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The Valuation The Valuation of Long-Term of Long-Term Securities Securities Bond & Stock Bond & Stock Valuation Valuation

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Page 1: Bonds and Stock Valuation

1

The Valuation of The Valuation of Long-Term Long-Term SecuritiesSecurities

The Valuation of The Valuation of Long-Term Long-Term SecuritiesSecurities

Bond & Stock Bond & Stock ValuationValuation

Bond & Stock Bond & Stock ValuationValuation

Page 2: Bonds and Stock Valuation

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The Valuation of The Valuation of Long-Term SecuritiesLong-Term Securities

The Valuation of The Valuation of Long-Term SecuritiesLong-Term Securities

Distinctions Among Valuation Concepts

Bond Valuation

Preferred Stock Valuation

Common Stock Valuation

Rates of Return (or Yields)

Distinctions Among Valuation Concepts

Bond Valuation

Preferred Stock Valuation

Common Stock Valuation

Rates of Return (or Yields)

Page 3: Bonds and Stock Valuation

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What is Value?What is Value?What is Value?What is Value?

Going-concern valueGoing-concern value represents the amount a firm could be sold for as a continuing operating business.

Going-concern valueGoing-concern value represents the amount a firm could be sold for as a continuing operating business.

Liquidation valueLiquidation value represents the amount of money that could be realized if an asset or group of assets is sold separately from its operating organization.

Liquidation valueLiquidation value represents the amount of money that could be realized if an asset or group of assets is sold separately from its operating organization.

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In general, the value of an asset is the price that a willing and able buyer pays to a willing and able seller

Note that if either the buyer or seller is not both willing and able, then an offer does not establish the value of the asset

What is Value?What is Value?What is Value?What is Value?

Page 5: Bonds and Stock Valuation

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Several Kinds of “ValueSeveral Kinds of “Value””

There are several types of value, of which we are concerned with three: Book Value - The asset’s historical cost less

its accumulated depreciation Market Value - The price of an asset as

determined in a competitive marketplace Intrinsic Value - The present value of the

expected future cash flows discounted at the decision maker’s required rate of return

Page 6: Bonds and Stock Valuation

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Bond ValuationBond ValuationBond ValuationBond Valuation

Important Terms

Types of Bonds

Valuation of Bonds

Handling Semiannual Compounding

Important Terms

Types of Bonds

Valuation of Bonds

Handling Semiannual Compounding

Page 7: Bonds and Stock Valuation

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BondsBonds

A bond is a tradeable instrument that represents a debt owed to the owner by the issuer. Most commonly, bonds pay interest periodically (usually semiannually) and then return the principal at maturity.

Most corporate, and some government, bonds are callable. That means that at the company’s option, it may force the bondholders to sell them back to the company.

(Refer to the bonds valuation handout)

Page 8: Bonds and Stock Valuation

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Bond TerminologyBond Terminology

There are several terms with which you must be familiar to solve bond valuation problems: Coupon Rate - This is the stated rate of interest on the bond.

It is fixed for the life of the bond. Also, this rate time the face value determines the annual interest payment amount.

Discount Rate - (capitalization rate or market required rate of return) is dependent on the risk of the bond and is composed of the risk-free rate plus a premium for risk.

Face Value - This is the principal amount (nominally, the amount that was borrowed). This is the amount that will be repaid at maturity

Maturity Date - This is the date after which the bond no longer exists. It is also the date on which the loan is repaid and the last interest payment is made.

Page 9: Bonds and Stock Valuation

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Calculating the Value of a BondCalculating the Value of a Bond

There are two types of cash flows that are provided by a bond investments: Periodic interest payments (usually every six months,

but any frequency is possible) Repayment of the face value (also called the principal

amount, which is usually $1,000) at maturity The following timeline illustrates a typical bond’s cash

flows:

0 1 2 3 4 5

100 100 100 100 1001,000

Page 10: Bonds and Stock Valuation

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We can use the principle of value additivity to find the value of this stream of cash flows

Note that the interest payments are an annuity, and that the face value is a lump sum

Therefore, the value of the bond is simply the present value of the annuity-type cash flow and the lump sum:

Calculating the Value of a Bond Calculating the Value of a Bond (Cont) (Cont)

DiscussedIn the

subsequent slides

Page 11: Bonds and Stock Valuation

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Different Types of BondsDifferent Types of BondsDifferent Types of BondsDifferent Types of Bonds

Perpetual Bond

Non-Zero Coupon Bond

Zero Coupon Bond

Perpetual Bond

Non-Zero Coupon Bond

Zero Coupon Bond

Page 12: Bonds and Stock Valuation

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Different Types of BondsDifferent Types of BondsDifferent Types of BondsDifferent Types of Bonds

A perpetual bondperpetual bond is a bond that never matures. It has an infinite life.

A perpetual bondperpetual bond is a bond that never matures. It has an infinite life.

(1 + kd)1 (1 + kd)2 (1 + kd)V = + + ... +I II

=

t=1(1 + kd)t

Ior I (PVIFA kd, )

V = II / kkdd [Reduced Form]

Page 13: Bonds and Stock Valuation

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Perpetual Bond ExamplePerpetual Bond ExamplePerpetual Bond ExamplePerpetual Bond Example

Bond P has a $1,000 face value and provides an 8% coupon. The appropriate discount rate is

10%. What is the value of the perpetual bondperpetual bond?

