PERPETUITIES NARRATIVE SCRIPT 2004 SOUTH-WESTERN, A THOMSON BUSINESS

Transcription

1 NARRATIVE SCRIPT 2004 SOUTH-WESTERN, A THOMSON BUSINESS

2 NARRATIVE SCRIPT: SLIDE 2 A good understanding of the time value of money is crucial for anybody who wants to deal in financial markets. It does not matter if we want to buy insurance, lease a car, buy or sell stocks or bonds or prepare for our eventual retirement. All of these actions are best undertaken armed with a good understanding of the time value of money. Time value of money principles underlie almost all facets of finance. Perhaps the most important of these is valuation. To value any asset, we need to know three things. These are the size of the cash flows, the timing of the cash flows and the discount rate to apply to the cash flows. If we know these three factors, we can calculate the value of almost any asset, be it a stock, a bond, an insurance policy or commercial real estate. During this tutorial, we will examine a type of cash flow known as a perpetuity. As the name implies, a perpetuity is a cash flow stream that continues forever. Does that mean that it lasts for your lifetime? No it is longer than that. A perpetuity has a cash flow stream that continues to infinity. At this point, you may be saying to yourself that no security that trades in financial markets could possibly act like a perpetuity. If you are saying this, you are incorrect, because the most common of all traded securities, the common stock, is one type of perpetuity. Preferred shares are also a perpetuity, as are some types of bonds. Perpetual bonds or bonds with no maturity date are often referred to as consol bonds. Copyright 2004 South-Western, a Thomson Business 1

3 NARRATIVE SCRIPT: SLIDE 3 There are three basic types of perpetuities, based on the nature of the cash flow stream. These are constant cash flows, increasing cash flows and decreasing cash flows. Let s look at each of these in turn, starting with constant cash flows. An example of a perpetuity with a constant cash flow stream is a preferred share. The dividend on a preferred share is usually a fixed percentage of the par value of the preferred. For example, if a $25 par value preferred share carries a 5% dividend, it will pay a fixed and constant dividend of $1.25 every year. How should we calculate the value of this type of security? Let s start by putting the cash flows on a time line. When we do this, we see that we receive $1.25 at the end of every year from now until infinity. To calculate the market price of this security, we need to calculate the present value of every cash flow and then add them up. Of course, this would be a huge amount of work, were it not for a little trick from high school algebra. What we need to remember is that the sum of an infinite geometric series easily collapses to a simple formula that looks like this. Within this formula, C is the cash flow that occurs at the end of the first year and r is the discount rate. Let s return to our example of the preferred share to see what the preferred might trade for in the market. For the purposes of our illustration, assume that our preferred share has a discount rate of 8%. At this time, we can think of the discount rate as being the yield on other securities with similar risk that trade in the market place. In that case, the market price of the preferred share should be $15.63, as shown on the screen. Copyright 2004 South-Western, a Thomson Business 2

4 NARRATIVE SCRIPT: SLIDE 4 Let s now discuss perpetuities with increasing cash flow streams. Can you think of any traded securities with cash flow streams that increase forever? If you said common stocks, you are absolutely correct. The cash flow stream generated by a common stock is the dividend and we hope that the dividend stream will increase over time, as the company and its profits grow. Let s assume that we are valuing a common stock with a dividend of $1.00 that was paid yesterday. We expect the dividend to grow at 4% per year and we know that stocks with similar risk are yielding 14%. What do you think the stock is worth? To answer this question, we must first calculate the value of the dividend at the end of the first year. We need to determine this amount because the perpetuity formula that we will use assumes that the first cash flow will always occur at the end of the first year. If this is not the case, you must adapt the formula accordingly. Knowing that we need to find the amount of the dividend at the end of the first year, how do you suppose we would determine that value? This is easily achieved. We solve for this value using the basic time value of money formula future value equals present value times one plus g raised to the exponent t. In this case, t is equal to one and g is equal to 4%, the growth rate of the dividend stream. Thus, the dividend at time one is equal to $1.04. We can now solve for the market price of the common stock using the formula for calculating the present value of a perpetuity with growth as illustrated on the screen. In this formula, C is the cash flow that occurs at the end of the first year, r is the discount rate and g is the expected growth rate of the dividend stream. Now that we know the variables of the formula, we can determine the stock price. By making the appropriate substitutions, we find this value to be $ Copyright 2004 South-Western, a Thomson Business 3

5 NARRATIVE SCRIPT: SLIDE 5 Let s now look at the final type of perpetuity. This is a perpetuity with declining cash flows. Unlike our previous perpetuity model discussions, there are no traded securities that are exactly like a perpetuity with a declining cash flow stream, but oil and cash income trusts come very close. Each year, the trust pumps a portion of the reserves. As the reserves diminish, the amount of energy extracted every year goes down. If the field is expected to diminish gradually over many, many years, we might view it as a perpetuity with a declining cash flow stream. To solve for the price of a perpetuity with a declining cash flow stream, we use the formula now illustrated on the screen. In this case, C is the cash flow at the end of the first year, r is the discount rate and g is the constant rate of decline in the cash flow stream. For purposes of illustration, let s assume that an oil trust distributed $2 yesterday. Assume further that the oil trust believes that its cash flow stream will decline by 2% per year and the yield of oil trusts with similar risk is 10%. What do you think is the value of the oil trust? To answer this question, we must first determine the cash flow at the end of period one. In this case, we find the cash flow as the present value of the distribution multiplied by one minus the decline rate of the cash flow raised to the exponent t. Following the mathematics on the screen, we find this amount to be $1.96. With this information, we can solve the value of the perpetuity. By substituting $1.96 for the cash flow at the end of year one, 10% for the discount rate and 2% for the decline rate of the cash flow stream, we find the value of the oil trust to be $16.33 per unit. Copyright 2004 South-Western, a Thomson Business 4

6 NARRATIVE SCRIPT: SLIDE 6 Perpetuities are all around us. They include investments like stocks, preferred shares, consol bonds and income trusts. Each investment alternative is considered a perpetuity as each represents a set of cash flows that lasts to infinity. Perpetuities are valued according to the nature of the cash flows received. The cash flows may be constant, they may increase at a constant rate, or they may decline at a constant rate. The one type of perpetual cash flow we have not addressed in this tutorial is cash flows that are irregular. In this case, each cash flow must be treated as a single payment and present valued back to time period zero. Regardless of the type of cash flow generated by a perpetuity, there is a primary principle that must be remembered. A perpetuity is valued based upon the cash flow that occurs at the end of the first period. This is not a particular problem when valuing a perpetuity with a constant payment. However, in the event that you are calculating the value of a perpetuity whose cash flows are increasing or declining, you must remember to revise the previous payment upward or downward to reflect the nature of the cash flow. Selecting the appropriate discount rate is also crucial. You must remember to select a yield that is reflective of the risk of the security. Thus, you should use the yield of similar risk securities in the market as the discount rate. Copyright 2004 South-Western, a Thomson Business 5