Multi-Stage Dividend Models - Finance

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Historical Context & Motivation

The quest to determine the intrinsic value of a share of stock has occupied financial economists for well over a century. Early approaches relied on rules of thumb and qualitative judgment, but the mid-twentieth century brought a decisive shift toward rigorous, present-value-based frameworks. In 1938, John Burr Williams published The Theory of Investment Value, arguing that a stock's worth equals the present value of all future dividends it will ever pay. This dividend discount model (DDM) became the cornerstone of equity valuation, yet its simplest form—assuming a single, constant growth rate forever—proved too restrictive for real-world application.

Myron Gordon and Eli Shapiro formalized the constant-growth variant in the 1950s, producing what we now call the Gordon Growth Model (GGM). The GGM works elegantly for mature, stable firms whose dividends grow at a predictable rate indefinitely—think regulated utilities or large consumer staples companies. However, companies rarely maintain a single growth rate throughout their entire existence. A high-growth technology firm, for instance, may expand dividends at 20% per year during its early phase, slow to 10% as competition intensifies, and eventually settle at 4% once the industry matures. Applying a single perpetual growth rate to such a firm produces either a gross overvaluation or an undervaluation, depending on which rate the analyst selects.

1938 • Williams's Dividend Discount Principle

John Burr Williams establishes that a stock's intrinsic value is the present value of its entire future dividend stream, laying the theoretical foundation for all DDM variants.

1956 • Gordon Growth Model

Myron Gordon and Eli Shapiro derive the closed-form solution V₀ = D₁ / (r − g), making equity valuation tractable but limiting analysis to a single perpetual growth rate.

1960s–70s • Emergence of Multi-Stage Extensions

Academics and practitioners begin adapting the DDM to accommodate two and three distinct growth phases, reflecting corporate lifecycle dynamics more accurately.

1980s–90s • H-Model & Continuous Transition

Fuller and Hsia propose the H-Model, allowing a gradual linear decline in growth rather than abrupt phase changes, improving realism for firms in mid-transition.

2000s–Present • Integration into Professional Practice

Multi-stage DDMs become standard in CFA curricula and equity research, often combined with free-cash-flow models and relative valuation to triangulate intrinsic value.

The central question these developments address is straightforward yet powerful: How do we value a stock when the rate at which its dividends grow is expected to change over time? Multi-stage dividend models answer this by breaking a company's future into distinct growth phases, discounting dividends in each phase appropriately, and summing the results to arrive at a single present-value estimate.

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Core Principles & Definitions

Before diving into formulas, it is essential to understand the conceptual pillars that support every multi-stage dividend model. These models rest on the same present-value logic as the basic DDM but introduce additional structure to capture the reality that corporate growth trajectories are not flat lines—they curve, decelerate, and eventually stabilize. Mastering the following principles will make the mathematical framework in later sections intuitive rather than mechanical.

Present Value of Dividends

A stock's intrinsic value is the sum of all future expected dividends, each discounted back to today at the investor's required rate of return (r). This discount rate typically derives from the Capital Asset Pricing Model or similar risk–return framework.

Distinct Growth Phases

Companies pass through identifiable lifecycle stages—high growth, transitional deceleration, and mature stability. Each phase has its own dividend growth rate (g), reflecting competitive dynamics, reinvestment needs, and market saturation.

Terminal Value

Once a firm reaches its mature, steady-state phase, the Gordon Growth Model can be applied to calculate a terminal value (TV)—the present value at that future point of all subsequent dividends growing at a constant rate forever.

Additivity of Present Values

Because present value is additive, the analyst can compute the PV of dividends during each explicit forecast period, then add the discounted terminal value. The sum equals the stock's intrinsic value (V₀).

g Must Be Less Than r in Perpetuity

For the terminal-value formula to converge, the long-run stable growth rate must remain strictly below the required return. If g ≥ r, the model yields an infinite or negative value—a signal that the growth assumption is unrealistic for the mature phase.

