Perturbational bond definition

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Introduction

This security type specifies a security whose returns are calculated using the security’s risk numbers (yield to maturity, modified duration, convexity) together with changes in the underlying yield curve. The security has a perturbational type.

Security description

The perturbational approach does not require any information about the cash-flow structure of the security. Instead, the user supplies a set of risk numbers (yield to maturity, modified duration, convexity) that describe how the security’s return is affected by the passage of time and by changes in the yield curve. In other words, risk numbers are used as a proxy for an explicit pricing model that represents the relationship between interest rate changes and return.

The return over a given interval is calculated using the following expression1:

<math>r = y \cdot \delta T - MD \cdot \delta y\,</math>

where

<math>r\,</math> is the return of the security over the interval;

<math>\delta T\,</math> is the elapsed time over the interval, expressed as a fraction of a year;

<math>y\,</math> is the yield to maturity of the security;

<math>MD\,</math> is the modified duration of the security, expressed in years;

<math>\delta y\,</math> is the change in the security’s zero coupon yield at the maturity date.

The perturbational security type may be used to model any type of security whose price is driven by the underlying market’s term structure. In practice, this means virtually all fixed income securities, including money market instruments (bills, FRNs, forwards) and longer-dated securities (bonds, MBS, inflation-linked securities).

However, there are some tradeoffs required in using this security type for attribution. Although no pricing model need be specified, more data is needed to use the perturbational model, and the results may not be as accurate, particularly when cash flows are not concentrated in a single bullet payment but are instead spread out over the security’s lifetime. This is because the perturbational approach implicitly assumes that the security’s price is determined by a single cash flow at some point during its lifetime.

The perturbational approach gives the most accurate results for securities with a single large cashflow, although it typically works well for the majority of coupon-paying bonds. However, the user can also use it in other cases, such as for securities where the cash flows are unknown but we have a good estimate of its interest rate sensitivity, or for complex securitized bonds.

For mortgage-backed securities (MBS) and securitized pools of MBS, it may be preferable to used effective duration as a measure of interest-rate sensitivity than the modified duration, which includes the effects of optionality and other security-specific effects.

Perturbational attribution can even be used for entire portfolios or benchmarks, if aggregate risk data is available.

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Perturbational bonds

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Contents

• 1Introduction
• 2Security description
• 3Security code
• 4Calculation of returns
• 5Security file setup
• 6Returns file setup
• 7Example 1
• 8Example 2
• 9See also

Introduction

This security type specifies a coupon-paying security whose returns are calculated using the security’s risk numbers (yield to maturity, modified duration, convexity) together with changes in the underlying yield curve. The security is referred to as a perturbational bond.

For coupon-paying bonds that are priced directly off the underlying yield curve, refer to the definition of a BOND.

Security description

The perturbational approach does not require any information about the cash-flow structure of the security. Instead, the user supplies a set of risk numbers (yield to maturity, modified duration, convexity) that describe how the security’s return is affected by the passage of time and by changes in the yield curve. In other words, risk numbers are used as a proxy for an explicit pricing model that represents the relationship between interest rate changes and return.

The return over a given interval is calculated using the following expression1:

<math>r = y \cdot \delta t - MD \cdot \delta y\,</math>

where

<math>r\,</math> is the return of the security over the interval;

<math>\delta t\,</math> is the elapsed time over the interval, expressed as a fraction of a year;

<math>y\,</math> is the yield to maturity of the security;

<math>MD\,</math> is the modified duration of the security, expressed in years;

<math>\delta y\,</math> is the change in the security’s zero coupon yield at the maturity date.

The perturbational security type may be used to model any type of security whose price is driven by the underlying market’s term structure. In practice, this means virtually all fixed income securities, including money market instruments (bills, FRNs, forwards) and longer-dated securities (bonds, MBS, inflation-linked securities).

However, there are some tradeoffs required in using this security type for attribution. Although no pricing model need be specified, more data is needed to use the perturbational model, and the results may not be as accurate, particularly when cash flows are not concentrated in a single bullet payment but are instead spread out over the security’s lifetime. This is because the perturbational approach implicitly assumes that the security’s price is determined by a single cash flow at some point during its lifetime.

The perturbational approach gives the most accurate results for securities with a single large cashflow, although it typically works well for the majority of coupon-paying bonds. However, the user can also use it in other cases, such as for securities where the cash flows are unknown but we have a good estimate of its interest rate sensitivity, or for complex securitized bonds.

For mortgage-backed securities (MBS) and securitized pools of MBS, it may be preferable to used effective duration as a measure of interest-rate sensitivity than the modified duration, which includes the effects of optionality and other security-specific effects.

Perturbational attribution can even be used for entire portfolios or benchmarks, if aggregate risk data is available.

Security code

Perturbatonal bonds have security type PERTURBATIONAL_BOND.

Calculation of returns

No pricing is carried out for this type of security. Instead, returns are calculated directly using the expression shown above.

Security file setup

A perturbational bond is set up as follows:

Unlike a conventional bond, a perturbational bond does not require entries for coupon or coupon frequency. If these quantities are supplied, they will be ignored. Otherwise, the definitions of a conventional bond and a perturbational bond are identical, apart from the security type.

Returns file setup

A bond requires the following information in the returns file:

Unlike a conventional bond, a perturbational bond must have values supplied for fields 7 (yield to maturity) and 8 (modified duration). If no convexity is supplied in field 9, the convexity of the bond is set to zero.

Example 1

A bond is issued in AUD with a 6% coupon, maturity date 15th February 2017, paying two coupons per year. The bond has a AAA credit rating and is priced off the AUD_CURVE yield curve.

This security is represented by a single entry in the security definition file:

The Start date field is left blank, indicating that all supplied characteristics remain unchanged during the bill's lifetime.

The security has the corresponding entries in the returns file:

These records show the weight, returns, yield to maturity (YTM), modified duration (MD) and convexity (C) of the security over successive days within the STF1 portfolio.

Example 2

A perturbational bond is issued in USD with a 9% coupon, maturity date 20th March 2015, two coupons per year, and is priced off the AUD_CURVE yield curve. At issue, the bond was assigned a AA credit rating but was downgraded to AA- on 15th June 2006. This security is represented by two entries in the security definition file:

Both rows are identical except for the entries in the Start date and Credit rating column.

Any of the entries apart from the ID code may be changed during the bond’s lifetime, including security type and currency.

See also

Security type BOND