Interest Rates, Bond Valuation, And Stock Valuation

Dr. Vinodh Madhavan

Interest Rates

Cost of Money

Factors affecting cost of money

Production Opportunities

Time preferences for consumption

Risk

Inflation

Interest rate is a function of

Producers’ expected rate of return on invested capital

Savers’ time preference for current vs. future consumption

Riskiness of loan

Expected future rate of inflation

Determination of Interest Rates

r = r* + IP + DRP + LP + MRP

r represents any nominal rate

r* represents the “real” risk-free rate of interest.

IP inflation premium

DRP is default risk premium

LP is liquidity premium

MRP and maturity risk premium

Premiums Added to r* for Different Types of

Debt

IP MRP DRP LP

S-T Treasury

L-T Treasury

S-T Corporate

L-T Corporate

Yield Curve and the Term Structure of

Interest Rates

Term structure – relationship

between interest rates (or yields)

and maturities.

The yield curve is a graph of the

term structure.

The October 2008 Treasury yield

curve is shown at the right.

0%

2%

4%

6%

8%

10%

12%

14%

0 10 20 30

Interest

Years to Maturity

March 1980

February 2000

October 2008

Constructing the Yield Curve: Inflation

N

INFL

IP

N

1t

t

N

Step 1 – Find the average expected inflation rate over Years 1 to N:

For the inflation premium to be precise and theoretically sound, it should be the geometric average of inflationary expectations for the residual life of the maturity.

Constructing the Yield Curve: Inflation

Assume inflation is expected to be 5% next year, 6% the following

year, and 8% thereafter.

Any financial security should earn atleast the estimated inflation

premium, for the holder of such a security to keep up with his/her

original purchasing power at the time of investment.

%75.720/)]18%(8%6%5[

%50.710/)]8%(8%6%5[

%00.51/%5IP

20

10

1

IP

IP

Constructing Yield Curve: Maturity Risk

Step 2 : Find the appropriate maturity risk premium (MRP).

For instance, the following simplistic equation could be a

mathematical representation of a security’s appropriate maturity

risk premium.

MRPt = 0.1% (t – 1)

Constructing Yield Curve: Maturity Risk

Using the given equation:

Notice that since the above equation is linear, the maturity risk

premium is increasing as the time to maturity increases, as it

should be.

%9.1)120(%1.0

%9.0)110(%1.0

%0.0)11(%1.0MRP

20

10

1

MRP

MPP

Step 3 – Adding the premiums to r*.

rRF, t = r* + IPt + MRPt

Assume r* = 3%,

%65.12%9.1%75.7%3

%4.11%9.0%5.7%3

%0.8%0.0%0.5%3r

20 ,

10 ,

1 RF,

RF

RF

r

r

Constructing Yield Curve: Construct Yield Curve

Hypothetical Yield Curve

An upward sloping yield

curve.

Upward slope due to an

increase in inflationary

expectations and increasing

maturity risk premium over

time.

Years to Maturity

Real risk-free rate

0

5

10

15

1

Interest Rate (%)

Maturity risk premium

Inflation premium

10 20

Treasury vs. Corporate Yield Curves

Corporate yield curves are higher than that of Treasury

securities, though not necessarily parallel to the Treasury curve.

The spread between corporate and Treasury yield curves widens

as the corporate bond rating decreases.

Bonds rated AAA (Aaa) are judged to have less default risk than

bonds rated AA (Aa), while AA bonds are less risky than bonds

rated A and so on.

Illustrating the Relationship Between

Corporate and Treasury Yield Curves

0

5

10

15

0 1 5 10 15 20

Years to Maturity

Interest Rate (%)

5.2% 5.9%

6.0% Treasury Yield Curve

BB-Rated

AAA-Rated

Pure Expectations Hypothesis

The PEH contends that the shape of the yield curve depends on

investor’s expectations about future interest rates.

If interest rates are expected to increase, long-term rates will

be higher than short-term rates, and vice-versa.

Thus, the yield curve can slope up, down, or even bow.

Assumptions of the PEH

Assumes that the maturity risk premium for Treasury securities

is zero.

Long-term rates are an average of current and future short-term

rates.

If PEH is correct, you can use the yield curve to “back out”

expected future short-term interest rates.

An Example: Observed Treasury Rates and

the PEH

If PEH holds, what does the market expect will be the interest rate on

one-year securities, one year from now? Three-year securities, two

years from now?

