6.12  The 2 Stage Model

The intrinsic value in the two stage dividend model is calculated as follows:

In the two stage model, dividends grow at rage g₁ for T years and then grow at rate g₂.  The dividends are discounted at rate k₁ in stage 1 and at rate k₂ thereafter.  This yields the following formula for the intrinsic value:

Where

The two stage growth model has another advantage.  It allows us to impose a constraint over second stage growth rates that are consistent across firms.  This constraint is that no firm can grow at a rate faster that than the economy as a whole in the second (perpetual) stage; otherwise, the firm would become larger than the economy as a whole, is a contradiction.  This constraint is imposed by restricting stage 2 growth (or normal growth) to be no larger than the estimated long term growth rate of the economy.  In the first stage, the growth rate (called the abnormal growth rate) is unrestricted.

Abnormal Growth:  Also known as “extraordinary growth,” this is the growth projected over the next finite number of years (typically 5-years- 10-years).   As an aside, when you are calculating the intrinsic value for a firm that has negative short run growth, you want to use a three stage model, where the first two stages provide a further breakdown of the growth rates into three phases --- a first phase of negative growth, followed by a second phase of abnormal positive growth and then ultimately a third phase of normal growth.

Normal Growth:  Ultimately in every multi-stage growth model of intrinsic value you must make some simplifying assumption that in the final stage the (going concern) firm grows in perpetuity at some normal or stable growth rate.  To avoid the limiting case implications discussed in relation to the Constant Growth we assume that Normal Growth is bound from above by the economy wide growth.  Therefore, to estimate Normal Growth we start with reviewing the behavior of long run economy wide growth behavior.

Estimating Normal (Stage 2) Growth

Normal growth in a 2-stage model is bound by the economy wide growth.  This number can be estimated by referring to long term average macroeconomic data for the US economy.  In Valuation Tutor’s information system, under “Papers and Reports,” you will see a link to a report on the long term growth rate of the US economy:

The following quote comes from this report:

We also observe over the last 100-year span that the rates of economic growth across the then emerging industrial nations were fairly tightly clustered around this 2.0% pace. At the high end was Japan with an annual rate of growth averaging about 2.7%, while at the low end was Great Britain with an annual growth rate averaging 1.4%. The United States, which grew at a 1.8% average annual rate, was slightly below average.

They also went on to observe:

For the United States, the long-term growth of real GDP per capita over the last 125 years has revealed remarkable steadiness, advancing decade after decade with only modest and temporary variation from the observed 1.8% annual rate of increase.

The estimates in these quotes are for the growth of real GDP, which does not take into account inflation.  In valuation exercises, we require the growth rate of nominal GDP, which takes into account inflation (otherwise, we would have to recast everything in terms of constant dollars!). Inflation numbers suggest that inflation compounded from 1913 to 2008 resulted in a cumulative rate of 2071.23%.  This implies an annual constant compounded rate of approximately 3.24%.

Combining the above we can make a reasonable estimate for one plus the long term nominal growth in the US, to be around 1.018*1.03 = 1.04854. 

In our examples, we will use 4.5% as the stage 2 growth rate, slightly below the economy wide rate.  For a company like IBM that has a strong track record on research and development as well as filed patents, it makes sense to think it mirror the long term growth rate of the economy.

The Length of Stage 1

The length of stage 1 will depend on two sets of factors.  The first set is firm specific.  For example, a firm may have developed a successful new product, and may enjoy a period of high growth before its competitors can slow its growth. Patents would give a company an advantage as well; for example, for a pharmaceuticals company, until a patent expires, the company can enjoy high growth.

The second set of factors concern the economy as a whole.  Growth will be affected by general economic conditions, such as a prolonged boom or a prolonged slump.  The National Bureau of Economic Research dates the start and end of a business cycle.  A business cycle measures fluctuations of economic growth around the long term growth rate. 

Generally, analysts are unwilling to use large numbers for stage 1 because it implies that current abnormal conditions are going to prevail for a long time.  Even if a firm is experiencing rapid growth, competition will limit the growth, so the time to market of competitors will limit the time.  One of the longest is patent protection to drug makers, which is twenty years.  But even in that case, it is unlikely that sales or earnings will grow at a rapid pace for the entire period.  And with economics fluctuations, the actual growth over the period may be tempered further.

If you search for examples of applications of the dividend model by analysts, you find that they typically use 5 or ten years as the length of stage 1.  Of course, if the company you are studying has specific characteristics that inform you about its growth period, you should certainly use that.