Returns on Common Stock As we saw in the preceding section, the value of a stock is the present

value of future cash flows, discounted at the required rate of return. If we know the future cash flows and the required rate of return, we can determine today’s value. Suppose that instead of determining today’s value, we wish to determine the return on a stock. For example, we may want to determine whether a particular stock provides a return over the next five years that is appropriate for its risk. In this case, we know the value of the stock today, we estimate its value in five years, and estimate any interme- diate cash flows (e.g., dividends). The missing piece is the return.

We can calculate the return on an investment in common stocks just as we did the internal rate of return in the preceding example. The return on stock is comprised of two components: (1) the appreciation (or depreciation) in the market price of the stock—the capital yield— and (2) the return in the form of dividends—the dividend yield:

Return on stock = Capital yield + Dividend yield Let’s ﬁrst ignore dividends. The return on common stock over a

period of time where there are no dividends is the change in the stock’s price divided by the beginning share price:

------------------------------------------------------------------------------------------------------------------------------- ( End-of-period price Beginning-of-period price – Return on stock ) =

Beginning-of-period price Suppose that at the beginning of 2000, Hype.com stock was $10 per

share, and at the end of 2001 Hype.com stock was $15 a share. The return on Hype.com during 2000–2001 was:

Return on Hype.com stock = -------------------------- = 50% $10

THE FUNDAMENTALS OF VALUATION

Hype.com stock appreciated $5.00 per share, providing a return of 50% for the two years. To make the return comparable to returns on other investments, we usually restate the return as a return per year. The return on Hype.com per year is calculated using the time value of money relationship:

PV = $10 FV = $15 N = 2

Solving for r, the return is 22.47% per year. Let’s work through another illustration. Suppose you bought one share of Berkshire Hathaway stock at the end of 1986 for $2,430. And suppose you sold this share of stock at the end of 1998 for $70,000. Over the twelve years, you earned over 2,700%! But in order to com- pare this return with other stocks’ returns, we need to place it on a common basis, a year. Given the following, we can translate the twelve-year return on Berkshire Hathaway stock into a return per year:

Future value (sales price) = FV = $70,000 Present value (cost) = PV = $2,430 Number of periods = N = 12 years

We can represent the return over the nine years on a per-year basis. Let r

be the annual return on the investment. Using the basic valuation equation, FV = PV (1 + r e ) N , we substitute the known elements into the basic valuation equation,

$70,000 = $2,430 (1 + r e ) 12

Next, we rearrange in terms of r e : $70,000

r e = 23 --------------------- – 1 = 32.32%

The return on your investment (not considering any commissions paid) is 32.32% per year.

If a stock pays dividends, we need to consider them as cash inﬂows, as well as the change in the share’s price, in determining the return. The simplest way to calculate the return is to assume that dividends are received at the end of the period:

Return on a stock

End-of-period price Beginning-of-period price ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ – + Dividends at end of period

Beginning-of-period price

Valuation of Securities and Options

Or, if we let: P 0 = beginning-of-period price

P 1 = end-of-period price

D 1 = dividends received at the end of period we can write:

0 Return on a stock D = -------------------------------- 1 – P + 1

We can break this return into one part representing the return due to the change in price and another part representing the return due to dividends:

D Return on a stock =

------------------ 0 +

------- 1 P 0 P 0

dividend yield The ﬁrst part is the capital yield and the second part is the dividend

capital yield

yield. If a company doesn’t pay dividends, the dividend yield is zero and the return on the stock is its capital yield.

When using this equation, be careful to specify the timing of the prices at the beginning and the end of the period and the timing of the dividends. Because we’re dealing with the time value of money, we have to be very careful to be exact about the timing of all cash ﬂows.

To simplify our analysis, let’s ignore our stockbroker’s commission, though we will discuss these costs later in this chapter. Suppose we bought 100 shares of Internet.com common stock at the end of 1997 at

× $35.25 = $3,525 in Internet.com stock. During 1997, Internet.com paid $0.43 per share in dividends, so we earned $43.00 in dividends. If we sold the Internet.com shares at the end of 1997 for 43 ($43.00 per share, or $4,300.00 for all 100 shares), what was the return on our investment? It depends on when the dividends were received. If we assume that the dividends were received at the end of 1997, our return was:

------------------------------------------------------------------------------------ $4,300.00 $3,525.00 – + Return on Internet.com for 1997 $43.00 =

$3,525.00 -------------------------- = $818.00 = 0.2321 or $23.21%

THE FUNDAMENTALS OF VALUATION

We can break this return into its capital yield and dividend yield components:

$4,300.00 $3,525.00 – Return on Internet.com for 1997 = ----------------------------------------------------------- + -------------------------- $43.00

Capital yield + Dividend yield

Most of the return on Internet.com stock was from the capital yield— the appreciation in the stock’s price.

