8-1

Chapter 5: Bond and Stock (Equity) Valuation

Bond valuation Zero coupon bond valuation and introduction to interest

rate/bond price changes. Valuation of coupon paying bonds, annual and semiannual Yield-to-Maturity (YTM) calculation

Bond terms and types Basics concerning stock valuation Valuation of constant growth (mature) stocks. Valuation of nonconstant growth stocks. Corporate value or Free Cash Flow model

8-2

Bond basics A bond is a debt security, where money/capital is borrowed

and is to be paid back along with interest. More specifically, bonds mature in more than 10 years, notes in less

than 10 years, and bills in less than one year. In this presentation we will use the term bond to refer to all maturities.

Bonds are known as fixed income securities. All of the future payments to be made on the bond are fixed or

predetermined, as stated in the bond contract. The current value of a bond is defined as the Present Value of

all the future cash flows to be received by the bondholder. A bond promises to pay a predetermined stream of future cash flows.

8-3

Example of a two-year zero coupon bond

Three years ago, a 5-year bond was issued. Today, this bond matures in 2 years. The par value is $100. Currently, this bond sells for $84.17 in the market. What annual rate of return do investor’s currently require on this two year bond? This bond must be competitively priced in the market with

similar bonds. This bond’s time line appears below:

t=0

FV2 = 100 par

t=2

PV0 = 84.17

t=1

8-4

Example of a two-year zero coupon bond, continued

From Chapter 4, we have the time value of money formulas that relate the PV0 and FVn for multiperiod applications: PV0 = FVn/(1+r)n, rearrange as → r = [FVn/PV0]1/n – 1

For this example, PV0=84.17, FVn=100, and n=2

r = [FVn/PV0]1/n – 1 = [100/84.17]1/2 – 1 = 0.09 or 9.0% On a financial calculator, enter FV=100, PV=-84.17, N=2,

P/Y=1, and compute the I/Y=8%.

8-5

Example of a 10 year bond that pays annual coupons

Assume that 10 years ago, a 20-year bond was issued. Today, this bond now matures in 10 years. The par value is $1000. It promises to pay the owner 9% (fixed rate) coupon interest each year. What is today’s bond price?

This bond will pay (0.09)($1000) = $90 coupon interest each year, and will also pay off the $1000 par value at t=10 years from today.

Currently, let’s assume that the 10 year market required rate of interest or return on this and comparable bonds is r=8.5% per year. Anyone that buys this bond today will expect to earn this rate over the next 10 years. The bond’s time line appears below:

t=0

1000 par+ 90 coupon=1090

t=1

PV0 = ?

kD=8.5%t=9 t=10

90 coupon 90 coupon

8-6

Example of a 10 year bond that pays annual coupons, continued

The 10 year, 9% annual fixed coupon, $1000 par bond promises to pay the following bundle of cash flows: 10 coupons, paid annually, of (0.09)($1000)=$90 each. The par value amount of $1000 in 10 years.

From a Chapter 4 TVM perspective, the bond’s promised cash flows can be represented as: A 10-year ordinary annuity of $90 annual cash flows. One lump sum cash amount of $1000 in 10 years.

8-7

Example of a 10 year bond that pays annual coupons, continued

The bond’s current price or value is thus the PV of all the promised future cash flows, discounted at r=8.5% per year.

To calculate this bond’s current price, add together the PVs of the annuity of coupons and the PV of the par value lump sum.

The coupon stream annuity PV0=590.52 and the lump sum PV0=442.29, and both sum up to $1032.81, which is therefore the bond’s current value or price. The TVM formulas are shown below:

1010nn0085.01

1000

0.085 1085.0

1 -

0.085

190

r1

PAR

r 1r

1 -

r

1C PV

8-8

Example of a 10 year bond that pays annual coupons, continued

On a financial calculator (or on the appropriate MS Excel function), the following items must be entered: A 10 year annuity of $90 annual cash flows. A lump sum of $1000 in 10 years.

On a financial calculator, enter FV=1000, PMT=90, N=10, I/Y=8.5%, P/Y=1, and compute the PV=-1032.81.

8-9

Bond prices and market interest rate changes, using the ten year bond

Interest rates (yields) and bond prices will change as time passes and economic conditions change.

What will happen to this ten year bond’s price if the one year market required yield suddenly either (1) decreases to r=8.0% or (2) increases to r=9.0%?

NOTE: the coupon rate and payment do not change and thus the cash flows paid by this bond will never change; however, the price that investors are willing to pay today will always change whenever current market interest rates or yields change.

