Talk:Duration (finance)

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Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 16:04, 16 January 2022 (UTC)

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):


 * $$MacD = \frac{\sum_{i=1}^{n}{t_i PV_i}} {V} = \frac{(D_0+V)\frac{1+g}{r-g}}{D_0\frac{1+g}{r-g}} = \frac{D_0+V}{D_0} = \frac{D_0+D_0\frac{1+g}{r-g}}{D_0} = 1+\frac{1+g}{r-g} = \frac{1+r}{r-g}$$