Equity Intrinsic Value Using Constant-Growth and Multistage Present Value Models

Company Life Cycle and Valuation

The equity valuation process involves:

• Forecasting cash flows using constant or multi-stage growth assumptions.
• Selecting an appropriate discount rate.
• Estimating a terminal value for cash flows beyond the explicit forecast period.

The usefulness of present value models depends on a firm’s life-cycle stage. Startups are difficult to value with these models because they often have little revenue, net losses, and a high probability of failure. Present value models become more applicable once a company has achieved initial success and entered the high-growth, mature, or declining stages, where cash flows are more predictable.

Constant Growth Model

This model is best suited for mature, well-established firms that have entered a stable phase of their life cycle. These companies typically share these characteristics:

• Stable Cash Flows – These companies generate predictable cash flow streams year after year.
• Lower Reinvestment Needs – Mature firms often have fewer new projects or growth opportunities.
• Higher Payouts – They have excess cash available, which is often returned to shareholders in the form of dividends or share buybacks.

The constant growth model calculates the Intrinsic Value \((IV)\) by treating its future cash flows as a stream of cash that lasts forever, growing at a fixed rate \((g)\). It uses the formula:

$$ IV_t = \frac{E(CF)_{t+1}}{r_e – g} $$

Where:

• \(E(CF)_{t+1}\) – Expected cash flow for the next period.
• \(r_e\) – Required rate of return.
• \(g\) – Constant growth rate.

The model only works if the growth rate is less than the discount rate. It can still be used with a negative growth rate. Additionally, the growth rate should not be higher than the growth rate of the overall economy. This is because no company can grow faster than the economy forever. A realistic long-term growth rate is often approximated using the economy’s nominal risk-free rate.

Example 1: Company Life Cycle and Valuation

An analyst is valuing the common stock of Bluetel Ltd., a mature company with stable cash flows. The company has just paid a Free Cash Flow to Equity (FCFE) of $4.80 per share. The analyst expects FCFE to grow at a constant rate of 5% per year indefinitely. The required rate of return on equity is 11%. The intrinsic value of Bluetel Ltd.’s stock is closest to?

The company just paid an FCFE of $4.80, this is \(CF_t\).

Calculate next year’s expected \(FCFE: CF_t= \$4.80 \times 1.05=\$5.04\)

Apply the Constant Growth Model:

$$ IV_t= \frac {\$5.04}{(11\%-5\%)}=\$84.00 $$

Implied Growth Rate

One can find the implied growth rate of a company by rewriting the constant growth valuation model. We take the current market price and figure out what growth rate the market is currently pricing into that price. This gives you the implied growth rate \((g_{\text{implied}})\):

$$ (g_\text{implied}) = r_e – \frac{E(CF)_{t+1}}{P_t} $$

Where:

• \(r_e\) – Required rate of return
• \(E(CF)_{t+1}\) – Expected cash flow for the next period
• \(P_t\) – Current price

The constant growth model is a massive oversimplification. Most companies experience shifting growth dynamics due to competition, technological shifts, and market forces. Analysts bypass this by using multi-stage models to forecast volatile growth over a set period before applying a stable constant growth rate at the end.

Example 2: Implied Growth Rate

An analyst is evaluating the market expectations for ABC Corporation, a mature company with stable cash flows. The stock currently trades at $60.00 per share. The analyst estimates that the company’s Free Cash Flow to Equity (FCFE) for the next year will be $3.60 per share, and the required rate of return on equity is 10%. Based on the constant growth model, the implied constant growth rate is closest to?

The implied growth rate is calculated as:

\begin{align*}
(g_\text{implied}) &= r_e – \frac{E(CF)_{t+1}}{P_t} \\
(g_\text{implied}) &= 10\% – \frac{3.60}{60} = 4\%
\end{align*}

The implied growth rate is 4%.

