Macaulay duration is a measure of sensitivity of cash flows, to interest rates. This is simply the weighted average of the terms of the cash flows, the weights being the present values of the cash flows. This is generally represented by the Greek letter \(\tau (Tau)\) .

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Nominal(face) value of the bond:    

Annual couopn rate: (%)  

Redemption amount:    

Annual effective rate of interest: (%)  

Term of the bond: (years)


Macaulay duration or DMT:   

Let \(c_{t_k}\) be a series of cash flows at times \(t_k\) , where \(k=1,2,3\dots .\). Here, \(t\) is the term of the cash flow and \(c_{t_k}{\left(1+i\right)}^{{-t}_k}\) is the present value, at rate \(i\), of a cash flow occurring at time \(t_k\).

Since DMT is the weighted average of the terms, the formula is
\[\tau =\frac{t_1c_{t_1}{\left(1+i\right)}^{-1}+t_2c_{t_2}{\left(1+i\right)}^{-2}\dots \dots \dots t_nc_{t_n}{\left(1+i\right)}^{-n}}{c_{t_1}{\left(1+i\right)}^{-1}+c_{t_2}{\left(1+i\right)}^{-2}\dots \dots \dots c_{t_n}{\left(1+i\right)}^{-n}}\] \[\tau =\frac{\sum^n_{k=1}{t_kc_{t_k}}{\left(1+i\right)}^{{-t}_k}}{\sum^n_{k=1}{c_{t_k}}{\left(1+i\right)}^{{-t}_k}}\]
DMT can also be represented in terms of the effective duration or volatility of the same set of cash flows, as follows:-
\[\tau =\nu\left(i\right)\times \left(1+i\right)\]
Where, \(\nu(i)\) is the effective duration of the cash flows.

Analysing DMT:

A series of cash flows with a higher DMT than another series would always be more sensitive to interest rates. So, the present value of a series of cash flows would have very high fluctuations even for small changes in interest rates.

Example 1

The amount of each coupon payment would be \(\textrm{₹}10\) , i.e. \((10\%\ of\ 100)\). The present value of each coupon payment is \(10{\left(1+i\right)}^{{-t}^k}\) and the present value of the final redemption amount would be \(100{\left(1+i\right)}^{\left(-10\right)}\) .

Since DMT is the weighted average of the terms it can be calculated as follows:-
\[\tau =\frac{\left[1\times 10{\left(1+i\right)}^{-1}+\dots \dots \dots 10\times 10{\left(1+i\right)}^{-10}\right]+10\times 100{\left(1+i\right)}^{-10}}{\left[10{\left(1+i\right)}^{-1}+10{\left(1+i\right)}^{-2}+\dots \dots \dots 10{\left(1+i\right)}^{-10}\right]+100{\left(1+i\right)}^{-10}}\] \[\tau =\frac{\left[\sum^{10}_{t=1}{t.10}{\left(1+i\right)}^{-t}\right]+\left[10\times 100{\left(1+i\right)}^{-t}\right]}{\left[\sum^{10}_{t=1}{10}{\left(1+i\right)}^{-t}\right]+\left[100{\left(1+i\right)}^{-t}\right]}\]
Using annuity notations, we can write this as follows:-
\[\tau =\frac{10{\left(Ia\right)}_{\overline{10|}}+1000v^{10}}{10a_{\overline{10|}}+100v^{10}}\]
Solving this at rate of interest \(10\%\) , the DMT for the above bond would therefore be \(6.76\).

It can be noted in the above example that the denominator in the formula was simply the price of the bond (present values of coupons plus the present value of the redemption amount). (Click here to know about bond pricing.)

So, the formula of DMT for a fixed interest bond with annual coupon \(D\) for \(n\) years and redemption amount \(R\) is
\[\tau =\frac{D{\left(Ia\right)}_{\overline{n|}}+R.n.v^n}{Da_{\overline{n|}}+Rv^n}\]
(Click here to know about annuities and annuity notations.)

Similarly the formula for DMT of a zero coupon bond can be written as follows:
\[\tau =\frac{R.n.v^n}{Rv^n}=n\]
Again, the denominator in the above formula represents the price of the zero-coupon bond. (Click here to know about zero coupon bnd pricing.) The DMT of a zero-coupon is simply the term of the bond, n. This is because, the average term of a series of cash flows consisting of only one cash flow (the final redemption payment) is obviously the term of that single cash flow.