# Stock duration

Stock duration of an equity stock is the average of the times until its dividends are received, weighted by their present values.[citation needed]

## Duration

As per Dividend Discount Model: Formula for the duration of stock is as follows-

${\displaystyle MacD_{ddm}={\frac {1+r}{r-g}}}$

where

• ${\displaystyle MacD_{ddm}}$ is the Macaulay duration of stock under the DDM model
• ${\displaystyle r}$ is the discount rate
• ${\displaystyle g}$ is the expected growth rate in perpetuity

The modified duration is the percentage change in price in response to a 1% change in the long-term return that the stock is priced to deliver. Per the relationship between Macaulay duration and Modified duration:

${\displaystyle ModD_{ddm}={\frac {1}{r-g}}}$

The other formula for the same is - D = saa[citation needed]

## Derivation

The Macaulay duration is defined as:

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

where:

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

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

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

The numerator in the Macaulay duration formula becomes:

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

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

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

Subtracting ${\displaystyle (4)-(3)}$:

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

Applying the Dividend Discount Model to the right side:

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

Simplifying:

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

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

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

## Modified duration

For the stock market as a whole, the modified duration is the price/dividend ratio, which for the S&P 500 was about 62 in February 2004.[citation needed]