Fixed Income Attribution With Minimum Raw Material
By: Andrew Colin
Security level approaches to fixed income attribution currently require all the information necessary to price an instrument from first principles or an external source of risk sensitivity measures. Both techniques require a supply of data for which a performance calculation oriented back office will probably not have access.
For many organizations, the sourcing and integration of this new data is one of the principle obstacles to implementing a detailed fixed income attribution capability.
However, by using recent advances in machine learning techniques, it is possible to obtain highly detailed attribution analyses with much less user-supplied information than has previously been the case.
Attribution On A Single Bond
The common feature between all fixed income attribution schemes is that they provide a link between sources of risk and performance. For instance, the main source of performance-generating risk for a vanilla corporate bond is just the bond’s yield to maturity – a single proxy measure that represents the current term structure.
Yield is a scalar-valued measure that can be decomposed into Treasury curve levels, credit spreads, and other effects such as liquidity, prepayment, and market noise. If, for example, credit spreads contract, the overall yield of the bond will fall, its price bond will rise, and extra performance will be generated by the exposure to credit spreads.
By measuring yield changes from various sources, the performance due to each source of risk may be measured. This return can then be aggregated over the entire portfolio and compared to the corresponding return from the benchmark. In this way, the skills of the portfolio manager at predicting and managing various types of exogenous risk can be measured and assessed.
Linking Risk To Return
To measure the effect of a change in yield on a bond’s price, we can either price the bond using a standard pricing formula or we can use some measure of price-interest rate sensitivity such as modified duration. These typically give results that are very close, but they both have well-documented data issues.
To calculate attribution returns, the change in yield is decomposed into a number of sub-changes such as treasury curve shift or credit spreads. This is known as first-principle pricing. The price of the security is calculated using the yield at the start of the calculation interval and then recalculated after including each successive yield change. The differences between the resulting prices can then be used to calculate the return arising from each source of risk.
Pricing of a security from first principles requires that we actually know a suitable pricing algorithm and the market’s pricing conventions. In addition, first-principle pricing requires that we have available the coupon, maturity date, coupon frequency, and other security-specific information for the bond. Obtaining this data is not easy, particularly where large benchmarks with many thousands of bonds are to be modeled, or for floating-rate notes where coupons may be reset on a monthly basis.
An alternative approach is to form a Taylor expansion for the price of the security. This provides a convenient and rapid means to calculate the effect of various changes in yield on performance as no pricing machinery is involved. However, a source of risk data is required, which has to be sourced externally. In addition, the implied linear relationship between pricing inputs may not always hold, in which case the accuracy of the analysis is likely to be poor.
Reduced to its simplest form, the common feature of both of the above approaches – and the main requirement of an attribution analysis – is simply a function that maps changes in yield (and other market- imposed quantities) to performance. This leads us to a third attribution approach.
By using machine intelligence algorithms, we can train a universal function approximator to learn the relationship between yield and price, without having to specify the form of this relationship. The algorithm considered is a neural network.
We define a neural network as an array of connected nodes. Nodes are grouped into separate layers and a network is organized as an input layer, one or more hidden layers, and an output layer. Each node in a given layer is connected to every other node in the next layer via a link, and each link has an associated weight. The information in each layer can be mapped to an ‘m’ (the number of nodes in the starting layer) and ‘n’ (the number in the ending layer) matrix. Every node has a scalar-valued activation level.
Suppose that each node in the input layer is activated to an externally set value. For instance, we might set the first and last nodes in the input layer to a level of ‘1’ and all the other nodes to a level of ‘-1.’ The signal is propagated through the network feeding the resulting signal from the hidden layer to the output layer so that the initial signal cascades through the network. This is an example of a ‘feed-forward’ network. At the end of the ‘feedforward’ process, the output node has an activation level that represents the network’s response to the initial activation vector.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -