 3% to 3.5%, and second would be from 3.5% to 3.8%. For the first part, we would use metrics at 3% to calculate value at 3.5% then we use metrics at 3.6% (one calculated immediately after 3.5%) to calculate the change in value from 3.5% to 3.8%. 

This will result in an approximated value at 3.5% of 84.59 (= 43.23 + 0.8627 * 50 – 0.001423 / 2 * 50²) to which we need to add 12.16 (= 0.4155 * 30 – 0.000685 / 2 * 30²) for the extra 30 bps from 3.5% to 3.8% using DV01 and CV01 at 3.6% to get to the total of 96.74. The dollar error becomes -0.27 (= 96.74 – 97.01) and the percentage error becomes -0.3% (= -0.27 / 97.1), obviously a tremendous improvement to the blind approximation.

Note that in this example, I simply used runs that did not require stochastic calculations. In practice, especially when metrics are calculated using stochastic runs the below steps could be used.

1. Calculate value of the instrument using stochastic scenarios.
2. Calculate value of the instrument using the central expectation of the stochastic scenarios including any additional spread used, CE, scenario.
3. Calculate option adjusted spread, OAS, that equates discounted value using CE + OAS to the one calculated in step 1.
4. Calculate DV01 and CV01 using stochastic runs represented with a superscript “s”.
5. Calculate 201 value of the instrument by running deterministic scenarios at CE + OAS and adding 1 bp shifts from -100 bps to +100 bps (note that these are deterministic runs).
6. Calculate DV01 and CV01 at various shifts as was done in table 1.1 denoted with a superscript “d”.
7. Perform approximation as was explained above by recognizing where the inflection point may occur.

Should you calculate OAS at +/-100 bps, in the above the OAS would be the interpolated OAS from OAS and those calculated at +/-100bp shifts.

Calculation of approximation up to the inflection point uses metrics with the superscript “s” while approximations beyond such points need the below adjustment to reflect that the metrics calculated using deterministic run is not identical to those using stochastic runs:

Where refers to the metric after the inflection point that is being used. Note that is the metric with no shift but using the deterministic run that uses OAS (i.e., the metric at 3% in the above example).

Appendix 2

To approximate the value of an instrument for a different rate curve one needs to quantify a level shift.  Since interest rates normally do not move in parallel, the below approach should be used.

The equivalent parallel shift in rates (spots) is the weighted average of change in the spot rate using the contribution of present value of each cash flow to the value as illustrated in the below formula.

where s_i is the spot rate at the period i, Δs_i is the change in spot rate at i and CF_i is the cash flow at i. Note that when using interest-rate sensitive products s should be replaced with CE + OAS and CFs are those from the deterministic run using CE.

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.

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Dariush Akhtari, FSA, FCIA, MAAA, is chief actuary at Converge RE. He can be contacted at dakhtari@converge-re.com.