# Macaulay Duration

This is an advanced page in Fixed Income. If you are unfamiliar with any of the terms, you can refer to the Fixed Income Glossary.

The Macaulay duration of a bond is the weighted average maturity of cash flows, which acts as a measure of a bond's sensitivity to interest rate changes. Bonds with a higher duration will carry more risk, and hence have a greater volatility in prices, when compared to bonds with lower durations.

There are many ways to calculate duration, and the Macaulay duration is the most common due to its simplicity.

## Macaulay Duration

The Macaulay Duration is given by

\[ MacD = \frac{ \frac{ n \times F } { ( 1 + y_n ) ^ n } + \sum_{i = 1 } ^ n \frac{ i \times C_i } { ( 1 + y_i) ^ i} } { P }, \]

where \(F\) is the face value of the bond,

\(C_i\) is the coupon payments at time \( i \),

\( y_i \) is the yield rate at time \( i \),

P is the price of the bond.Note: For practical purposes, we often use the yield to maturity \(y\), if the values of \( y_i \) are not known.

If we let \( PV_i \) be the present value of the cashflow at time \( i \), then we have

\[ P = \sum_{i = 1 } ^ T PV_i \Rightarrow 1 = \sum_{i = 1 } ^ T \frac{ PV_i } { P } . \]

Then, the Macaulay Duration can be written as

\[ MacD = \sum_{i = 1 } ^ T i \frac { PV_i } { P }. \]

As such, the Macaulay duration is the weighted average maturity of the cash flows. From the definition of weighed average, if the positive cash flows occur at times \( t_i \), then we must have \( t_ 1 \leq MacD \leq t_n \). In particular, \( Mac D = t_n \) if and only if the bond is a zero-coupon bond. This makes sense, since that is the amount of time that it would take for the

What is the Macaulay Duration for a 5 year bond with fixed coupon payments of 5% and a face value of $1000, if the current price is $1100?

First, we have to find the yield to use. We will use the Yield to Maturity. Solve for \(y \) satisfying

\[ 1100 = \frac{ 50} { (1+y) ^ 1 } + \frac{ 50} { (1+y) ^ 2 } + \frac{ 50} { (1+y) ^ 3 } + \frac{ 50} { (1+y) ^ 4 } + \frac{ 1050} { (1 + y) ^ 5 } \]

this gives us \( y = 2.82 \% \).

Next, we substitute this into the formula for MacD, obtaining

\[ \begin{align} MacD & = \frac{ 1 \times \frac{ 50} { (1+y) ^ 1 } + 2 \times \frac{ 50} { (1+y) ^ 2 } + 3 \times \frac{ 50} { (1+y) ^ 3 } + 4 \times \frac{ 50} { (1+y) ^ 4 } + 5 \times \frac{ 1050} { (1 + y) ^ 5 } } { 1100 } \\ & = 4.571 . \\ \end{align} \]

**Cite as:**Macaulay Duration.

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