Bond P has a $1,000 face value and provides an 8% coupon. The appropriate discount rate is

10%. What is the value of the perpetual bondperpetual bond?

II = $1,000 ( 8%) = $80$80.

kkdd = 10%10%.

VV = II / kkdd [Reduced Form]

= $80$80 / 10%10% = $800 $800.

II = $1,000 ( 8%) = $80$80.

kkdd = 10%10%.

VV = II / kkdd [Reduced Form]

= $80$80 / 10%10% = $800 $800.

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Different Types of BondsDifferent Types of BondsDifferent Types of BondsDifferent Types of Bonds

A non-zero coupon-paying bondnon-zero coupon-paying bond is a coupon-paying bond with a finite life.A non-zero coupon-paying bondnon-zero coupon-paying bond is a coupon-paying bond with a finite life.

(1 + kd)1 (1 + kd)2 (1 + kd)nn

V = + + ... +I I + MVI

= nn

t=1(1 + kd)t

I

V = I (PVIFA kd, nn) + MV (PVIF kd, nn)

(1 + kd)nn

+ MV

Page 15: Bonds and Stock Valuation

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Bond C has a $1,000 face value and provides an 8% annual coupon for 30 years. The

appropriate discount rate is 10%. What is the value of the coupon bond?

Bond C has a $1,000 face value and provides an 8% annual coupon for 30 years. The

appropriate discount rate is 10%. What is the value of the coupon bond?

Coupon Bond ExampleCoupon Bond ExampleCoupon Bond ExampleCoupon Bond Example

VV = $80 (PVIFA10%, 30) + $1,000 (PVIF10%, 30) = $80 (9.427) + $1,000 (.057)

[[Table IVTable IV] ] [[Table IITable II]]

= $754.16 + $57.00= $811.16 $811.16 (D to R)

VV = $80 (PVIFA10%, 30) + $1,000 (PVIF10%, 30) = $80 (9.427) + $1,000 (.057)

[[Table IVTable IV] ] [[Table IITable II]]

= $754.16 + $57.00= $811.16 $811.16 (D to R)

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Coupon Bond Example Coupon Bond Example ( Ms Excel Approach)( Ms Excel Approach)

Present Value CalculationsFuture Value 1000.00Years 30Rate 10%Present Value $57.31

Present Value of an AnnuityPayment 80Interest Rate 10%Number of Payments 30Present Value $754.15

VALUE $811.46

Page 17: Bonds and Stock Valuation

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Different Types of BondsDifferent Types of BondsDifferent Types of BondsDifferent Types of Bonds

A zero-coupon bondzero-coupon bond is a bond that pays no interest but sells at a deep

discount from its face value; it provides compensation to investors in the form

of price appreciation.

A zero-coupon bondzero-coupon bond is a bond that pays no interest but sells at a deep

discount from its face value; it provides compensation to investors in the form

of price appreciation.

(1 + kd)nn

V =MV

= MV (PVIFkd, nn)

Page 18: Bonds and Stock Valuation

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VV = $1,000 (PVIF10%, 30)= $1,000 (.057)= $57.00 (D to R) $57.00 (D to R)

VV = $1,000 (PVIF10%, 30)= $1,000 (.057)= $57.00 (D to R) $57.00 (D to R)

Zero-Coupon Bond Zero-Coupon Bond ExampleExample

Zero-Coupon Bond Zero-Coupon Bond ExampleExample

Bond Z has a $1,000 face value and a 30-year life. The appropriate

discount rate is 10%. What is the value of the zero-coupon bond?

Bond Z has a $1,000 face value and a 30-year life. The appropriate

discount rate is 10%. What is the value of the zero-coupon bond?

Page 19: Bonds and Stock Valuation

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Non-Zero Coupon Bond Example Non-Zero Coupon Bond Example ( Ms Excel Approach)( Ms Excel Approach)

Present Value CalculationsFuture Value 1000.00Years 30Rate 10%Present Value $57.31

VALUE $57.31

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Semiannual CompoundingSemiannual CompoundingSemiannual CompoundingSemiannual Compounding

(1) Divide kkdd by 22

(2) Multiply nn by 22

(3) Divide II by 22

(1) Divide kkdd by 22

(2) Multiply nn by 22

(3) Divide II by 22

Most bonds in the U.S. pay interest twice a year (1/2 of the annual

coupon).

Adjustments needed:

Most bonds in the U.S. pay interest twice a year (1/2 of the annual

coupon).

Adjustments needed:

Page 21: Bonds and Stock Valuation

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(1 + kd/2 2 ) 22*nn(1 + kd/2 2 )1

Semiannual CompoundingSemiannual CompoundingSemiannual CompoundingSemiannual Compounding

A non-zero coupon bondnon-zero coupon bond adjusted for semiannual compounding.

A non-zero coupon bondnon-zero coupon bond adjusted for semiannual compounding.

