# Alternative views on extrapolated yield curves: A fundamental question remains unanswered

*by John Hibbert*

*The **Modeling of Financial Markets for Financial Institutions seminar** will take place in conjunction with and the day after the **2012 Investment Symposium**. Attendees of the seminar will discuss a variety of topics related to investment and modeling, including alternative views on extrapolated yield curves. *

The valuation of ultra long-term cash flows is surely one of the most basic challenges faced by accountants and actuaries. Yet the resulting debate grinds on about how to extrapolate observable yield curves generating considerable heat in Solvency II implementation. At its heart is a fundamental choice. On the one hand, extrapolated prices could represent where we might truly expect to trade a promised cash flow. These prices will be uncomfortably volatile and can imply relatively high levels of solvency capital for insurance firms who are unable to match liability duration in asset portfolios. Alternatively, extrapolated prices can be stabilized for the purposes of avoiding this variability. Prices produced by the two methods can be quite different.

The extrapolation of market yield curves has gained the attention of insurance firms, accountants and regulators over the past few years as a consequence of a move towards a market-based approach to valuation. This basic valuation principle is embedded in FASB rules for fair valuation (including insurance contracts) and a worldwide move by insurance regulators towards modernization of risk-based capital to support an economic balance sheet. Here we will discuss one fundamental question – the basic purpose of any extrapolation. There are two separate ways of thinking about extrapolation which turn out to be quite different.

**View #1: Accountants and traders – the Traders’ yield curve**

Consider the perspective of accountants and dealers. The FASB definition of fair value says:

“Fair value is the price that would be received to sell an asset or paid to transfer a liability in an orderly transaction … between market participants … under current market conditions.”

FASB/IASB anticipates using a range of market information and judgment to arrive at a valuation. Where prices are unavailable, valuations are to be based on expected values with risk adjustments. In order to give insight into these risk adjustments, consider the question of where a transaction would take place and ask yourself what factors would determine where you would be prepared to trade? The trader’s approach is to ask: firstly, how much of the resulting risk in my position can be hedged? Secondly, what is the cost of the hedge over the possible life of the position (itself often highly uncertain – especially if the hedge requires dynamic action)? Thirdly, how to adjust a price for the unhedgeable risk for which additional trading risk capital is required i.e. what is the necessary amount and required return on the trader’s risk capital. For a cash flow falling beyond the traded market, we could say:

- The cost of the hedge will be closely linked to the availability of similar tradable bonds.
- The risk charge is likely to be unstable. In stressed market conditions, the risk charge will probably rise in line with other risk capital costs.

This line of argument suggests that a fair value approach will produce volatile extrapolated prices. For anyone who must trade an asset or liability, these are the values that really matter and pretending otherwise could be costly. Let’s call the results produced by hedging and taking account of capital costs the traders’ yield curve. You might expect the extrapolated curves produced to contain flat forward curves where variability in traded rates is transferred to extrapolated maturities.

As an aside, you should note the European regulators’ view is that risk adjustments for unhedgeable market risk can be ignored – either because they are immaterial or just too difficult to calculate.

**View #2: Actuaries & insurance regulators – the Stabilized yield curve**

Volatile market-based yield curves result in volatile insurance balance sheets and, if you choose to use a short-term VaR capital measure, relatively high and volatile capital requirements. This volatility is seen as unhelpful by many in a long-term business where extrapolated cash flows are rarely traded. A consensus has emerged in Europe to stabilize the curve and methods essentially attempt to answer three questions:

1) What is the limiting (“ultimate”) forward interest rate (UFR)?

2) Which observable market prices should be fitted?

3) What is the appropriate speed of convergence between observed rates and the ultimate rate?

The way these questions are answered could deliver curves that are indistinguishable from the trader’s curve. Alternatively, highly stable rates could be generated that bear little resemblance to the trader’s curve. For all three questions, there are quite different views. The discarding of market prices – due to their lack of liquidity – runs the risk of disconnecting liability values from markets and creating a disincentive to hedge – probably not the regulators’ intention nor in the long-term interests of shareholders and policyholders.

**Where next?**

So, is it possible to simultaneously stabilize the extrapolated curve whilst respecting market prices and the trader’s pricing models? I doubt it. The stability that regulators and firms seek simply is not consistent with a trader’s view of pricing dynamics.

Does this matter? In Europe at least, it seems a simple fact of life that the answers to these questions will now be determined by political compromise rather than cold, objective economic analysis. However, there is a fine balance here which now risks the creation of two sets of yield curves – one for accountants’ fair valuation which makes use of all reliable market information and an alternative which stabilizes for the purposes of regulatory valuation. That would be a disappointing outcome for the architects of Solvency II.

*Learn more from John Hibbert at the **The Modeling of Financial Markets for Financial Institutions** seminar taking place in New York, March 28. Get more expert information by arriving two days early and attending the **2012 Investment Symposium.*

I believe there is a conceptual approach that can partially reconcile the trader’s view with the long term stabilized view of the yield curve. Assume for discussion that the risk free yield curve consists of the realistic expectation of forward rates plus term premiums based on the the risk of locking in an interest rate over time. The risk of locking in an interest rate over time is related to the variance of the distribution of the present value of the fixed maturity benefit, when discounted over stochastic scenarios of the realistic forward rates. If one uses this measure of term premiums, one can find that the risk free yield curve has a hump, because the calculated term premiums first increase then decrease by term to maturity. This provides theoretical justification for the idea that the very long end of the risk free yield curve will converge towards the mean reversion point.

The difficulty with this approach is that it relies on a fixed mean reversion point. In a trading environment, the market’s estimate of the mean reversion point could be constantly changing. However, since the mean reversion point is not observable, it can be possible to fit the observable part of the yield curve reasonably well by adjusting parameters for mean reversion strength, volatility, and the realistic expected path of the short rate in the short term in ways that will calibrate to the observed yield curve, even using a fixed mean reversion point.

Starting on page 6 of this link is an article I wrote on the subject.

http://www.soa.org/library/newsletters/the-actuary/2004/april/act0404.pdf

Stability, as John Hibbert uses the term, is the opposite of volatility. As discussed in my article linked above, arithmetic mean return is a linear function of volatility. The more volatility, the greater the return. Geometric mean return, however, increases to a maximum at a certain degree of volatility and declines from there becoming a negative number with high volatility. High durations correspond to high volatility (i.e., instability) and negative geometric mean return. The published yield curve is geometric mean return.

If zero-coupon bonds (e.g., STRIP’s) were traded at all durations, then the yield curve would stabilize, i.e., interest rates would be positive. Contrariwise, as long as the higher durations are not traded, the yield curve can extrapolate to be a negative rate of interest (an excessively high cost to hedge).

It is possible to derive a formula the yield curve should obey, in theory, using certain assumptions. You can curve-fit the formula to actual data. Likely, the formula will extrapolate to be a negative interest rate (i.e., to be unstable) at the higher durations where no trading occurs. You can constrain the formula to be non-negative at all durations and again curve-fit the formula to actual data. Likely the formula will not fit the data as well. John Hibbert asks, “is it possible to simultaneously stabilize the extrapolated curve whilst respecting market prices and the trader’s pricing models?” Technically, you can. You just don’t fit the data as well as you do if you respect the market prices and don’t stabilize.

To answer the questions the consensus raises—as Mr. Hibbert puts in—a good approach would be to derive a formula for the yield curve from first principles and then to curve-fit it to the data. This would then imply the UFR and the speed of convergence.