Assume there are two stocks with returns that are perfectly inversely (negatively) cor-related. When the price of one increases by a certain percent, the price of the other decreases by the same percent. For example, SwimmingPools, Inc., pays a 16 percent return each year when temperatures are above average and zero percent in other years. SkiResorts, Inc., pays a 16 percent return each year when temperatures are be-low average and zero percent in other years. Assuming that half the time the tem-peratures are above average and half the time they are below average (a reasonable assumption given the nature of averages!), the expected return to holding either stock is 8 percent. If an investor chooses to hold one or the other, but not both, the ex-pected return will be 8 percent, but the risk involves the fact that the return will fl uctuate between either 0 or 16 percent. As the following table illustrates, smart in-vestors will split their surplus funds equally between SwimmingPools and SkiResorts and earn an 8 percent return all of the time, thus eliminating all risk!
The expected return to owning a share of shock is the sum of each possible out-come multiplied by the probability of that outout-come.
Stock
(a) Possible Outcomes
(b)
Probability (a) × (b) SwimmingPools, Inc.
(above average temps) 16% .50 8%
(below average temps) 0% .50 0
Expected return to owners - - 8
Return fl uctuates between 0% and 16%
SkiResorts, Inc.
(above average temps) 0% .50 0%
(below average temps) 16% .50 8
Expected return to owners - - 8
Return fl uctuates between 0% and 16%
If an investor’s surplus funds are split between SwimmingPools and SkiResorts, when temperatures are above average, half of the portfolio (the half invested in Swim-mingPools) will earn 16 percent and half (the half invested in SkiResorts) will earn zero percent. As noted above, the expected return is the sum of the possible outcomes multiplied by their probabilities. In this case, the overall portfolio will earn 8 percent (.16 × .5 + 0 × .5 = .08). The reverse is true for below- average temperatures, with SkiRe-sorts earning 16 percent and SwimmingPools earning zero percent. Again, the portfo-lio split between the two fi rms earns 8 percent. Thus, the risk or the fl uctuation of returns between zero and 16 percent has virtually been eliminated by diversifi cation, or owning both companies. The portfolio earns 8 percent when temperatures are above average and 8 percent when temperatures are below average. Risk has been
A Closer Look
together. For instance, if, as the economy improves, two stocks also tend to improve, and vice versa, they are positively correlated. Note that the price of one stock may be increasing 6 percent while the other is increasing 2 percent but that they are still positively correlated even though the magnitude of the increase is different. To be posi-tively correlated, the only thing that matters is that the change to each be in the same direction.
Returns are negatively (inversely) correlated if the return on one asset increases while the return on the other tends to decrease. To be negatively correlated, the only thing that matters is that the change to each be in the opposite direction. Some fi nancial instruments do better in recessions, while most perform better in expansions. Returns on fi nancial instruments that improve over the business cycle are negatively correlated with those that lose value. As long as returns among fi nancial instruments are not per-fectly correlated (that is, they do not change by the same magnitude and direction all the time), then the risk or fl uctuation of a combination or basket of assets with a given expected return will be less than the risk for any one asset with the same expected re-turn. Thus, one can earn a higher return for any level of risk or be exposed to less risk for any given return by diversifying. Now would be a good time to read “A Closer Look”
on p. 139 to learn more about the benefi ts of diversi fi cation.
Assuming that an individual opts for a portfolio of various fi nancial instruments with different risks and returns, how would you go about managing such a portfolio?
More specifi cally, how would you decide whether to purchase and hold stocks or bonds or some combination of both, and so on? We hope that you would compare the expected rates of return on the different types of fi nancial assets, selecting those with the highest expected return consistent with the risk you are willing to take, while realizing that by diversifying you can improve the expected per for mance or reduce the risk of your portfolio.
eliminated because the returns to the two assets were perfectly inversely corre-lated. In the real world, few assets exhibit this property. It is also true that returns are usually not perfectly positively (directly) correlated, meaning that the prices of two stocks usually do not always change by the exact same percent. If stock prices are perfectly correlated, risk is not reduced by diversifi cation. In the case in which re-turns are correlated (either directly or inversely), but not perfectly so, risk can be reduced (although not eliminated as in the case of perfect inverse correlation) through diversifi cation.
Thus, we can conclude the following:
• If returns to two assets held are perfectly negatively correlated, risk can be eliminated through diversifi cation.
• If returns to two assets held are perfectly positively correlated, risk can not be reduced through diversifi cation.
• If returns to two assets held are positively or negatively correlated, risk can be reduced for any given expected return but not eliminated through diversifi cation.
