8

Pricing, risk and the attribution equation

8.1 INTRODUCTION

A central requirement for fixed income attribution is the ability to translate changes in market risks into return.

Since the majority of securities can be priced using explicit formulae, one might think that this would be a straightforward task. In fact, the process can be fraught with difficulty. This chapter covers some of the more effective and widely used techniques for calculating attribution returns.

8.2 PRICING SECURITIES FROM FIRST PRINCIPLES

The fundamental way to price a security is to calculate its individual cash flows, to price them using the appropriate discount rate and to add them together. The security is priced with and without the current risk in place, and the return due to that risk is then given by the difference between the two prices, divided by the starting price.

Central banks and corporate issuers usually provide detailed documentation on the pricing of the bonds they issue. For instance, the pricing of a UK gilt bond is described in detail in UK DMO (2005).

8.3 CALCULATING RETURN USING THE PERTURBATIONAL EQUATION

Assuming that the price p of an arbitrary security is a function of time t and yield y, we can write δp in terms of a Taylor expansion to give

We expand to one term in time but two in yield. This is because virtually all securities have small second-order dependence on time, but can have appreciable second-order yield dependence.

Dividing throughout by p and defining quantities r (return), y (yield to maturity), MD (modified duration), C (convexity) as

equation (8.1) then becomes

This is the full version of the fundamental attribution equation, first introduced in equation (6.7) in Chapter 6. It is also called the perturbational equation.

The perturbational equation is essentially a proxy for any pricing function. Instead of requiring the details of how an arbitrary security is priced, it replicates the results of a returns calculation by using the security’s risk numbers: yield to maturity, modified duration and convexity.

In Table 8.1:

• P is market price;
• AI is accrued interest;
• Y is yield to maturity;
• MD is modified duration;
• C is coupon;
• r_M is market return, calculated from market price and coupon payments;
• δy is change in yield over the current interval;
• r_C is calculated return, using the expression r = y × δt − MD × δy;
• δr is the difference between market and calculated return, also called the residual for the calculation.

Table 8.1 illustrates the use of the attribution equation by calculating the market return of a UK gilt-edged security and compared it to the return calculated from the perturbational equation, omitting the convexity term.

Note that a coupon was paid on 28 November.

This calculation illustrates several practical points for attribution on individual bonds:

• It is often difficult to work out exactly when a coupon payment took place, based solely on a knowledge of the bond’s terms and conditions. However, in this case we have a value for the accrued interest. The coupon was paid at the date on which the accrued interest was reset.
• The value of the coupon is very close to the difference between the two values of the accrued interest before and after this reset. This is the value that is used for the coupon cash flow.
• No convexity value was available, so I have used the simpler expression r_C = y × δt − MD × δy. This makes little difference to the overall results.
• Even for this gilt-edged security, there is appreciable day-to-day noise in the attribution calculation. However, over time the market and calculated returns follow each other closely. Useful ways to see this are either by constructing a scatter chart of true against calculated returns, or by graphing cumulative returns over time.

Table 8.1 Attribution on a 4¼% Treasury Gilt 2049

Source: Compiled with data from the UK Debt Management Office, http://www.dmo.gov.uk

The use of expression (8.6) to perform attribution can be extremely appealing. If risk numbers for all the securities in a portfolio and benchmark are available, their use makes attribution:

• flexible;
• straightforward – the same expression can be used for all securities (but see comments below);
• simple – no pricing machinery is required;
• fast – the expression is simple to implement and calculate;
• futureproof – if new security types are introduced, then they can probably be modelled in this framework.

It is tempting to view (8.6) as a ‘one size fits all’ approach to attribution, and numerous commercial systems have been built on this basis. Unfortunately, this is not always a valid assumption:

• Some securities have other sources of return, such as inflation for TIPS (Treasury Inflation Protected Securities) and inflation-linked gilts.
• Many securities have multiple risk sensitivities. For instance, an FRN (floating rate note) has two duration measures: one for risk-free curve movements and one for credit spread movements.
• The model assumes a security with a single cashflow. For securities with large bullet payments, such as government and corporate bonds, the payment of coupons that are small relative to the final principal repayment does not affect this assumption too much. However, for sinking securities where the bond’s principal is repaid in large tranches over its lifetime, the cash flow structure is much more complex and the accuracy of modelling the bond’s return in terms of a single risk measure becomes questionable. In this case, many practitioners turn to the use of key rate durations, which increase the amounts of data needed by orders of magnitude.
• Some specialised types of securities, such as Australian and New Zealand bond futures, do not generate carry.

Any system that offers perturbational-based attribution should also offer the ability to customise the perturbational equation according to the type of security.¹

8.4 RESIDUALS

No matter how good your attribution model may be, it is unlikely to account exactly for all the market return generated by a security. The reason is that the market price of a security usually differs slightly from the theoretical price.

The difference between the true return and the theoretical return is called the residual.² Generally, a low residual is desirable and expected, both at the security and the portfolio level, but there are cases in which higher residuals are expected. For instance, pricing a corporate bond using a sovereign curve ignores all the return generated except for changes in the credit curve, and this will generate a substantial residual that is actually equivalent to credit return.

8.5 STAND-ALONE PORTFOLIOS

An important difference between top-down attribution and bottom-up attribution at the security level is that bottom-up attribution does not require a benchmark.

The Brinson attribution algorithms are expressed in terms of differences between the sector weights and returns for a portfolio and a benchmark. In contrast, bottom-up fixed income attribution may be run on a portfolio without any benchmark being present, or vice versa.

If a benchmark is available, the active returns from each source of risk may be derived in the same way. For instance, if the performance contributions to active return from each source of risk are as shown in the C_P and C_B columns in Table 8.2, the active performance contribution from each is the difference in the third column δC. The portfolio and benchmark analyses could have been run separately and only combined for this report.

Table 8.2 Absolute and relative performance contributions for simple fixed income portfolio and benchmark

Note that the requirement for a benchmark still applies if any part of the calculation is top-down. For instance, balanced attribution (Chapter 13) combines elements of both Brinson and bottom-up attribution, and so requires benchmark data.