V = + + ... +I / 22 I / 22 + MV

= 22*nn

t=1(1 + kd /2 2 )t

I / 22

= I/22 (PVIFAkd /2 2 ,22*nn) + MV (PVIFkd /2 2 , 22*nn)

(1 + kd /2 2 ) 22*nn+

MV

I / 22(1 + kd/2 2 )2

Page 22: Bonds and Stock Valuation

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VV = $40 (PVIFA5%, 30) + $1,000 (PVIF5%, 30) = $40 (15.373) + $1,000 (.231)

[[Table IVTable IV] ] [[Table IITable II]]

= $614.92 + $231.00= $845.92 ( D to R) $845.92 ( D to R)

VV = $40 (PVIFA5%, 30) + $1,000 (PVIF5%, 30) = $40 (15.373) + $1,000 (.231)

[[Table IVTable IV] ] [[Table IITable II]]

= $614.92 + $231.00= $845.92 ( D to R) $845.92 ( D to R)

Semiannual Coupon Semiannual Coupon Bond ExampleBond Example

Semiannual Coupon Semiannual Coupon Bond ExampleBond Example

Bond C has a $1,000 face value and provides an 8% semiannual coupon for 15 years. The

appropriate discount rate is 10% (annual rate). What is the value of the coupon bond?

Bond C has a $1,000 face value and provides an 8% semiannual coupon for 15 years. The

appropriate discount rate is 10% (annual rate). What is the value of the coupon bond?

Page 23: Bonds and Stock Valuation

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Semi Annual Coupon Bond Example Semi Annual Coupon Bond Example

( Ms Excel Approach)( Ms Excel Approach)Present Value CalculationsFuture Value 1000.00Years 30Rate 5%Present Value $231.38

Present Value of an AnnuityPayment 40Interest Rate 5%Number of Payments 30Present Value $614.90

VALUE $846.28

Page 24: Bonds and Stock Valuation

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Different Types of StocksDifferent Types of Stocks

•Preferred Stock

•Common Stock

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Preferred StockPreferred Stock is a type of stock that promises a (usually) fixed dividend, but at

the discretion of the board of directors.

Preferred StockPreferred Stock is a type of stock that promises a (usually) fixed dividend, but at

the discretion of the board of directors.

Preferred Stock ValuationPreferred Stock ValuationPreferred Stock ValuationPreferred Stock Valuation

Preferred Stock has preference over common stock in the payment of dividends and claims on assets.

Preferred Stock has preference over common stock in the payment of dividends and claims on assets.

Page 26: Bonds and Stock Valuation

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Preferred Stock ValuationPreferred Stock ValuationPreferred Stock ValuationPreferred Stock Valuation

This reduces to a perpetuityperpetuity!This reduces to a perpetuityperpetuity!

(1 + kP)1 (1 + kP)2 (1 + kP)VV = + + ... +DivP DivPDivP

=

t=1 (1 + kP)t

DivP or DivP(PVIFA kP, )

VV = DivP / kP

Page 27: Bonds and Stock Valuation

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Preferred Stock ExamplePreferred Stock ExamplePreferred Stock ExamplePreferred Stock Example

DivDivPP = $100 ( 8% ) = $8.00$8.00.

kkPP = 10%10%.

VV = DivDivPP / kkPP = $8.00$8.00 / 10%10% = $80 $80

DivDivPP = $100 ( 8% ) = $8.00$8.00.

kkPP = 10%10%.

VV = DivDivPP / kkPP = $8.00$8.00 / 10%10% = $80 $80

Stock PS has an 8%, $100 par value issue outstanding. The appropriate

discount rate is 10%. What is the value of the preferred stockpreferred stock?

Stock PS has an 8%, $100 par value issue outstanding. The appropriate

discount rate is 10%. What is the value of the preferred stockpreferred stock?

Similar toPerpetual

Bonds

Page 28: Bonds and Stock Valuation

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Common Stock ValuationCommon Stock ValuationCommon Stock ValuationCommon Stock Valuation

Pro rata share of future earnings after all other obligations of the firm (if any remain).

Dividends maymay be paid out of the pro rata share of earnings.

Pro rata share of future earnings after all other obligations of the firm (if any remain).

Dividends maymay be paid out of the pro rata share of earnings.

Common stock Common stock represents a residual ownership position in the corporation.Common stock Common stock represents a residual ownership position in the corporation.

Page 29: Bonds and Stock Valuation

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Common Stock ValuationCommon Stock ValuationCommon Stock ValuationCommon Stock Valuation

(1) Future dividends

(2) Future sale of the common stock shares

(1) Future dividends

(2) Future sale of the common stock shares

What cash flows will a shareholder receive when owning shares of

common stockcommon stock?

Page 30: Bonds and Stock Valuation

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Dividend Valuation ModelDividend Valuation ModelDividend Valuation ModelDividend Valuation Model

Basic dividend valuation model accounts for the PV of all future dividends.

Basic dividend valuation model accounts for the PV of all future dividends.

(1 + ke)1 (1 + ke)2 (1 + ke)V = + + ... +

Div1DivDiv2

=

t=1(1 + ke)t

Divt Divt: Cash dividend at time t

ke: Equity investor’s required return

Divt: Cash dividend at time t

ke: Equity investor’s required return

Page 31: Bonds and Stock Valuation

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Adjusted Dividend Adjusted Dividend Valuation ModelValuation Model

Adjusted Dividend Adjusted Dividend Valuation ModelValuation Model

The basic dividend valuation model adjusted for the future stock sale.

The basic dividend valuation model adjusted for the future stock sale.

(1 + ke)1 (1 + ke)2 (1 + ke)nn

V = + + ... +Div1 Divnn + PricennDiv2

nn: The year in which the firm’s shares are expected to be sold.