8-10

Case 1: market interest rates (yields) decrease from 8.5% to 8.0%

What are investors now willing to pay for a 10 year bond that pays $90 annual coupons and $1000 par value exactly ten years from today at t=10, after rates (yields) fall today by 0.5%?

On a financial calculator, enter FV=1000, PMT=90, N=10, I/Y=8%, P/Y=1, and compute the PV=-1067.10. Bond price rises by $34.29

When market interest rates or yields decrease, the price of all existing fixed rate coupon bonds will rise.

8-11

Case 2: market interest rates (yields) increase from 8.5% to 9.0%

What are investors now willing to pay for a 10 year bond that pays $90 annual coupons and $1000 par value exactly ten years from today at t=10, after rates (yields) rise today by 0.5%?

On a financial calculator, enter FV=1000, PMT=90, N=10, I/Y=9%, P/Y=1, and compute the PV=-1000. Bond price falls by $32.81

When market interest rates or yields increase, the price of all existing fixed rate coupon bonds will fall.

8-12

Finding the “Yield-to-Maturity” or YTM of an existing bond (not in lecture notes)

An existing bond matures in 5 years. The par value is $1000. It pays an annual coupon payment of 8% or (0.08)($1000) = $80 each year. Currently the bond sells for $1050.

What must r be, here called the YTM? On a financial calculator, enter FV=1000, PMT=80,

PV=-1050, N=5, P/Y=1, and compute the I/Y=6.787%. Note the FV and PMT are of opposite sign than the PV!!!

8-13

Yield to Maturities of bonds, continued

What we actually observe in the bond market are a bond’s current price, and also the promised future coupon and par value payments. The bond’s yield (YTM) or r is something that is calculated or extrapolated from these observable items.

This is where the bond yields (YTMs) we see in the financial press, e.g., Wall Street Journal, come from; they are extrapolated from the bond market data. In these notes, my use of the term yield is synonymous with

YTM, not the term current yield or CY as sometimes mention the textbooks. CY=(coupon payment/bond price).

8-14

Bonds that pay semiannual coupon interest payments

Most coupon paying bonds pay the interest every six months or semiannually.

An example: a U.S. government Treasury Bond matures 11.5 years from today. The par value is $1000. The coupon interest is 14% per year (always stated on annual basis), paid semiannually.

Currently, on such bonds, the r or YTM is 8% per year. What is this bond’s current value?

Here, we return to the Chapter 4 section on other than annual compounding of interest.

8-15

Bonds that pay semiannual coupon interest payments, continued

The fixed semiannual coupon payments: The 14% annual coupon is (0.14)(1000) = $140 per year,

actually paid as ($140/2) = $70 each six months. There are 23 of these semiannual payments of $70 each

over the following 11.5 years (23 semiannual periods). The par value consists of the $1000 payment in 11.5

years (at maturity). With semiannual bonds, the YTM of 8% means 8%

annual nominal, compounded semiannually. The bond’s true effective yield is actually (8%/2) or 4%

semiannually or every six months.

8-16

Bonds that pay semiannual coupon interest payments, continued

On a financial calculator, enter FV=1000, PMT=70, N=23, I/Y=8%, P/Y=2, and compute the PV=-1445.71. The coupon annuity must be entered as done above, 23

payments of $70 each (never as 11.5 payments of $140 each). The par value repayment occurs 23 semiannual periods from today.

The calculator takes I/Y, divides by P/Y, and solves the problem using a semiannual effective rate of 8%/2 = 4%.

t=0 t=0.5

PV0 = ?

YTM=8%t=11 t=11.5

70 coupon 70 coupon 70 couponand 1000 par

8-17

Types and terms of bonds Callable bond: the issuer has right to retire the bond

before maturity, at a predetermined price that is always specified in the bond contract. Almost all corporate bonds are callable. If interest rates

then fall in the future, firms can retire these existing bonds and replace them with new lower rate bonds.

Callable bonds will command a higher interest rate or yield (lower price) than a comparable risk non-callable bond.

Mortgage bond: bond is secured or collateralized by some physical asset in case the issuer defaults. Commonly used in the transportation industry.

8-18

Types and terms of bonds, continued

Convertible bond: bond can be converted into a predetermined number of shares of common stock. Investors are willing to accept a lower yield on such bonds. The right to convert may become very valuable. A convertible bond thus has the opportunity to become an

exciting investment if the firm does unexpectedly well. Debenture bond: bond is backed by the issuer’s

ability to generate future cash flow to make the promised payments. There is no collateral.

