Barrie & Hibbert

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Yield Curve Extrapolation: Longest reliable point

Posted on 19-04-2011 | 0 comments

It is profoundly worrying that the most basic valuation question we might ask of an insurer -- what is the value today of a fixed liability cash flow at some future date?-- is still unanswered less than two years ahead of Solvency II implementation.

Policymakers, regulators, firms, actuaries and accountants have belatedly focussed on a number of difficult questions including the choice of risk-free asset, the method for extending valuation beyond traded maturities and the possible impact of market liquidity premia on the way the valuation question is answered. This comment focuses on one specific aspect of the extrapolation – the reliability of forward rate estimates and whether we should switch focus from the liquidity of assets to the reliability of forward rates derived from market prices.

It’s apparent that there is considerable discomfort with potential variation in yield curves and their impact on the balance sheet where firms are exposed to large duration mismatches. Putting aside the rights and wrongs of exposing the economic mismatch between assets and liabilities (and I’m in the accounting camp here), I think there is a lack of appreciation of the difficulty and estimation error in establishing a reliable forward interest rate curve even where markets are Active and meet the ‘Deep/Liquid/Transparent’ (DLT) requirement (if we could decide what ‘DLT’ really means).  This is important because I think the preference for a reduced longest liquid point (‘LLP’) is driven by the desire to stabilise the forward curve between valuation dates rather than any objective view on liquidity. This objective is not unreasonable given that, in practice, for durations beyond 15-years there can be a reasonably wide range of credible forward rates that will closely fit market prices. So, if your objective is not simply to produce the best fit to market prices at a single point in time but to produce a good fit and manage changes between valuation dates in a credible fashion, it may be more useful to think in terms of a ‘longest reliable point’ (LRP). Here, we might choose to extend the curve from 15 or 20 years, not because of liquidity considerations but because we can better manage the trade-off between providing a good fit to market instruments out to – say – 30 years and, additionally, ensuring that the behaviour of the curve beyond the LRP is credible through time.

Here’s an example. This compares the government forward curves produced by Barrie & Hibbert and the US Fed for end-December 2008:

A few points to note:

• Both curves provide an extremely close fit to bond prices
• The Fed (in common with other central banks) use methods which focus only on the traded part of the curve (why would they do otherwise?)
• Generally, the spline approach allows for a richer description of the market and provides a marginally better fit
• However, the B&H curve is additionally influenced by what we want to happen in the extrapolated part of the curve (which is not shown) and so the terminal gradient is managed, but this does not have a material impact on the overall fit to observed prices.

My conclusion is that we may want to think about shifting the discussion from a narrow focus on liquidity (which could be argued is a ‘red herring’) to a longest reliable observation on forward rates and management of the trade-off between fitting instruments beyond this LRP with stability of the curve over time.
 

The path really matters - Part III: A de-accumulation example

Posted on 11-03-2011 | 0 comments

Previous entries have highlighted the sensitivity of savings accumulation to the path or order of a set of investment returns.  We concluded that – for accumulation – savers benefit from avoiding poor returns late in the accumulation period and that these returns will do less damage early in a savings plan because the invested fund is small. The analysis presented below demonstrates that the reverse is true for de-accumulation. In the example we analyse the same set of returns (mean 7% and standard deviation / volatility of 16.5%) over a 30-year horizon where the initial fund is €100,000 and an annual withdrawal of €6,000 is made. As before we now experiment only with the order in which the returns are delivered but leave the magnitude of returns unchanged. What difference does this make to results? Let us examine the terminal fund and the probability and timing of fund exhaustion i.e. where the fund is wiped out before year 30.

The chart below plots some possibilities for the growth in the underlying investment asset price. As before, you can see that all paths start and finish in the same place for a unit investment. The top (green) profile shows the path where we place returns in order from best to worst. The bottom (red) profile shows the reverse with returns ordered from worst to best. In contrast to accumulation strategies, de-accumulation will benefit from ‘early returns’ as under the green path. In this case, for the same set of returns, the ‘early return’ green path delivers a final fund of €+389K whilst the worst ordered outcome (‘early damage’) produces €-1.3M (although this result rather unrealistically assumes we ‘borrow’ at the unit rate of return). These differences are the exact opposite of what we saw for accumulation. Two additional random-ordered paths are also plotted on the chart.

Now consider two more obvious questions: firstly, what is the probability of the fund being exhausted before year 30?; secondly, what is the chance of the fund being wiped out in any particular year? The chart below helps to answer the question by analysing 10,000 possible (randomized) orderings of the returns. You can see that the assumptions suggest that the strategy delivers the €6,000 withdrawal over the full 30 years in approximately 70% of simulations. In the 30% of simulations where the fund is wiped out prematurely, there is a very high probability of the fund surviving beyond year 10 but around a 10% chance it is exhausted by year 20.

In these past three articles, I’ve highlighted some of the problems faced by savers. In future articles, we’ll take a look at some of the strategies used by product designers to help tackle the questions raised.

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