Pricenn: The expected share price in year nn.

nn: The year in which the firm’s shares are expected to be sold.

Pricenn: The expected share price in year nn.

Page 32: Bonds and Stock Valuation

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Dividend Growth Dividend Growth Pattern AssumptionsPattern Assumptions

Dividend Growth Dividend Growth Pattern AssumptionsPattern Assumptions

The dividend valuation model requires the forecast of all future dividends. The

following dividend growth rate assumptions simplify the valuation

process.

Constant GrowthConstant Growth

No GrowthNo Growth

The dividend valuation model requires the forecast of all future dividends. The

following dividend growth rate assumptions simplify the valuation

process.

Constant GrowthConstant Growth

No GrowthNo Growth

Page 33: Bonds and Stock Valuation

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Constant Growth ModelConstant Growth ModelConstant Growth ModelConstant Growth Model

The constant growth model constant growth model assumes that dividends will grow forever at the rate g.

The constant growth model constant growth model assumes that dividends will grow forever at the rate g.

(1 + ke)1 (1 + ke)2 (1 + ke)V = + + ... +

D0(1+g) D0(1+g)

=(ke - g)

D1

D1: Dividend paid at time 1.

g : The constant growth rate.

ke: Investor’s required return.

D1: Dividend paid at time 1.

g : The constant growth rate.

ke: Investor’s required return.

D0(1+g)2

Page 34: Bonds and Stock Valuation

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Constant Growth Constant Growth Model ExampleModel Example

Constant Growth Constant Growth Model ExampleModel Example

Stock CG has an expected growth rate of 8%. Each share of stock just received an

annual $3.24 dividend per share. The appropriate discount rate is 15%. What

is the value of the common stockcommon stock?

DD11 = $3.24$3.24 ( 1 + .08 ) = $3.50$3.50

VVCGCG = DD11 / ( kkee - g ) = $3.50$3.50 / ( .15.15 - .08 ) = $50 $50

Stock CG has an expected growth rate of 8%. Each share of stock just received an

annual $3.24 dividend per share. The appropriate discount rate is 15%. What

is the value of the common stockcommon stock?

DD11 = $3.24$3.24 ( 1 + .08 ) = $3.50$3.50

VVCGCG = DD11 / ( kkee - g ) = $3.50$3.50 / ( .15.15 - .08 ) = $50 $50

ke e = D1/V + g

Page 35: Bonds and Stock Valuation

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Zero Growth ModelZero Growth ModelZero Growth ModelZero Growth Model

The zero growth model zero growth model assumes that dividends will grow forever at the rate g = 0.

The zero growth model zero growth model assumes that dividends will grow forever at the rate g = 0.

(1 + ke)1 (1 + ke)2 (1 + ke)

VZG = + + ... +D1 D

=ke

D1 D1: Dividend paid at time 1.

ke: Investor’s required return.

D1: Dividend paid at time 1.

ke: Investor’s required return.

D2

Page 36: Bonds and Stock Valuation

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Zero Growth Zero Growth Model ExampleModel ExampleZero Growth Zero Growth

Model ExampleModel Example

Stock ZG has an expected growth rate of 0%. Each share of stock just received an

annual $3.24 dividend per share. The appropriate discount rate is 15%. What

is the value of the common stockcommon stock?

Stock ZG has an expected growth rate of 0%. Each share of stock just received an

annual $3.24 dividend per share. The appropriate discount rate is 15%. What

is the value of the common stockcommon stock?

DD11 = $3.24$3.24 ( 1 + 0 ) = $3.24$3.24

VVZGZG = DD11 / ( kkee - 0 ) = $3.24$3.24 / ( .15.15 - 0 ) = $21.60 $21.60

DD11 = $3.24$3.24 ( 1 + 0 ) = $3.24$3.24

VVZGZG = DD11 / ( kkee - 0 ) = $3.24$3.24 / ( .15.15 - 0 ) = $21.60 $21.60

Page 37: Bonds and Stock Valuation

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Practice QuestionsPractice Questions

1. Beta Budget Brooms will pay a big $2 dividend next year on its common stock, which is currently selling at $50 per share. What is the market's required return on this investment if the dividend is expected to grow at 5% forever?

2. What is the intrinsic value of a $1,000 face value, zero-coupon bond that matures in 20 years if an investor's required rate of return for the bond is 8%? (Assume annual discounting.)

3. What is the intrinsic value of a $1,000 face value, 8% coupon paying perpetual bond if an investor's required rate of return is 6%? (Assume annual interest payments and discounting.)

4. A $250 face value share of preferred stock, pays a $20 annual dividend and investors require a 7% return on this investment. If the security is currently selling for $276, what is the difference (overvaluation) between its intrinsic and market value (rounded to the nearest whole dollar)?

Page 38: Bonds and Stock Valuation

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Practice QuestionsPractice Questions5. You have just purchased a 15-year, $1,000 par value bond. The

coupon rate on this bond is nine percent (9%) annually, with interest being paid each six months. If you expect to earn a 12 percent nominal rate of return on this bond, how much did you pay for it?

6. WeeWiggly Woodcarvers (WWW) common stock is currently expected to maintain its current earnings level indefinitely (earnings will stay at the current $4.00 per share). Investors in WWW expect to earn 10% on this investment since the firm anticipates retaining only 10% of earnings. What is the intrinsic value of a share of WWW?

