By Cheng F. Lee Rutgers University, USA John Lee Center for PBBEF Research, USA

Financial Analysis, Planning and ForecastingTheory and ApplicationChapter 6Valuation of Bonds and Stocks By Cheng F. Lee Rutgers University, USA John Lee Center for PBBEF Research, USA

Outline • 6.1. Introduction • 6.2. Bond Valuation • 6.3. Stock Valuation • 6.4. Growth Rate Estimation and Its Application • 6.5. Preferred Stock Valuation • 6.6. Risk, Return, and Market Efficiency • 6.7. Exchange Rates and Investing Overseas • 6.8. Application Examples • 6.9. Summary • Appendix 6.A. The Relationship between Exchange Rates • and Interest Rates • Appendix 6.B.Derivation of Dividend Discount Model • Appendix 6.C.Duration Analysis

6.1. Introduction • The main purpose of this chapter is to discuss how to use time value of money concepts and methods to determine the valuation of bonds and stocks. Some bonds pay coupon interest annually, some semiannually, and some pay no coupon interest at all. Some stocks pay no dividends, others pay the same dollar amount each year as a dividend, and some pay dividends that change every year. How much should investors be willing to pay for these securities? And what price can issuing corporations expect to receive in selling them? Fortunately, basic techniques exist to help us to value, or determine a price on these assets. In the next chapter, we will use the valuation of bonds and stocks to discuss capital structure and other related issues.

6.2. Bond Valuation • A bond represents a liability of a firm or government entity that arises from borrowing money. A typical bond pays a set amount of interest each year and repays the principal amount of the loan at maturity. The valuation principle states that the bond’s value equals the present value of these expected future cash flows. • Four factors affect bond valuation: The YTM is the effective annual rate-of-return demanded by investors on bonds of a given maturity and risk. The face value of a bond is called the par value. In general, this is the amount of money that the issuer has initially borrowed and promised to repay at a future maturity date. Most U.S. corporate bonds have a par value of $1,000 per bond. The coupon interest rate is the percentage of the par value to be paid annually, as interest, to the bond holder.

6.2. Bond Valuation • Most U.S. corporate bonds make interest payments semiannually, that is, every 6 months. Eurobonds, which are bonds denominated in U.S. dollars issued by firms in financial markets outside the United States, typically pay interest annually. A bond’s maturity date, par value, and coupon rate are all disclosed to investors prior to purchase; for the most part, these items are fixed and do not change over the life of a bond issue. The YTM changes, however, depending upon economic and financial trends, market expectations, and firm-specific information affecting risk.

6.2. Bond Valuation • Suppose that a bond with par (face) value F is purchased today and that the bond matures in N years. Let us assume that interest payments of dollar amount I are to be made at the end of each of the next N years. The bondholder will then receive a stream of N annual payments of I dollars, plus a payment of F dollars at the end of the Nth year. Using the rate of interest k to discount future receipts, the present value of the bond is • The first term on the right-hand side of Eq. (6.1) is the present value of the stream of interest payments, while the second term is the present value of the future receipt of the par amount.

6.2.1. Present Value of Future Cash Flows • Cash flows arise from the receipt of coupon interest every 6 months and from the par value at maturity. Thus, cash flows from a bond have two components. First, the coupon interest payments resemble an annuity that occurs every 6 months over the bond’s life. Second, the par value payment is a lump sum or single payment made when the bond matures. • The annual interest payments on a bond equal the coupon rate multiplied by the par value. Thus, the amount of each semiannual coupon payment is half of this amount. For example, a 10% coupon bond pays annual interest of 0.10 times $1,000, or $100; each semiannual coupon payment is half of $100, or $50.

6.2.1. Present Value of Future Cash Flows • The easiest way to compute the present value, or price, of a bond is to divide the analysis into two simpler present value problems that we already know how to solve. The complex cash flow pattern shown can be separated into two familiar cash flow patterns: (1) an annuity (the coupon payments), and (2) a lump sum (the principal payment). Compute the present value of the coupon annuity; the second part of the analysis involves computing the present value of the par value. The price of the bond will be the sum of these two present value calculations, as shown in Eq. (6.2):

Q: A Eurobond with a coupon rate of 8% pays interest annually, has a par value of $1,000, and matures 7 years from now. What current market price would we expect for this bond, assuming that bonds of similar risk and maturity have a 10% YTM? A: To find the price of the bond, first find the present value of the coupon annuity and add it to the present value of the par value. With an 8% coupon rate, annual interest payments for the bond are 0.08 times $1,000, or $80; with interest paid annually, the size of the coupon annuity is $80 per year. With 7 years until maturity and a 10% YTM, the present value of the coupon annuity is: EXAMPLE 6.1 - Eurobonds–Annual Coupons

EXAMPLE 6.1 - Eurobonds–Annual Coupons (cont’d) • The present value of the par value is: • Thus, the current market price of the bond should be $389.47 plus 513.16, or $902.63.

