Proof of DCF model by mathematical induction

Question

I am confident with the concept of DCF. However, I wanted to check the following proof given that if investors hold a share in $ t $, sell it at $t+1$, receive dividend $\ D_{t+1} $ and the price at $t+1$, $\ P_{t+1}$. Plus, the current value of share is the present value of all future cash flows (dividends).

Note that:

$\ D_{t+j} $ stands for dividend at period $ t+j $ $\ P_{t+j} $ stands for price at period $ t+j $

Starting from:

1) $\ P_t = \frac{D_{t+1}}{(1+k)} + \frac{P_{t+1}}{(1+k)} $

Induction Hypothesis:

2) $\ P_{t+j} = \sum\limits_{s=1}^{j} \frac{D_{t+s}}{(1+k)^s} + \frac{P_{t+j}}{(1+k)^s} $ with $j > 1 $

Induction Step:

3) $\ P_{t+j+1} = \sum\limits_{s=1}^{j+1} \frac{D_{t+s}}{(1+k)^s} + \frac{P_{t+j+1}}{(1+k)^{s+1}} $

Now, I was provided with the above proof which does not look right to me. My doubt is the following: why in 1 and 2 the price term on the right side is the present value of the price we are measuring and not the PV of the future price? The same is happening for the dividend terms in the summation. This seems inconsistent with 1 and counterintuitive.

Any help in assessing the rightness of the above proof or providing an entirely new one will be greatly appreciated! Also this is first post, happy to join the community!

Answer 1

I suspect your induction hypothesis should be an expression for $P_t$ not $P_{t+j}$ , and should be discounting $P_{t+j}$ not $s$ times but $j$ times i.e.

$$P_{t} = \left(\sum\limits_{s=1}^{j} \frac{D_{t+s}}{(1+k)^s}\right) + \frac{P_{t+j}}{(1+k)^j}$$

for $j \ge 1$, which reduces to your initial statement $P_t = \frac{D_{t+1}}{(1+k)} + \frac{P_{t+1}}{(1+k)}$ when $j=1$

Now consider your initial statement replacing $t$ by $t+j$ to give $P_{t+j}= \frac{D_{t+j+1}}{(1+k)} + \frac{P_{t+j+1}}{(1+k)}$ and substitute this into your induction hypothesis to give

$$P_{t} = \left(\sum\limits_{s=1}^{j+1} \frac{D_{t+s}}{(1+k)^s}\right) + \frac{P_{t+j+1}}{(1+k)^{j+1}}$$

which proves that the corrected induction hypothesis follows from your original statement