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KEY TAKEAWAY

Think of a multi-stage DDM like planning a cross-country road trip with different speed limits. On the open highway (high-growth phase), you cover ground quickly; as you enter suburban zones (transition phase), you slow down; once you reach city streets (mature phase), your speed stabilizes at a lower limit. To estimate your total arrival time, you cannot simply assume one speed for the entire journey—you must account for each segment's distinct pace. Similarly, a multi-stage model sums the present value of dividends at each phase's specific growth rate to arrive at a realistic stock price.

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Visual Explanation — The Multi-Stage Growth Path

The diagram below illustrates how a company's dividend growth rate evolves over time through three distinct stages. During the high-growth phase, the firm reinvests aggressively and dividends expand rapidly. The transition phase sees competition erode abnormal returns, causing growth to decelerate linearly. Finally, the stable-growth phase represents perpetuity, where the firm grows roughly in line with the broader economy.

The solid cyan line represents the high-growth phase where g₁ = 20%. The dashed violet segment shows the transition period in which the growth rate declines linearly from 20% to the stable rate. The solid emerald line marks the perpetual stable phase at gₙ = 4%. The vertical markers at Year N₁ and Year N₂ delineate the boundaries between phases.

Notice that a two-stage model would eliminate the dashed transition segment entirely—growth would jump abruptly from 20% to 4% at a single switchover point. A three-stage model, by contrast, introduces the gradual slope, which more realistically represents how competitive advantages erode over time. The H-Model is a special shortcut that approximates the three-stage path by assuming the growth rate declines linearly from an initial supernormal level to the long-run stable rate, with the halfway point of the transition period (H) serving as the key parameter. Each variant offers a different trade-off between analytical simplicity and descriptive accuracy.

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Mathematical Framework

All multi-stage dividend models begin from the same bedrock equation: the general dividend discount model. The key distinction across the two-stage, three-stage, and H-Model variants lies in how the analyst partitions the dividend stream into growth phases and how the terminal value is computed. We present the three most widely used formulations below, starting from the general DDM and progressing to each specialized form.

GENERAL DIVIDEND DISCOUNT MODEL

V₀ = Σ (t=1 to ∞) Dₜ / (1 + r)ᵗ

V₀ = intrinsic value today; Dₜ = expected dividend in year t; r = required rate of return. This infinite sum is the theoretical foundation, but practical application requires simplifying assumptions about the growth trajectory of Dₜ.

Two-Stage DDM

The two-stage DDM assumes dividends grow at a supernormal rate gₛ for the first N years, after which growth drops instantaneously to a stable long-run rate gₗ that persists forever. The stock's value is the sum of two components: the present value of dividends during the high-growth phase plus the present value of the terminal value calculated at year N.

TWO-STAGE DDM

V₀ = Σ (t=1 to N) [D₀ × (1 + gₛ)ᵗ / (1 + r)ᵗ] + [D₀ × (1 + gₛ)ᴺ × (1 + gₗ)] / [(r − gₗ) × (1 + r)ᴺ]

D₀ = current annual dividend; gₛ = supernormal growth rate (Phase 1); gₗ = long-run stable growth rate (Phase 2); N = number of years in the high-growth phase; r = required rate of return. The second term is the Gordon Growth Model applied at year N, then discounted back to today.

H-Model (Linear Decline)

The H-Model was developed by Fuller and Hsia as a pragmatic approximation. Instead of assuming an abrupt growth-rate change, it posits that the growth rate declines linearly from an initial high rate gₛ to the long-run rate gₗ over a period of 2H years. The parameter H represents the half-life of the transition—the midpoint at which the growth rate equals the average of gₛ and gₗ. The elegance of the H-Model lies in its compact closed-form solution.