Multistage Growth Model

Multistage growth models are used when a firm’s expected cash flow growth varies over time, e.g., a rapid expansion phase followed by a transition to stable growth. A company can experience a single high growth rate for n periods, or multiple different rates, before settling into a stable, constant long-term growth rate. Once a firm reaches its stable growth phase, a terminal value is calculated to capture all remaining future cash flows.

Two-stage Growth Model

The simplest multistage model assumes an initial finite period \((n)\) of constant high growth \((g_s)\), followed by an indefinite period of lower, sustained long-term growth \((g_l)\).

The two-stage growth model calculates the intrinsic value as:

$$IV_t = \sum_{t=1}^{n} \frac{FCFE_t(1+g_s)^i}{(1+r_e)^t} + \frac{IV_{t+n}}{(1+r_e)^n}$$

Where:

• \(FCFE_t\) – Expected cash flow in period \(t\).
• \(r_e\) – Required rate of return
• \(IV_{t+n}\) – Terminal intrinsic value at the end of the stable growth horizon

Where the terminal intrinsic value \(IV_{t+n}\) at period \((n)\) is calculated as:

$$IV_{t+n} = \frac{FCFE_{t+n+1}}{r_e – g_l}$$

Where:

• \(FCFE_{t+n+1}\) – Expected cash flow for the first year after the high-growth period ends.
• \(r_e\) – Required rate of return
• \(g_l\) – Long-term constant growth rate

Analysts typically rely on two primary methods to forecast future cash flows:

• Historical Growth Rates – These are derived from past dividends or calculated from historical FCFE/FCFF. The exact year you choose as your starting point drastically changes the calculated average growth rate.
• Sustainable Growth Rate – This is the maximum rate at which a company can grow its sales, earnings, or assets using only its own retained profits. It is the long-term growth rate a firm can sustain internally. It is calculated as:

$$SGR = \text{Retention Ratio} \times \text{Return on Equity (ROE)}$$

Dividend-Based SGR can be calculated as:

$$SGR_t = \left( 1 – \frac{\text{Dividends}_t}{\text{Net income}_t} \right) \times \left( \frac{\text{Net income}_t}{\text{Book value of equity}_{t-1}} \right)$$

FCFE-Based SGR can be calculated as:

$$SGR_t = \left( 1 – \frac{FCFE_t}{\text{Net income}_t} \right) \times \left( \frac{\text{Net income}_t}{\text{Book value of equity}_{t-1}} \right)$$

FCFE typically exceeds dividends, resulting in a lower FCFE sustainable growth rate compared to the dividend-based SGR.

FCFF-Based SGR can be calculated as:

$$SGR_t = \left( 1 – \frac{FCFF_t}{NOPAT_t} \right) \times \left( \frac{NOPAT_t}{\text{Book value of equity}_{t-1}} \right)$$

Where \(\text{NOPAT} = \text{Operating Income} \times (1 – \text{Tax Rate})\)

Example 3: Sustainable Growth Rate(SGR)

Let’s assume we are looking at Vanguard Manufacturing’s financial statements and have gathered the following metrics:

• Net Income: $5,000,000
• Operating Income/EBIT: $8,000,000
• Tax Rate: 25% (or 0.25)
• Free Cash Flow to Firm/FCFF: $2,000,000
• Prior Year Book Value of Equity: $24,000,000

SGR can be calculated as:

$$ \begin{align*} \text{NOPAT} & = \text{Operating Income} \times (1 – \text{Tax Rate}) \\
\text{NOPAT} & = \$8,000,000 \times (1 – 0.25) = \$6,000,000 \\
\text{SGR}_t & = \left( 1 – \frac{\text{FCFF}_t}{\text{NOPAT}_t} \right) \times \left( \frac{\text{NOPAT}_t}{\text{Book value of equity}_{t-1}} \right) \\
\text{SGR}_t & = \left( 1 – \frac{\$2,000,000}{\$6,000,000} \right) \times \left( \frac{\$6,000,000}{\$24,000,000} \right) \\
\text{SGR}_t &= 0.6667 \times 0.25 = 0.1667 = 16.67\% \end{align*} $$

Discount Rates and Terminal Value

The discount rate selected for a valuation model must be consistent with the type of cash flow being discounted. The discount rate should reflect the required return of the investors who have a claim to those cash flows.

