Valuing Securities Stocks and Bonds

Bond Cash Flows, Prices, and Yields Bond Terminology Face Value Notional amount used to compute the interest payments Coupon Rate Determines the amount of each coupon payment, expressed as an annual percentage rate Coupon Payment

Zero-Coupon Bonds Zero-Coupon Bond (Zero) Does not make coupon payments Always (almost always) sells at a discount (a price lower than face value), so they are also called pure discount bonds Treasury Bills are U.S. government zero-coupon bonds with a maturity of up to one year. “Strips”

Zero-Coupon Bonds Suppose that a one-year, risk-free, zero-coupon bond with a $100,000 face value has an initial price of $97,323.60. The cash flows would be: Although the bond pays no “coupon,” your compensation is the difference between the initial price and the face value. 1 -$97,323.60 $100,000

Zero-Coupon Bonds The difference between the price and the face value (here $2,676.40) is the compensation you receive for buying and holding the bond for a year. The compensation is commonly talked about as a return on the initial investment: This is the return you would receive from investing in (buying) the bond and holding it till maturity.

Zero-Coupon Bonds Yield to Maturity The single discount rate that sets the present value of the remaining promised bond payments equal to the current market price of the bond. For a Zero-Coupon bond: For a zero, this is also the actual annualized return (actual spot rate).

Zero-Coupon Bonds Yield to Maturity For the risk free one-year zero coupon bond presented above: Thus, the YTM is 2.75%. In bond markets the standard presumption is that a “period” is six months. Why must this equal the one year risk free rate?

Example Suppose the following three zero-coupon bonds are trading at the prices shown. Each has a face value of $1,000 and the time to maturity shown in the table. What is the yield to maturity for each bond? Maturity 1 Year 2 Years 3 years Price 980.39 942.60 901.94

Example The general formula is found by re-arranging the present value equation: The specific solutions in our example are: This is how the spot rates on the yield curve are found. 1 Year – YTM $1,000/$980.39 -1 2.0% 2 Years – YTM ($1,000/$942.60)1/2 - 1 3.0% 3 Years – YTM ($1,000/$901.94)1/3 - 1 3.5%

Coupon Bonds Yield to Maturity The YTM is the single discount rate that equates the present value of the bond’s remaining cash flows to its current price. Yield to Maturity of a Coupon Bond For a coupon bond the YTM is an artificial yield intrinsic to the bond itself. An IRR for the bond.

YTM – Example On 9/1/95, PG&E bonds with a maturity date of 3/01/25 and a coupon rate of 7.25% were selling for 92.847% of par, or $928.47 per $1,000 of face value. What is their YTM? Semiannual coupon payment = 0.0725*$1,000/2 = $36.25. Number of semiannual periods to maturity = 30*2 – 1 = 59.

YTM - Example r/2 can only be found by trial and error. However, calculators and spread sheets have algorithms to speed up the search. Searching reveals that r/2 = 3.939% or a stated annual rate (YTM) of r = 7.878%. This is (sort of) an effective annual rate of:

Example Suppose that now the yield to maturity for the PG&E bonds has changed to 7% (on a stated annual basis, APR, with semi-annual compounding). What must be the current price of the bond? Note that the bond itself has not changed (assume for simplicity that the change occurs instantaneously) only its pricing.

Example We find the price using the present value of the bond payments discounted at the YTM. Does it make sense that the price changed like this?

Discounts and Premiums If a coupon bond trades at a discount, an investor will earn a return both from receiving the coupons and from receiving a face value that exceeds the price paid for the bond. If a bond trades at a discount, its yield to maturity will exceed its coupon rate. If a coupon bond trades at a premium it will earn a return from receiving the coupons but this return will be diminished by receiving a face value less than the price paid for the bond. If a bond trades at a premium, its coupon rate will exceed its yield to maturity. Most coupon bonds have a coupon rate set so that the bonds will initially trade at, or very close to, par.

Interest Rate Changes and Bond Prices The sensitivity of a bond’s price to changes in interest rates is measured by the bond’s duration. The prices of bonds with high durations are highly sensitive to interest rate changes. The prices of bonds with low durations are less sensitive to interest rate changes. A bond’s term or time to maturity is a major determinant of duration. All else equal, the longer the term the more sensitive to changes in the interest rates is bond’s price. However, that is not the whole story:

Interest Rate Changes and Bond Prices Duration is essentially a weighted average of the times to receipt of cash flows from the bond. The face value is usually a comparatively very large cash flow making the maturity date a very important “time to receipt” of cash. However if the coupons are large they will also play an important role in lowering the duration of a long-term coupon bond.

The Yield Curve and Bond Arbitrage Using the Law of One Price and the yields of default-free zero-coupon bonds, one can determine the price and yield of any other default-free bond. The yield curve provides sufficient information to evaluate (price) all such bonds.

Replicating a Coupon Bond Replicating a two-year $1,000 bond that pays a 5% coupon using four zero-coupon bonds: time 1/2 1 1 ½ 2 coupon bond - Price $25 $1,025 6 month zero -Price 1 year 18 month 2 year portfolio - sum of prices

Replicating a Coupon Bond Yields and Prices (per $100 Face) for Zero Coupon Bonds Again these are the spot rates on the yield curve. Maturity 6 months 1 year 1 ½ years 2 years YTM 2.0% 3.00% 3.50% 4.00% Price $99.01 $97.07 $94.93 $92.38

Replicating a Coupon Bond Zero Face Value Needed Cost/Price 6 month $25 $99.01/4 = $24.75 1 year $97.07/4 = $24.27 18 month $94.93/4 = $23.73 2 year $1,025 $92.38x41/4 = $946.90 Total Cost $1,019.65 By the Law of One Price, the three-year coupon bond must trade for a price of $1,019.65.

