Talk:Duration (finance)

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Macaulay duration derivation

This section was removed from the article on Stock duration as being inappropriate there, and better placed here. But I'll leave it to the editors here to decide where to include it. Sbalfour (talk) 17:01, 14 March 2022 (UTC)

Derivation

The Macaulay duration is defined as:

$(1)\ \ \ \ MacD={\frac {\sum _{i}{t_{i}PV_{i}}}{V}}$

where:

• $i$ indexes the cash flows,
• $PV_{i}$ is the present value of the $i$th cash payment from an asset,
• $t_{i}$ is the time in years until the $i$th payment will be received,
• $V$ is the present value of all future cash payments from the asset.

The present value of dividends per the Dividend Discount Model is:

$(2)\ \ \ \ V=\sum _{t=1}^{\infty }{D_{0}}{\frac {(1+g)^{t}}{(1+r)^{t}}}={\frac {D_{0}(1+g)}{r-g}}$

The numerator in the Macaulay duration formula becomes:

$(3)\ \ \ \ \sum _{i}t_{i}PV_{i}=\sum _{t=1}^{\infty }t{D_{0}}{\frac {(1+g)^{t}}{(1+r)^{t}}}=D_{0}{\frac {(1+g)}{(1+r)}}+2{D_{0}}{\frac {(1+g)^{2}}{(1+r)^{2}}}+3{D_{0}}{\frac {(1+g)^{3}}{(1+r)^{3}}}+...$

Multiplying by ${\frac {1+r}{1+g}}$:

$(4)\ \ \ \ {\frac {1+r}{1+g}}\sum _{i}t_{i}PV_{i}=D_{0}+2{D_{0}}{\frac {(1+g)}{(1+r)}}+3{D_{0}}{\frac {(1+g)^{2}}{(1+r)^{2}}}+...$

Subtracting $(4)-(3)$:

${\frac {1+r}{1+g}}\sum _{i}t_{i}PV_{i}-\sum _{i}t_{i}PV_{i}=D_{0}+D_{0}{\frac {(1+g)}{(1+r)}}+D_{0}{\frac {(1+g)^{2}}{(1+r)^{2}}}+...$

Applying the Dividend Discount Model to the right side:

$\left({\frac {1+r}{1+g}}-1\right)\sum _{i}t_{i}PV_{i}=D_{0}+{\frac {D_{0}(1+g)}{r-g}}=D_{0}+V$

Simplifying:

${\frac {r-g}{1+g}}\sum _{i}t_{i}PV_{i}=D_{0}+V$

$(5)\ \ \ \ \sum _{i}t_{i}PV_{i}=(D_{0}+V){\frac {1+g}{r-g}}$

Combining (1), (2) and (5):

$MacD={\frac {\sum _{i=1}^{n}{t_{i}PV_{i}}}{V}}={\frac {(D_{0}+V){\frac {1+g}{r-g}}}{D_{0}{\frac {1+g}{r-g}}}}={\frac {D_{0}+V}{D_{0}}}={\frac {D_{0}+D_{0}{\frac {1+g}{r-g}}}{D_{0}}}=1+{\frac {1+g}{r-g}}={\frac {1+r}{r-g}}$