Maturity Yield

1 year 6.0%

2 years 6.2%

3 years 6.4%

4 years 6.5%

5 years 6.5%

One-Year Forward Rate

(1.062)2 = (1.060) (1 + X)

1.12784/1.060 = (1 + X)

6.4004% = X

0 1 2

6.0% x%

6.2%

PEH says that one-year securities will yield 6.4004%, one year from now.

Notice, if an arithmetic average is used, the answer is still very close. Solve: 6.2% = (6.0% + X)/2, and the result will be 6.4%.

Three-Year Security, Two Years from Now

(1.065)5 = (1.062)2 (1 + X)3

1.37009/1.12784 = (1 + X)3

6.7005% = X

0 1 2 3 4 5

6.2% x%

6.5%

PEH says that three-year securities will yield 6.7005%, two years from now.

Conclusions about PEH

Some would argue that the MRP ≠ 0, and hence the PEH is incorrect.

Most evidence supports the general view that lenders prefer short-

term securities, and view long-term securities as riskier.

Hence, investors demand a premium to persuade them to hold

long-term securities (i.e., MRP > 0).

Macroeconomic Factors That Influence

Interest Rate Levels

Monetary policy

Federal budget deficits or surpluses

International factors / foreign trade deficit

Level of business activity

Bond Valuation

What is value?

The term value is used in different senses in the finance literature.

Liquidation Value vs. Going Concern Value

Liquidation value is the amount that can be realized if part of

a firm or the firm as a whole is sold separately from the

operating organization to which it belongs.

Going concern value is the amount that can be realized

should the firm be sold as a continuing operating entity.

Book Value vs. Market Value

Book Value of an asset is the carrying value of any asset, which is

calculated as the original cost-base of the asset minus the accumulated

depreciation accounted for the specific asset

Book value for a firm as a whole is the difference between book value of

all assets of the reporting entity minus the book value of all liabilities of

the reporting entity (SHE = TA-TL)

Market Value of an asset is the price at which the asset trades in the

market place. Almost always, market value of equity is higher than its

book (par) value. However this is not the case with bonds.

What is value?

Market Value vs. Intrinsic Value

The intrinsic value of an asset is the present value of all cash

flows expected from the asset, discounted at a rate of return

that is appropriate for the risk associated with the security.

Intrinsic value is economic value of an asset

Should the market be reasonably efficient, the market price of

an asset should hover around its intrinsic value.

What is value?

Bonds

A bond is a contract wherein a borrower promises to pay interest

and principal on specific dates to the holders of the bond.

In India, the principal issuers of bonds are

Central Government (Treasury Bonds)

State Government (State Government Bonds)

Public Sector Undertakings (PSU Bonds)

Private sector companies (Corporate Bonds)

Bonds issued by PSUs and private sector companies, generally

have a maturity ranging from 1 year to 15 years, and they pay

coupons on a semi-annual basis, unless stated otherwise.

Key Features of a Bond

Par value – face amount of the bond, which is paid at maturity (assume $1,000 if not specified).

Coupon interest rate – stated interest rate (generally fixed) paid by the issuer.

Multiply by par value to get dollar payment of interest.

Maturity date – years until the bond must be repaid.

Issue date – when the bond was issued.

Yield to maturity – rate of return earned on a bond held until maturity (also called the “promised yield”).

The Bond Pricing Equation

T

T

)(1

FV

R

R)(1

1-1

C Value BondR

Pure Discount Bonds

Make no periodic interest payments (coupon rate = 0%)

The entire yield to maturity comes from the difference between the purchase price and the par value.

Cannot sell for more than par value

Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs)

Treasury Bills and principal-only Treasury strips are good examples of zeroes.

Pure Discount Bonds

Information needed for valuing pure discount bonds: Time to maturity (T) = Maturity date - today’s date Face value (F) Discount rate (r)

TR

FVPV

)1(

Present value of a pure discount bond at time 0:

0

0$

1

0$

2

0$

1T

F$

T

Pure Discount Bond: Example

Find the value of a 30-year zero-coupon bond with a $1,000 par

value and a YTM of 6%.

11.174$)06.1(

000,1$

)1( 30

TR

FVPV

0

0$

1

0$

2

0$

29

000,1$

30

0

0$

1

0$

2

0$

29

000,1$

30

Level Coupon Bonds

Make periodic coupon payments in addition to the maturity value

The payments are equal each period. Therefore, the bond is just a combination of an annuity and a terminal (maturity) value.

Coupon payments are typically semiannual.

Consols

Not all bonds have a final maturity.

British consols pay a set amount (i.e., coupon) every period forever.

These are examples of a perpetuity.