7. You have been asked to determine the intrinsic value of a share of Quick Quilters, Inc. (QQ) common stock. The stock most recently paid a $2.00 annual dividend (D0). You expect dividends to grow at a supernormal rate of 15% for the next three years. You then expect that dividends will grow at a normal 5% rate thereafter (indefinitely). As a potential investor, you would expect to earn 10% on this investment. What is the intrinsic value of a share of QQ?

Page 39: Bonds and Stock Valuation

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ANSWERSANSWERS

1. 9%

2. $214.55

3. $1333.33

4. $9.71; Hence $10 ( due to rounding)

5. $793.53

6. $36.00

7. $54.55

Page 40: Bonds and Stock Valuation

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Bonds &

Stocks

Yields (Rates of Return )Yields (Rates of Return )

Page 41: Bonds and Stock Valuation

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Yield to MaturityYield to Maturity

The yield to maturity is an important number in bond valuation.

It is the rate which equates the market price of the bond with the value of the discounted cash flows.

That is, YTM is the kd such that the bond equation holds.

Page 42: Bonds and Stock Valuation

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Determining Bond YTMDetermining Bond YTMDetermining Bond YTMDetermining Bond YTM

Determine the Yield-to-Maturity (YTM) for the coupon-paying bond

with a finite life.

Determine the Yield-to-Maturity (YTM) for the coupon-paying bond

with a finite life.

P0 = nn

t=1(1 + kd )t

I

= I (PVIFA kd , nn) + MV (PVIF kd , nn

)

(1 + kd )nn

+ MV

kd = YTM

Page 43: Bonds and Stock Valuation

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Page 44: Bonds and Stock Valuation

44

Determining the YTMDetermining the YTMDetermining the YTMDetermining the YTM

Julie Miller want to determine the YTM for an issue of outstanding bonds at Basket Wonders (BW). BW has an issue of 10% annual coupon bonds with 15 years left to maturity. The

bonds have a current market value of $1,250$1,250.

What is the YTM?What is the YTM?

Julie Miller want to determine the YTM for an issue of outstanding bonds at Basket Wonders (BW). BW has an issue of 10% annual coupon bonds with 15 years left to maturity. The

bonds have a current market value of $1,250$1,250.

What is the YTM?What is the YTM?

Page 45: Bonds and Stock Valuation

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YTM Solution (Try 9%)YTM Solution (Try 9%)YTM Solution (Try 9%)YTM Solution (Try 9%)

$1,250$1,250 = $100(PVIFA9%,15) + $1,000(PVIF9%, 15)

$1,250$1,250 = $100(8.061) + $1,000(.275)

$1,250$1,250 = $806.10 + $275.00

= $1,081.10$1,081.10[[Rate is too high!Rate is too high!]]

Page 46: Bonds and Stock Valuation

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YTM Solution (Try 7%)YTM Solution (Try 7%)YTM Solution (Try 7%)YTM Solution (Try 7%)

$1,250$1,250 = $100(PVIFA7%,15) + $1,000(PVIF7%, 15)

$1,250$1,250 = $100(9.108) + $1,000(.362)

$1,250$1,250 = $910.80 + $362.00

= $1,272.80$1,272.80[[Rate is too low!Rate is too low!]]

Page 47: Bonds and Stock Valuation

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.07 $1273

.02 YTMYTM $1250$1250 $192

.09 $1081

($23)(0.02) $192

YTM Solution (Interpolate)YTM Solution (Interpolate)YTM Solution (Interpolate)YTM Solution (Interpolate)

$23X

X = X = .0024

YTMYTM = .07 + .0024 = .0724 or 7.24%7.24%

Page 48: Bonds and Stock Valuation

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YTM Solution ( Ms Excel Approach)YTM Solution ( Ms Excel Approach)YTM (kd) of a NON-ZERO COUPOIN BOND

Present Value Calculations (PVIF) for "M.V."Future Value $1,000.00Years 15

YTM (kd) 7.22%

Present Value $351.45

Present Value of an Annuity for "INTEREST"Future Value $1,000.00Coupon Rate 10%Payment $100.00

YTM (kd) 7.22%

Number of Payments 15Present Value $898.27

VALUE $1,249.72

Page 49: Bonds and Stock Valuation

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YTM = INT + {(FV – C.P)/n}

(FV + C.P)/2

= 100 + {(1000 – 1250)/15}

(1000+1250)/2

= 0.074 = 7.4%(Approximate)

Yield Approximation FormulaYield Approximation FormulaYield Approximation FormulaYield Approximation Formula

Page 50: Bonds and Stock Valuation

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Determining Semiannual Determining Semiannual Coupon Bond YTMCoupon Bond YTM

Determining Semiannual Determining Semiannual Coupon Bond YTMCoupon Bond YTM

P0 = 2nn

t=1 (1 + kd /2 )t

I / 2

= (I/2)(PVIFAkd /2, 2nn) + MV(PVIFkd /2 , 2nn)

+ MV

[ 1 + (kd / 2) ]2 -1 = YTM

Determine the Yield-to-Maturity (YTM) for the semiannual coupon-

paying bond with a finite life.

Determine the Yield-to-Maturity (YTM) for the semiannual coupon-

paying bond with a finite life.

