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Discounted cash flow and the price of capital

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Start with the idea

Discounting asks how much cash today would grow into a future payment at the required rate. A terminal value is a compact representation of a long sequence of payments; it is often the most assumption-sensitive part of a valuation.

Symbols, units & horizon

PV or V₀: value today in currency
CFₜ: cash flow at period t
k: required decimal return per period
g: perpetual decimal growth per period
T: explicit forecast horizon
TV_T: terminal value measured at time T
A: first discounted perpetuity payment
q: geometric-series ratio, below 1
EV: operating enterprise value
E: equity value
FCFF: cash flow to all capital providers
WACC: weighted average cost of capital

When and why to use this

Use discounted cash flow for fundamental forecasts, merger consideration and long-horizon price scenarios. Use the enterprise-to-equity bridge to avoid comparing the value of an entire business with only its equity claim.

Valuation relates a price today to uncertain future cash flows. Separate the cash-flow forecast from the discount rate: changing both to express the same risk can double count it. A model is useful when its assumptions and sensitivities are visible.

P V = \sum_{t = 1}^{T} \frac{C F_{t}}{(1 + k)^{t}}, T V_{T} = \frac{C F_{T + 1}}{k - g}, k > g

Algebra: geometric series

Reverse compounding; sum a growing perpetuity

From $C F_{t} = P V_{t} (1 + k)^{t}$ , divide by $(1 + k)^{t}$ . Add the discounted payments because values in today’s currency are additive.
At T, let $A = \frac{C F_{T + 1}}{(1 + k)}$ and $q = \frac{(1 + g)}{(1 + k)}$ . Then $T V = A (1 + q + q^{2} + \dots)$ . Subtract qTV from TV to obtain $T V (1 - q) = A$ .
Substitute q and simplify: $T V = \frac{C F_{T + 1}}{(k - g)}$ . The geometric series converges only when $q < 1$ , or k>g. Discount TV from T to today separately.

Work it by hand

Next year’s perpetuity payment $10, k=8%, g=2% gives 10/(.08−.02)=$166.67 at the valuation date.

Cash flow $C F_{t}$ , annual discount rate $k$ , and growth $g$ must use consistent nominal or real conventions. The terminal-value formula assumes perpetual constant growth. Its sensitivity becomes extreme when $k$ approaches $g$ .

E V_{firm} = P V (F C F F), E = E V_{firm} - debt + e x c e s s c a s h

Algebra and arithmetic

Bridge enterprise value to the equity claim

With excess cash separated from operating assets, $E V = E + debt - e x c e s s c a s h$ .
Subtract debt from both sides and add excess cash: $E = E V - debt + e x c e s s c a s h$ . EV is obtained by discounting FCFF at the appropriate capital-provider rate.

Work it by hand

Operating value $150m, debt $40m, excess cash $10m implies equity $120m; with 6m shares that is $20 per share.

Free cash flow to the firm belongs to debt and equity providers and is discounted at a weighted cost of capital. Cash flow to equity is discounted at the cost of equity. Here EV means enterprise value; in the probability lessons EV means expected value. Always read the units and context.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def dcf(cashflows, discount_rate, terminal_growth):
    """End-of-year cash flows; last grows once to form next year's payment."""
    if discount_rate <= terminal_growth or discount_rate <= -1:
        raise ValueError("Need k > g and k > -1")
    horizon = len(cashflows)
    terminal = cashflows[-1]*(1+terminal_growth)/(discount_rate-terminal_growth)
    pv = sum(cf/(1+discount_rate)**t for t, cf in enumerate(cashflows, 1))
    return pv + terminal/(1+discount_rate)**horizon

def equity_value(enterprise_value, debt, excess_cash):
    return enterprise_value - debt + excess_cash

print(equity_value(150e6, 40e6, 10e6))

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