6.2.1.1. Semiannual coupon payments • Semiannual coupon payments would change our analysis in Example 6.1. We know that when cash flows occur more frequently than once a year, adjustments must be made to n, the number of periods, and to r, the discount rate. The number of periods, n, becomes the number of years multiplied by the number of cash flows per year. The interest rate also will need to be adjusted to determine the appropriate periodic interest rate. The method by which we compute the periodic interest rate depends, however, upon which type of annual interest rate is quoted. Bonds can have two interest rates that reflect their annual returns: they can be based on the effective annual rate (EAR), or they can be quoted based upon the annual percentage rate (APR) concept. We discuss each of these in turn.

6.2.1.1. Semiannual coupon payments • EAR or YTM. When the EAR is quoted for bonds, it is called the bond’s YTM or market interest rate. To properly discount the semiannual coupons, we must determine the periodic interest rate that corresponds to the EAR. • We can calculate the effective annual rate as: • in order to solve for the periodic interest rate:

If the YTM is given as 10% for a bond that makes coupon payments semiannually, the appropriate discount rate is: • Let’s use this in an example.

EXAMPLE 6.2 - Computing Price Using the YTM • Revise Example 6.1 to assume semiannual coupon payments, which is the norm with most bonds issued in the U.S. Let’s assume a YTM of 10%. • Recall that the bond has an 8% coupon rate and par value of $1,000; it matures in 7 years. • What is the present value (i.e., the current market price) of this bond? • Annual interest payments for this bond are $80, so the periodic cash flow is $40. The number of periods is 14 (7 years times 2). Since the YTM is given, the periodic interest rate is:

The present value of the coupon annuity is: • The present value of the par value is:

Thus, the current market price of the bond should be $398.98 plus 513.16, or $912.14. This is higher than the price of the bond in Example 6.1, since one-half of each year’s coupon payments occurs earlier. Since some cash flows arrive sooner, their present value is higher when they are discounted at the same YTM. • It is important to note that the present value of the par value is $513.16 in both Examples 6.1 and 6.2. Regardless of the frequency of the coupon cash flows, the present value of the part value remains the same. When the effective annual rate is 10%, a $1,000 lump sum cash flow seven years from now is worth $513.16 in present value terms, regardless of the frequency of the coupon payments.

Annual Percentage Rate or Stated Annual Interest Rate. When the bond interest rate is quoted as an APR, it is called a stated annual interest rate or nominal interest rate. Given an annual percentage rate, the periodic interest rate is APR/m, where m represents the number of periods or cash flows in a year. Recall that the APR assumes no period by- period compounding of cash flows, so it fails to account for interest on- interest. If a bond has a coupon rate of 8% with semiannual coupon payments and has a nominal interest rate of 10%, the bond’s periodic cash flows should be discounted at 10%/2% or 5% per 6-month period.

EXAMPLE 6.3 - Computing Price with a Stated Annual Rate • Suppose the bond in Example 6.2 had 10% stated annual interest rate. What should its price be? • With no other facts changing, the bond will still pay $40 every 6 months over the next 14 periods (7 years). It will repay its $1,000 par value at maturity. Since the stated annual rate is equivalent to an APR, the periodic interest rate will be APR/m, that is, 10%/2% or 5%. The present value of the coupon annuity is:

The present value of the par value is: • The current market price of the bond will be $395.94 plus $505.10, or $90,104. Why is the market price of the bond in Example 6.3 less than the $912.14 price found in Example 6.2? Because in Example 6.2, the 10% market interest rate represents an effective annual rate. In Example 6.3, the 10% stated rate (or APR) is equivalent to a YTM (or effective annual rate) of: • It is well known that higher discount rates lead to lower present values; in Example 6.3, a higher effective annual rate or YTM of 10.25% leads to a lower price.

