First things first
Discounted cash flow models
The dividend discount model: Theory
The dividend discount model: Versions
The dividend discount model: An example
The big picture
Excel section
Challenge section
T
here exist many models of equity valuation, some based on discounted cash flow and others based on multiples. Many academics and practitioners portray the dividend discount model as the simplest of the discounted cash flow models. That’s a big mistake. If you learn one thing from this chapter, let it be this: the dividend discount model is deceptively simple, it can easily be misused, and its proper implementation is much more difficult than is usually believed.First things first
There are three important distinctions to keep in mind when discussing valuation. Two are specific to equity valuation, the first a distinction between fundamental analysis and technical analysis, and the second between absolute valuation and relative valuation. The third is more general, applying to the valuation of all assets, and is a distinction between price and value.
Fundamental analysis refers to a valuation technique that focuses on the drivers of a company’s value. It involves the analysis of the company’s financial statements, financial ratios, market, competitors, and many other factors in order to determine its value. Technical analysis, on the other hand, largely focuses on the trading history of a stock. It does not really attempt to value a company; rather, it involves the extrapolation of trends and patterns from past prices in order to extract clues as to how the price may behave in the future.
Models of equity valuation can be grouped into two types. Models of absolute (or intrinsic) valuationestimate the value of a company on the basis of its own fundamentals; models of relative valuation assess it in relation to the value of comparable companies. The former estimate value by discounting expected cash flows at a rate that reflects their risk; the latter do it by comparing ratios usually called multiples. Both types of models belong to the category of fundamental (as opposed to technical) analysis.
The dividend discount model (DDM), the weighted-average cost of capital (WACC) model, the adjusted present value (APV) model, and the flows-to-equity (FTE) model are all discounted cash flow (DCF) models. Price-to-earnings (P/E) ratios, price-to-book (P/B) ratios, price-to-cash flow (P/CF), and price-to- dividend (P/D) ratios are some of the most widely used multiples in relative valuation.
Finally, it is important to keep in mind a critical distinction between price and value. The former simply indicates the number of dollars an investor has to pay for a share of a company, just like the number of dollars we pay for a dinner or an airplane ticket. The latter is much more subtle. In a way, it is the number
of dollars an investor shouldpay for a share of a company. This means that given the fundamentals of the company and what it is expected to deliver, there is an appropriate price to pay for its shares. This appropriate price, which may or may not be equal to the market price, is called ‘value’ or ‘intrinsic value.’
Note that allpricing models yield an estimate of intrinsic value, that is, they all yield the price investors should pay for a company or one of its shares. In fact, the whole concept of market efficiency is based on whether market prices appropriately reflect intrinsic values; the more this is the case, the more efficient markets are.
Discounted cash flow models
All DCF models are based on the calculation of a present value (discussed in Chapter 21). The various versions of this model differ in the type of cash flows discounted and therefore in the discount rate. Other than that, all DCF models require the analyst to estimate expected cash flows and the appropriate rate at which they should be discounted.
In this chapter we’ll deal with dividends, the ‘simplest’ of all cash flows.
Perhaps one reason for which the DDM is considered the simplest of all DCF models is because dividends are observed directly but other types of cash flows have to be calculated from financial statements. This obviously applies to the past values of these magnitudes. Looking forward, all three magnitudes need to be forecasted and it’s not at all clear that forecasting dividends is any easier than forecasting other types of cash flow.
The underlying idea behind the discount rate is that it should capture the risk of the cash flows discounted. That’s why DCF models that differ in their definition of cash flow also differ in their discount rate. Some DCF models discount cash flows at the cost of equity whereas some others do it at the cost of capital. The discount rate for the DDM we discuss in this chapter is the former.
Finally, in the typical implementation of DCF models, analysts make some assumptions about the expected growth rate of the cash flows to be discounted.
These assumptions may range all the way from constant growth to two or more stages of growth. It is also typical for analysts to estimate a terminal value; this is the last cash flow to be discounted and attempts to summarize in a single number all the cash flows from that point on. The two most widely used alternatives for estimating the terminal value are a growing perpetuity (an infinite sequence of cash flows growing at a constant rate) or a multipleof some fundamental variable (such as earnings or cash flow).