H-MODEL

V₀ = [D₀ × (1 + gₗ) / (r − gₗ)] + [D₀ × H × (gₛ − gₗ) / (r − gₗ)]

H = half-life of the high-growth period (in years); gₛ = initial supernormal growth rate; gₗ = long-run stable growth rate; D₀ = current dividend; r = required return. The first term is the stable-growth value; the second term captures the additional value contributed by the above-normal growth during the transition.

Three-Stage DDM

The three-stage DDM combines the precision of explicit forecasting during the high-growth phase with a linearly declining growth transition and a terminal Gordon Growth calculation. It is the most realistic—and the most computationally intensive—of the standard variants. In practice, the analyst forecasts dividends year by year during the first two phases (high growth and transition), computing a unique growth rate for each year of the transition, and then applies the GGM at the start of the stable phase.

THREE-STAGE DDM (CONCEPTUAL)

V₀ = Σ (t=1 to N₁) [Dₜ / (1 + r)ᵗ] + Σ (t=N₁+1 to N₂) [Dₜ / (1 + r)ᵗ] + [TVₙ₂ / (1 + r)ᴺ²]

Phase 1 (years 1 to N₁): dividends grow at gₛ. Phase 2 (years N₁+1 to N₂): growth declines linearly each year from gₛ to gₗ. Phase 3 (year N₂ onward): TVₙ₂ = Dₙ₂₊₁ / (r − gₗ), discounted back to today. Dₜ in the transition phase uses gₜ = gₛ − [(gₛ − gₗ) × (t − N₁) / (N₂ − N₁)].

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Critical Constraint

In every variant, the terminal-phase growth rate gₗ must satisfy gₗ < r. If the stable growth rate equals or exceeds the discount rate, the Gordon Growth formula produces an undefined or negative result. A common rule of thumb is to cap gₗ at the long-run nominal GDP growth rate of the economy (typically 3–5%).

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Comparing Multi-Stage Variants

Choosing the right multi-stage variant depends on the firm's growth profile and the analyst's tolerance for complexity. A two-stage model fits companies whose competitive advantage is expected to erode quickly after a defined period—such as a pharmaceutical firm approaching patent expiration. The H-Model suits firms undergoing a prolonged, gradual competitive transition. The full three-stage DDM is most appropriate for companies with a clearly defined high-growth runway followed by a visible transition period before settling into maturity. The following diagram and table compare these variants side by side.

The pink line shows the two-stage DDM's abrupt step-down in growth at year N. The amber line depicts the H-Model's linear decline from gₛ to gₗ with the half-life H marked at the midpoint. The emerald line illustrates the three-stage DDM, featuring a flat high-growth segment, a linearly declining transition, and a stable terminal phase. Notice that the three-stage model is offset slightly for visual clarity.

Feature	Two-Stage DDM	H-Model	Three-Stage DDM
Number of phases	2 (high + stable)	2 (linear decline + stable)	3 (high + transition + stable)
Growth-rate transition	Abrupt step-down	Linear decline over 2H years	Linear decline over N₂ − N₁ years
Closed-form solution?	Yes	Yes (approximate)	No — requires year-by-year computation
Best suited for	Patent-expiry firms; clear-cut phase shifts	Gradual competitive erosion	Firms with visible high-growth runway + gradual slowdown
Complexity	Low	Low–Medium	High

Comparison of the three major multi-stage dividend model variants

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Worked Example — Two-Stage DDM

Consider TechGrow Inc., a mid-cap technology company that currently pays a dividend of $2.00 per share. Analysts expect its earnings and dividends to grow at 15% per year for the next 5 years as it capitalizes on a new product line. After year 5, growth is expected to stabilize at 4% indefinitely as the market matures. An investor requires a 10% return on TechGrow's equity. What is the stock's intrinsic value today?

Two-Stage DDM: TechGrow Inc.

