Review

Bond Yields and PricesBond Yields and Prices

Learning ObjectivesCalculate the price of a bond.Calculate major bond yield measures,

including yield to maturity, current yield, coupon rate

Account for changes in bond prices.Explain and apply the concept of duration.

Bond Valuation PrincipleIntrinsic value

Is an estimated value Present value of the expected future cash flowsRequired to compute intrinsic value

Expected future cash flows Timing of expected cash flows Discount rate, or required rate of return by

investors

Bond ValuationValue of a coupon bond with semi-annual

payments:

2n

2n

1tt

t

/2)r(1

MV

r/2)(1

/2C V

2n

2n

1tt

t

/2)r(1

MV

r/2)(1

/2C V

• Biggest problem is determining the discount rate or required yield

• Required yield is the current market rate earned on comparable bonds with same maturity and credit risk

Yield to MaturityYield to maturity (YTM)

Rate of return on bonds most often quoted for investors

Promised compound rate of return received from a bond purchased at the current market price and held to maturity

Equates the present value of the expected future cash flows to the initial investment Similar to internal rate of return

Yield to MaturitySolve for YTM (semi-annual coupons):

2t

2n

1tt

t

YTM/2)(1MV

YTM/2)(1

/2CP

2t

2n

1tt

t

YTM/2)(1MV

YTM/2)(1

/2CP

• Investors earn the YTM if the bond is held to maturity and all coupons are reinvested at YTM

For: (1) longer-term bonds

(2) bonds with higher coupon rates

(i.e., have more money to reinvest)

NO reinvestment risk for “Zeroes”

Bond Price ChangesOver time, bond prices that differ from face

value must changeBond prices move inversely to market yieldsThe change in bond prices due to a yield

change is directly related to time to maturity and inversely related to coupon rate

Measuring Bond Price Volatility: Duration

Important considerationsDifferent effects of yield changes on the prices

and rates of return for different bondsMaturity inadequate measure of a bond’s

economic lifetimeA measure is needed that accounts for both

size and timing of cash flows

DurationA measure of a bond’s lifetime, stated in

years, that accounts for the entire pattern (both size and timing) of the cash flows over the life of the bond

The weighted average maturity of a bond’s cash flowsWeights determined by present value of cash

flows

Calculating DurationNeed to time-weight present value of cash

flows from bond

tPriceMarket

)PV(CFD

n

1t

t

tPriceMarket

)PV(CFD

n

1t

t

• Duration depends on three factors Maturity of the bond Coupon payments Yield to maturity

Duration RelationshipsDuration increases with time to maturity,

but at a decreasing rateFor coupon paying bonds, duration is always

less than maturityFor zero coupon-bonds, duration equals time

to maturityDuration increases with lower couponsDuration increases with lower yield to

maturity

Why is Duration Important?Allows comparison of effective lives of bonds

that differ in maturity, couponUsed in bond management strategies,

particularly immunizationMeasures bond price sensitivity to interest

rate movements, which is very important in any bond analysis

ConvexityRefers to the degree to which duration

changes as the yield to maturity changesPrice-yield relationship is convex

Duration equation assumes a linear relationship between price and yield

Convexity largest for low coupon, long-maturity bonds, and low yield to maturity

Duration ConclusionsTo obtain maximum price volatility, investors

should choose bonds with the longest duration

Duration is additivePortfolio duration is just a weighted average

Duration measures volatility, which is not the only aspect of risk in bonds

Review part 2

Common Stock ValuationCommon Stock ValuationCompany AnalysisCompany Analysis

Learning ObjectivesThe dividend discount model to estimate

stock pricesExplain the P/E ratio approach.Fundamental Analysis : estimate share’s

intrinsic value

Estimated intrinsic value compared to the current market priceWhat if market price is different than

estimated intrinsic value?

