Stock duration

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.