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Step 1 — Identify Given Values

D₀ = $2.00, gₛ = 15% (years 1–5), gₗ = 4% (year 6 onward), r = 10%, N = 5 years. We need to check the convergence condition: gₗ (4%) < r (10%) ✓.
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Step 2 — Forecast Dividends During the High-Growth Phase

D₁ = $2.00 × 1.15 = $2.30; D₂ = $2.30 × 1.15 = $2.645; D₃ = $2.645 × 1.15 = $3.042; D₄ = $3.042 × 1.15 = $3.498; D₅ = $3.498 × 1.15 = $4.023.
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Step 3 — Discount Each High-Growth Dividend to Present Value

PV(D₁) = $2.30 / 1.10¹ = $2.091; PV(D₂) = $2.645 / 1.10² = $2.186; PV(D₃) = $3.042 / 1.10³ = $2.285; PV(D₄) = $3.498 / 1.10⁴ = $2.389; PV(D₅) = $4.023 / 1.10⁵ = $2.498.

Sum of PV (Phase 1) = $2.091 + $2.186 + $2.285 + $2.389 + $2.498 = $11.449
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Step 4 — Calculate the Terminal Value at Year 5

D₆ = D₅ × (1 + gₗ) = $4.023 × 1.04 = $4.184. Terminal Value at year 5: TV₅ = D₆ / (r − gₗ) = $4.184 / (0.10 − 0.04) = $4.184 / 0.06 = $69.733.

TV₅ = $69.733
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Step 5 — Discount the Terminal Value to Today

PV(TV₅) = $69.733 / (1.10)⁵ = $69.733 / 1.61051 = $43.298.

PV of Terminal Value = $43.298
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Step 6 — Sum to Obtain Intrinsic Value

V₀ = PV of high-growth dividends + PV of terminal value = $11.449 + $43.298 = $54.747. TechGrow's intrinsic value is approximately $54.75 per share. Notice that the terminal value accounts for roughly 79% of total value, underscoring the importance of the stable-growth assumption.

V₀ ≈ $54.75

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Sensitivity Check

Because nearly 80% of the value comes from the terminal calculation, small changes in gₗ or r have an outsized impact on V₀. If gₗ rises from 4% to 5%, TV₅ jumps to $83.68 and V₀ climbs to approximately $63.40—a 16% increase. Always perform sensitivity analysis on your terminal-phase assumptions.

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Strengths & Limitations

Multi-stage dividend models represent a significant upgrade over the single-stage Gordon Growth Model, but no valuation tool is without constraints. Understanding the strengths and limitations of these models enables the analyst to deploy them wisely and to supplement them with alternative approaches when necessary.

Strengths	Limitations
Captures lifecycle dynamics—high growth, transition, and maturity—yielding more realistic valuations than a single-rate model.	Highly sensitive to terminal growth rate (gₗ) and required return (r); small input changes can dramatically shift the output.
Grounded in first-principles present value theory, making valuations internally consistent and auditable.	Only applicable to dividend-paying firms; excludes growth companies that retain all earnings (e.g., early-stage tech firms).
Multiple variants (two-stage, H-Model, three-stage) allow the analyst to match model complexity to the firm's growth profile.	Assumes a deterministic dividend path; ignores stochastic factors like recession-driven dividend cuts or special dividends.
Forces explicit assumptions about growth, payout, and risk—promoting disciplined fundamental analysis.	Estimating the length and magnitude of each growth phase requires subjective judgment, introducing analyst bias.
Terminal value calculation leverages the well-understood and analytically tractable Gordon Growth Model.	Terminal value often dominates total value (70–90%), meaning the model's precision depends heavily on the least certain input.

Strengths and limitations of multi-stage dividend models

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KEY TAKEAWAY

Multi-stage DDMs are like high-resolution telescopes: they bring distant cash flows into sharper focus than a simple model, but the image quality still depends on the clarity of your lens—that is, the accuracy of your growth and discount rate assumptions. Always pair your DDM output with a sensitivity table and cross-validate against peer multiples or a discounted-cash-flow model to ensure robustness.