R

CPV

Bond Concepts

Bond prices and market interest rates move in opposite directions.

When coupon rate = YTM, price = par value

When coupon rate > YTM, price > par value (premium bond)

When coupon rate < YTM, price < par value (discount bond)

YTM with Annual Coupons

Consider a bond with a 10% annual coupon rate, 15 years to

maturity, and a par value of $1,000. The current price is $928.09.

Will the yield be more or less than 10%?

YTM = {C+(MV – Price )/n}/{0.4*MV + 0.6*Price}

Effect of a Call Provision

Allows issuer to refund the bond issue if rates decline (helps the

issuer, but hurts the investor).

Borrowers are willing to pay more, and lenders require more, for

callable bonds.

Most bonds have a deferred call and a declining call premium.

What is a sinking fund?

Provision to pay off a loan over its life rather than all at maturity.

Similar to amortization on a term loan.

Reduces risk to investor, shortens average maturity.

But not good for investors if rates decline after issuance.

How are sinking funds executed?

Call x% of the issue at par, for sinking fund purposes.

Likely to be used if rd is below the coupon rate and the bond sells at

a premium.

Buy bonds in the open market.

Likely to be used if rd is above the coupon rate and the bond sells at

a discount.

Definitions

CGY Expected

CY Expected YTM return total Expected

price Beginning

price in Change (CGY) yieldgains Capital

priceCurrent

payment coupon Annual (CY) eldCurrent yi

7-38

Other Types (Features) of Bonds

Convertible bond – may be exchanged for common stock of the

firm, at the holder’s option.

Warrant – long-term option to buy a stated number of shares of

common stock at a specified price.

Putable bond – allows holder to sell the bond back to the company

prior to maturity.

Income bond – pays interest only when income is earned by the

firm.

Indexed bond – interest rate paid is based upon the rate of inflation.

Stock Valuation

The Present Value of Common Stocks

The value of any asset is the present value of its expected future cash

flows.

Stock ownership produces cash flows from:

Dividends

Capital Gains

Valuation of Different Types of Stocks

Zero Growth

Constant Growth

Differential Growth

Case 1: Zero Growth

Assume that dividends will remain at the same level forever

RP

RRRP

Div

)1(

Div

)1(

Div

)1(

Div

0

3

3

2

2

1

10

321 DivDivDiv

Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity:

Case 2: Constant Growth

)1(DivDiv 01 g

Since future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity:

gRP

1

0

Div

Assume that dividends will grow at a constant rate, g, forever, i.e.,

2

012 )1(Div)1(DivDiv gg

3

023 )1(Div)1(DivDiv gg .

. .

Case 3: Differential Growth Assume that dividends will grow at different rates in the foreseeable

future and then will grow at a constant rate thereafter.

To value a Differential Growth Stock, we need to:

Estimate future dividends in the foreseeable future.

Estimate the future stock price when the stock becomes a Constant Growth Stock (case 2).

Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate.

Case 3: Differential Growth

)(1DivDiv 101 g

Assume that dividends will grow at rate g1 for N years and grow at rate g2 thereafter.

2

10112 )(1Div)(1DivDiv gg

N

NN gg )(1Div)(1DivDiv 1011

)(1)(1Div)(1DivDiv 21021 ggg N

NN

. . .

. . .


)(1Div 10 g

Dividends will grow at rate g1 for N years and grow at rate g2 thereafter

2

10 )(1Div g

Ng )(1Div 10 )(1)(1Div

)(1Div

210

2

gg

g

N

N

…

0 1 2

…

N N+1

…


We can value this as the sum of:

an N-year annuity growing at rate g1

T

T

AR

g

gR

CP

)1(

)1(1 1

1

plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1

NBR

gRP

)1(

Div

2

1N


Consolidating gives:

NT

T

R

gR

R

g

gR

CP

)1(

Div

)1(

)1(1

2

1N

1

1

Or, we can “cash flow” it out.

Estimates of Parameters

The value of a firm depends upon its growth rate, g, and its discount rate, R.

Where does g come from? g = Retention ratio × Return on retained earnings

Stock Valuation Problems

1. Ezzel Corporation issued perpetual preferred stock with a 10%

annual dividend. The stock currently yields 8% and its par value

is $100.

a. What is the preferred stock’s value?

b. Should the interest rates in the broader economy increase,

and in-turn pull the preferred stock’s yield up to 12%, what is

the new market value of preferred stock?

2. Bruner Aeronautics has perpetual preferred stock outstanding

with a par value of $100. The stock pays a quarterly dividend of

$2 and its current price is $80.

a. What is its nominal annual rate of return?

b. What is its effective annual rate of return?

