Articles - Is it the end of the game for cash flow matching

One of the few computer games from the 1980s that remains popular today is Tetris, where falling ‘tetrominoes’ must be lined up to fill the available space. There may be more sophisticated games available today but it is still played because, like all good games, it has a clear objective with simple rules and is difficult to master. In 2003, mathematicians from the Massachusetts Institute of Technology (MIT) revealed that Tetris is so challenging because it belongs to a class of problems beloved by code-breakers.

 By Guy Cameron, Director, Cameron Hume
 A good code is one that is easy to unlock if you have the key, but where the task of finding that key is hard. For a cryptologist a hard task is one that takes a long time to complete.
 Cash flow matching has features in common with Tetris - it’s one of the oldest techniques for matching assets to liabilities. The objective of cash flow matching is clear but finding a perfect solution is difficult and, despite more sophisticated approaches being available, it remains popular.
 Cash flow matching involves putting together a portfolio of bonds with cash flow receipts that match the timing and scale of expected cash flow payments - for example, when the time comes for a pension fund to pay a group of investors. A graph showing the proportion of cash flows payable by an annuity provider within 90 years would require a cash flow matching portfolio of bonds that, together, have cash flows distributed similarly. While approximate matches are achievable, exact matches are hard to find.
 Where assets exceed liabilities, the asset curve on a graph will bulge through the liability curve but, over time, the situation can reverse and in those circumstances the asset curve would fail to reach the liability curve. Long term investment managers seek to achieve a state where the value of assets is equal to that of liabilities and achieving perfection is hard in the sense meant by cryptologists. The liability curve consists of 1,000 cash flows and to match each of those exactly would be a laborious task.
 The question of how good a match is has long exercised bond managers and actuaries. Writing in 1952, a British actuary Frank Redington suggested that rather than seeking to achieve a perfect cash flow match, a good way to control the economic exposures of a portfolio is to require the blue areas on a graph (surpluses) and red areas (deficits) to be equal in extent. He termed this approach ‘immunisation’ and showed that the difference in the size of the blue and red areas was the net duration of the portfolio.
 His argument was instrumental in ‘duration’ being adopted as a control measure in fixed-income, portfolio management. Where the red areas are equal to the blue areas, we say that the net assets of an annuity business are ‘duration neutral’, that is, the net assets will not be changed by a small increase or decrease in interest rates.
 Redington’s proposal suggests, not only a way to judge the goodness of our fit, but also a way to improve it. He summarised the thousand liability cash flows with a single number - an audacious simplification. For Star Trek fans, Redington’s approach to cash flow matching was like James T Kirk’s approach to the Kobayashi Maru scenario. Like Kirk, Redington opted not to play the game as presented but to write his own rules. If he were to match the thousand plus cash flows of the liabilities, then he would need an equivalent number of zero coupon bonds. His approach would allow him to match his approximate curve with a single instrument.
 Redington borrowed a technique often used by mathematicians: if we cannot solve a problem exactly, then the next step should be to find a similar but simpler problem that we can solve? Rather than seek to achieve a perfect match to the liabilities, his approach was to match an approximation of the liability curve. He provided a practical way to approximate the liabilities for a time before we had computers. If we could precisely describe every bond with a single number, then we could perfectly match Redington’s curve.
 Unfortunately, the only bonds that can be so described are zero coupon bonds, which are as rare as hen’s teeth.
 Today, a common practice is to approximate the curve by a stepped function. If we can precisely describe each bond in a similar fashion, using cash flow ‘buckets’, then we can perfectly match the stepped curve. This description is a better reflection of the reality of a bond’s cash flows, but because we generally fix the location of the steps we introduce a further complication. As time passes, cash flows fall from one step to the next, moving suddenly from one bucket to one spanning shorter tenors. This causes measures of the quality of the fit to jump around. For this reason, at Cameron Hume we have moved to using so-called key rate curves, which both describe the liability curve better and do not suffer from sudden transitions. In fact, approximations are nearly indistinguishable from the actual cash flows.
 The strategy developed first by Redington and elaborated upon since offers a simpler approach to asset liability management than cash flow matching by reducing the number of features that we need to match. Redington’s strategy had far fewer features than the thousand or so cash flows of the full description and is easily and readily solvable with a simple portfolio optimiser.
 In a real-world, asset liability management problem, the quality of the cash flow match is only one of many criteria that an asset liability management policy must satisfy. The task is complicated by the need to meet diversity and credit quality constraints; to achieve an investment return target subject to a limit on capital and to take into account any ‘under’ or ‘over’ funding of the business. It is in these circumstances that the simplicity of the Redington approach becomes a disadvantage. We might ask the optimiser to choose bonds that maximise the expected return of assets, while matching the characteristics of our various approximations to the liability curve, subject to mild additional constraints and that no bond may be more than 20% of the assets. By adding a single constraint – for example that no bond may represent more than 1% of the assets – this causes the cash flow fit from Redington’s approach to change, but has less effect on the other two approaches.
 Cash flow matching puts a disproportionate focus on the fine detail of the interest rate exposures and crowds-out consideration of other factors. Redington’s approach is at the opposite extreme and imposes too little control on the interest rate exposures. More modern approaches seek an optimal, intermediate way that combines sufficient control, stable solutions and intuitively simple means of monitoring the net exposures of the assets and liabilities. The key rate approximation solves the cash flow matching problem and more elaborate approximations allow greater control and better oversight of other exposures.
 The ease with which players can learn the rules of Tetris is one explanation for its continuing popularity, but another is that it is simply challenging. Cash flow matching is also simple to understand and it is challenging to implement, but what is addictive in Tetris is a hindrance in asset liability management. It is time to trade in cash flow matching for more modern approaches that are designed for today’s technology, not that of 1950s Britain.

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