8-19

Types and terms of bonds, continued

Subordinated bonds: the bond’s claim on the issuer is junior to one or more senior bond issues. The more senior bonds have the higher priority in bankruptcy and/or liquidation.

Sinking fund provision: issuer may be required to retire a certain amount of an issue each year. For example, having to retire 10% of a 20 year bond issue each year from year 11 to year 20.

Bond contract (indenture): a legal contract between the issuer and bondholders that specifies all of the terms and conditions of the bond issue.

8-20

Evaluating default risk:Bond ratings

Bond ratings are designed to reflect the probability of a bond issue going into default. The lower the rating (the higher the default risk), the higher the required yield.

AAA or Aaa bonds have the highest rating. Depository institutions, e.g., commercial banks and Savings &

Loans may only own Investment Grade bonds.

Investment Grade Junk Bonds

Moody’s Aaa Aa A Baa Ba B Caa C

S & P AAA AA A BBB BB B CCC D

8-21

Common stock basics Common stock represents the ownership of a corporation.

The holders of debt or bonds have a senior claim on the firm. Stockholders have a residual claim, what remains after other

obligations met, including any new asset investment in the firm. Stocks are risky investments; however, we seek to understand

the basics of stock valuation and how to price the risk. Current stock prices reflect today’s expectations of future cash flow

performance of firms and the risk of these cash flows. Expectations concerning future performance can never be proven in the

present. Firms pay out excess (residual) cash to shareholders primarily

as: (1) cash dividends and (2) share repurchases.

8-22

Common stock basics

The primary focus here is placed on Intrinsic Value. Intrinsic Value is the Present Value of all future forecasted cash flows. We define Free Cash Flow to Equity (FCFE) as the firm’s

excess cash flow that can be paid out through both dividends and stock repurchases.

We calculate the PV of all future forecasted FCFE at a discount rate or cost of equity capital r that is (assumed to be) estimated using the Capital Asset Pricing Model (CAPM) which will be covered in Chapter 10.

8-23

Common stock basics Many tend to either overcomplicate the mechanics of stock

valuation or unfortunately insert misconceptions and/or pseudoscience into the analysis.

For simplicity here, we will assume that all the FCFE is paid as a cash dividend, and thus the stock’s intrinsic value today (V0) is the PV of all future forecasted dividends. The timeline and TVM valuation equation always resembles the following.

t=0 t=1

V0 = ?

t=9

D1

t=2 t=10 t=11

D2 D9 D10 D11

......

r 1

D .. ...

r 1

D

r 1

D

r 1

D V t

t3

32

21

10

8-24

Intrinsic value (V) versus actual market prices (P)

Intrinsic values are usually privately obtained estimates of value, here using discounted cash flow (DCF) analysis. The term V (usually designated as V0) is used extensively

here since stock valuation is a private effort. V0 is thus something we can estimate but not prove.

In efficient capital markets, on average, the market value or price P0 should equal the intrinsic value V0.

Note: the total value of any firm’s equity is always the value per share times the total number of shares. Most of our analysis here is done on a per share basis.

8-25

Valuation of a Constant Growth common stock

The term constant growth indicates that a firm is mature and is expected to grow at an assumed constant rate g throughout the future. The term growth rate typically refers to the growth of the

firm’s cash dividends; however, everything associated with the firm is also assumed to grow at the same rate g.

If a firm is expected to have a variable rate of growth in the coming years, then constant growth valuation is not appropriate. However, we will always assume that constant growth does begin somewhere out in the future.

8-26

Example: valuation of a Constant Growth common stock

A mature firm just paid a dividend of D0=$5 per share today and is expected to have a constant growth rate of g=5% per year forever. Based on the stock’s perceived risk, the stock has a required return of r=14% per year.

8-27

Example: valuation of a Constant Growth common stock, continued

Given the dividend growth rate g=5% per year, now forecast the dividends for the following years: D0 = $5.00 (given with example) D1 = D0(1+g) = (5.00)(1+0.05) = $5.25 D2 = D0(1+g)2 = (5.00)(1+0.05)2 = $5.5125 Dn = D0(1+g)n

The D0=$5.00 per share has already been paid out and is no longer part of the firm.

The intrinsic value V0 of the stock will be the Present Value of all the future forecasted dividends, beginning with D1.

