Stock Valuation Models

Gordon's growth model, 'no growth' discount dividend model, valuation with investment opportunities approach and discounted cash flow model.

Jan 14, 2023

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Before diving in, I would like to note that the article includes numerous mathematical expressions. Also, since the editor I used doesn't support adding subscripts directly to variables outside the embedded Latex Block, I've used square brackets for this purpose. For instance, 'P[0]' should be interpreted as 'P' with a subscript '0'.

Introduction

When an investor purchases a stock, they typically expect to receive two (2) types of cashflows. They earn dividends during the holding period and receive additional income when the stock is sold. The expected sales price of the stock in the future is the sum of discounted dividends from that point through infinity so the value of a stock that is bought today at t = 0 is the summation of an infinite stream of dividends discounted at an investors’ opportunity cost of capital. This is the dividend discount model (DDM) for valuing a stock and it is mathematically summarized in [Eqn 1]

$P_{0} = \sum_{n = 1}^{\infty} \frac{DPS_{t}}{(1 + k)^{t}}\;\; \; [Eqn\: 1]$

Where DPS[t] is the expected dividend per share across an infinite time period (t = 1 to ∞) and k is the cost of equity.

J.B Williams is credited as the originator of the DDM for stock valuation1. In his book published in 1938, Williams expressed the price of a stock as a function of dividend payments similar to [Eqn 1]2. A few iterations of stock valuation models have been developed afterwards and the purpose of this article is to show that under certain conditions, they are equivalent to the DDM.

Specifically, we start with deriving different varieties of DDM under unique growth assumptions. Furthermore, we present the investment opportunities approach and discounted cash flow procedure for stock valuation and prove their equivalence to the DDM.

Dividend Approach with Unique Growth Rates

Constant Growth

Myers Gordon developed a discount dividend model to value the stock of a corporation that is expected to grow at a constant rate indefinitely. The derivation of the model was predicated on 4 assumptions:

The corporation doesn’t issue new shares.
There is no debt financing.
The firm retains a fixed portion of its income in every future period. The retention rate is denoted as b.
The corporation earns a permanent return on its investment in every future period. The return on invested capital is represented with r.

Gordon argued the rationality of dividend payments that grow at a constant rate by noting that when investors purchase stocks, they form a subjective probability distribution of growth of future cash receipts. The mean of the distribution could be considered the constant growth rate. Lintner J, a renowned Harvard Professor had deduced in a 1956 research on payout policy that corporations tend to follow a regime of paying a stable fraction of their income3. Hence, an assumption of a fixed retention rate was plausible.

Gordon suggested that the worth of a stock P[0] with growing annual payments can be mathematically expressed as follows4

$P_{0} = \int_{0}^{{\infty}} DPS_{t}\;e^{-kt}dt \;\;\;\; [Eqn\: 2]$

With the assumption of a fixed investment strategy, we know the fraction of the net income the firm will retain in every future period. Hence, DPS[t] can easily be derived at any point in time by the equation presented below

$DPS_{t} = (1 - b) Y_{t} \;\;\;\; [Eqn\: 3]$

In other words, DPS[t] is a residual amount that is available to be paid out to shareholders after a portion of the net earnings bY[t] is retained to expand the asset base; the added benefit produced by the invested capital in perpetuity is rbY[t]. Consequently, the net income that is generated in period t is the sum of net profit from a previous year t - 1 and the incremental earnings that is obtained as a result of the additional investment made at t - 1. The mathematical relationship is as follows

$Y_{t} = Y_{t-1} + rbY_{t-1} \;\;\;\; [Eqn\: 4]$

For ease of illustration, we can use [Eqn 4] to derive the earnings for 3 consecutive years (t = 1,2 and 3)

$Y_{1} = Y_{0} + rbY_{0} \;\;\;\; ; \;\;\;\; Y_{2} = Y_{1} + rbY_{1} \;\;\;\; ; \;\;\;\; Y_{3} = Y_{2} + rbY_{2} $

$Y_{1} = Y_{0} (1 + rb) \;\;\;\; ; \;\;\;\; Y_{2} = Y_{0} (1 + rb)^{2} \;\;\;\; ; \;\;\;\; Y_{3} = Y_{0}(1 + rb)^{3}$

A look at the expressions for obtaining Y across the arbitrarily selected periods shows that a compound interest formula is adequate to determine the net profit at any time t. Hence, [Eqn 4] can be rewritten as

$Y_{t} = Y_{1}(1 + rb)^{t} \;\;\;\; [Eqn\: 5]$

Sidenote: Y[1] is used in [Eqn 5] instead of Y[0] because investors capitalize cash receipts that are expected in the future. So, if I buy a stock P today at t = 0, I would be interested in the cash flows it will generate in the future starting from t = 1.