The dividend discount model: Theory
The underlying idea of the DDM is both very simple and very plausible: an investor should pay for a share of a company the present value of all the cash flows he expects to receive from the share. And what does an investor pocket from a share? Dividends (if the company pays them) for as long as he holds the stock, and a final cash flow given by the price at which the investor expects to sell the share. It can’t really get much simpler than that, and that’s one of the reasons why, mistakenly, the DDM is sold as a ‘simple’ model. However, as we’ll discuss below, the devil is in the detail.
But let’s leave the devil for later. The formal expression of the DDM is given by
(13.1)
where p0 denotes the intrinsic value of (or, misusing the word, the price an investor should pay for) a share of the company, E(Dt) the expected dividend per share in period t, E(pT) the expected share price at time T, Rthe discount rate, and Tthe number of periods for which dividends are forecast.
Note that the cash flows we’re discounting, dividends, end up in the pocket of shareholders, who are then the ones bearing their risk. The discount rate must then reflect the return shareholders require from holding the shares of the company. This required return on equity, sometimes also called the cost of equity, can be estimated with many models, the most popular of which is the capital asset pricing model, CAPM (discussed in Chapter 7). Therefore, the discount rate is usually estimated as R= Rf+ MRP · ß, where Rf, MRP, and ß denote the risk-free rate, the market risk premium, and the company’s beta, respectively.
It is possible, though in no way essential, to forecast E(pT) as a function of the dividends expected to be received from time T on, that is, E(pT) = f{E(DT+1), E(DT+2), E(DT+3) . . .}. In that case, equation (13.1) turns into the present value of an infinite sequence of dividends. That is,
(13.2) p0= E(D1)
+ E(D2)
+ E(D3)
+ . . . + E(DT) + E(pT) (1 + R) (1 + R)2 (1 + R)3 (1 + R)T
p0= E(D1)
+ E(D2)
+ E(D3)
+ . . .+ E(DT) (1 + R) (1 + R)2 (1 + R)3 (1 + R)T + E(DT+1)
+ E(DT+2)
+ . . . (1 + R)T+1 (1 + R)T+2
You may find the idea of an ‘infinite’ sequence of dividends a bit hard to grasp. But although no investor is going to hold a share ‘for ever,’ the life of the company is in principle unlimited. If it makes your life easier, just think of equation (13.2) as the present value of a ‘very long’ sequence of dividends. You can even drive this thought home by noting that dividends that are very far away add very little to p0.
The dividend discount model: Versions
The DDM is not used in practice as stated in equation (13.2). Its usual implementation imposes some structure on the expected growth of dividends, with different assumptions generating different versions of the DDM. It is important to keep in mind that an assumption about the way dividends are expected to evolve is a statement about the company’s expected dividend policy.
This policy, in turn, depends not only on the expected profitability of the company but also on the existence of alternative uses for the company’s profits (investment opportunities). This is one of the reasons that, however simple some versions of the DDM may look, its proper implementation is far from trivial. Again, the devil is in the detail.
No growth
The simplest assumption we could make about expected dividends is that they will remain constant at the level of the last dividend paid by the company (D0), that is, E(D1) = E(D2) = E(D3) = . . . = D0. Substituting this stream of dividends into (13.2) we get
(13.3)
Now, if you’ve never dealt with this model before or are not a bit trained in math, the second equality may surprise you. But it is indeed the case that if we discount an infinite sequence of a constant magnitude (D0 in our case), then the sum of the infinite terms collapses into the constant magnitude divided by the discount rate. Mathematically, this is called a perpetuity.
Whether or not it makes for a good pricing model we’ll discuss later.
p0= D0
+ D0
+ D0
+ . . . = D0 (1 + R) (1 + R)2 (1 + R)3 R
Constant growth
A second possibility is to assume that dividends will grow at a constant rate g beginning from the last dividend paid by the company, that is, E(D1) = D0· (1 + g), E(D2) = D0 · (1 + g)2, E(D3) = D0 · (1+g)3, and so on. Substituting this stream of dividends into equation (13.2) we get
(13.4) Again, if you haven’t dealt with this model before or are not a bit trained in math the second equality may surprise you. But again it is the case that an infinite sum of terms collapses into something relatively simple. Mathematically, this is called a growing perpetuityand holds as long as R> g.
Whether the assumption that dividends will grow at a constant rate in perpetuity is a plausible one we’ll discuss later. At this point, it’s important to note that this assumption shouldn’t be thought of as implying that dividends are expected to grow exactlyat the rate g; that would be naive. Rather, gshould be though of as an average growth rate of dividends. That is, in some periods dividends may grow at more than g% and in some others at less than g%, but on average we do expect them to grow at g%.