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Connection to Free Cash Flow & Advanced Valuation

Multi-stage dividend models are one expression of a broader family of discounted cash flow (DCF) approaches. In fact, the logic of partitioning a firm's future into distinct growth phases and computing a terminal value applies equally to free cash flow to equity (FCFE) models and free cash flow to the firm (FCFF) models. The only difference is the numerator of each period's cash flow and the corresponding discount rate. When a firm pays no dividends at all, multi-stage FCFE or FCFF models become the natural alternative, preserving the same structural insight—that growth rates evolve over the corporate lifecycle—while substituting a broader cash flow measure.

Dimension	Multi-Stage DDM	Multi-Stage FCFE/FCFF
Cash flow measure	Dividends per share	FCFE (equity) or FCFF (firm)
Discount rate	Cost of equity (r)	Cost of equity (FCFE) or WACC (FCFF)
Applicability	Dividend-paying firms only	All firms, including non-dividend payers
Terminal value formula	TV = Dₙ₊₁ / (r − gₗ)	TV = FCFₙ₊₁ / (discount rate − gₗ)
When DDM ≈ FCFE	When payout ratio ≈ 100%	When dividends ≈ FCFE

Multi-Stage DDM versus Multi-Stage Free Cash Flow Models

Advanced practitioners also incorporate residual income models and economic value added (EVA) frameworks, which also use multi-stage structures but focus on excess earnings above the cost of capital rather than raw dividends or cash flows. The unifying insight across all these models is that a company's value creation story unfolds in chapters, and the analyst's job is to assign the right growth assumptions to each chapter while anchoring the perpetuity calculation in economically sustainable long-run conditions.

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Practice Problems

Question 1: CONCEPTUAL

Explain why the single-stage Gordon Growth Model may produce a misleading valuation for a young, rapidly growing technology company, and describe how a two-stage DDM addresses this limitation.

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The Gordon Growth Model assumes a single, perpetual growth rate. For a young tech firm growing dividends at, say, 25% per year, plugging this rate into V₀ = D₁ / (r − g) will either produce an impossibly large value (if g < r) or an undefined result (if g ≥ r), because no company can sustain supernormal growth indefinitely. A two-stage DDM resolves this by explicitly forecasting high-growth dividends for a finite period, then switching to a sustainable long-run growth rate for the terminal value, ensuring the model converges to a finite, economically meaningful price.

Question 2: BASIC CALCULATION

A company pays a current dividend of $1.50. Dividends are expected to grow at 12% for 3 years and then at 5% forever. If the required return is 9%, what is the stock's intrinsic value using the two-stage DDM?

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First, forecast high-growth dividends: D₁ = $1.50 × 1.12 = $1.680; D₂ = $1.680 × 1.12 = $1.882; D₃ = $1.882 × 1.12 = $2.108. Discount each: PV(D₁) = $1.680/1.09 = $1.541; PV(D₂) = $1.882/1.09² = $1.585; PV(D₃) = $2.108/1.09³ = $1.628. Sum = $4.754. Terminal value at year 3: D₄ = $2.108 × 1.05 = $2.213; TV₃ = $2.213/(0.09 − 0.05) = $55.325. PV(TV₃) = $55.325/1.09³ = $42.727. V₀ = $4.754 + $42.727 ≈ $47.48.

Question 3: INTERMEDIATE

Using the H-Model, estimate the value of a stock with D₀ = $3.00, an initial supernormal growth rate gₛ = 18%, a stable growth rate gₗ = 3%, a half-life H = 6 years, and a required return r = 11%. Then explain what the H parameter conceptually represents.

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Apply the H-Model: V₀ = [D₀ × (1 + gₗ)/(r − gₗ)] + [D₀ × H × (gₛ − gₗ)/(r − gₗ)]. Stable component: [$3.00 × 1.03]/(0.11 − 0.03) = $3.09/0.08 = $38.625. Growth premium: [$3.00 × 6 × (0.18 − 0.03)]/0.08 = [$3.00 × 6 × 0.15]/0.08 = $2.70/0.08 = $33.75. V₀ = $38.625 + $33.75 = $72.375 ≈ $72.38. The H parameter represents the half-life of the transition period: if the total transition from gₛ to gₗ takes 2H = 12 years, then H = 6 is the point at which the growth rate has declined halfway to gₗ. It compresses the linear-decline transition into a single parameter for computational convenience.

