Learning Module 11: Yield-Based Bond Duration Measures and Properties

Fixed Income

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Modified Duration (ModDur)

Using Macaulay Duration (equation 2) from Learning module 10

Recall that the price, \(PV\), of an option-free bond is the present value of the bond’s cash flows.

\[ PV = \frac{PMT}{(1+r)^1} + \frac{PMT}{(1+r)^2} + \ldots + \frac{PMT}{(1+r)^N} + \frac{FV}{(1+r)^N} \]

First, we take the derivative with respect to \(r\), the yield-to-maturity:

\[ \frac{dPV}{dr} = \frac{(-1)PMT}{(1+r)^2} + \frac{(-2)PMT}{(1+r)^3} + \ldots + \frac{(-N)PMT}{(1+r)^{N+1}} + \frac{(-N)FV}{(1+r)^{N+1}} \]

Then, we factor out \(-\frac{1}{(1+r)}\) to get

\[ \frac{dPV}{dr} = -\frac{1}{(1+r)} \left[ \frac{(1)PMT}{(1+r)^1} + \frac{(2)PMT}{(1+r)^2} + \ldots + \frac{(N)PMT}{(1+r)^N} + \frac{(N)FV}{(1+r)^N} \right] \]

While this gives us the change in a bond’s price for a change in yield, it does so in terms of \(PV\), but percentage change in price would be more useful. To get percentage change, we divide by \(PV\) (price):

\[ \frac{\frac{dPV}{dr}}{PV} = \frac{ -\frac{1}{(1+r)} \left[ \frac{(1)PMT}{(1+r)^1} + \frac{(2)PMT}{(1+r)^2} + \ldots + \frac{(N)PMT}{(1+r)^N} + \frac{(N)FV}{(1+r)^N} \right] }{PV}. \]

Look closely at the term in brackets. Each \(\frac{PMT}{(1+r)^t}/PV\) is the present value of that cash flow expressed as a percentage of the bond price, which is then multiplied by the time to receipt of that cash flow. In other words, the term in brackets divided by \(PV\) is the Macaulay duration, \(MacDur\), introduced in prior lessons. We can substitute \(MacDur\) in the equation to obtain

\[ \frac{\frac{dPV}{dr}}{PV} = -\frac{1}{(1+r)} \times \text{MacDur} \]

or

\[ \frac{\frac{dPV}{dr}}{PV} = -\frac{MacDur}{(1+r)} \tag{1} \]

Without the negative sign, this is known as a bond’s modified duration, or \(ModDur\):

\[ ModDur = \frac{MacDur}{(1+r)} \tag{2} \]

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### Modified Duration (ModDur)

*Using Macaulay Duration (equation 2) from Learning module 10*

Recall that the price, $PV$, of an option-free bond is the present value of 
the bond’s cash flows.

$$
PV = \frac{PMT}{(1+r)^1} + \frac{PMT}{(1+r)^2} + \ldots + 
\frac{PMT}{(1+r)^N} + \frac{FV}{(1+r)^N}
$$

First, we take the derivative with respect to $r$, the yield-to-maturity:

$$
\frac{dPV}{dr}
=
\frac{(-1)PMT}{(1+r)^2}
+
\frac{(-2)PMT}{(1+r)^3}
+
\ldots
+
\frac{(-N)PMT}{(1+r)^{N+1}}
+
\frac{(-N)FV}{(1+r)^{N+1}}
$$

Then, we factor out $-\frac{1}{(1+r)}$ to get

$$
\frac{dPV}{dr}
=
-\frac{1}{(1+r)}
\left[
\frac{(1)PMT}{(1+r)^1}
+
\frac{(2)PMT}{(1+r)^2}
+
\ldots
+
\frac{(N)PMT}{(1+r)^N}
+
\frac{(N)FV}{(1+r)^N}
\right]
$$

While this gives us the change in a bond’s price for a change in yield, it 
does so in terms of $PV$, but percentage change in price would be more useful. 
To get percentage change, we divide by $PV$ (price):

$$
\frac{\frac{dPV}{dr}}{PV}
=
\frac{
-\frac{1}{(1+r)}
\left[
\frac{(1)PMT}{(1+r)^1}
+
\frac{(2)PMT}{(1+r)^2}
+
\ldots
+
\frac{(N)PMT}{(1+r)^N}
+
\frac{(N)FV}{(1+r)^N}
\right]
}{PV}.
$$