If Y[t] is being evaluated at a distant time in the future, [Eqn 5] can be restated as

$Y_{t} = Y_{1}\;e^{rbt} \;\;\;\; [Eqn\: 6]$

If it is assumed that the growth rate rb or g is less than the discount rate (which is expected in the long run), we can substitute [Eqn 6] and [Eqn 3] into [Eqn 2] and solve the integration expression. The equation simplifies to

$P_{0} = \int_{0}^{{\infty}} (1 - b) Y_{t}\;e^{-kt}dt = \int_{0}^{{\infty}}Y_{1}\;e^{rbt}e^{-kt}dt \;\;\;\; $

$P_{0} = \frac{(1 - b) Y_{1}}{k - rb} = \frac{DPS_{1}}{k - g} \;\;\;\; [Eqn\: 7] $

[Eqn 7] is Gordon’s growth model and it shows that the value of a share that gives its owner the right to receive dividend payments growing at a constant rate in perpetuity is the first cash receipt a year into the holding period (t = 1) divided by the difference between the rate of profit of investors and the rate of growth of the dividend.

[Eqn 7] could also have been derived from [Eqn 1] using algebra. We shall briefly highlight the process.

From [Eqn 1], the price of a share with dividends growing at a stable percentage g can be written as

$P_{0} = \frac{DPS_{1}}{(1 + k)^{1}} + \frac{DPS_{1}(1 + g)}{(1 + k)^{2}} + \frac{DPS_{1}(1 + g)^{2}}{(1 + k)^{3}}......\;\; \; [Eqn\: 8]$

If we let a = DPS[1]/(1+k) and x = (1+g)/(1+k), [Eqn 8] can be restated as

$P_{0} = a(1 + x + x^{2}......)\;\; \; [Eqn\: 9]$

[Eqn 9] is an infinite geometric series and since x < 1 we can derive the sum of the series with simple mathematical manipulation. If we multiply both sides of the series with x, we have

$P_{0}x = a(x + x^{2} + x^{3}......)\;\; \; [Eqn\: 10]$

Subtracting [Eqn 10] from [Eqn 9] gives

$P_{0}(1 - x) = a\;\; \; [Eqn\: 11]$

Divide both sides of [Eqn 11] by (1-x) and the right-hand side of the ensuing expression is the sum of an infinite geometric series.

$P_{0} = \frac{a}{(1-x)}\;\;\;[Eqn\;12] $

We can now proceed to substitute a and x into [Eqn 12] and simplify the resulting expression

$P_{0} = \frac{DPS_{1}}{(1 + k)}/(1 - \frac{(1+g)}{(1+k)})$

$P_{0} = \frac{DPS_{1}}{(1 + k)} \times (\frac{(1+k)}{(k-g)})$

$P_{0} = \frac{DPS_{1}}{(k-g)}\;\;\; [Eqn\: 13]$

[Eqn 13] is the same as [Eqn 7] so we have derived Gordon’s growth dividend model with an alternative method.

No Growth

If a corporation is in a steady state and pays the same amount of dividends in perpetuity, then the price of its stock will be equal to the annual payment divided by the investor’s cost of equity. That is P[0] = DPS[t]/k where DPS[t] is the same at t = 1,2,3…..∞. The derivation of this shorthand formula from DDM is presented below.

For a constant perpetuity payment, [Eqn 1] can be rewritten as

$P_{0} = \frac{DPS_{1}}{(1 + k)} + \frac{DPS_{1}}{(1 + k)^{2}} + \frac{DPS_{1}}{(1 + k)^{3}}......\;\; \; [Eqn\: 14] $

If we let c = DPS[1]/(1+k) and z = 1/(1+k), [Eqn 14] can be restated as

$P_{0} = c + cz + cz^{2}......\;\; \; [Eqn\: 15]$

[Eqn 15] is an infinite geometric series similar to [Eqn 9] and we previously obtained the sum of an infinite series. Therefore, [Eqn 15] simplifies to

$P_{0} = \frac{c}{(1-z)}\;\; \; [Eqn\: 16]$

If we substitute c and z into [Eqn 16], we have a final expression presented below in [Eqn 17]

$P_{0} = \frac{{DPS_{1}/(1+k)}}{1 - \frac{1}{(1+k)}}$

$P_{0} = {\frac{DPS_{1}}{k}}\;\;\; [Eqn\: 17] $

A firm in a steady state is unable to find investment opportunities with an excess return above shareholders’ cost of equity (that is r = k), and in a rational world, the management will be expected to pay out all the net profits as dividends so equity owners can buy the stocks of other corporations that can invest in value-creating opportunities. Thus, [Eqn 17] can be revised as

