Yield Curve
A yield curve is effectively a representation of the rate structure that can be locked in today, going out into the future. Interest rate derivative markets rely heavily on yield curves for pricing, as they are essentially a snapshot of current market conditions across the tenor of the instrument.
Yield curves are constructed using a Yield Curve Construction Methodology. Such methodologies come in all levels of complexity, from the simplest bond yield curve to a smoothed composite curve covering all markets. The composition of a curve and its construction method depend on what the curve is to be used for.
Single-Market Curves
These are curves such as the bond yield curve or the swap yield curve. They are very useful for pricing securities within these markets. For example, if we were pricing a bond issue, we could interpolate the appropriate maturity from the government or semi-government bond yield curve and add a suitable credit margin (to cover the extra credit risk for a corporate bond over government). This would give us an indicative yield for the pricing of our bond issue. While such curves are useful for such comparative analysis, they are not suitable for pricing instruments from different markets (for example, we cannot reliably price a swap from a bond curve).
Linear Interpolation
This is the simplest of all of the composite curves that cover several markets. Such curves can be used to price instruments across a range of markets. Linear interpolation is the simplest of all composite curve construction techniques, and has the lowest computational intensity. This makes it suitable for real-time curve construction applications. A linear interpolation curve is also ideally suited for derivative markets where market convention includes such a method. Forward Rate Agreement (FRA) markets are an example of this. The convention for pricing FRAs is linear interpolation from futures contracts. In fact, you can price most FRAs without a yield curve, by simply interpolating between futures, but a curve can be handy for pricing ‘off-run’ FRA contracts such as 1 and 2 month instruments.
Linear Interpolation curves tend to be jagged with sharp ‘corners’. These corners can be good and bad. On the up side, these corners mean that the input rates, or ‘observed rates’ are preserved, and hence will be returned as an implied forward rate. On the down side, such sudden fluctuations in the rate can mean that daily revaluations jump around with short-term rate changes.
An example of where linear interpolation curves suffer serious problems is revaluing an option-based interest rate derivative. If we are pricing off a jagged curve, we can face the situation where today’s revaluation will be markedly different to yesterdays. This is compounded in the volatility surface as rates are generally used in the surface construction process and in any necessary yield-to-price-volatility conversions. Since we are not pricing off a smooth curve, we will not get smooth time decay (otherwise referred to as ‘theta’) and therefore we face daily fluctuations in our portfolio revaluations.
Cubic Splines
A common method of smoothing out the corners is to apply cubic splines to our interpolated curve. This method works by dividing the curve into sections and overlaying each ‘corner’ with a smooth curve defined by a cubic mathematical function. This method has the effect of smoothing out the peaks and troughs but also suffers from a form of ‘clipping’ where we actually lose information. After splines have been applied to a curve, it is generally not possible to interpolate a forward rate that exactly matches an input rate, which means we are losing accuracy.
Other Methods
Necessity is the mother of invention, and many institutions with deep pockets have developed various alternatives to the above methods. Some of these methodologies have been publicised and others are regarded as proprietary by their inventors. Continued development of yield curves tend to follow the law of diminishing returns, as even the most advanced curve construction methodology cannot hope to exactly model real-world conditions. This is because different markets work under different conventions and have differing levels of liquidity. There is no silver bullet.
Hybrid Techniques
Such techniques involve using a variety of methods to generate the curve, usually separated along the division of short end / long end, which occurs around 2 years out. For example, you could use linear interpolation for the short end, while smoothing the long end in an attempt to combine the market-matching nature of interpolation with the benefits of smoothing for longer-tenor instruments.
Generally speaking, however, there are a few rules of thumb to follow when choosing a curve methodology. Ideally, the implied forward rates given by a yield curve should be smooth. This is to filter out fluctuations when producing daily revaluations, especially for option positions. While linear interpolation may imply that rates rise and fall with sharp corners, the reality is that rate movements are a generally a continuous process. Such a process is interrupted only by occasional jumps (such as RBA intervention and irregular market movements). The curve should also be capable of returning its inputs – exactly. If an interpolated forward rate does not match the input rate of the same period, then the curve construction process is losing information. While smoothing peaks and troughs is desirable, it must be remembered that such extremes come from observed market rates.
The overall consideration should be the input data. The most sophisticated yield curve construction methodology running off bad data will under perform the simplest linear interpolation from good data. Advance yield curve methodologies are mostly of benefit to the price makers, as they have the resources to create a ‘view’ of the various spreads between markets and enter these as inputs.
