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Stock duration

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

Contents

Duration

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

M a c D d d m = 1 + r r g

where

  • M a c D d d m 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:

    M o d D d d m = 1 r g

    The other formula for the same is - D = saa

    Derivation

    The Macaulay duration is defined as:

    ( 1 )         M a c D = i t i P V i V

    where:

  • i indexes the cash flows,
  • P V 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 = t = 1 D 0 ( 1 + g ) t ( 1 + r ) t = D 0 ( 1 + g ) r g

    The numerator in the Macaulay duration formula becomes:

    ( 3 )         i t i P V i = t = 1 t D 0 ( 1 + g ) t ( 1 + r ) t = D 0 ( 1 + g ) ( 1 + r ) + 2 D 0 ( 1 + g ) 2 ( 1 + r ) 2 + 3 D 0 ( 1 + g ) 3 ( 1 + r ) 3 + . . .

    Multiplying by 1 + r 1 + g :

    ( 4 )         1 + r 1 + g i t i P V i = D 0 + 2 D 0 ( 1 + g ) ( 1 + r ) + 3 D 0 ( 1 + g ) 2 ( 1 + r ) 2 + . . .

    Subtracting ( 4 ) ( 3 ) :

    1 + r 1 + g i t i P V i i t i P V i = D 0 + D 0 ( 1 + g ) ( 1 + r ) + D 0 ( 1 + g ) 2 ( 1 + r ) 2 + . . .

    Applying the Dividend Discount Model to the right side:

    ( 1 + r 1 + g 1 ) i t i P V i = D 0 + D 0 ( 1 + g ) r g = D 0 + V

    Simplifying:

    r g 1 + g i t i P V i = D 0 + V ( 5 )         i t i P V i = ( D 0 + V ) 1 + g r g

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

    M a c D = i = 1 n t i P V i V = ( D 0 + V ) 1 + g r g D 0 1 + g r g = D 0 + V D 0 = D 0 + D 0 1 + g r g D 0 = 1 + 1 + g r g = 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.

    References

    Stock duration Wikipedia


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