Value a Coupon Bond Using Zero-Coupon Yields The price of a coupon bond must equal the present value of its coupon payments and its face value discounted at the relevant spot rate. Price of a Coupon Bond Where the YTMn‘s are the yields to maturity of an n-period zero.

Coupon Bond Yield to Maturity Knowing the price we can of course also find the yield to maturity of the coupon bond. Annualized this is 3.968%, compare this to the zero- coupon yields.

Stock Valuation: An Example 3M is expected to pay a dividend of $1.92 per share in the coming year. You expect the stock price to be $85 per share at the end of the year. Investments with equivalent risk have an expected return of 11% (the cost of equity capital, rE). What is the most you would pay today for 3M stock? What dividend yield and capital gain rate would you expect at this price? 24

Example Solution: the law of one price implies that to value a security we must determine the discounted expected cash flow associated owning it. Total Return = 2.45% + 8.54% = 11% (but for rounding) so at this price the investor is receiving an appropriate return. 25

A Multi-Year Investor What is the price if we plan on holding the stock for two years? Again the price must be the present value of the future cash flows or one side (i.e. either the buyer or the seller) will refuse to trade.

A Multi-Year Investor What is the price if we plan on holding the stock for N years? This is known as the Dividend Discount Model. Note that the above equation holds for any horizon. Thus all investors (with the same beliefs) will attach the same value to the stock, independent of their investment horizons. The price of any stock is equal to the present value of all of the expected future dividends it will pay.

The Dividend-Discount Model Constant Dividend Growth The simplest forecast for the firm’s future dividends states that they will grow at a constant rate, g, forever. The value of the firm depends on the current dividend level, the cost of equity, and the growth rate. The expected return is from dividend yield and the expected capital gain (g by assumption).

Example Problem AT&T plans to pay $1.44 per share in dividends in the coming year. Its equity cost of capital is 8%. Dividends are expected to grow by 4% per year in the future. Estimate the value of AT&T’s stock. 29

Example Solution Note that its price after one year is expected to be: The expected capital gain equals the dividend growth rate, g. 30

Changing Growth Rates We cannot use the constant dividend growth model to value a stock if the growth rate is not constant. For example, young firms often have very high initial earnings growth rates. During this period of high growth, these firms often retain 100% of their earnings to exploit profitable investment opportunities. As they mature, their growth slows. At some point, their earnings exceed their investment needs and they begin to pay dividends.

Changing Growth Rates Dividend-Discount Model with Constant Long-Term Growth

Example Batesco Inc. just paid a dividend of $1. The dividends of Batesco are expected to grow by 50% next year (time 1) and 25% the year after that (year 2). Subsequently, Batesco’s dividends are expected to grow at 6% in perpetuity. The proper discount rate for Batesco is 13%. What is the fair price for a share of Batesco stock?

Example cont… First, determine the dividends. Draw the timeline! D0 = $1 g1 = 50% D1 = $1(1.50) = $1.50 g2 = 25% D2 = $1.50(1.25) = $1.875 g3 = 6% D3 = $1.875(1.06) = $1.9875 3 2 1 4 ...... 1.50 1.875 1.9875 2.107 g1=50% g2=25% g3=6% g4=6%

Example cont… Supernormal growth period: Constant growth period. Value at time 2: Discount Pc to time 0 and add to Ps: What if supernormal growth lasted 8 yrs at 50%?

Limitations of the Dividend-Discount Model There is a tremendous amount of uncertainty associated with forecasting a firm’s dividend growth rate and its future dividends (particularly those many periods from now). Compounding the issue, dividends are discretionary. Small changes in the assumed dividend growth rate can lead to large changes in the estimated stock price. Many firms pay no dividends (especially true recently).

Discounted Free Cash Flow Model The Discounted Free Cash Flow Model Determines the value of the firm to all investors, including both equity and debt holders The enterprise value can be interpreted as the net cost of acquiring control of the firm, buying its equity, taking its excess cash, paying off all debt, and owning the then unlevered business or “enterprise”.

Discounted Free Cash Flow Model Valuing the Enterprise Free Cash Flow Cash flow available to pay both debt holders and equity holders Discounted Free Cash Flow Model

Discounted Free Cash Flow Model Implementing the Model Since we are discounting cash flows to both equity holders and debt holders, the free cash flows should be discounted at the firm’s weighted average cost of capital, rwacc. If the firm has no debt, rwacc = rE.

Discounted Free Cash Flow Model Implementing the Model Often, the terminal value is estimated by assuming a constant long-run growth rate gFCF for free cash flows beyond year N, so that:

Example: Stock Price from FCF A proforma forecast of the (simplified) income statement for Clive.com appears below. As Clive.com is a mature firm it is forecast that future capital expenditures will be for the replacement of worn equipment. The firm’s experience is that necessary levels of net working capital are 12% of sales in any year. The firm has $75 Million in excess cash on hand, $50 Million in outstanding debt, 100 Million shares of stock outstanding, and faces a 35% tax rate. After 2014 it is expected that free cash flows for the firm will increase at a constant rate of 3% and due to its risk and use of debt a WACC of 15% is appropriate. What is the value of Clive.com’s equity and its share price at the start of 2010?

Solution First we need a forecast of future free cash flow. What about the rest? FCF in 2015 is forecast to be $153.31x1.03 = $157.91 Terminal value is then $157.91/(0.15-.03) = $1,315.92 (end of 2014). PV of TV is $1,315.92/(1.15)5 = $654.25

Solution The present value of the free cash flow for the forecast period is: Now just add the present values: PV(forecast period) + PV(Terminal Value) Enterprise value plus Excess Cash less existing Debt = Equity value Share price is Equity value divided by the number of shares