(1 + kd /2 )2nn

Page 51: Bonds and Stock Valuation

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Determining the Semiannual Determining the Semiannual Coupon Bond YTMCoupon Bond YTM

Determining the Semiannual Determining the Semiannual Coupon Bond YTMCoupon Bond YTM

Julie Miller want to determine the YTM for another issue of outstanding

bonds. The firm has an issue of 8% semiannual coupon bonds with 20 years left to maturity. The bonds

have a current market value of $950$950.

What is the YTM?What is the YTM?

Julie Miller want to determine the YTM for another issue of outstanding

bonds. The firm has an issue of 8% semiannual coupon bonds with 20 years left to maturity. The bonds

have a current market value of $950$950.

What is the YTM?What is the YTM?

Page 52: Bonds and Stock Valuation

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Determining Semiannual Determining Semiannual Coupon Bond YTMCoupon Bond YTM

Determining Semiannual Determining Semiannual Coupon Bond YTMCoupon Bond YTM

[ 1 + (kd / 2) ]2 -1 = YTM

Determine the Yield-to-Maturity (YTM) for the semiannual coupon-

paying bond with a finite life.

Determine the Yield-to-Maturity (YTM) for the semiannual coupon-

paying bond with a finite life.

[ 1 + (.042626) ]2 -1 = .0871 or 8.71%

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Determining Semiannual Determining Semiannual Coupon Bond YTMCoupon Bond YTM

Determining Semiannual Determining Semiannual Coupon Bond YTMCoupon Bond YTM

[ 1 + (kd / 2) ]2 -1 = YTM

This technique will calculate kd. You must then substitute it into the

following formula.

This technique will calculate kd. You must then substitute it into the

following formula.

[ 1 + (.0852514/2) ]2 -1 = .0871 or 8.71% (same result!)

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YTM Solution ( Ms Excel Approach)YTM Solution ( Ms Excel Approach)YTM (kd) - Semi Annual Compounding

Present Value Calculations (PVIF) for "M.V."Future Value $1,000.00Years 40

YTM (kd/2) 4.263%

Present Value $188.31

Present Value of an Annuity for "INTEREST"Future Value $1,000.00Coupon Rate 4%Payment $40.00

YTM (kd/2) 4.263%

Number of Payments 40Present Value $761.70

VALUE $950.00

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Note: The investor uses the bond's computed YTM by comparing it to his/her required rate of return on the bond after considering all risk factors. 1) If the investor's required return is

greater than the YTM, the investor should not buy the bond

2) If the investor's required return is less than the YTM, the investor should buy the bond

Guideline to YTMGuideline to YTM

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Bond Price-Yield Bond Price-Yield RelationshipRelationship

Bond Price-Yield Bond Price-Yield RelationshipRelationship

Discount BondDiscount Bond -- The market required rate of return exceeds the coupon rate (Par > P0 ).

Premium BondPremium Bond ---- The coupon rate exceeds the market required rate of return (P0 > Par).

Par BondPar Bond ---- The coupon rate equals the market required rate of return (P0 = Par).

Discount BondDiscount Bond -- The market required rate of return exceeds the coupon rate (Par > P0 ).

Premium BondPremium Bond ---- The coupon rate exceeds the market required rate of return (P0 > Par).

Par BondPar Bond ---- The coupon rate equals the market required rate of return (P0 = Par).

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Bond Price Par Bond Price Par ( Ms Excel Approach)( Ms Excel Approach)Bond Price (Par)

Coupon Rate = 10%

Present Value Calculations (PVIF) for "M.V."Future Value 1,000.00$ Years 15

YTM (kd) 10.00%

Present Value $239.39

Present Value of an Annuity for "INTEREST"Future Value $1,000.00Coupon Rate 10%Payment $100.00

YTM (kd) 10.00%

Number of Payments 15Present Value $760.61

VALUE $1,000.00

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Bond Price Discount Bond Price Discount ( Ms Excel Approach)( Ms Excel Approach)Bond Price (Discount)

Coupon Rate = 10%

Present Value Calculations (PVIF) for "M.V."Future Value 1,000.00$ Years 15

YTM (kd) 12.00%

Present Value $182.70

Present Value of an Annuity for "INTEREST"Future Value $1,000.00Coupon Rate 10%Payment $100.00

YTM (kd) 12.00%

Number of Payments 15Present Value $681.09

VALUE $863.78

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Bond Price Premium Bond Price Premium ( Ms Excel Approach)( Ms Excel Approach)Bond Price (Premium)

Coupon Rate = 10%

Present Value Calculations (PVIF) for "M.V."Future Value 1,000.00$ Years 15

YTM (kd) 8.00%

Present Value $315.24

Present Value of an Annuity for "INTEREST"Future Value $1,000.00Coupon Rate 10%Payment $100.00

YTM (kd) 8.00%

Number of Payments 15Present Value $855.95

VALUE $1,171.19

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Bond Price-Yield Bond Price-Yield RelationshipRelationship

Bond Price-Yield Bond Price-Yield RelationshipRelationship

Coupon RateCoupon Rate

MARKET REQUIRED RATE OF RETURN (%)

Coupon RateCoupon Rate

MARKET REQUIRED RATE OF RETURN (%)

BO

ND

PR

ICE

($)

1000 Par

1600

1400

1200

600

00 2 4 6 8 1010 12 14 16 18

5 Year5 Year

15 Year15 Year

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62

Bond Price-Yield Bond Price-Yield RelationshipRelationship

Bond Price-Yield Bond Price-Yield RelationshipRelationship

Assume that the required rate of return on a 15-year, 10% coupon-

paying bond risesrises from 10% to 12%. What happens to the bond price?