$P_{0} = {\frac{DPS_{1}}{k}} = {\frac{Y_{1}}{k}}\;\;\; [Eqn\: 18]$

If the management decides otherwise and retains a fixed portion of the earnings to grow the assets base for an infinite period of time, the residual amount that will be available to be disbursed to shareholders as dividends will grow at a constant rate. However, the price of the stock will not change. This can be proven by substituting r with k in [Eqn 7]

$P_{0} = \frac{(1 - b) Y_{1}}{k - rb} = \frac{(1 - b) Y_{1}}{k - kb}$

$P_{0} = \frac{(1 - b) Y_{1}}{k(1 - b)} = \frac{Y_{1}}{k} \;\;\;\; [Eqn\: 19]$

The equivalence of [Eqn 18] and [Eqn 19] shows that a firm’s valuation doesn’t necessarily increase when there is growth in revenue and profit. Value is only created when the return on invested capital r exceeds investors’ rate of profit k (r >k).

Discounted Cash Flow Model

In the discounted cash flow (DCF) model, the price of a stock is equal to the present value of future cash flows that accrues to a share. Under the assumption that a corporation is unlevered (no debt) and doesn’t issue new securities, then the only source of cash generation is from the revenue obtained during the normal course of operations. The expected use of funds is for investment and payout to shareholders. The mathematical depiction of the firm’s source and use of funds is presented in [Eqn 20]

$E_{t} = I_{t} + DPS_{t}\;\;\;[Eqn\;20]$

DPS[t] represents the free cash flow (FCF) available for distribution to equity owners and we can rearrange [Eqn 16] to make it the subject as follows

$DPS_{t} = E_{t} - I_{t}\;\;\;[Eqn\;21]$

Hence, the price P[0] of a share with the DCF model is

$P_{0} = \sum_{n = 1}^{\infty} \frac{E_{t} - I_{t}}{(1 + k)^{t}} = \;\; \sum_{n = 1}^{\infty} \frac{DPS_{t}}{(1 + k)^{t}} \;\;\;[Eqn\;22]$

[Eqn 22] is equivalent to [Eqn 1], the mathematical representation of the DDM. If it is further assumed that FCFs will grow at a constant rate, then [Eqn 22] can be simplified to correspond to Gordon’s dividend model.

It is important to point out that in deriving the equivalence of DCF to DDM, we have implied that all the FCFs of a firm are paid out as dividends but in reality, that is not always the case. When DDM was proposed, the predominant means of returning cash to shareholders was through dividends. However, in recent decades, stock buyback has become prevalent. To the extent that a firm prefers to pay out cash in form of stock repurchase, then using DDM will understate its actual worth.

Additionally, there is an assumption in DDM that the retained capital is invested in fixed assets and working capital but that could be incorrect. There is an accepted rule of thumb that a matured firm’s operational cash requirement is only 2% of revenue5. To the extent that this is true, an examination of the balance sheets of corporations with positive cash flow will reveal that most of them carry more cash than what is needed and it could potentially be paid out as dividends. The argument is not that holding excess cash is bad especially if there is potential to invest in value-creating opportunities in the future. The takeaway is that if we are solely relying on dividend payout to determine valuation and not adding ‘underutilized’ and non-operational assets to the market cap of a firm, its stock will be undervalued.

Investment Opportunities Approach

In the investment opportunities model (IOM), the price of a share is obtained from the earnings generated by a firm’s existing asset and the premium expected to be earned from the ability of the corporation to invest in future opportunities with returns that exceed shareholders’ opportunity cost of equity.

$P_{0} = \frac{Y_{1}}{k} + PVGO \;\;\;[Eqn\;23]$

Where PVGO is the present value of growth opportunities. If we assume the same parameters that were used to deduce Gordon’s dividend model, we can derive a mathematical expression for PVGO that will help show the equivalence of [Eqn 23] with [Eqn 13].

If Y[1] is income from a firm’s existing assets at t = 1 and b is the percentage of the income that is retained to expand the asset base, then the invested capital can be represented as bY[1]. Moreover, the extra income that will be earned in perpetuity (starting at t = 2) from the reinvestment is bY[1]r.

