Copyright 1997 by Campbell R. Harvey and Stephen Gray. All rights reserved. No part of this lecture may be reproduced without the permission of the authors.
Latest Revision: January 5, 1997
2.0 Introduction
This lecture provides an overview of equity securities (or stocks or shares). These securities provide and ownership interest in the firm whereas debt securities (or loans or bonds or fixed-interest securities) provide a creditor relationship with the firm. After a brief overview of some of the institutional details of these securities, this module focuses on valuing the securities in a world of relative certainty. Equity securities can be valued as the present value of the future dividend stream. The analysis is extended to evaluating investment proposals with a focus on determining which cash flows are relevant in deciding whether a particular proposal should be undertaken.
2.1 Introduction to Stocks
Stocks represent ownership interest in a company and entitle the holder to vote at company meetings, collect periodic dividend payments, and to sell it at his discretion. Stocks are issued by companies to finance their operations and are purchased by investors whose gains come in the form of:
Stocks are first issued to investors through what is known as the primary or new issue market. However, investors will be reluctant to purchase a stock unless there is a mechanism available for the speedy resale of these stocks. Provision of a resale mechanism is the function of the stock exchange (also known as the secondary market). Investors are able to buy and sell stocks through the stock exchange. Investors trade between themselves on these exchanges. The company is not a party to the transaction and receives no funds as a result of these transactions.
The major global secondary markets are:
Various stock indexes are also maintained and are closely watched by investors. When we think of how the stock market performed in a particular period, we invariably refer to one of these indexes. The major stock exchange indexes are:
2.2 Stock Transactions
There are three ways of transacting in stocks:
Buy - we believe that the stock will appreciate in value over time. We are expecting a bullish market for the stock. It is also said that we are long in the stock.
Sell - we believe that the stock will depreciate in value over time or we require funds for another purpose (liquidity selling).
Short Sell - here we do not own the stock, but we borrow it from another investor, sell it to a third party, and, in theory, receive the proceeds. We are obligated to pass on to the lender of the stock any dividends declared on the stock and also to pay to the lender the market price of the stock if he himself should decide to sell. When we short sell, we believe that the stock will decline in value thus enabling us to buy it back at a low price later on to make up our obligations to the lender. We are expecting a bearish market for the stock. It is also said that we are short the stock.
When a short sale is executed, the brokerage firm must borrow the shorted security from its own inventory or that of another institution. The borrowed security is then delivered to the purchaser on the other side of the short-sale. The purchaser then receives dividends paid out by the corporation. The short-seller must pay out any dividends declared by the firm to the original owner from which the security was borrowed during the period in which the short-sale is outstanding. To close out the short sale, the short seller must buy the stock in order “return” the security originally borrowed. Note that borrowing fees can be significant for “hard-to-borrow” securities because these securities are in high demand due to a high level of short-selling (e.g., Netscape immediately after it went public).
In modeling finance problems we often assume that the investor receives the full proceeds of a short-sale. There are a number of practical mechanics which limit the investors ability to access these funds. The proceeds from a short sale are usually held by an investor’s brokerage firm as collateral. The investor usually does not receive the interest from the short sale proceeds, and will likely have to meet a margin requirement. In practice, short sales require a cash outlay. They do not provide a cash inflow.
2.3 Valuation of Stocks
In this section, we determine the value of a stock in a world of certainty. We make three assumptions regarding the stock (We will subsequently relax these assumptions):
Assume that a stock has just paid a dividend so that the series of future periodic dividends (d_{t} ) can be represented as:
Period | 0 | 1 | 2 | ... | n |
Dividend | d_{1} | d_{2} | ... | d_{n} |
If we denote the required rate of return as r_{e} (for return on equity), what is the ex-dividend (just after the current dividend) price of the stock at time zero, p_{0} ? The table above represents a series of cash flow and we can obtain the current price, p_{0} , simply by finding the present value of these flows using the investors required rate of return, r_{e}. Thus:
or, using more compact notation:
We no consider two special cases of this equation: Constant dividends and constant growth.