8-28

Example: valuation of a Constant Growth common stock, continued

We use the Constant Growth model (introduced in Chapter 4) to calculate the Present Value. The intrinsic value of any currently assumed constant growth stock or investment is: V0=D1/(k-g), plugging in the numbers we have:

V0=D1/(r-g) = 5.25/(0.14 – 0.05) = 5.25/0.09 = $58.33 If D0=$5 has not yet been paid out, then the stock value

would be 58.33 + 5.00 = $63.33 per share (cum dividend). Thus this stock should be worth $58.33 today if the firm is

expected to have a permanent growth rate of 5% per year and next year’s dividend at t=1 years is $5.25 per share.

8-29

The constant growth model A more general form of the constant growth model is

given below: Vt=Dt+1/(r-g); assuming that capital markets are efficient

we often reexpress this relation as Pt=Dt+1/(r-g) For the equation to work: (1) r must exceed g and (2) all

dividends following the dividend in the equation’s numerator must grow at a constant rate g.

This equation above will always give you the stock value, exactly one year before the dividend that you plug into the model. If you plug in the dividend expected at t=30 years, then the equation gives you the value at t=29 years.

8-30

What will be the value of this stock exactly one year from today?

From previously, we know that r=14%, g=5%, and D0=$5, D1=$5.25, and D2=$5.5125.

The constant growth equation, Vt=Dt+1/(r-g), calculates the stock’s value, exactly one year before the dividend that is plugged into the equation. The dividend exactly two years from today is estimated to be D2=$5.5125 at t=2 years. V1 = D2/(r-g) = 5.5125/(0.14-0.05) = $61.25

This stock is predicted to rise in value (or perhaps price) from $58.33 today to $61.25 in exactly one year (t=1 years). We thus forecast that in one year (t=1), the stock will be

worth $61.25 per share just after it pays out D1=$5.25.

8-31

What will be the stock’s estimated value in exactly one year? A second approach.

An alternate method to estimate the future price of a constant growth stock: Everything associated with the firm is expected to grow at the rate g=5% per year forever, including the stock’s value! Therefore, V1 = V0(1+g) = 58.33(1+0.05) = $61.25

8-32

The two components of a stock’s total return on investment

The return on the stock comes in two components: Cash dividends The change in stock price (capital gain or loss)

Let’s assume efficient markets for this case (where on average, P0=V0,): for any constant growth stock we have the following relation: P0 = D1/(r-g).

Rearrange the equation to yield the following relation in terms of total return, we have: r = (D1/P0) + g The first part is D1/P0, the dividend yield The second part is g, the capital gains yield

8-33

The two components of a stock’s total return: r = (D1/P0) + g

We have the following (previously): D1=$5.25, P0=$58/33, and g=5%. Solving the above equation, we have a known result: k = (D1/P0) + g = (5.25/58.33) + 0.05 = 0.09 + 0.05 = 14%

If we pay $58.33 today for this stock, then the expected 14% return comes to us as: (1) a 9% dividend yield and (2) a 5% capital gains yield,

which is a 5% increase in stock price from $58.33 to $61.25.

8-34

How today’s stock values (or stock prices) can change

Example 1: Assume that r increases from 14% to 16% because investors demand a higher risk premium from the stock. V0=D1/(r-g) = 5.25/(0.16 – 0.05) = $47.73

Example 2: Assume that r decreases from 14% to 12% because investors demand a lower risk premium from the stock. V0=D1/(r-g) = 5.25/(0.12 – 0.05) = $75.00

What really changed above? It was not the future cash flow amounts, but rather the required return, due to risk premium changes.

8-35

The valuation of nonconstant growth stocks (most stocks!)

Most stock analysts using an Intrinsic Value analysis will forecast the following for most stocks that they cover:

Ten (10) future years of individual cash flows that can be paid out to stockholders. Refer to the valuation model at bottom of slide.

A terminal value, i.e., what the stock will be worth in exactly 10 years (V10), assuming constant growth (maturity) at rate g following year 10.

The stock’s intrinsic value is then the sum of the PVs of D1 through D10 and the PV of the terminal value V10=D11/(r-g).

A good approximation for the constant growth g (at maturity) for a firm is expected future inflation plus the real expected rate of economic growth in GDP.

r 1

1

g -r

D

r 1

D .. ...

r 1

D

r 1

D

r 1

D V 10

1110

103

32

21

10

8-36

An example of nonconstant growth valuation

Cirrus Corp. is expected to pay out the following dividends, per share: D0=D1=D2=D3=$0, D4=$0.50, D5=$0.65, D6=0.80,

D7=$0.90, ad D8=$1.00. Timeline appears on next slide. All dividends following year 8 or D8 will grow at g=6%

per year forever. This means that D9 = D8(1+g) = 1.00(1+0.06) = $1.06, although this amount won’t be needed. We are also simplifying the example by assuming that maturity begins at t=8 years.