6.2.2. Interest Rate, YTM, and Bond Price • A bond that sells below par value, such as the one in Example 6.2, is said to be selling at a discount and is called a discount bond. Someone who purchases the discount bond in Example 6.2 today and holds it to maturity will receive, in addition to the stream of coupon interest payments, a gain of $87.86, the difference between the bond’s price ($912.14) and its principal repayment ($1,000). • A bond’s price will reflect changes in market conditions while it remains outstanding with its fixed 8% coupon rate. The bond in the above example will no longer be attractive to investors when alternative investments yield 10%. The bond’s market price will have to fall in order to offer buyers a combined return of 10% from the coupon payments and the par value.

6.2.2. Interest Rate, YTM, and Bond Price (cont’d) • If the market YTM falls, to say 6.09%, the price of the 7-year bond in Example 6.2 will rise to $1,112.96 (check this on your own!), a price above the par value. The price of a discount bond will rise as it nears maturity if the market rate remains the same, since at maturity its price will equal par value. When a bond’s price exceeds its par value, it is said to be selling at a premium, and it is called a premium bond. The price of a premium bond will fall as it nears maturity if the market rate remains the same, since at maturity its price will equal its par value. The investor who holds the bond until maturity will receive the above-market coupon payments of 8% per year, offset by a loss of $112.96 (the difference between its purchase price and par value). In most cases when the bond sells at a premium, interest rates have fallen after the bond’s issue. This bond’s 8% coupon rate makes it very attractive to investors; buying pressure increases its price until its overall yield matches the YTM of 6.09% of other bonds of similar risk and maturity.

6.2.2. Interest Rate, YTM, and Bond Price (cont’d) • When they originally are issued, most bonds sell at prices close to par and offer coupon rates close to the market rates on bonds of similar maturity and risk. Over the life of a bond its price will fluctuate as a result of interest rate fluctuations in the economy. Our discussion of discounts and premiums shows that bond prices will vary inversely to interest rates. As interest rates rise in the Economy, bond prices fall; as interest rates fall, bond prices rise.

Bond price quotes are available in the marketplace, either from bond dealers or from the daily price listings found in secondary sources, such as The Wall Street Journal. Both investors and financial managers must calculate the YTM on bonds, given known par values, coupon rates, times to maturity, and current prices. The YTM can be determined from the PVIFA and PVIF formulas we used to compute the bond price: 6.2.2.1. Calculating a YTM

[The Relationship Between Interest Rates and Bond Prices — The Seesaw Effect], • where k represents the periodic interest rate and n is the number of semiannual periods until the bond matures. The YTM equals (1 + k)2 − 1; the stated annual rate equals k×2. But mathematics offers no simple technique for computing r.

6.2.3. Credit Risk, Interest Rate Risk, and Reinvestment Rate Risk • Basically, investors in domestic bonds face three types of risks: credit risk, interest rate risk, and reinvestment rate risk. Inventors in foreign bonds are subject to two additional risks: political risk and exchange rate risk.

6.2.3.1. Credit risk • The cash flows received by bond market investors are not certain; corporate debtors may pay interest payments late or not at all. They may fail to repay principal at maturity. To compensate investors for this credit or default risk, rates-of-return on corporate bonds are higher than those on government securities with the same terms to maturity. Government securities are presumed to be free of credit risk. Since perceptions of a bond’s default risk may change over its term, the bond’s YTM also may change, even if all else remains constant. Firms provide information on the riskiness of individual bond issues.

Higher bond ratings imply a lower risk of default and, given the risk/return tradeoff, lower expected return (ER) to investors. Lower bond ratings imply a higher level of default risk and higher ER. Investment quality bonds have ratings of BBB (or Baa) or higher. They are called investment quality as some institutional investors, such as pension funds and insurance companies, restrict themselves to investing only in these low-default risk issues. Non-investment quality bonds are called junk bonds or high-yield bonds to reflect their higher risk and higher ERs.

For those seeking a quick estimate of the return, the approximation method may be used. Here, the average annual dollar return to the investor of a bond that matures in n years is the coupon payment plus a straight line amortization of the bond’s premium (or discount): • The average amount invested in the bond is the average of it purchase price and par value:

The approximate YTM is the ratio of the average annual dollar return to the average investment. • Calculators and spreadsheets usually use a trial and error method for computing values of r. Various values of k are used until one is found, such that the present value of the cash flows equals the price within some (pre-specified) error range.