Assume that dividends are constant through time. That is, d_{t} = d for all t
This assumption would be more appropriate in the case of preferred stock. Preferred stock (or preference shares) promises a constant dollar dividend forever. Dividends must be paid on preferred stock before any dividends can be paid to common stockholders.
If dividends are constant through time and the firm has an infinite life and r_{e} is constant, the dividend stream is simply a perpetuity. Therefore, the price of the stock is given by:
p_{0} = d / r_{e}
In this case, the dividends of the firm will grow at a constant rate, g, indefinitely. Our time line of dividend payments now becomes:
Period | 0 | 1 | 2 | ... | n |
Dividend | d_{1} | d_{1}(1+g) | ... | d_{1}(1+g)^{n} |
Once again, the table above is just a series of cash flows and we can obtain the current price, p_{0} , simply by finding the present value of these cash flows assuming a constant r_{e+}.
Writing the present value of the cash flows out in full yields:
Taking out the common factor of yields:
Define and we have:
Recall from Class 1 that:
Thus:
Now since the dividend stream is infinite, we must take the limit of the above equation as n gets large. For b<1, b^{n} will go to zero as n becomes large. Recall that b is defined as
Thus, b will be less than 1 if r_{e}>g. In this case the term b^{n} will vanish and we are left with:
We now substitute for b to obtain:
Therefore, with constant growth, the stock price is given by:
where:
Example 2.6
A company has just paid a dividend of US $2 per share. The dividend is expected to grow at a rate of 10 percent per year indefinitely. What is the stock price if the required rate of return is 12 percent? Here we apply the constant growth rate formula. We note that since the current dividend d_{0} = 2, the period 1 dividend, d_{1} , will be d_{0} (1+g). Thus: |
Example 2.7
Suppose the company in the previous example has undertaken new management. It is now expected that the rate of growth in dividends will be 15 percent per year for 3 years, and then 10 percent per year thereafter. What is the new stock price if the required return remains at 12 percent. The time line of the cash flows to stockholders is:
Since the stock pays dividends at a constant rate indefinitely after year 3, find the price of the stock at the end of year 3 using the constant growth formula: Finally, compute the current stock price by taking the present value of each of the first three dividends and the present value of the stock price in year 3: |
Example 2.8a) You’re interested in buying the stock of company WWWFinance. Analysts expect earnings, and consequently dividends, to increase at a rate of 10% per year. The next dividend payout is expected to be 185 ¥ (yen). Your required rate of return is 15%. What should the stock price be? b) Assume that you have paid the price from a) for company WWWFinance stock. Suddenly, analysts revise their future earnings estimates for the company upwards to 11%. What is the new stock price? What absolute gain have you realized on your investment? |
2.9 Present Value of Growth Opportunities
You often hear the buzz words growth firm. We will explore what this means in the context of our stock valuation model. From the formulas already developed, we can separate the price of a company’s stock into two components: a no growth component and a growth component.
Before we work out the details, we could write the price of the share as:
where PVGO equals the present value of growth opportunities. Let’s think about what this means. The first term is the present value of an earnings flow if all earnings in the future were constant. Note that the first term of our formula represents the present value of all future earnings assuming an infinite constant earnings stream and a constant required rate of return. If the firm was not expected to grow, all earnings would be paid in dividends (d_{t}=E_{t} ) and the value of the stock would be the present value of the “dividend annuity” given above.
It is unlikely that the firm will not have any growth opportunities. The more realistic scenario is that some of the earnings are plowed back into the firm’s operations and there is an additional term which represents the present value of growth opportunities. In terms of the formula above, you can consider the PVGO to be a remainder term. We can determine the value of the no growth portion of the stock price. If the stock price is available, then the difference must be the present value of the growth opportunities. We will be a little more precise about what is contained in PVGO in a few minutes.