Stocks
The size of a shareholder’s own ership position depends on the number of stock shares owned. For example, if 1,000 shares are outstanding, a stockholder who owns 100 shares in effect owns 10 percent of the fi rm. The value of each share— and, therefore, the value of the stockholder’s holdings in the corporation— depends on the prevailing price of the fi rm’s stock. If, for example, the stock’s price is $50 per share, the total value of the stockholder’s 100 shares is $5,000, and the total value of the fi rm is $50,000 ($50 × 1,000 shares). If the stock price rises to $60 per share, then the value of the stockholder’s 100 shares increases to $6,000 and the value of the fi rm increases to $60,000. Likewise, given an unforeseen fall in the stock’s price to $40 per share, the values of the stock-holder’s shares and the fi rm decrease to $4,000 and $40,000, respectively. The key ques-tion, then, is, What determines the price per share?
Outstanding shares of stocks of publicly held companies are traded (bought and sold) on or ga nized exchanges such as the New York Stock Exchange or by networks of brokers and dealers around the country. (More on this in Chapter 13.) Stock prices fl uc-tuate daily, some going up, some going down, as fi nancial investors buy and sell shares of various corporations. In part, those fl uctuations occur because a share of stock repre-sents a claim on the earnings of a fi rm. Tangible evidence of this sharing of earnings comes in the form of dividends, which are a distribution of profi ts to stockholders. If earnings prospects are improving, the share price and dividend paid per share may also be rising.2 Financial investors will be attracted by the improved outlook (profi tability) for the fi rm.
In general, as current and expected future earnings rise, stock prices also rise, and as current and prospective earnings decline, stock prices also decline. A growing econ-omy means that sales, production, and incomes are expanding, while a declining economy means the opposite. Because expected earnings also rise when the economy is expected to grow and tend to fall when the economy is expected to contract, there is often a positive correlation between the growth of real national income and stock prices.3
What is the expected return on stocks? Generally speaking, the expected return on a share of stock, say, over a year, is the expected dividend plus the expected change in the price of the stock, all divided by the share price at the time of purchase. For example, if you pay $50 a share, the expected dividend is $1 per share, and you expect the price to rise $3 over the year, the expected return is 8 percent [($1 + $3)/$50 = .08 = 8 percent].4 An 8 percent expected return would also result if the expected dividend is $4 and the expected capital gain is $0 because [($4 + $0)/$50 = .08 = 8 percent]. Now would be a good time to look at Exhibit 7- 1, which examines this relationship.
Bonds
With regard to bonds, the expected return to a newly issued bond is the current interest rate. Recall from Chapter 5 that bonds represent long- term debt and pay a fi xed annual coupon payment.5 The coupon payment is the product of the face value of the bond multiplied by the coupon rate. Bondholders are entitled to be paid the coupon payment before dividends are paid to stockholders. The coupon rate is the interest rate at the time the bond is originally issued and usually appears on the bond itself. The coupon rate is not the same thing as the current interest rate if interest rates have changed since the bond was issued.
For example, if the face value of a bond is $1,000 and the coupon rate is 6 percent, then the coupon payment is $60, since $60 divided by $1,000 is equal to 6 percent
($60 / $1,000 = .06). This coupon payment does not change even if interest rates change after the bond has been issued. However, the bond’s price will change whenever interest rates change or if the issuer’s ability to make the agreed- upon interest or principal pay-ments comes into question.
As portrayed in Exhibit 7- 2, the expected return on previously issued bonds is the coupon rate plus the expected percentage change in the bond’s price over the course of the year.
Let’s assume that you purchase a $1,000 newly issued 30- year bond described above with a 6 percent coupon rate. One year after the bond is issued, interest rates fall to 4 percent. The price of the bond with 29 years to maturity would increase to $1,339.67, because the present value of the 29 coupon payments of $60 plus the present value of the repayment of the $1,000 principle at maturity would equal $1,339.67.6 In other words, as you saw in Chapter 5, prices of previously issued bonds adjust so that they pay the new prevailing interest rate. The expected return on the bond is equal to the coupon rate (6 percent) plus the expected percentage capital gain from the change in interest rates.
In our example, the new interest rate is 4 percent, and the expected percentage capital gain is 34 percent [($1,339.67 − $1,000) / $1,000 = .34 = 34 percent]. Thus, the expected return to owning the bond over the year is the coupon rate (6 percent) plus the expected percentage capital gain (34 percent), or 40 percent.