Two stages of growth
A third possibility is to assume that dividends will grow at a rate g1over the first T periods, and at the rate g2 from that point on, usually (but not necessarily always) with g1> g2. Imposing this assumption on equation (13.2) we get the rather-scary expression
(13.5) Let’s think about equation (13.5) a bit. The first T terms of the right-hand side simply show a sequence of dividends growing at the rate g1 during T
p0=D0· (1 + g)
+D0· (1 + g)2
+D0· (1 + g)3
+ . . . = D0· (1 + g)
(1 + R) (1 + R)2 (1 + R)3 R – g
p0=D0· (1 + g1)
+D0· (1 + g1)2
+. . .+ D0· (1 + g1)T
+ R – g2
(1 + R) (1 + R)2 (1 + R)T (1 +R)T
{D0· (1 + g1)T} · (1 + g2)
periods. The numerator of the last term is the terminal value and therefore an estimate of the stock price at time T. Note that this numerator is basically the same as equation (13.4) but set at time T rather than at time 0: if dividends grow at the rate g1 over T periods, the dividend in period Twill be D0 · (1 + g1)T; and if beginning from that level dividends grow at g2from that point on (in perpetuity), then at time Tthe stock price should be
{D0· (1 + g1)T} · (1 + g2) R– g2
The denominator of the last term is simply the discount factor for a cash flow expected at the end of period T.
The first growth rate (g1) is usually thought of as a period of fast growth in dividends; the second (g2) as the growth in dividends after the company matures. The number of periods for which dividends are expected to grow at g1 (T) in principle depends on each individual company, its stage of growth, and its dividend policy. However, in practice, T = 5 and T = 10 are popular choices (perhaps for no particularly good reason).
Other possibilities
Analysts might consider it appropriate for a company at some point in time to model three or even more stages of growth in dividends. Or they may consider one or more stages of growth in dividends and a terminal price estimated with a multiple. The possibilities are, of course, endless.
The dividend discount model: An example
In 2003, General Electric (GE) delivered a profit of $15 billion on revenues of
$133 billion. Its market cap at the end of the year was $312 billion and its stock price $30.98. Its earnings per share (EPS) and dividends per share (DPS) were
$1.49 and $0.76, respectively, giving it a price to earnings (P/E) ratio of 21 and a healthy dividend payout ratio (DPR) of 51%. How much shouldan investor have paid for a share of GE at that time? This is the question we’ll attempt to answer using the DDM. (Throughout our analysis we’ll assume that we’re valuing GE at the end of the year 2003.)
Before we get to the numbers bear this in mind: our goal here is to go over different versions of the DDM and briefly discuss their pros and cons, not to
make a strong statement about GE’s intrinsic value. This implies that we’ll be making different assumptions about how dividends are expected to grow over time, effectively making an ‘if–then’ analysis (if the dividends evolve this way, thenthe price should be that). This is obviously notthe way analysts implement the DDM (or any other model). Analysts derive their estimates from what they consider their most plausible scenario, perhaps complementing it with some sensitivity analysis, but they do not go over widely different scenarios as we’ll do here for illustrative purposes.
It’s also important to keep in mind that it’s fundamentally wrong to make a set of assumptions, get an estimate of intrinsic value, compare that with the market price, and determine from the comparison whether our assumptions are right. That defeats the very purpose of the analysis. Stock pricing is about coming up with what we believe is a plausible set of assumptions, getting an estimate of intrinsic value that follows from those assumptions, and then deciding whether to buy, hold, or sell based on the comparison between our estimate of intrinsic value and the market price. Nevercompare an estimate of intrinsic value to a market price to assess the plausibility of your assumptions;
always use assumptions that you believe to be plausible to start with.
The discount rate
As discussed above, the discount rate for the DDM is the required return on equity and is typically estimated with the CAPM. At year-end 2003 the yield on ten-year notes was 4.3% and GE’s beta was 1.1. For the market risk premium we can use the popular estimate of 5.5% (see Chapter 7). Putting these three numbers together we get a cost of equity for GE of 0.043 + 0.055 · 1.1 = 10.4%.
That will be our discount rate.
No growth
Let’s start by assuming that our best estimate of GE’s expected dividends is that they will remain constant at the level of the last dividend paid by the company ($0.76). According to equation (13.3), then, our best estimate of GE’s intrinsic value would be $0.76/0.104 = $7.3. The calculation is trivial, but is this version of the DDM plausible?