Now suppose instead that Internet.com is not sold at the end of 1997, but rather sold at the end of 1999 at $50 per share. This is a more complicated problem to solve because we not only have to consider each cash ﬂow—the purchase price, any dividends paid during the 1997– 1999, and the sale price—and the time value of money.

D P = --------------------- 1997

+ --------------------- 1998 + --------------------- 1999

0 + --------------------- 1999

+ ( 2 1 r e ) ( 1 + r 3 e ) ( 1 + r 3 e ) If dividends in 1998 and 1999 are the same as those in 1997, --------------------- $3,525 $43 = + --------------------- $43

$43 + --------------------- ---------------------

$3,525 = --------------------- + --------------------- + ---------------------

1 + r ( 1 e ) ( 1 + r 2 e ) ( 1 + r 3 e ) Where do we begin? We can solve this using a ﬁnancial calculator or

trial and error. Using trial and error, we want to ﬁnd the return that equates the present value of the investment—the $3,525—with the

present value of the future cash ﬂows. For example, if we try an r e of 10%, the present value of the future cash ﬂows is $3,863.51. Since this value is not equal to $3,525, 10% is not the rate we seek.

To reduce the present value, we must use a larger value of r e . If we try 15%, the present value is $3,385.76 and therefore 15% is not the

Valuation of Securities and Options

value we seek. So we know that the return is between 10% and 15%. Using a ﬁnancial calculator we would ﬁnd that the answer is 13.45%.

You can see that we can compute returns on investments whether or not we have sold them. In the cases where we do not sell the asset repre- sented in the investment, we compute the capital yield (gain or loss) based on the market value of the asset at the point of time we are evalu- ating the investment. It becomes important to consider whether or not we actually realize the capital yield only when we are dealing with taxes. We must pay taxes on the capital gain only when we realize it. As long we don’t sell the asset, we are not taxed on its capital appreciation.

VALUATION OF PREFERRED STOCK The value of preferred stock is the present value of all future dividends.

If a share of preferred stock has a 5% dividend (based on a $100 par value), paid at the end of each year, today’s price is the present value of the stream of $5’s forever, discounted at the rate r p :

Present value of preferred stock $5

= ---------------------- + ---------------------- + ---------------------- + … + ----------------------- = ------

If the discount rate is 10%, the present value of the preferred stock is $50. That is, investors are willing to pay $50 today for the promised stream of $5 per year since they consider 10% to be sufﬁcient compen- sation for both the time value of money and the risk associated with the perpetual stream of $5s.

Let’s rephrase this relation, letting P p indicate today’s price, D p indicate the perpetual dividend per share per period, and r p indicate the discount rate, (i.e., the required rate of return on the preferred stock). Then:

We can make some generalizations about the value of preferred stock:

■ The greater the dividend rate, the greater the value of a share of preferred stock. ■ The greater the required rate of return—the discount rate—the lower value of a share of preferred stock.

THE FUNDAMENTALS OF VALUATION

Here is another example of valuing a share of preferred stock. Con- sider a share of preferred stock with a par value of $100 and a dividend rate of 12%. If the required rate of return is 15%, the value of the preferred stock is less than $100:

P p = $12/0.15 = $80

If the required rate of return declines to 10%, the price would rise to $120. Let’s look at a feature of preferred stock that may affect its value: the call feature. If preferred stock has a call feature, the issuer has the right to call it—buy it back—at a speciﬁed price per share, referred to as the call price.

Suppose the dividend rate on preferred stock is $6 per share and the preferred stock is callable after three years at par value, $100. If the preferred stock has a required rate of return of 5%, the value of a share of preferred stock without the call is:

P p =$6/0.05 = $120

Considering the call feature and assuming the issue is called in three years, we need to alter our valuation equation so that we ﬁnd the present value of the ﬁrst three dividends and the present value of the call price:

P $6.00

p = ----------------------------- + ----------------------------- + ----------------------------- + ----------------------------- $100.00

If the preferred shares did not have a call feature, they would be worth more—the call feature reduces the value of the shares. What is the likeli- hood that the firm will call in the preferred shares? If the required rate of return is 5%—that is, investors demand a 5% return—and the stock pays $6 on the par of $100, or 6%, the firm can call in the 6% preferred shares and issue 5% shares. Since calling in the preferred shares makes sense—the firm can lower its costs of raising capital—it is very likely the firm will call in the preferred shares when they can.

Returns on Common Stock As we saw in the preceding section, the value of a stock is the present