Weighted Average Cost of Capital (WACC)

This is a market-value-weighted average of the after-tax cost of debt \((r_d)\) and the cost of equity \((r_e)\). It represents the required return for all capital providers. It is used to discount Free Cash Flow to the Firm (FCFF). FCFF calculates the total enterprise value, therefore, you must subtract the market value of debt from the enterprise value calculated to determine the intrinsic value of equity.

$$IV_t = \left[ \sum_{i=1}^{n} \frac{FCFF_{t+i}}{(1+\text{WACC})^i} + \frac{E(P)_{t+n}}{(1+\text{WACC})^n} \right] – MV_{\text{Debt}}$$

Cost of Equity (r_e)

It is used to discount cash flows available solely to equity holders, such as dividends, Free Cash Flow to Equity (FCFE), and residual income.

$$IV_t = \sum_{i=1}^{n} \frac{FCFE_{t+i}}{(1+r_e)^i} + \frac{E(P)_{t+n}}{(1+r_e)^n}$$

Debt carries less risk than equity, therefore, the cost of debt is lower than the cost of equity. Consequently, WACC is usually less than the cost of equity \((r_e)\). They are only equal if a firm uses zero debt. FCFE and FCFF models should yield similar equity values if assumptions are applied consistently.

The terminal value \(IV_{t+n}\) represents the expected share price at the end of the explicit forecast horizon, capturing all subsequent cash flows. It makes up a larger percentage of total intrinsic value when the forecast horizon is short or when the short-term growth rate is close to the long-term rate.

Analysts use standardized approaches for terminal values, focusing on long-run averages. Analysts should consider:

• The constant long-term growth rate generally falls within the range of expected inflation and nominal economic GDP growth.
• The different geographic markets where a firm operates, not just where it is headquartered.
• Terminal growth rates may drop near zero or even turn negative when evaluating companies in a declining life-cycle phase.

Valuation models are highly sensitive to estimated growth rates and time horizons. Rather than relying on a single baseline figure, analysts adjust these key variables to map out a range of alternative scenarios around the base case, establishing realistic potential valuation outcomes.

Example 4: Value of Equity

Nova Tech Corp is a mature company looking to forecast its cash flows for the next 2 years \((n=2)\) before reaching its terminal state. Analysts have the following information on the company:

• Current Market Value of Debt (\(MV_{\text{debt}}\)): $2,000,000
• Cost of Equity (\(r_e\)): 10%
• After-Tax Cost of Debt (\(r_d\)): 4%
• Capital Structure: 60% Equity / 40% Debt
• Sustainable Growth Rate (SGR): 3%
• Nominal GDP Growth: 4.0%
• Year 1 FCFF: $500,000
• Year 2 FCFF: $530,000
• Terminal Firm Value: $8,000,000

Using the information provided, the intrinsic value of equity can be calculated as:

1. Calculate the WACC
   
   $$ \begin{align*} WACC & = (\text{Weight of Equity} \times r_e) + (\text{Weight of Debt} \times r_d) \\
   
   WACC & = (0.60 \times 10\%) + (0.40 \times 4\%) = 7.6\% \end{align*} $$The Sustainable Growth Rate (3%) will serve as the long-term constant growth rate (\(g\)) for the terminal phase.
2. Apply the Valuation Formula
   
   $$ \begin{align*} IV_t & = \left[ \sum_{i=1}^{n} \frac{FCFF_{t+i}}{(1+\text{WACC})^i} + \frac{E(P)_{t+n}}{(1+\text{WACC})^n} \right] – MV_{\text{Debt}} \\
   
   IV_{\text{Equity}} & = \left[ \frac{\$500,000}{(1.076)^1} + \frac{\$530,000}{(1.076)^2} + \frac{\$8,000,000}{(1.076)^2} \right] – \$2,000,000 \\
   
   IV_{\text{Equity}} &= [\$464,684 + \$457,770 + \$6,909,742] – \$2,000,000 \\
   
   IV_{\text{Equity}} &= \$7,832,196 – \$2,000,000 = \$5,832,196 \end{align*} $$The intrinsic value of equity for Nova Tech Corp is $5,832,196.