Not really. Note that a constant nominal dividend implies that the real dividend will eventually be 0, that is, owing to inflation, the dividend will gradually lose purchasing power. That doesn’t sound like a plausible dividend
policy for a company to follow. Constant nominal dividends may be plausible for a few years, but not in the long term.
Unsurprisingly, then, our estimate of GE’s intrinsic value is much lower than its price. Given that GE has a long history of increasing its dividend (by roughly 13% a year over the past 20 years), the market is plausibly factoring some growth in dividends into GE’s price. In short, because our assumption is not very plausible, neither is our estimate of intrinsic value.
Constant growth
Let’s now assume a more plausible dividend policy for GE. Let’s assume that we expect the company to keep the purchasing power of its dividend constant over time. If we expect inflation to run at an average of 3% a year (the historical annual rate), and GE to increase its annual dividend at that rate in the long term, then according to equation (13.4), our best estimate of GE’s intrinsic value would be $0.76 · (1.03)/(0.104 – 0.03) = $10.7. If we believe our assumption of 3% constant growth in dividends, then we should also believe that GE should be trading at $10.7 and therefore that at $31 it’s overpriced.
What if we expected GE to increase its dividend at the rate of 6% in the long term instead? Then according to equation (13.4), GE’s intrinsic value would be
$0.76 · (1.06)/(0.104 – 0.06) = $18.5, and we would still conclude that GE is overpriced. Note that under both assumptions our estimate of intrinsic value is much lower than the market price. Should we then conclude that the market is expecting a much higher long-termgrowth in dividends?
Hard to believe. The reason is that the constant-growth model makes an assumption about the long-term growth of dividends. In the long term, the growth in dividends, the growth in earnings, and the growth of the company’s value must align. In addition, this rate of growth cannot outpace the growth of the overall economy, simply because the growth of a component factor cannot for ever outpace that of the aggregate.
Historically, the US economy has grown at an annual rate of roughly 6% (3%
in real terms plus 3% inflation). That becomes an upper boundary for any plausible estimate of the long-term growth of dividends. In other words, whenever we use the constant-growth model, or any model in which the terminal value is expressed as a growing perpetuity, it is simply not plausible to assume a long-term growth beyond, roughly, 6%. Other economies may of course have different long-term rates of growth, but it would be hard to make a plausible case for rates much higher than 6% or so.
Two stages of growth
The main problem with the constant-growth model is its lack of flexibility.
Perhaps we plausibly expect GE to increase its dividend at a much higher rate than 6% in the short term, but we cannot accommodate that in the constant growth model. The extra flexibility of the two-stage model then becomes valuable.
At the end of 2003, analysts expected GE to increase its EPS at the annual rate of 10% for the following five years. Let’s then assume that dividends will grow at the same 10% rate during those five years. And let’s also assume that from that point on dividends will grow at a long-term rate of 6% a year. According to equation (13.4) then, our best estimate of GE’s intrinsic value would be
p0= $0.76 · (1.10)
+ . . . + $0.76 · (1.10)5
+ 0.104 – 0.06
= $22.0
(1.104) (1.104)5 (1.104)5
Note that, if we believe our assumptions, given its price of $31, GE at $22 is overpriced. Note, also, that our assumptions lead us to a terminal price of $29.8, whose present value is roughly $18. Therefore, some 80% of our estimated value of $22 comes from the terminal value. Although this proportion is unusually high, it’s not uncommon for a terminal value to be around 50–60% of the estimated intrinsic value. This fact makes sensitivity analysis on the terminal value a critical part of any valuation.
Terminal value as a multiple
Finally, let’s consider a DDM in which we model the terminal value as a multiple (rather than as a growing perpetuity). Let’s assume, first, that over the next five years dividends will grow at the annual rate of 13%, that is, the rate at which they’ve been growing over the past 20 years. Let’s also assume that over the next five years EPS will grow at the 10% annual rate expected by analysts; that would imply EPS of $2.40 = ($1.49) · (1.105) five years down the road. Finally, let’s assume that at that point in time GE’s P/E ratio remains at its current 21, which would give us a terminal value of $50.4 = (21) · ($2.40). According to equation (13.1), then, our estimate of intrinsic value would be
{$0.76 · (1.10)5} · (1.06)