6.2.3.2. Interest rate risk • The general level of interest rates in an economy does not remain fixed; it fluctuates. For example, interest rates will change in response to changes in investors’ expectations about future inflation rates. A rise in interest rates renders the fixed coupon interest payments on a bond less attractive, lowering its price. Therefore, bond holders are subject to the risk of capital loss from such interest rate changes should the bonds have to be sold prior to maturity.

A long term to maturity, all else being equal, increases the sensitivity of a bond’s price to a given change in interest rates, as the discount rate change compounds over a longer time period. Similarly, a lower coupon rate also increases the sensitivity of the bond’s price to market interest rate changes. This occurs since lower coupons bonds have most of their cash flow occurring further into the future, when the par value is paid.

Because of interest rate risk, investors will demand a larger risk premium for a bond whose price is especially sensitive to market interest rate changes. Hence, we would expect higher YTM for long-term bonds with low coupon rates than for short-term bonds with high coupon rates. A bond’s duration is a measure of a bond’s sensitivity to market rate changes that takes maturity coupons and other factors into account..

Duration is computed by means of a complicated-looking formula. The concept of duration is now discussed in the following section

6.2.3.3. Bond returns and duration • For a bond, YTM and “total return” are two different concepts. Total return for a fixed-income investment includes not only the income yield but also the interest on reinvested interest and price changes. • Compounded interest is the largest return component for investors that purchase and hold on to long-term bonds until they mature. For short-term investors that buy a bond hoping to sell it quickly at a higher price, fluctuations in a bond’s price caused by market interest rate or bond rating changes are the largest return component.

As we’ve seen in this chapter, a bond’s price moves inversely to prevailing interest rates. A measure of the price sensitivity of a bond to interest rate changes is called the bond’s duration. Higher duration bonds have larger price reactions to interest rate changes; lower duration bonds have smaller price reactions. One method of computing duration calculates it as the weighted average of the time to receive a bond’s cash flows; the weights are present values of each cash flow (coupon interest and par value) divided by the bond’s price.

The concept of duration has many applications; areas in which duration is used frequently include bank asset and liability management, investment analysis, and pension fund management. A common risk management technique in these areas is to set the duration of a pension fund’s asset equal to the duration of its liabilities. This way, as market rates fluctuate, the market value of the asset and liabilities will rise or fall together. By trying to set asset and liability durations equal to each other, the equity, or net worth will be little affected by changing interest rates. Further detail analysis of duration can be found in the Appendix C of this chapter.

6.2.3.4. Reinvestment rate risk • The return that an investor receives from a bond investment equals the bond’s YTM or effective annual rate only if the coupon payments can be reinvested at a rate equal to the bond’s YTM. Recall the form of the interest factor in bond price Equation 6.3: (1 + k)n. This assumes that all the cash flows are reinvested at the periodic rate k. Should future coupons be reinvested at a lower rate, the investor’s actual yield will be less than the bond’s YTM. Thus, reinvestment rate risk occurs when fluctuating interest rates cause coupon payments to be reinvested at different interest rates. Another illustration of reinvestment rate risk occurs when maturing bank CDs are rolled over into new CDs. The risk benefits the investor when the new CD rate is higher than the maturing CD rate; it works against the investor when the new CD rate is lower.

6.2.3.5. Risk of non-domestic bonds • Investors in domestic securities face a number of risks beyond those of domestic securities. Among these are political risk and exchange rate risk. Political risk can affect a bond investor in a number of ways. A foreign government may block currency exchanges, preventing the investor from repatriating coupon income. Social unrest may lead a foreign corporation to default on its bonds. Of course, exchange rate changes will cause fluctuations in the values of cash flows in terms of U.S. dollars; this is called exchange rate risk.

6.2.4. Zero Coupon Bonds • Zero coupon bonds pay no coupon interest and provide only one cash flow: payment of their par value upon maturity. Treasury bills are a form of zero coupon debt. An investor purchases a T-bill at a price below par and receives no interest or other cash flows until maturity. At that time, the investor receives the par value of the T-bill. The return on the security is the difference between its discount price and its par value.

There are two reasons for the popularity of zero coupon bonds: (1) the investor does not face a reinvestment rate risk (2) for certain investors (for example, those with IRAs) the tax of the bond return can be deferred until some future date. However, under IRS regulations, investors must pay yearly taxes on the implicit interest paid by the bonds; the IRS has special rules for determining this value. In essence, investors must pay taxes on income they have not received. Thus, zero coupon bonds are mainly purchased by tax-exempt investors who pay no tax on their investment returns, such as pension funds.