Look closely at the term in brackets. Each $\frac{PMT}{(1+r)^t}/PV$ is the 
present value of that cash flow expressed as a percentage of the bond price, 
which is then multiplied by the time to receipt of that cash flow. In other 
words, the term in brackets divided by $PV$ is the Macaulay duration, 
$MacDur$, introduced in prior lessons. We can substitute $MacDur$ in the 
equation to obtain

$$
\frac{\frac{dPV}{dr}}{PV} = -\frac{1}{(1+r)} \times \text{MacDur}
$$

or

$$
\frac{\frac{dPV}{dr}}{PV} = -\frac{MacDur}{(1+r)} \tag{1}
$$

Without the negative sign, this is known as a bond’s modified duration, or 
$ModDur$:

$$
ModDur = \frac{MacDur}{(1+r)} \tag{2}
$$

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Estimate the percentage price change for a bond

\[ \%\Delta PV^{\text{Full}} \approx -\text{AnnModDur} \times \Delta \text{AnnYield} \tag{3} \]

• Since \(ModDur\) captures the relationship between a bond’s price and its yield, we can use it to estimate the percentage price change for a bond given a change in its yield-to-maturity, if we substitute \(-AnnModDur\) for the right side of Equation 1 and multiply both sides by \(dr\), or the change in annualized yield-to-maturity

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### Estimate the percentage price change for a bond

$$
\%\Delta PV^{\text{Full}} \approx -\text{AnnModDur} \times 
\Delta \text{AnnYield} \tag{3}
$$


- Since $ModDur$ captures the relationship between a bond’s price and its 
  yield, we can use it to estimate the percentage price change for a bond 
  given a change in its yield-to-maturity, if we substitute $-AnnModDur$ for 
  the right side of Equation 1 and multiply both sides by $dr$, or the change 
  in annualized yield-to-maturity

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Annualized Modified Duration

\[ AnnModDur \approx \frac{(PV_{-}) - (PV_{+})}{2 \times (\Delta Yield) \times (PV_0)} \tag{4} \]

• \(PV_0\) = The quoted full price of the bond
  
• \(\Delta Yield\) = Changed in yield-to-maturity
  
• To estimate the slope, the yield-to-maturity is changed up and down by the same amount—the \(\Delta Yield\)—and is used to calculate corresponding bond prices \(PV_{+}\) and \(PV_{-}\). We can use these variables to find the slope of the line tangent to the price-yield curve: the difference between \(PV_{+}\) and \(PV_{-}\) divided by twice the assumed change in the yield-to-maturity. To find the slope in terms of percentage change in \(PV_0\), we further divide by \(PV_0\). This is shown as Equation 4.

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### Annualized Modified Duration

$$
AnnModDur \approx 
\frac{(PV_{-}) - (PV_{+})}{2 \times (\Delta Yield) \times (PV_0)}
\tag{4}
$$

- $PV_0$ = The quoted full price of the bond  
- $\Delta Yield$ = Changed in yield-to-maturity  
- To estimate the slope, the yield-to-maturity is changed up and down by the 
  same amount—the $\Delta Yield$—and is used to calculate corresponding bond 
  prices $PV_{+}$ and $PV_{-}$. We can use these variables to find the slope 
  of the line tangent to the price-yield curve: the difference between 
  $PV_{+}$ and $PV_{-}$ divided by twice the assumed change in the 
  yield-to-maturity. To find the slope in terms of percentage change in 
  $PV_0$, we further divide by $PV_0$. This is shown as Equation 4.

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Annualized Modified Duration

\[ AnnModDur \approx AnnModDur \times (1+ r) \tag{5} \]

• The Macaulay duration also can be approximated by multiplying the approximate modified duration by 1 plus the yield per period.

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### Annualized Modified Duration
$$
AnnModDur \approx AnnModDur \times (1+ r) \tag{5}
$$

- The Macaulay duration also can be approximated by multiplying the 
  approximate modified duration by 1 plus the yield per period.

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Money Duration (MoneyDur)

\[ MoneyDur = AnnModDur \times PV^{Full}. \]

• \(MoneyDur\) is the product of the annualized modified duration and the full price (\(PV^{Full}\)) of the bond, in either percent of par or the currency value of the position.

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### Money Duration (MoneyDur)

$$
MoneyDur = AnnModDur \times PV^{Full}.
$$

- $MoneyDur$ is the product of the annualized modified 
  duration and the full price ($PV^{Full}$) of the bond, in either percent 
  of par or the currency value of the position.

