A perpetuity is a constant stream of identical cash flows with no end. The formula for determining the present value of a perpetuity is as follows:

A delayed perpetuity is perpetual stream of cash flows that starts at a predetermined date in the future. For example, preferred fixed dividend paying shares are often valued using a perpetuity formula. If the dividends are going to originate (start) five years from now, rather than next year, the stream of cash flows would be considered a delayed perpetuity.

Although it may seem a bit illogical, an infinite series of cash flows can have a finite present value. Because of the time value of money, each payment is only a fraction of the last.

The net present value (NPV) of a delayed perpetuity is less than a comparable ordinary perpetuity because, based on time value of money principles, the payments have to be discounted to account for the delay. Retirement products are often structured as delayed perpetuities.

Examples of Perpetuities
The perpetuity is not as abstract a concept as you may think. The British-issued bonds, called consols, are a great example of a perpetuity. By purchasing a consol from the British government, the bondholder is entitled to receive annual interest payments forever.

Another example is a type of government bond called an undated issue that has no maturity date and pays interest in perpetuity. While the government can redeem an undated issue if it so chooses, since most existing undated issues have very low coupons, there is little or no incentive for redemption. Undated issues are treated as equity for all practical purposes due to their perpetual nature, but are also known as perpetual bonds.

Perhaps the best-known undated issues are the U.K. government's undated bonds or gilts, of which there are eight issues in existence, some of which date back to the 19th century. The largest of these issues presently is the War Loan, with an issue size of £1.9 billion and a coupon rate of 3.5% that was issued in the early 20th century.

Perpetuities and the Dividend Discount Model
The concept of a perpetuity is used often in financial theory, particularly with the dividend discount model (DDM). Unfortunately, the theory is the easy part. The model requires a number of assumptions about a company's dividend payments, growth patterns and future interest rates. Difficulties spring up in the search for sensible numbers to fold into the equation. Here we'll examine this model and show you how to calculate it.

The basic idea is that any stock is ultimately worth no more than what it will provide investors in current and future dividends. Financial theory says that the value of a stock is worth all of the future cash flows expected to be generated by the firm, discounted by an appropriate risk-adjusted rate. According to the DDM, dividends are the cash flows that are returned to the shareholder.

To value a company using the DDM, calculate the value of dividend payments that you think a stock will generate in the years ahead. Here is what the model says:

P= the price at time 0
r= discount rate

For simplicity's sake, consider a company with a $1 annual dividend. If you figure the company will pay that dividend indefinitely, you must ask yourself what you are willing to pay for that company. Assume the expected return (or the required rate of return) is 5%. According to the dividend discount model, the company should be worth $20 ($1.00 / .05).

How do we get to the formula above? It's actually just an application of the formula for a perpetuity:

The obvious shortcoming of the model above is that you'd expect most companies to grow over time. If you think this is the case, then the denominator equals the expected return less the dividend growth rate. This is known as the constant growth DDM or the Gordon model after its creator, Myron Gordon. Let's say you think the company's dividend will grow by 3% annually. The company's value should then be $1 / (.05 - .03) = $50. Here is the formula for valuing a company with a constantly growing dividend, as well as the proof of the formula:

The classic dividend discount model works best when valuing a mature company that pays a hefty portion of its earnings as dividends, such as a utility company.

The Problem of Forecasting
Proponents of the dividend discount model say that only future cash dividends can give you a reliable estimate of a company's intrinsic value. Buying a stock for any other reason - say, paying 20 times the company's earnings today because somebody will pay 30 times tomorrow - is mere speculation.

In truth, the dividend discount model requires an enormous amount of speculation in trying to forecast future dividends. Even when you apply it to steady, reliable, dividend-paying companies, you still need to make plenty of assumptions about their future. This model is only as good as the assumptions it is based upon. Furthermore, the inputs that produce valuations are always changing and susceptible to error.

The first big assumption that the DDM makes is that dividends are steady or grow at a constant rate indefinitely. But even for steady, utility-type stocks, it can be tricky to forecast exactly what the dividend payment will be next year, never mind a dozen years from now. (Find out some of the reasons why companies cut dividends in Your Dividend Payout: Can You Count On It?)

Multi-Stage Dividend Discount Models
To get around the problem posed by unsteady dividends, multi-stage models take the DDM a step closer to reality by assuming that the company will experience differing growth phases. Stock analysts build complex forecast models with many phases of differing growth to better reflect real prospects. For example, a multi-stage DDM may predict that a company will have a dividend that grows at 5% for seven years, 3% for the following three years and then at 2% in perpetuity.

However, such an approach brings even more assumptions into the model - although it doesn't assume that a dividend will grow at a constant rate, it must guess when and by how much a dividend will change over time.

What Should Be Expected?
Another sticking point with the DDM is that no one really knows for certain the appropriate expected rate of return to use. It's not always wise simply to use the long-term interest rate because the appropriateness of this can change.

The High-Growth Problem
No fancy DDM model is able to solve the problem of high-growth stocks. If the company's dividend growth rate exceeds the expected return rate, you cannot calculate a value because you get a negative denominator in the formula. Stocks don't have a negative value. Consider a company with a dividend growing at 20% while the expected return rate is only 5%: in the denominator (r-g) you would have -15% (5%-20%)!

In fact, even if the growth rate does not exceed the expected return rate, growth stocks, which do not pay dividends, are even tougher to value using this model. If you hope to value a growth stock with the dividend discount model, your valuation will be based on nothing more than guesses about the company's future profits and dividend policy decisions. Most growth stocks do not pay out dividends. Rather, they reinvest earnings into the company with the hope of providing shareholders with returns by means of a higher share price.

Consider Microsoft, which did not pay a dividend for decades. Given this fact, the model might suggest the company was worthless at that time, which is completely absurd. Remember, only about one-third of all public companies pay dividends. Furthermore, even companies that do offer payouts are allocating less and less of their earnings to shareholders.

The dividend discount model is by no means the be-all and end-all for valuation. However, learning about the dividend discount model does encourage critical thinking. It forces investors to evaluate different assumptions about growth and future prospects. If nothing else, the DDM demonstrates the underlying principle that a company is worth the sum of its discounted future cash flows. Whether or not dividends are the correct measure of cash flow is another question. The challenge is to make the model as applicable to reality as possible, which means using the most reliable assumptions available.

The Effect Of Compounding

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