Issuing a zero coupon bond also helps lower borrowing costs for the firm. The original discount can be expensed for tax purposes on a straight-line basis over the life of the bond. Thus, rather than cash outflows from coupon interest payments, the issuing firm receives annual cash inflows from tax savings. However, the issuer must plan for a large capital requirement at the maturity of these bonds. Present value calculations to determine the price of a zero coupon bond are straight-forward, as the following example illustrates:

EXAMPLE 6.4 - Zeroing in on Bonds • A zero coupon bond with a par value of $1,000 has a maturity date 7 years from now. At what price would this bond provide a YTM to match the current market rate of 10%? • This problem requires finding the present value of a single future cash flow. The price will equal: • The current price of this bond should be $513.16.

6.3. Stock Valuation • We now turn to the valuation of common stock. A stockholder receives income from two sources: (1) dividend payments, and (2) capital gain from the change in stock price (which can, of course, be negative). The price of common stock reflects investors’ expectations of these two income sources. • The principle for determining an appropriate stock price is the same as that for determining a bond price: find the present value of expected future cash flows. With bonds, this is a relatively straightforward process; the typical corporate bond has a definite life, fixed coupon payments and par value, and an easily discernible YTM or stated annual interest rate.

Equity offers no such certainty. Common and preferred stocks generally are assumed to have infinite lives. For common stock, relevant cash flows (divided payments) are variable and depend upon firm growth, profitability, and investment opportunities. Finally, each firm can have its own required rate-of-return. Despite these difficulties, in this section we shall see that the present value of all future dividends should equal a stock’s current price, and that some simplifying assumptions can make the task of determining stock value much easier. Our discussion in this section focuses on common stock. As we shall see, the method for valuing preferred stock is a special case of common stock valuation.

It may seem rather strange to treat the stock price as nothing more than the present value of all future dividends. Investors generally buy stock with the intention of selling it at some future time, ranging from a few hours to 30 years or longer. Despite the length of any one investor’s time horizon, we can show that the current price of common stock should equal the present value of all future dividends.

Let the present value, or current market price, of a share of common stock be denoted by P0. Let d0, d1, d2, . . . be the successive annual dividends and P1, P2, . . . be the price per share of stock at the end of successive year. Suppose that a share of stock is held for 1 year. At this year’s end, after receiving dividend payment d1, the stockholder could sell the stock for an amount P1. If these future ERS are discounted at the required rate-of return, k, the present value of the investor’s cash flows comes to:

Suppose that someone else purchases the stock at price P1 at the end of the first year and holds it for 1 year. The purchase price should equal: • Substituting this information into Eq. (6.4) gives us:

Continuing in this way, looking N years into the future, we have: • If we continue this process indefinitely, the time horizon N becomes infinitely large. In that case, the final term on the right-hand side of Eq. (6.5) approaches zero, and we have the final result:

The present value of a share of stock is the sum of all future dividend payments, discounted to the present. It does not depend on the investment horizon of any individual investor. • To finance rapid growth, young firms often retain all their earnings; when they mature, they often begin paying out a portion of earnings as dividends. Second, at the very least, the firm’s stock should be worth the per-share liquidation value of its assets. Otherwise, an arbitrageur could buy all the outstanding stocks to purchase the firm and then make a profit by selling its real assets piece-by-piece. Most firms’ values as going concerns exceed their liquidation values; the price of any such firm will reflect its going-concern value. • As it stands, Eq. (6.6) is difficult to use, since it requires estimation of all future dividend payments and investors’ required return on the stock. Matters can be simplified considerably if the analyst can assume that the firm’s dividends will remain constant or will growat a constant rate over time.

6.3.1. Constant Dividend Growth Rate Model • If the firm’s dividends are expected to remain constant, so that d0 = d1 = d2 . . . , we can treat its stock as a perpetuity. This is how we value preferred stock. If a common stock is expected to have constant dividends, the valuation process is identical to that of preferred stock. The present value of a perpetuity is the cash flow divided by the discount rate. For stocks with constant dividends, this means Eq. (6.6) becomes P0 = d0/k.

By Cheng F. Lee Rutgers University, USA John Lee Center for PBBEF Research, USA