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

In this notebook we only consider examples where the stochastic structure of the cash flows is not used. This excludes financing by cash flows (chapter 2.6.1) and default (chapter 2.8).

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 = 1, 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 2.1.

(Subscript[k, u]) = 0.2;

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

Autonomous financing or APV-formula

When all future amounts of debt are already determined at valuation time t=0, we are speaking of
autonomous financing. In this case, the firm is following a completely fixed redemption plan.

Notice that due to some limitations of Mathematica we will use a slightly different time index for debt than in the DCF-book. For more details see DCF, chapter 2.3.

Array[Dbt,  T - 1, 0];
For[t = 0, t < T, Dbt[t] = 100; t++]          

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

Financing based on market values or WACC-formula

If a firm's debt ratios and not its amount of debt are available, then autonomous financing is no
longer the case. In the case of such a leverage policy, there are three different calculation procedures, which all
lead to the same firm value. Which of the three calculations should be used by the valuator depends upon, among other things, the state of information that she has on hand. 

Notice that due to some limitations of Mathematica we will use a slightly different time index for WACC than in the DCF-book. We will use the WACC approach. For more details see DCF, chapter 2.4.

Clear[lev];  
      Array[lev,  T - 1, 0];   Array[WACC,  T - 1, 
        0]; 
For[t = 0, 
        t < T - 1, {lev[t] = 0.5,  
          WACC[t] = (1 + 
                    (Subscript[k, u])) (1 - ((τ (Subscript[r, f]) lev[t])/(1 + (Subscript[r, f])))) - 1} ; t++];  (* leverage and WACC index runs from 0 to T-1 *)

(Subscript[V, l]) = 
        Sum[FCF[t]/Product[(1 + WACC[s]),  {s, 0,  t - 1}],  {t, 1,  
            T}]

Financing based on book values (cash flow investment policy)

If the managers of a firm announce that they are planning on lowering or raising the leverage ratio, then they are normally not measuring the leverage ratio in market values, but in book values. In this case they follow a different financing policy. In order to get any further, we have to characterize the investment policy of the firm more  in detail. 

We will an investment policy based on cash flows, and we assume that there are no changes in subscribed capital.  For more details see DCF, chapter 2.5, in particular section 2.5.4.

Clear[α];  (Subscript[Y, 0]) = 150; n = 2; 
Array[α,  T + n - 1, 
      0];   α[1] = 0.5; α[2] = 0;
Array[(Subscript[lev, b]),  T + n - 1, 0];  (Subscript[lev, b])[0] = 0.5;  
        (Subscript[lev, b])[1] = 0.2;   (Subscript[lev, b])[2] = 0;  
        For[t = T, t < T + n,  (Subscript[lev, b])[t] = 0;  t++];   

Array[Inv,  n - 1,  1 - n];                                  
 For[t = 1 - n,  t ≤ 0,  Inv[t] = 0;  t++];

Array[Φ,  0,  T - 1]; 
For[t = 0,  t < T,  Φ[t] = 0;  t++];


(Subscript[V, l]) = 
      (Subscript[V, u]) + τ (Subscript[r, f]) Sum[
            (Subscript[lev, b])[t] ((((Subscript[Y, 0]) + Φ[t] - Sum[Min[n + s, t]/n Inv[s],  {s,  1 - n,  0}])/(((1 + (Subscript[r, f])))^(t + 1)))),  {t,  
              0,  T - 
                1}] + τ (Subscript[r, f]) Sum[((α[t] FCF[
                      
                      t])/(((1 + (Subscript[k, u])))^t)) (Sum[((((n - s)/n) (Subscript[lev, b])[t + s])/(((1 + (Subscript[r, f])))^(s + 1))),  {s,  0,  n - 1}]),  {t,  
              1,  T - 1}]

Financing based on dividends

It is apparent with many corporations, that the managers retain the dividends to be paid to the shareholders over a longer period of time. We assume that the management determines the distribution. If the managers subordinate the redemption of debt under the implemented policy of dividends in the way described, then we want to speak of financing based on dividends.
It does not make sense to assume that a policy of constant dividends can be carried out for a very long time. Firstly, that does not agree with the picture that can be empirically observed, and secondly, you fall into a logical contradiction if you assume that constant dividends can be paid for all eternity from a taxable, levered firm. That is why we will assume that a financing policy based on dividends is carried out only for a determined finite time of n periods. We suppose that the amount of debt remains constant beyond time t=n, and the expected cash flows of the unlevered firm grow constantly with the rate g.

For more details see DCF, section 2.6.2.

n = 2; g = 0.05; (Subscript[D, 0]) = 100; Clear[Div];  
    Div = 50;
γ = ((1 + (Subscript[r, f]) (1 - τ))/(1 + (Subscript[r, f])));  δ = ((1 + g)/(1 + (Subscript[k, u])));
(Subscript[V, l]) = (1 - (γ^n) (1 - τ (1 - 
                          (1/(((1 + (Subscript[r, f])))^(T - n)))))) 
          (Subscript[D, 0]) + (1 - (γ^n) (1 - τ (1 - 
                          (1/(((1 + (Subscript[r, f])))^(T - n))))) - τ (1 - 
                    (1/(((1 + (Subscript[r, f])))^T)))) 
          (Div/((Subscript[r, f]) (1 - τ))) + ((δ^n) - (δ^T) + (((γ^n) - (δ^n))/((γ/δ) - 1)) (((Subscript[k, u]) - 
                      g)/(1 + 
                      g)) (1 - τ (1 - 
                          (1/(((1 + (Subscript[r, f])))^(T - n)))))) 
          ((Subscript[V, u])/(1 - (δ^T)))

Financing based on debt-cash flow ratio

Der dynamische Verschuldungsgrad ist eine Kennzahl, bei der der Cash flow ins Verhältnis zu den Schulden des Unternehmens gesetzt wird. Diese Kennzahl dient als (vereinfachter) Maßstab für die Zeitdauer, in der das Unternehmen bei ausschließlicher Verwendung des Cashflows zur Schuldentilgung völlig eigenfinanziert werden würde. Wir wollen diese Kennzahl verwenden, um eine sechste Finanzierungspolitik zu untersuchen.

For more details see DCF, section 2.6.3.

Array[(Subscript[L, d]),  T];  
    For[t = 1, t ≤ T,  
      (Subscript[L, d])[t] = 
        0.5;  t++];

(Subscript[V, l]) = 
      (Subscript[V, u]) + Dbt[0] 
          Sum[(((((τ (Subscript[r, f]))/(1 + (Subscript[r, f])))))^(s + 1)) 
              Product[(Subscript[L, d])[u],  {u,  1,  s}],  {s,  0,  T - 1}] + 
        Sum[FCF[s]/(((1 + (Subscript[k, u])))^s) 
            Sum[(((((τ (Subscript[r, f]))/(1 + (Subscript[r, f])))))^(u + 1 - s)) 
                Product[(Subscript[L, d])[w],  {w,  u,  s}],  {u,  s,  
                T - 1}],  {s,  1,  T - 1}]