Learning Module 10: Interest Rate Risk and Return
Fixed Income
Duration Gap
\[ \text{Duration gap} = \text{Macaulay duration – Investment horizon} \]
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### Duration Gap
$$
\text{Duration gap} = \text{Macaulay duration – Investment horizon}
$$Macaulay Duration
The general calculation of Macaulay duration, \(MacDur\), that also accounts for partial coupon periods if the calculation is done between coupon dates is shown in Equation 2.
\[ MacDur = \left\{ \begin{aligned} (1 - t/T)\left[\frac{\frac{PMT}{(1+r)^{1-t/T}}}{PV^{Full}}\right] + (2 - t/T)\left[\frac{\frac{PMT}{(1+r)^{2-t/T}}}{PV^{Full}}\right] + \\ \cdots + (N - t/T)\left[\frac{\frac{PMT + FV}{(1+r)^{N-t/T}}}{PV^{Full}}\right] \end{aligned} \right\} \tag{2} \]
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
- \(t/T\) is the fraction of the coupon period that has passed since the last payment
- \(PMT\) is the coupon payment per period
- \(FV\) is the future value paid at maturity, or the par value of the bond
- \(r\) is the yield-to-maturity per period; and
- \(N\) is the number of evenly spaced periods to maturity as of the beginning of the current period
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### Macaulay Duration
The general calculation of Macaulay duration, $MacDur$, that also accounts for partial coupon periods if the calculation is done between coupon dates is shown in Equation 2.
$$
MacDur =
\left\{
\begin{aligned}
(1 - t/T)\left[\frac{\frac{PMT}{(1+r)^{1-t/T}}}{PV^{Full}}\right]
+
(2 - t/T)\left[\frac{\frac{PMT}{(1+r)^{2-t/T}}}{PV^{Full}}\right]
+ \\
\cdots +
(N - t/T)\left[\frac{\frac{PMT + FV}{(1+r)^{N-t/T}}}{PV^{Full}}\right]
\end{aligned}
\right\} \tag{2}
$$
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
- $t/T$ is the fraction of the coupon period that has passed since the last
payment
- $PMT$ is the coupon payment per period
- $FV$ is the future value paid at maturity, or the par value of the bond
- $r$ is the yield-to-maturity per period; and
- $N$ is the number of evenly spaced periods to maturity as of the beginning
of the current period Macaulay Duration: Closed-Form
\[ MacDur = \left\{ \frac{1+r}{r} - \frac{1 + r + [N \times (c - r)]}{c \times \left[(1+r)^N - 1\right] + r} \right\} - \frac{t}{T} \tag{3} \]
Where:
- \(r\) is the yield-to-maturity per period
- \(N\) is the number of evenly spaced periods to maturity as of the beginning of the current period
- \(c\) is the coupon rate per period
- \(t\) is the number of days from the last coupon payment to the settlement date and
- \(T\) is the number of days in the coupon period
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### Macaulay Duration: Closed-Form
$$
MacDur =
\left\{
\frac{1+r}{r}
-
\frac{1 + r + [N \times (c - r)]}{c \times \left[(1+r)^N - 1\right] + r}
\right\}
-
\frac{t}{T} \tag{3}
$$
Where:
- $r$ is the yield-to-maturity per period
- $N$ is the number of evenly spaced periods to maturity as of the beginning
of the current period
- $c$ is the coupon rate per period
- $t$ is the number of days from the last coupon payment to the settlement
date and
- $T$ is the number of days in the coupon period