It is also useful to write the above formula in terms of an earnings-price ratio. All we have done is to rearrange a few terms:
Note that for a constant required rate of return, a higher PVGO implies a lower earnings-price ratio. Correspondingly, a lower earnings-price ratio implies a higher price-earnings ratio (V_{0}/E_{1}). This explains why young growth firms often have high PE ratios while more mature firms have lower PE ratios. It also helps us to understand what goes into a PE ratio. We often hear that a particular stock has a PE that is lower than the industry average, and hence it is a good buy. It is possible that this firm’s growth opportunities are not as favorable as the other firms in the industry. Having a low PE ratio relative to the market does not imply that a profitable opportunity is present.
The splitting of the value of the stock into a growth and a no growth component is an insightful exercise. It is now time to reconcile our dividend growth formula with this split. We will use a to indicate the constant fraction of earnings that are paid out as dividends. Correspondingly, (1-a), represents that fraction of earnings that are plowed back into the firm.
We can rearrange the formula in the following steps:
So, the present value of the growth opportunities is:
Note that the constant stream of earnings, E_{1} , is pulled out of the PVGO to avoid double counting.
To illustrate some of these points, it is useful to go through a few examples. Suppose that TECHCO, Inc. will have earnings per share next year of US $4. The company will plow back 60% of its earnings into continuing operations. These operations and investment projects will have a rate of return on equity of 20%. The required rate of return for the firm, r_{e} , is 16%. Let’s calculate: 1) the stock price, 2) the PE ratio, 3) the PVGO, and 4) the amounts of investment that produce the PVGO.
The stock price calculation is straightforward. We know the next period’s earnings, so the dividend will be:
d_{1} = aE_{1} = (0.4)(4.00) = $1.60
We can deduce what the growth in earnings is going to be. The company is going to plow back 60% of its earnings to be used in projects that are expected to earn 20% per year. So the dividend growth will be given by:
g = (1 - a)(ROE) = (0.6)(0.20) = 12%
Now we can plug directly into the constant dividend growth model:
The PE ratio is also fairly elementary. We know that:
Since earnings will be growing at 12%, we can calculate the current PE:
We can use our formula for the PVGO to calculate this quantity:
So, $25 of the stock price is related to the present value of no growth opportunities, and $15 of the stock price is related to the growth opportunities of the firm.
Now the hardest part of the problem is to show the amounts of investment that produce the PVGO. Let’s take it in steps. In the first year, the firm plows back (0.6)(4.00), or $2.40 of their earnings back into the firms operations. This plowback will earn 20% or $0.48 in perpetuity. The value of this perpetuity is:
0.48 / 0.16 = $3.00
The net present value of the investment is $0.60 or $3.00 - $2.40. Further investment through the years leads to the net present values increasing at 12%. So, we could write:
A PVGO of $15 is the same result that we arrived at earlier.
2.10 Introduction to Capital Budgeting
In order to make good investment decisions it is important to be able to distinguish between profitable investment alternatives and unprofitable investment alternatives. One of the most important requirements in business is to be able to recognize a profitable investment.
Consider the following investment proposal:
Year |
0 |
1 |
2 |
... |
25 |
Cash Flow |
-100 |
11 |
11 |
... |
11 |
That is, the firm can pay out US $100 million now and receive US $11 million at the end of each of the new 25 years. If the required return for this project is 10%, is this a worthwhile investment? To analyze this decision, we compared the present value of the cash inflows in years 1 to 25 with the required outlay of $100 million and concluded that, in fact, this project was not a good investment since the present value of the cash inflows was less than the cost of the investment.
In our consideration of capital budgeting issues and investment decisions, we examine this approach in more detail, considering the following types of questions:
2.11 Determination of Cash Flows
When examining an investment proposal we are interested only in the marginal or incremental cash flows associated with the project in question. The incremental net cash flow of an investment proposal is defined to be the difference between the firm’s cash flows if the investment project is undertaken and the firm’s cash flows if the investment project is not undertaken. Thus, when considering the determination of cash flows, we are concerned only with the incremental net cash flows of the investment proposal.