Assume that the required rate of return on a 15-year, 10% coupon-

paying bond risesrises from 10% to 12%. What happens to the bond price?

When interest rates riserise, then the market required rates of return riserise

and bond prices will fallfall.

When interest rates riserise, then the market required rates of return riserise

and bond prices will fallfall.

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Bond Price-Yield Bond Price-Yield RelationshipRelationship

Bond Price-Yield Bond Price-Yield RelationshipRelationship

Coupon RateCoupon Rate

MARKET REQUIRED RATE OF RETURN (%)

Coupon RateCoupon Rate

MARKET REQUIRED RATE OF RETURN (%)

BO

ND

PR

ICE

($)

1000 Par

1600

1400

1200

600

00 2 4 6 8 1010 12 14 16 18

15 Year15 Year

5 Year5 Year

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Bond Price-Yield Bond Price-Yield Relationship (Rising Rates)Relationship (Rising Rates)

Bond Price-Yield Bond Price-Yield Relationship (Rising Rates)Relationship (Rising Rates)

Therefore, the bond price has fallen fallen from $1,000 to $864.

Therefore, the bond price has fallen fallen from $1,000 to $864.

The required rate of return on a 15-year, 10% coupon-paying bond

has risenrisen from 10% to 12%.Refer

Slide # 55

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Bond Price-Yield Bond Price-Yield RelationshipRelationship

Bond Price-Yield Bond Price-Yield RelationshipRelationship

Assume that the required rate of return on a 15-year, 10% coupon-

paying bond fallsfalls from 10% to 8%. What happens to the bond price?

Assume that the required rate of return on a 15-year, 10% coupon-

paying bond fallsfalls from 10% to 8%. What happens to the bond price?

When interest rates fallfall, then the market required rates of return fallfall

and bond prices will riserise.

When interest rates fallfall, then the market required rates of return fallfall

and bond prices will riserise.

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Bond Price-Yield Bond Price-Yield RelationshipRelationship

Bond Price-Yield Bond Price-Yield RelationshipRelationship

Coupon RateCoupon Rate

MARKET REQUIRED RATE OF RETURN (%)

Coupon RateCoupon Rate

MARKET REQUIRED RATE OF RETURN (%)

BO

ND

PR

ICE

($)

1000 Par

1600

1400

1200

600

00 2 4 6 8 1010 12 14 16 18

15 Year15 Year

5 Year5 Year

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Bond Price-Yield Relationship Bond Price-Yield Relationship (Declining Rates)(Declining Rates)

Bond Price-Yield Relationship Bond Price-Yield Relationship (Declining Rates)(Declining Rates)

Therefore, the bond price has risenrisen from $1,000 to $1,171.

Therefore, the bond price has risenrisen from $1,000 to $1,171.

The required rate of return on a 15-year, 10% coupon-paying bond

has fallenfallen from 10% to 8%.Refer

Slide # 56

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The Role of Bond MaturityThe Role of Bond MaturityThe Role of Bond MaturityThe Role of Bond Maturity

Assume that the required rate of return on both the 5- and 15-year, 10% coupon-paying bonds fallfall from 10% to 8%. What happens to the changes in bond prices?

Assume that the required rate of return on both the 5- and 15-year, 10% coupon-paying bonds fallfall from 10% to 8%. What happens to the changes in bond prices?

The longer the bond maturity, the greater the change in bond price for a

given change in the market required rate of return.

The longer the bond maturity, the greater the change in bond price for a

given change in the market required rate of return.

Page 69: Bonds and Stock Valuation

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Bond Price-Yield Bond Price-Yield RelationshipRelationship

Bond Price-Yield Bond Price-Yield RelationshipRelationship

Coupon RateCoupon Rate

MARKET REQUIRED RATE OF RETURN (%)

Coupon RateCoupon Rate

MARKET REQUIRED RATE OF RETURN (%)

BO

ND

PR

ICE

($)

1000 Par

1600

1400

1200

600

00 2 4 6 8 1010 12 14 16 18

15 Year15 Year

5 Year5 Year

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70

Bond Maturity Bond Maturity ( Ms Excel Approach)( Ms Excel Approach)

Coupon Rate = 10%Present Value Calculations (PVIF) for "M.V."Future Value 1000.00Years 5

YTM (kd) 8.00%

Present Value $680.58

Present Value of an Annuity for "INTEREST"Future Value $1,000.00Coupon Rate 10%Payment $100.00

YTM (kd) 8.00%

Number of Payments 5Present Value $399.27

VALUE $1,079.85

5-YEAR MATURITY

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Bond Maturity Bond Maturity ( Ms Excel Approach)( Ms Excel Approach)

Coupon Rate = 10%Present Value Calculations (PVIF) for "M.V."Future Value 1000.00Years 15

YTM (kd) 8.00%

Present Value $315.24

Present Value of an Annuity for "INTEREST"Future Value $1,000.00Coupon Rate 10%Payment $100.00

YTM (kd) 8.00%

Number of Payments 15Present Value $855.95

VALUE $1,171.19

15-YEAR MATURITY

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The Role of Bond MaturityThe Role of Bond MaturityThe Role of Bond MaturityThe Role of Bond Maturity

The 5-year bond price has risenrisen from $1,000 to $1,080 for the 5-year bond

(+8.0%).