Let’s just assume here that the firm’s stock has r=10% per year.

8-37

An example of nonconstant growth valuation, continued

A timeline of the stock’s dividends is shown below. The salient item here is D8, since all dividend growth

after t=8 years will be at g=6% per year forever. We can use this information to forecast the stock’s value exactly three years from now (at t=7 years). V7 = D8/(r-g) = 1.00/(0.10 – 0.06) = $25.00

t=0 t=1 t=3

D1=0

t=2 t=4 t=5

D2=0 D3=0 D8=1.00D5=0.65

g=6%

t=6 t=7 t=8

D4=0.50 D6=0.80 D7=0.90

8-38

An example of nonconstant growth valuation, continued

The current intrinsic value V0 will be the Present Value of D1, D2, D3, D4, D5, D6, D7 and V7 (Terminal Value). As given previously, V7 = D8/(r-g) = 1.00/(0.10-0.06) = $25.00

r 1

1

g -r

D

r 1

D

r 1

D

r 1

D

r 1

D V 7

87

76

65

54

40

shareper $14.49 V

2)(25)(0.513 0.0.4618 0.4516 0.4036 0.3415 V

0.1 1

1

0.06 - 0.1

1.00

0.1 1

0.90

0.1 1

0.80

0.1 1

0.65

0.1 1

0.50 V

0

0

776540

8-39

Nonconstant growth: another example

XYZ Corp. currently pays no dividends. XYZ’s first forecasted dividend is 18 years from

today at t=18 years, and is expected to be D18=$6.00 per share. Note that D0 through D17 are all forecasted to be zero. All dividends past t=18 years are forecasted to grow at g=7% per year.

The stock has a required return r=14%.

t=0 t=1 t=17

D1=0

t=2 t=18 t=19

D2=0 D17=0 D18=6.00

g=7%

D19

8-40

Nonconstant growth: another example, continued

XYZ pays the first dividend at t=18 years. Using the constant growth formula, we can estimate the value of XYZ shares at t=17 years, since constant growth occurs following year 18. Step 1: V17 = D18/(k-g) = 6.00/(0.14 – 0.07) = $85.7143

Step 2: V0 = V17/(1+k)17 = 85.7143/(1+0.14)17 = $9.24

The stock is forecasted to be worth $85.71 per share exactly 17 years from today (t=17). Today’s PV0 of this year 17 value of $85.71 is $9.24

8-41

The Corporate Valuation Model or Free Cash Flow (FCF) Model

Most financial analysts use the FCF model. FCF is the cash that can be paid out to the firm’s investors, both the debt and equity holders.

The FCF model will give a value that is the total value of the firm’s capital, i.e., the sum of both debt and equity. Note the following items:

Earnings before interest and taxes: EBIT = Revenues - Costs Net operating profit after tax: NOPAT = EBIT(1 - Tax Rate) FCF = NOPAT - net new investment in operating capital.

The appropriate TVM discount rate is the firm’s total cost of capital – both debt and equity. In Chapter 12, we will cover the Weighted Average Cost of Capital or WACC.

8-42

The Corporate Valuation Model or Free Cash Flow (FCF) Model

......

wacc 1

FCF .. ...

wacc 1

FCF

wacc 1

FCF

wacc 1

FCF V t

t3

32

21

10

The above model looks very similar to the dividend model we covered. However, the V0 estimated here is the total firm value or enterprise value of the firm. To obtain the equity value, the debt value (and preferred

stock value) must then be subtracted from the total value. To obtain value per share, divide by the number of shares.

Many assumptions enter into valuation, so equity estimates using the FCF method may differ from those using the FCFE/Dividend model we covered.

8-43

How new stock is usually issued in the U.S. capital markets

The firm usually goes to an Investment Banker such as Merrill Lynch, Salomon/Smith Barney, etc.

The investment banker usually underwrites the issue – purchasing the entire stock issuance from the firm and reselling it to the initial investors (The markup averages around 7%).

Initial Public Offering (IPO): a privately held firm issues publicly traded stock for the first time. Needless to say, there is a lot of uncertainty in valuing many of IPO firms.

Seasoned Equity Offering (SEO): an already publicly traded firm issues additional stock, which we refer to as external equity.