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4.3 Bond Duration Measures

The duration of a bond measures the sensitivity of the bond's full price (including accrued interest) to changes in the bond's yield-to-maturity or, more generally, to changes in benchmark interest rates.

Duration estimates changes in the bond price, assuming that variables other than the yield-to-maturity or benchmark rates are held constant. The duration of zero-coupon bond equals maturity, while that of a coupon bond is less than maturity.

There are several types of bond duration. In general, these can be divided into yield duration and curve duration.

  • Yield duration is the sensitivity of the bond price with respect to the bond's own yield-to- maturity. They include Macaulay duration, Modified duration, Money duration, and Price value of a basis point (PVBP)
  • Curve duration is the sensitivity of the bond price (or more generally, the market value of a financial asset or liability) with respect to a benchmark yield curve, for example effective duration, Key rate duration.

4.3.1 Yield Duration Measures

4.3.1.1 Macaulay Duration The Macaulay duration (named after Frederick Macaulay, an economist who developed the concept in 1938) is a measure of a bond's sensitivity to interest rate changes.

Technically, duration is the weighted average number of years the investor must hold a bond until the present value of the bond's cash flows equals the amount paid for the bond.

The Macaulay duration (D) formula (for the period)

Where t is the number of days from the last coupon payment to the settlement date; T is the number of days in the coupon period; PMT is the coupon payment per period; FV is par value; r is YTM/discount rate per period; and N is the number of coupon periods to maturity.

The denominator in the equation is the full price (PVFull) of the bond, including accrued interest.

Alternative Macaulay Duration (D) Formula

  • This formula is derived from the general formula using calculus.

Example: A 6% annual payment bond matures on 14 February 2022 and is purchased for settlement on 11 April 2014. The YTM is 4%. Calculate the bond's Macaulay duration (actual/actual convention):

Solution:

Period Time to Receipt CF (cash flow)

PV of CF Time-Weighted PV of CF 1 309/365 = 0.8466 6 6/(1 + 0.04)^0.8466 = 5.80 0.8466 x 5.80 = 4.91 2 1.8466 6 5.58 10.31 3 2.8466 6 5.37 15.28 4 3.8466 6 5.16 19.85 5 4.8466 6 4.96 24.05 6 5.8466 106 84.28 492.74 111.15 567.13 D = 567.13/111.15 = 5.1 years

  • If the 30/360 convention is used, the time to receipt will be 303/360 = 0.841667.

Calculating the Macaulay Duration with the Alternative Formula

  • Microsoft Excel users can obtain the Macaulay duration using the DURATION financial function: DURATION ("4/11/2014," "2/14/2022," 0.06, 0.06, 2, 0). The inputs are the settlement date, maturity date, annual coupon rate as a decimal, annual yield-to-maturity as a decimal, periodicity, and the code for the day count (0 for 30/360, 1 for actual/actual).

4.3.1.2 Modified Duration Modified duration (MD) is a direct measure of the interest rate sensitivity of a bond. It assumes that yield changes do not change the expected cash flows.

  • MD is expressed in annual terms.
  • To get the % change in bond price, the % change must be multiplied by the original bond price.

Modified duration provides a linear estimate of the percentage price change for a bond given a change in its yield-to-maturity.

The DeltaYield term is the change in the annual yield-to-maturity. The ~ sign indicates that this calculation is estimation. The minus sign indicates that bond prices and yields-to- maturity move inversely.

Approximate Modified Duration (AMD)

  • The objective of the approximation is to estimate the slope of the line tangent to the price/yield curve. The price/yield relationship is convex.
  • The smaller the change in yield, the more accurate is approximation.

Approximate Macaulay Duration (AD) from Approximate Modified Duration The approximate Macaulay duration (AD) is calculated from the approximate modified duration (AMD).

The approximation formulas produce results for annualized modified and Macaulay durations. The frequency of coupon payments and the periodicity of the yield-to- maturity are included in the bond price calculations.

4.3.1.3 Efficient Durations The difference between approximate modified duration and effective duration is in the denominator. Modified duration is a yield duration statistic in that it measures interest rate risk in terms of a change in the bond's own yield-to-maturity (DeltaYield). Effective duration is a curve duration statistic in that it measures interest rate risk in terms of a parallel shift in the benchmark yield curve (DeltaCurve).

4.3.1.4 Portfolio Duration Bonds are typically held in a portfolio.

There are two ways to calculate the duration of a bond portfolio.

  • The weighted average of time to receipt of the aggregate cash flows – This method is the theoretically correct approach, but it is difficult to use in practice.

First, the cash flow yield is not commonly calculated for bond portfolios.

Second, the amount and timing of future coupon and principal payments are uncertain if the portfolio contains callable or putable bonds or floating-rate notes. Third, interest rate risk is usually expressed as a change in benchmark interest rates, not as a change in the cash flow yield. Fourth, the change in the cash flow yield is not necessarily the same amount as the change in the yields- to-maturity on the individual bonds.

  • The weighted average of the individual bond durations that comprise the portfolio – This method is commonly used by fixed-income portfolio managers, but it has its own limitations – This measure of portfolio duration implicitly assumes a parallel shift in the yield curve. A parallel yield curve shift implies that all rates change by the same amount in the same direction. In reality, interest rate changes frequently result in a steeper or flatter yield curve.

4.3.2 Money Duration of a Bond and Price Value of a Basis Point (PVBP)

Money Duration Calculation The Money duration of a bond is a measure of the price change in units of currency in which the bond is denominated.

The money duration can be stated per 100 of par value or in terms of the actual position size of the bond in the portfolio. Money duration (MoneyDur) is calculated as follows:

The estimated change in the bond price in the currency units is calculated by the following:

DeltaPVFull ~ -MonD x DeltaYield Price Value of a Basis Point (PVBP)

The PVBP is also called the "PV01," standing for the "price value of an 01" or "present value of an 01," where "01" means 1 bp.

The Price Value of a Basis Point is an estimate of the change in the full price given a 1 bp change in the yield-to-maturity.

The PVBP is calculated as follows:

Example: Assume a T-note is priced at 99.561006 and yields 0.723368%. An increase and decrease in 1 bp results in the price changing to 99.512707 and 99.609333 respectively.

Calculate the PVBP.

Solution: PVBP = (99.609333 – 99.512707)/2 = 0.04831

  • A related statistic, sometimes called a "basis point value" (or BPV), is the money duration times 0.0001 (1 bp).
  • Another money duration statistic reported on the Bloomberg YAS (yield and spread analysis) page is "risk." It would be 4.831. Bloomberg's risk statistic is simply the PVBP (or PV01) times 100.

Lesson Wrap-Up

This lesson should leave you able to explain the bond duration measures in a fixed-income context and connect it to the decisions made by issuers, investors, or analysts.

Review Prompts

  1. Explain yield duration measures in your own words.
  2. Explain money duration of a bond and price value of a basis point (pvbp) in your own words.
  3. State one exam-style risk, valuation, or market implication of the bond duration measures.