To see how bond prices fi t into the picture, assume that the current interest rate on bonds is 6 percent and that the expected return on stocks is 8 percent, with the typi-cal stock costing $50 and the expected dividend equal to $4. We also assume for simplic-ity that (1) the expected capital gain is zero, (2) stocks and bonds have the same degree of liquidity,7 (3) stocks are riskier than bonds, and (4) the portfolio managers must be
Expected Dividend
Expected Price Change
[Capital Gain (+) or Loss (−)] Expected Return
$3 −$2 ($3 + (−$2))/$50 = 2 percent
$3 $0 ($3 + $0)/$50 = 6 percent
$3 $2 ($3 + $2)/$50 = 10 percent
$3 $4 ($3 + $4)/$50 = 14 percent
$4 −$2 ($4 + (−$2))/$50 = 4 percent
$4 $0 ($4 + $0)/$50 = 8 percent
$4 $2 ($4 + $2)/$50 = 12 percent
$4 $4 ($4 + $4)/$50 = 16 percent
7-1
The Expected Return to Owning Stock
As the body of the text explains, the expected return of owning a share of stock for, say, one year is the expected dividend plus the expected capital gain or loss, divided by the share price at the time of pur-chase. Thus, when either factor changes, the expected return will change.
Assume that the stock originally costs $50 per share. The preceding table shows the rate of return for owning the stock given various expected dividends and expected price changes (the capital gains or losses).
If actual dividends or actual capital gains and losses turn out to be different from those expected, the actual return will be different from the expected. Needless to say, all investors hope that actual divi-dends and capital gains turn out to be higher than expected rather than the reverse.
compensated 2 percent for the additional risks of owning stocks. Under these condi-tions, when bonds pay a 6 percent return and stocks pay an 8 percent return, the typical portfolio manager is indifferent between stocks and bonds. He or she will presumably hold some of each because the risk- adjusted returns are equalized. Equation (7- 1) de-picts this situation:
(7-1) Risk-adjusted return on stocks risk-adjusted return on bonds
Nominal return on stocks – compensation for higher risk of owing stocks risk-adjusted return on bonds (8 percent 2 percent) 6 percent
=
= − =
Now suppose that the Fed decides to pursue a more expansionary monetary policy.
The initial result of this is a decline in the interest rate on bonds to 4 percent and a re-duction in the risk- adjusted return on bonds from 6 percent to 4 percent.8 The fall in the interest rate will tend to raise stock prices through two channels.
First, the expected return on bonds is now below the risk- adjusted expected return on stocks. Given the substitutability of stocks for bonds in investors’ portfolios and the higher expected return on stocks, the demand for stocks will rise, tending to raise stock prices. Within the confi nes of our simple example, we can even say how high stock prices will rise: stock prices will rise until the expected return on stocks is again 2 percent higher than the expected return on bonds (4 percent). This will occur when the price of our typical share of stock rises to $66.67, because the $4 expected dividend divided by
$66.67 equals 6 percent ($4/$66.67 = .06).
Second, the fall in the interest rate will be expected to raise the demand for goods and ser vices and increase the sales and earnings of fi rms. With earnings expected to rise, dividends will also be expected to rise. This reinforces the fi rst effect. For example, if the dividend is expected to rise to $5 per share, then fi nancial investors will be willing to bid up the price per share even further to $83.33 because $5 divided by $83.33 is equal to 6 percent ($5 / $83.33 = .06 = 6 percent).9 Again, after stock prices have adjusted to the change in interest rates, the risk- adjusted return on stocks will be equal to the risk- adjusted return on bonds.10
Assuming that you and other portfolio managers would like to have owned the stock before all of this occurred, you can see now why actual and expected changes in the interest rate get so much attention in the stock market.
In the real world, many types of long- term fi nancial instruments offer varying degrees of risk and liquidity. Because of the substitutability of various fi nancial instru-ments, prices of fi nancial instruments will adjust so that returns to owning different in-struments are equalized after adjustments have been made for differences in risk and liquidity. In other words, in fi nancial markets, risk- and liquidity- adjusted rates of re-turn are equalized.
Expected Percentage Return on Bonds
= Coupon rate + Expected percentage change in the bond price
= (coupon payment/bond price at the beginning of the year) + (expected bond price at the end of the year − bond price at the beginning of the year)/bond price at the begin-ning of the year)
7-2
The Expected Return on Bonds
Prices of long- term financial instruments change as current and future expected earnings change. If interest rates fall, prices of previously issued bonds rise, and vice versa. If cur-rent and expected future earnings rise, ceteris paribus, stock prices also rise, and vice versa. In managing a portfolio, market participants compare expected rates of return and select those financial assets with the highest expected return consistent with varying degrees of risk and liquidity. As long as returns among various financial instruments are not perfectly correlated, diversification reduces risk for any given expected return. Stock and bond prices adjust until the portfolio manager is indifferent between stocks and bonds. If interest rates change, ceteris paribus, stock prices also change. When full ad-justment has occurred, differences in returns on various financial instruments reflect dif-ferences in only risk and liquidity.
To reiterate, it is the expected return on bonds and the expected return on stocks that determine stock and bond prices. It should not surprise you that this is true of all fi nancial instruments. Because expectations play such a central role, we turn now to a general theory of how price expectations are formed, which will then be applied to fi -nancial instruments.