Question 4: APPLIED

PharmaCo just paid a $4.00 dividend. Its blockbuster drug has patent protection for 4 more years, during which dividends are expected to grow at 20%. Over the following 4 years (years 5–8), growth will decline linearly to a stable rate of 3%. From year 9 onward, dividends grow at 3% perpetually. The required return is 12%. Calculate PharmaCo's intrinsic value using the three-stage DDM.

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Phase 1 (years 1–4, g = 20%): D₁ = $4.80, D₂ = $5.76, D₃ = $6.912, D₄ = $8.294. PVs: $4.286, $4.592, $4.919, $5.272. Sum = $19.069. Phase 2 (years 5–8, g declines linearly): g₅ = 20% − (20%−3%) × (1/4) = 15.75%, so D₅ = $8.294 × 1.1575 = $9.600; g₆ = 11.50%, D₆ = $10.704; g₇ = 7.25%, D₇ = $11.480; g₈ = 3.00%, D₈ = $11.824. PVs: $5.448, $5.424, $5.198, $4.778. Sum = $20.848. Phase 3 terminal: D₉ = $11.824 × 1.03 = $12.179; TV₈ = $12.179/(0.12 − 0.03) = $135.322. PV(TV₈) = $135.322/(1.12)⁸ = $54.667. V₀ = $19.069 + $20.848 + $54.667 ≈ $94.58. The terminal value still represents a significant share (about 58%) of total value, but the explicit Phase 1 and Phase 2 forecasts capture substantial near-term dividend income as well.

Question 5: CRITICAL THINKING

An analyst applies a two-stage DDM to Company X and obtains V₀ = $80, but the stock trades at $110. Before concluding the stock is overvalued, what key assumptions should the analyst stress-test, and under what circumstances might the multi-stage DDM systematically undervalue or overvalue a firm?

Show answer

The analyst should stress-test several inputs. First, the length of the high-growth phase: if Phase 1 lasts 7 years instead of 5, more dividends are compounded at the higher rate, increasing V₀. Second, the stable growth rate gₗ: raising it from 4% to 5% can boost the terminal value by 25% or more given the sensitivity of the GGM. Third, the required return r: a lower r—perhaps justified by declining risk premiums—raises V₀. Fourth, the payout ratio: if the firm is retaining earnings now but plans to increase its payout later, current dividends understate future cash flows, and a DDM will systematically undervalue the stock. Conversely, if dividends exceed sustainable free cash flow, the DDM may overvalue. The analyst should also consider whether the market is pricing in option value, strategic assets, or M&A premiums that a DDM—which only captures dividend streams—cannot reflect. Triangulating with a FCFE model or relative multiples analysis would help determine whether the discrepancy reflects genuine overpricing or model limitations.

SUMMARY

Summary — Multi-Stage Dividend Models

Summary

Multi-stage dividend models extend the foundational dividend discount model by partitioning a company's future into distinct growth phases—typically high growth, transition, and stable maturity. The two-stage DDM handles an abrupt shift from supernormal to stable growth, the H-Model approximates a gradual linear decline using a half-life parameter, and the three-stage DDM offers the most granular year-by-year forecast across all phases.

In every variant, the stock's intrinsic value equals the sum of discounted near-term dividends plus the discounted terminal value, which is derived from the Gordon Growth Model applied at the onset of the mature phase. The stable growth rate must remain below the required return to ensure convergence. Because the terminal value frequently dominates total value, sensitivity analysis on gₗ, r, and phase duration is essential. Analysts should cross-validate DDM outputs with free cash flow models and relative valuation multiples to ensure a robust estimate of fair value.