The ‘no growth DDM’ can be employed to find the present value of the added benefit created from the retention of net profit at t = 1

$PV\;of\;extra\;income\;at\;t_{1} = \sum_{n = 1}^{\infty} \frac{bY_{1}r}{(1 + k)^{t}} = \frac{bY_{1}r}{k} \;\;\;[Eqn\;24]$

We are now able to calculate the net present value (NPV) of the investment at t = 1

$NPV\;of\;investment\;at\;t_{1} = \frac{bY_{1}r}{k} - bY_{1} = \frac{bY_{1}(r - k)}{k} \;\;\;[Eqn\;25] $

Furthermore, we can determine the NPV of reinvestment at t = 2 but to do that, we start off by stating the expected income in the period

$Y_{2} = Y_{1} + bY_{1}r = Y_{1}(1 + br) \;\;\;[Eqn\;26]$

Since the firm will maintain its retention ratio b and reinvests a percentage of the earnings, the additional income that will be obtained starting from t = 3 till perpetuity will be bY[1]r(1+br). Like we did at t = 1, we can find the present value of this new benefit and subsequently compute the NPV

$NPV\;of\;at\;t_{2} = \frac{bY_{1}r(1+br)}{k} - bY_{1}(1+br) = \frac{bY_{1}(1+br)(r - k)}{k} \;\;\;[Eqn\;27] $

We can extend the same logic used at t = 2 to find the reinvestment, additional income and NPV at t = 3 and by doing that, we will observe an interesting pattern; NPV of investment at t = 1 (bY[1](r-k)/k) grows at a constant rate (1+br) indefinitely. Hence, we can find PVGO at t = 0 by applying the Gordon formula from [Eqn 7] on the stream of NPVs. In this case, we simply replace DPS[t] with bY[1](r-k)/k.

$PVGO = (\frac{bY_{1}(r - k)}{k})(\frac{1}{(k - br)}) \;\;\;[Eqn\;28]$

Substituting [Eqn 28] into [Eqn 25] gives

$P_{0} = \frac{Y_{1}}{k} + (\frac{bY_{1}(r - k)}{k})(\frac{1}{(k - br)}) \;\;\;[Eqn\;29]$

Under the assumptions of a fixed investment policy b and permanent return on investment r, [Eqn 29] is the complete mathematical depiction of the IOM. For brevity, further simplification of [Eqn 29] will be skipped but if the common factor Y[1]/k is taken out, the equation can be reduced to [Eqn 7]

Proof of Equivalence between Growth Models of IOM and DDM

To further establish the equivalence of growth models in DDM and IOM, we use a hypothetical company with manufactured numbers to compute the price of a share under both methods and compare them. From our previous derivations, the price should be the same.

Let’s assume Valuation Collective (VC) has an earning per share Y[0] of $100 at t = 0 with retention rate b at 20% of the income. Then the firm’s payout ratio is 80%. Let’s further assume that the permanent return on invested capital r is 20% and the discount rate is 10%.

Method 1 - DDM

With the given parameters, if the firm is expected to maintain its investment policy in perpetuity without external financing, we can use [Eqn 7] to derive its price P[0]. To do that, we need to obtain values for all the required variables.

Since the firm is retaining 20% of Y[0], then its additional invested capital at t = 0 is $20 and the asset will earn the corporation $4 (since r is 20%) in perpetuity starting from t = 1. Therefore, the net income expected in Y[1] is $104 and the percentage growth is 4%. 4% also represents the growth rate of the dividends so DPS[1] will be $104 times 80% which is $83.2. We now have all the parameters for [Eqn 7].

$P_{0} \;with\;DDM(Gordon's\;model) = \frac{\$ 83.2}{0.06} = \$1386.67$

P[0] with Gordon’s discount model is $1386.67

Method 2 - IOM

We have already computed the values of the variables required in [Eqn 29] and we can substitute them into the formula. P[0] is $1386.67, the same result as what was calculated from Gordon’s model.

$P_{0} = \frac{\$104}{0.10} + (\frac{\$20.8(0.10)}{0.10})(\frac{1}{(0.06)}) = \$1386.67$

Conclusion

The fundamental axiom for the valuation of an asset is the capitalization of cash flows expected to be generated in the future with an investor’s opportunity cost of capital. While some valuation frameworks could be preferred over others, under certain conditions and with proper modifications, the intrinsic worth of the asset that is being valued will pan out to be approximately the same regardless of the model that is selected.

Smith, P.J., ‘An investigation of Gordon’s Common Stock Valuation Model’, 1958

Williams, J.B., ‘The Theory of Investment Value’, 1938

Lintner, J., ‘Distribution of Incomes of Corporation Among Dividends, Retained Earnings and Taxes’, 1956

Gordon, M.J and Shapiro, E., ‘Capital Equipment Analysis: The Required Rate of Profit’, 1956

Koller T., Goedhart M., and Wessels, D., ‘Measuring and Managing the Value of Companies, 2020

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