Stock duration

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

Duration

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

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

where

$MacD_{ddm}$ is the Macaulay duration of stock under the DDM model
$r$ is the discount rate
$g$ 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:

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

The other formula for the same is - D = saa

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}}

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.

External links

Hussman Funds - Weekly Market Comment: February 23, 2004 - Buy-and-Hold For the Duration?

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Stock duration

Duration

Derivation

Modified duration

See also

External links