Question

Aura Electronics Inc. is evaluating its financial trajectory for the next 2 years \((n=2)\) before reaching a stable, terminal growth phase. An analyst needs to determine the company’s Intrinsic Value of Equity using an FCFF-based multistage valuation model. The analyst has gathered the following data from the company’s financial records:

• Net Income: $10,000,000
• Operating Income (EBIT): $15,000,000
• Tax Rate: 20%
• Free Cash Flow to Firm (FCFF) in the Current Year \((t)\): $4,000,000
• Prior Yearbook Value of Equity \((t – 1)\): $100,000,000
• Year 1 Forecasted FCFF \((FCFF_1)\): $1,200,000
• Year 2 Forecasted FCFF \((FCFF_2)\): $1,300,000
• Cost of Equity \((r_e)\): 12%
• After-Tax Cost of Debt \((r_d)\): 5%
• Capital Structure: 70% Equity / 30% Debt
• Current Market Value of Debt \((MV_{\text{debt}})\): $5,000,000

The Intrinsic Value of Equity for Aura Electronics Inc. is closest to:

1. $34,902,765
2. $49,902,765
3. $58,349,456

The correct answer is C.

Calculate the Sustainable Growth Rate (SGR)

$$ \begin{align*} \text{NOPAT} & = \text{Operating Income} \times (1 – \text{Tax Rate}) \\
\text{NOPAT} & = \$15,000,000 \times (1 – 0.20) = \$12,000,000 \\
g_l &= SGR = \left( 1 – \frac{FCFF_t}{NOPAT_t} \right) \times \left( \frac{NOPAT_t}{\text{Book Value of Equity}_{t-1}} \right) \\
g_l & = \left( 1 – \frac{\$4,000,000}{\$12,000,000} \right) \times \left( \frac{\$12,000,000}{\$100,000,000} \right) \\
g_l &= 0.6667 \times 0.12 = 0.08 = 8.0\%
\end{align*} $$

Calculate the Weighted Average Cost of Capital (WACC)

$$ \begin{align*} {WACC} & = (\text{Weight of Equity} \times r_e) + (\text{Weight of Debt} \times r_d) \
{WACC} & = (0.70 \times 12\%) + (0.30 \times 5\%) = 8.4\% + 1.5\% = 9.9\% \end{align*} $$

Calculate the Terminal Value \( (IV_{t+2}) \)

$$ \begin{align*} \text{Terminal Value } (IV_{t+2}) &= \frac{{FCFF}_2 \times (1+g_l)}{{WACC}-g_l} \\ \text{Terminal Value} (IV_{t+2}) &= \frac{\$1,300,000 \times (1+8.0\%)}{9.9\% – 8.0\%} = \frac{\$1,404,000}{0.019} = \$73,894,737 \end{align*} $$

Apply the Multistage Valuation Formula

$$ \begin{align*} IV_{\text{Equity}} &= \left[ \frac{FCFF_1}{(1+\text{WACC})^1} + \frac{FCFF_2}{(1+\text{WACC})^2} + \frac{E(P)_{t+2}}{(1+\text{WACC})^2} \right] – MV_{\text{Debt}} \\&= \left[ \frac{1,200,000}{(1.099)^1} + \frac{1,300,000}{(1.099)^2} + \frac{73,894,737}{(1.099)^2} \right] – 5,000,000 \\&= \$63,349,456 – \$5,000,000 = \$58,349,456 \end{align*} $$