The 15-year bond price has risenrisen from $1,000 to $1,171 (+17.1%). Twice as fastTwice as fast!!

The 5-year bond price has risenrisen from $1,000 to $1,080 for the 5-year bond

(+8.0%).

The 15-year bond price has risenrisen from $1,000 to $1,171 (+17.1%). Twice as fastTwice as fast!!

The required rate of return on both the 5- and 15-year, 10% coupon-paying bonds has fallenfallen from 10% to 8%.

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The Role of the The Role of the Coupon RateCoupon Rate

The Role of the The Role of the Coupon RateCoupon Rate

For a given change in the market required rate of return, the price of a bond will change

by proportionally more, the lowerlower the coupon rate.

For a given change in the market required rate of return, the price of a bond will change

by proportionally more, the lowerlower the coupon rate.

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Example of the Role of Example of the Role of the Coupon Ratethe Coupon Rate

Example of the Role of Example of the Role of the Coupon Ratethe Coupon Rate

Assume that the market required rate of return on two equally risky 15-year

bonds is 10%. The coupon rate for Bond H is 10% and Bond L is 8%.

What is the rate of change in each of the bond prices if market required

rates fall to 8%?

Assume that the market required rate of return on two equally risky 15-year

bonds is 10%. The coupon rate for Bond H is 10% and Bond L is 8%.

What is the rate of change in each of the bond prices if market required

rates fall to 8%?

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Example of the Role of the Example of the Role of the Coupon RateCoupon Rate

Example of the Role of the Example of the Role of the Coupon RateCoupon Rate

The price for Bond H will rise from $1,000 to $1,171 (+17.1%).

The price for Bond L will rise from $848 to $1,000 (+17.9%). It rises fasterIt rises faster!!

The price for Bond H will rise from $1,000 to $1,171 (+17.1%).

The price for Bond L will rise from $848 to $1,000 (+17.9%). It rises fasterIt rises faster!!

The price on Bonds H and L prior to the change in the market required rate of

return is $1,000 and $848, respectively.

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Lower Coupon Rate Lower Coupon Rate ( Ms Excel Approach)( Ms Excel Approach)

Present Value Calculations (PVIF) for "M.V."Future Value 1000.00Years 15

YTM (kd) 10.00%

Present Value $239.39

Present Value of an Annuity for "INTEREST"Future Value $1,000.00Coupon Rate 10%Payment $100.00

YTM (kd) 10.00%

Number of Payments 15Present Value $760.61

VALUE $1,000.00

BOND H (COUPON RATE = 10%)

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Lower Coupon Rate Lower Coupon Rate ( Ms Excel Approach)( Ms Excel Approach)

Present Value Calculations (PVIF) for "M.V."Future Value 1000.00Years 15

YTM (kd) 10.00%

Present Value $239.39

Present Value of an Annuity for "INTEREST"Future Value $1,000.00Coupon Rate 8%Payment $80.00

YTM (kd) 10.00%

Number of Payments 15Present Value $608.49

VALUE $847.88

BOND L (COUPON RATE = 8%)

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Determining the Yield on Determining the Yield on Preferred StockPreferred Stock

Determining the Yield on Determining the Yield on Preferred StockPreferred Stock

Determine the yield for preferred stock with an infinite life.

P0 = DivP / kP

Solving for kP such that

kP = DivP / P0

Determine the yield for preferred stock with an infinite life.

P0 = DivP / kP

Solving for kP such that

kP = DivP / P0

Recallfrom

slide #26

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Preferred Stock Yield Preferred Stock Yield ExampleExample

Preferred Stock Yield Preferred Stock Yield ExampleExample

kP = $10 / $100.

kkPP = 10%10%.

kP = $10 / $100.

kkPP = 10%10%.

Assume that the annual dividend on each share of preferred stock is $10.

Each share of preferred stock is currently trading at $100. What is

the yield on preferred stock?

Assume that the annual dividend on each share of preferred stock is $10.

Each share of preferred stock is currently trading at $100. What is

the yield on preferred stock?

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Determining the Yield on Determining the Yield on Common StockCommon Stock

Determining the Yield on Determining the Yield on Common StockCommon Stock

Assume the constant growth model is appropriate. Determine the yield

on the common stock.

P0 = D1 / ( ke - g )

Solving for ke such that

ke = ( D1 / P0 ) + g

Assume the constant growth model is appropriate. Determine the yield

on the common stock.

P0 = D1 / ( ke - g )

Solving for ke such that

ke = ( D1 / P0 ) + g

Recallfrom

slide #34

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Common Stock Common Stock Yield ExampleYield Example

Common Stock Common Stock Yield ExampleYield Example

ke = ( $3 / $30 ) + 5%

kkee = 15%15%

ke = ( $3 / $30 ) + 5%

kkee = 15%15%

Assume that the expected dividend (D1) on each share of common stock is $3. Each share of common stock

is currently trading at $30 and has an expected growth rate of 5%. What is

the yield on common stock?

Assume that the expected dividend (D1) on each share of common stock is $3. Each share of common stock

is currently trading at $30 and has an expected growth rate of 5%. What is

the yield on common stock?

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End of ChapterEnd of Chapter