A cash flow is money paid or received by the firm. The firm’s net cash flow in period t, X_{t}, is determined by the difference between cash inflows and cash outflows:
(1) X_{t} = Cash Inflows_{t} - Cash Outflows_{t} = R_{t} - E_{t} - I_{t} - T_{t}
The income tax paid is determined by:
2. T_{t} = t (R_{t}-E_{t}-D_{t})
where t is the corporate tax rate and D_{t} is the depreciation charge in period t. Note that depreciation is not a cash expense and only affects cash flows through its effect on taxes
Substituting equation (2) into equation (1) yields an expression for the firm’s cash flow in period t:
X_{t} = (1-t)(R_{t} - E_{t}) + D_{t} - I_{t}.
The term tD_{t} is sometimes referred to as the depreciation tax shield.
Using this terminology, we define the incremental net cash flows of an investment proposal (delta(X_{t})) as:
delta(X_{t}) = X_{t} (with
investment) - X_{t} (without investment)
delta(X_{t}) = (1-t )(delta(R_{t})
- delta(E_{t}) + t delta(D_{t})
- delta(I_{t}).
delta(X_{t}) = (1-t )(DR_{t}
- DE_{t}) + t delta(D_{t})
- delta(I_{t})
2.12 Depreciation
With the enactment of the U.S. Tax Reform Act of 1986, depreciable assets generally fall into one of three classes:
The five-year recovery class includes (i) automobiles and trucks, (ii) computers and peripheral equipment, calculators, typewriters and copiers, and (iii) items used for research and experimentation. The seven-year recovery class includes books, furniture and office equipment not included in the five-year class.
For items in the five-year and seven-year classes, you can deduct the entire cost in the first year of the item’s life, up to $10,000. For items exceeding this amount, there are two choices of depreciation methods - the double declining balance method and the straight-line method. A mid-year convention is followed in the year of acquisition and the year of disposal. Under this convention, a half-year of depreciation is taken in the year of acquisition and the year of disposal. Real estate is depreciated using the straight-line method and a mid-month convention. Residential real estate is 27.5 year property, and non-residential real estate is 31.5 year property.
For the purposes of this course, we will use straight-line depreciation to facilitate computations. Practically speaking, the firm would not elect to use the straight-line method if an accelerated method is allowed. However, straight-line depreciation simplifies the calculations and conveys all the necessary intuition.
There are a number of points to note regarding the depreciation charge:
1. Sales tax, delivery charges and installation are regarded as part of the cost of the equipment and are added to the cost of the asset for depreciation purposes.
2. If the equipment has an estimated salvage value at the end of its
useful life, the
asset is depreciated down to this value. For example,
if the asset costs $20,000
and has an estimated salvage value of $5,000 at
the end of its useful life of n years,
annual straight-line depreciation is given by:
Thus, if this asset had a useful life of n
= 5 years, then the depreciation charge per year
would be:
3. If an asset is sold at an amount in excess of its depreciated (also
known as book)
value, then this excess is a taxable gain. Similarly,
if the asset is sold for an amount
less than its depreciated value, then this difference
is an allowable deduction for
tax purposes.
4. Any costs associated with old equipment, such as removal expenses,
do not count
as part of the cost of the new machinery and are
expensed immediately (i.e., at time 0).
Example 2.13
A corporation is considering installing a machine that costs $60,000 plus installation costs of $2,000. It will generate revenues of $155,000 annually and cash expenses annually of $100,000. It will be depreciated to a salvage of $6,000 over a seven-year life using the straight-line method. Assuming the firm has a marginal cost of capital of 12 percent and is in the 34 percent marginal tax bracket, determine the incremental cash flows of this investment project. What is the present value of this project? Year 0: The incremental cash flows associated with the project in year 0 are:
Years 1-7:
We can summarize this information in three tables. The first computes taxable income and the tax expense. The second table computes the net cash flow. The third table calculates the project’s present value. Computation of Taxable Income:
Computation of Net Cash Flows:
Present Value of Project
So, the present value of the project is $118,793. |