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Estimated change in the bond price in currency units

\[ \Delta PV^{\text{Full}} \approx - \text{ MoneyDur} \times \Delta \text{Yield} \tag{7} \]

• The estimated change in the bond price in currency units is very similar to Equation 4. The difference is that for a given change in the annual yield-to-maturity (\(\Delta Yield\)), modified duration estimates the percentage price change while money duration estimates the change in currency units.

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###  Estimated change in the bond price in currency units

$$
\Delta PV^{\text{Full}} \approx - \text{ MoneyDur} \times \Delta 
\text{Yield} \tag{7}
$$

- The estimated change in the bond price in currency units is very similar to 
  Equation 4. The difference is that for a given change in the annual 
  yield-to-maturity ($\Delta Yield$), modified duration estimates the 
  percentage price change while money duration estimates the change in 
  currency units.

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Price Value of a Basis Point (PVBP)

\[ PVBP = \frac{(PV_{-}) - (PV_{+})}{2} \tag{8} \]

• \(PVBP\) = an estimate of the change in the full price of a bond given a 1 bp change in its yield-to-maturity
  
• 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
  
• \(PV_{-}\) and \(PV_{+}\) are the full prices calculated by decreasing and increasing the yield-to-maturity by 1 bp
  
• The PVBP is particularly useful for bonds for which future cash flows are uncertain, such as callable bonds.

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### Price Value of a Basis Point (PVBP)

$$
PVBP = \frac{(PV_{-}) - (PV_{+})}{2} \tag{8}
$$


- $PVBP$ = an estimate of the change in the full price of a bond given a 1 bp 
  change in its yield-to-maturity  
- 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  
- $PV_{-}$ and $PV_{+}$ are the full prices calculated by decreasing and 
  increasing the yield-to-maturity by 1 bp  
- The PVBP is particularly useful for bonds for which future cash flows are 
  uncertain, such as callable bonds.  

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Macaulay Duration: Non-callable perpetuities (MacDur)

\[ \text{MacDur} = \frac{(1+r)}{r} \tag{9} \]

• A perpetuity or perpetual bond is a bond that does not mature, so there is no face or maturity value received at time T. The investor receives a fixed coupon payment forever unless the bond is called. Non-callable perpetuities are rare, but they have an interesting Macaulay duration

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### Macaulay Duration: Non-callable perpetuities (MacDur)

$$
\text{MacDur} = \frac{(1+r)}{r} \tag{9}
$$

- A perpetuity or perpetual bond is a bond that does not mature, so there is 
  no face or maturity value received at time T. The investor receives a fixed 
  coupon payment forever unless the bond is called. Non-callable perpetuities 
  are rare, but they have an interesting Macaulay duration

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Macaulay duration for a floating-rate note or bond (\(MACDur_{Floating}\))

\[ MacDur_{Floating} = \frac{(T - t)}{T} \tag{10} \]

• As described in an earlier lesson, interest on floating-rate instruments varies depending on the level of a market reference rate (\(MRR\)) plus a quoted margin. At predetermined dates, payment amounts are reset to reflect changes in the \(MRR\). Therefore, interest rate risk arises only between reset dates, because at the next reset date, coupon payments will adjust to the new \(MRR\). Therefore, the Macaulay duration for a floating-rate note or bond is simply the fraction of a period remaining until the next reset date
  
• If there are \(180\) days in the coupon period and \(57\) days have passed since the last coupon, the Macaulay duration is
  • \(MacDur_{Floating} = \frac{(180 - 57)}{180} = 0.683333\)
    
• Floating-rate instruments typically have very low duration because coupon periods are typically less than six months in length. As a result, they are commonly used by investors to reduce duration in fixed-income portfolios.

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### Macaulay duration for a floating-rate note or bond ($MACDur_{Floating}$)

$$
MacDur_{Floating} = \frac{(T - t)}{T} \tag{10}
$$

- As described in an earlier lesson, interest on floating-rate instruments 
  varies depending on the level of a market reference rate ($MRR$) plus a 
  quoted margin. At predetermined dates, payment amounts are reset to reflect 
  changes in the $MRR$. Therefore, interest rate risk arises only between 
  reset dates, because at the next reset date, coupon payments will adjust to 
  the new $MRR$. Therefore, the Macaulay duration for a floating-rate note or 
  bond is simply the fraction of a period remaining until the next reset date  
- If there are $180$ days in the coupon period and $57$ days have passed since 
  the last coupon, the Macaulay duration is  
  - $MacDur_{Floating} = \frac{(180 - 57)}{180} = 0.683333$  
- Floating-rate instruments typically have very low duration because coupon 
  periods are typically less than six months in length. As a result, they are 
  commonly used by investors to reduce duration in fixed-income portfolios.  

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