3. A stock is expected to pay a dividend of $0.50 one year hence, and it should continue to grow at a constant rate of 7% a year. If its required rate is 12%, what is the stock’s expected price 4 years from today?

4. Microtech Corporation is expanding rapidly and currently needs to retain all of its earnings, hence it does not pay dividends. However investors expect Microtech to begin paying dividends, beginning with a dividend of $1.00 coming 3 years from today. The dividend should grow rapidly –at the rate of 50% per year- during years 4 and 5; but after year 5, growth should be a constant 8% per year. If the required return on Microtech is 15%, what is the value of the stock today?


5. Mitts Cosmetics Co’s stock price is $58.88, and it recently paid a

$2.00 dividend. This dividend is expected to grow by 25% for the

next 3 years , then grow forever at a constant rate, g and r = 12%. At

what constant rate is the stock expected to grow after year 3?

6. Your broker offers to sell you some shares of Bahnsen and Co.

common stock that paid a dividend of $2.00 yesterday. Bahnsen’s

dividend is expected to grow at 5% per year for the next 3 years.

a. If you buy the stock, plan to hold it for 3 years, and then sell it at

$34.73, what is the most you should be willing to pay for this

stock, assuming a discount rate of 12%.

b. If the holding period is 5 years rather than 3 years, would this

affect the value of stock today?


7. Taussing Technologies Corporation (TTC) has been growing at a

rate of 20% per year in recent years. This same growth rate is

expected to last for another 2 years, and then decline to 6%. If

current dividend (at t=0) is $1.60, and if discount rate is 10%

a. What is TTC’s stock worth today?

b. What are the expected dividend yields and capital gains yield

for years of supernormal growth (years 1 and 2)?

c. Should TTC’s supernormal growth rate last for 5 years rather

than 2 years, calculate the price of TTC’s stock today, and the

dividend yield and capital gains yield for the years of

supernormal growth.


Q7 continued: d: Suppose TTC recently introduced a new line of products

that has been wildly successfully. On the basis of this success and anticipated

future success, the following free cash flows (in millions) were projected.

After the tenth year, TTC’s financial planners anticipate FCF to grow by 6%

every year. Further, this new project has reduced overall enterprise risk,

which in-turn has reduced enterprise cost of capital to 9%.

Assuming (a) market value of TTC’s debt to be 1200 million, and (b) 20

million common shares outstanding (no preferred shares), what is value of

TTC’s stock as of today (use corporate valuation model).


Year FCF Year FCF

1 5.5 6 88.8

2 12.1 7 107.5

3 23.8 8 128.9

4 44.1 9 147.1

5 69.0 10 161.3

8. A company’s annual dividends have increased from $1.25 for 1990 to $1.75 for 1995.

a. What is the average annual rate of growth of dividends from 1990 to 1995?

b. If an investor’s required rate of return is 12%, how much should he be willing to pay for a share of the company’s stock at the beginning of 1996, assuming that the rate of growth will continue at the same rate as during the preceding five years?

c. What would be the required rate of growth of the annual dividends for the stock to be worth a selling price of $40 per share at the beginning of 1996?


9. Barrett Industries invests a large sum of money in R&D, as a result,

it retains and reinvests all of its earnings. In other words, Barrett

does not pay any dividends and it has no plans to pay any dividends

in the near future. A major pension fund is interested in purchasing

Barrett’s stock. The pension fund manager has estimated Free Cash

Flows for the next four years as follows: $3 million, $6 million, $10

million, and $15 million. After the fourth year, Barrett’s cash flow is

projected to grow at a constant rate of 7%.

I. If Barrett’s enterprise cost of capital is 12%, what is the firm’s

value as of today?

II. What is the estimate of Barrett’s price per share, if Barrett

Industries has (a) cumulative debt and preferred stock totaling

$60 million and (b) 10 million outstanding common shares


10. Assume that today is December 31st, 2008 and that the following

information (estimation) applies to Vermeil Airlines:

• After-tax operating income for 2009 is expected to be $500

million

• Depreciation expense for 2009 is $100 million

• Capital Expenditures for 2009 are expected to be $200 million

• No change in net working capital

• FCFs are expected to grow at a constant rate of 6% going forward

• Enterprise cost of capital is 10%

• Market Value of company’s debt is $3 billion.

• Number of common shares outstanding: 200 million

• What should be the company’s stock price today?


Documents

Interest Rates, Bond Valuation, And Stock Valuation