DCF Example (finite case, personal tax) by Kruschwitz/Löffler

Parameters of the Model

We are dealing here with an (unlevered) company,which will bear three years of payments to its owners.They, however, cannot be predicted with certainty. We know that the firm will live for T years and the free cash flow after tax will form a geometrical series starting at (Subscript[C, 1]): The free cash flow of year 1 will be (Subscript[C, 1]), the free cash flow of year 2 will be (1 + g) (Subscript[C, 1])etc.

T = 3;  (Subscript[C, 1]) = 100;  g = 0.1;  τ = 0.5;  
    (Subscript[r, f]) = 0.1;
Array[FCF,  T];

For[t = 0, t ≤ T, 
      FCF[t] = (((1 + g))^(t - 1)) (Subscript[C, 1]); t++]

Valuation concept: the unlevered company

To value the firm we need to know the cost of capitel of the (unlevered) firm. These cost of capital are expected returns of holding the firm's share for one period. This allows us to value the unlevered company.

Notice that due to some limitations of Mathematica we will use a slightly different notation for the cost of capital than in the DCF-book. For more details see DCF, chapter 3.1.

(Subscript[k, u]) = 0.15;

    
(Subscript[V, u]) = Sum[FCF[t]/(((1 + (Subscript[k, u])))^t),  {t,  1,  T}]

Autonomous retention

With autonomous retention, the retetntion is deterministic.

Notice that due to some limitations of Mathematica we will use a slightly different notation for the cost of capital than in the DCF-book. For more details see DCF, chapter 3.3.1.

Array[A,  T - 1, 0];
A[0] = 10; A[1] = 20; A[2] = 0;
(Subscript[V, l]) = 
      (Subscript[V, u]) + (1 - τ) A[0] + 
        Sum[((τ (1 - τ) (Subscript[r, f]) A[t])/(((1 + (Subscript[r, f]) (1 - τ)))^(t + 1))),  {t, 0,  T - 1}]

Retention based on cash flows

A firm is following an earnings retention policy based on cash flows if the retention is a determinate multiple of the free cash flow of the unlevered firm.

Clear[α];  
    Array[α,  T - 1, 0];
For[t = 0, t < T, α[t] = 0; t++];        (* α runs from 0 to T - 1 *)
α[0] = 0;  α[1] = 0.1; α[2] = 0.2;

(Subscript[V, l]) = 
      (Subscript[V, u]) + (((1 + (Subscript[r, f])) (1 - τ) α[0] FCF[0])/(1 + (Subscript[r, f]) (1 - τ))) + ((τ (Subscript[r, f]) (1 - τ))/(1 + (Subscript[r, f]) (1 - τ))) Sum[((α[t] FCF[t])/(((1 + (Subscript[k, u])))^t)),  {t, 1,  T - 1}]

Retention based on dividends

A firm is following a retention policy based on dividends if a prescribed certain dividend before taxes should be payed in the first n<T periods.

Clear[Div];  Array[Div,  1, T - 1];  
    n = 2;
(Subscript[A, 0]) = 0;  Div[1] = 40;  
    Div[2] = 40;

(Subscript[V, l]) = 
      (Subscript[V, u]) + (1 - τ + (((((((1 + (Subscript[r, f]))/(1 + (Subscript[r, f]) (1 - τ)))))^(n - 2)) - 1)/((Subscript[r, f]) τ))) (Subscript[A, 0]) + τ (Subscript[r, f]) 
          Sum[(FCF[t]/(((1 + (Subscript[k, u])))^t) - (((1 - τ) Div[t])/(((1 + (Subscript[r, f]) (1 - τ)))^t))) (1 + (((((1 + (Subscript[r, f]))/(1 + (Subscript[r, f]) (1 - τ)))))^(n + 1 - t))),  {t, 1,  
              n}]

Retention based on market values

The firm is following a retention policy based on market values if the relation of retention and value of the firm.

Clear[lev];  
    Array[lev, 1, T - 1];   Array[k,  1, T];
For[t = 0, 
          t ≤ T - 1, {lev[t] = 0.1,  
            k[t] = (1 + 
                      (Subscript[k, u])) (1 - (((1 + (Subscript[r, f]))  (1 - τ) lev[t])/(1 + (Subscript[r, f]) (1 - τ)))) - 1} ; t++];  (* retainment and k index run from 0 to T-1 *)
(Subscript[V, l]) = 
      Sum[((Product[(1 - (1 - τ) lev[h]),  {h,  1,  s - 1}] FCF[s])/Product)[(1 + k[h]),  {h, 0,  s - 1}],  {s, 1,  
          T}]