**Q2 Let the value of a firm’s equity be equal to the discounted stream of deliverable cash flows**

**(“DCF”) and assume that firm****‐**

**wide DCF grows at rate g, so Vt = DCFt+1 / (k**

**‐**

**g). Let Nt be the**

**number of shares of stock at time t, so the per****‐**

**share stock price Pt is simply Vt/Nt. Each year, the**

**company pays a proportion d of DCF to existing shareholders as dividends, and then uses the****remaining proportion (1****‐**

**d) to repurchase shares. Assume that shares are repurchased at price**

**Pt, which simultaneously determines Nt.**

**a) Derive a simple expression for the per**

**‐**

**share dividend growth rate g*, in terms of k, d and g**

**only, and give a simple interpretation (Hint: Write down expressions involving DCF, don’t be****afraid to substitute equivalent expressions, and notice that all growth rates will be constant****given the assumptions above. In particular, Nt/Nt+1 = Nt****‐**

**1/Nt).**

**b) Show algebraically that if the per**

**‐**

**share dividend growth rate g* is used, the standard**

**dividend discount model holds even in the presence of share repurchases, so that adding in****repurchases as if they were a separate payment to shareholders would actually represent****double****‐**

**counting. Specifically, show that Vt/Nt = Dt+1 / k**

**‐**

**g*.**

a) From the problem statement we have the following equation:

P

_{0}= V

_{0}/ N

_{0}= (DCF

_{1}/N

_{0}) / (k-g)

I will use g’ instead of g* as * is generally used as multiplication sign and may confusion i.e. g’ = g*

We can also drive the price per share using dividend discount method i.e. if Dividends D

_{t}grow at constant rate g’ then

P

_{0}= D

_{1}/k-g’

Also, from the problem statement D

_{t}= d*DCF

_{t}

Therefore, P

_{0}= D

_{1}/k-g’ = (d*DCF

_{1}/N

_{0}) / (k-g’)

The price per share obtained from both the methods should be same.

Therefore, k-g’ = d*(k-g) or

**g’ = g* = k - d*(k-g)**

I am not sure if the question expected me to prove that P

_{0}obtained from two methods is same or not.

However, there is an entire paper that goes on to show the equivalence of Cash Flow Method and Dividend Discount Method.

b) From the result above: d * (k-g) = (k-g’)

V

_{t}/N

_{t}= DCF

_{t+1}/(k-g) = D

_{t+1}/ d*(k-g) = D

_{t+1}/ (k-g’)

This makes sense since we assumed P

_{t}(V

_{t}/N

_{t}) is equivalent to derive g’ in the first case.

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