n

1ttk) (1

Flows Cash security of Value

n

1ttk) (1

Flows Cash security of Value

• Intrinsic value of a security isPresent Value Approach DDM

Expected cash flows Dividends paid out of earnings

Earnings important in valuing stocksRetained earnings enhance future earnings

and ultimately dividends Retained earnings imply growth and future

dividends Produces similar results as current dividends in

valuation of common shares

Required Inputs

Current value of a share of stock is the discounted value of all future dividends

1tt

cs

t

cs2

cs

21

cs

1cs

)k(1

D

)k1(D

...)k1(

D

)k1(D

P

1tt

cs

t

cs2

cs

21

cs

1cs

)k(1

D

)k1(D

...)k1(

D

)k1(D

P

Dividend Discount Model

Problems:Need infinite stream of dividendsDividend stream is uncertain

Must estimate future dividendsDividends may be expected to grow over time

Must model expected growth rate of dividends and need not be constant

Dividend Discount Model

Assume constant growth rate in dividendsDividends expected to grow at a constant rate,

g, over time

gkD

P 10 gk

D P 1

0

Dividend Discount Model

D1 is the expected dividend at end of the first period

D1 = D0 x (1+g)

Multiple growth rates: two or more expected growth rates in dividendsUltimately, growth rate must equal that of the

economy as a wholeAssume growth at a rapid rate for n periods,

followed by steady growth

nt

t

)k1(

1g-k

)g(1D

k)(1

)g(1D P cn

n

1t

100

nt

t

)k1(

1g-k

)g(1D

k)(1

)g(1D P cn

n

1t

100

Dividend Discount Model

To estimate share value

1o1

o

/EP E ratio P/E justified

earnings estimatedP

1o1

o

/EP E ratio P/E justified

earnings estimatedP

• P/E ratio can be derived from

g - k/ED

/EP or g - k

DP 11

1o1

o g - k/ED

/EP or g - k

DP 11

1o1

o

Indicates the factors that affect the estimated P/E ratio

P/E Ratio Approach

Market-to-book ratio (M/B)Ratio of share price to per share

shareholder’s equity as measured on the balance sheet

Price paid for each $1 of equityPrice-to-sales ratio (P/S)

Ratio of company’s market value (price times number of shares) divided by sales

Market valuation of a firm’s revenues

Other Valuation Techniques

Company AnalysisCompany Analysis

g-k

DP̂ valueIntrinsic 1

0 g-k

DP̂ valueIntrinsic 1

0

Earnings multiple could also be usedP0 = estimated EPS justified P/E ratio

Stock is under- (over-) valued if intrinsic value is larger (smaller) than current market price

Focus on earnings and P/E ratioDividends paid from earningsClose correlation between earnings and

stock price changes

Fundamental Analysis

A function of the riskless rate of return and a risk premium

k = RF + RPConstant growth version of dividend discount

model can be rearranged so thatk = (D1/P0) + g

Growth forecasts are readily available

Required Rate of Return

Risk premium for a stock regarded as a composite of business, financial, and other risks

If the risk premium rises (falls), then k will rise (fall) and P0 will fall (rise)

If RF rises (falls), then k will rise (fall) and P0 will fall (rise)

Discount rates and P/E ratios move inversely to each other

Required Rate of Return

Five types of ratios used to analyze a firm:1. Liquidity: ability to generate cash and

meet short-term debt2. Asset Management: ability to effectively

manage its assets to generate sales and profits

3. Debt Management: ability to effectively handle its debt

4. Profitability: ability to generate profits5. Value: market value versus accounting

values

Appendix 17-A Financial Ratio Analysis

DuPont AnalysisXYZ (2004)

ROE = (NI/Sales) (Sales/TA) ((TA/Equity) = (.0299)(1.042)(4.339) = 13.51%

Industry averages (2004) ROE = (NI/Sales) (Sales/TA) ((TA/Equity)

= (.0568)(1.23)(1.74) = 12.16% This analysis suggests that XYZ displays an

above average ROE due to its higher leverage factor, and despite the fact it has below average profitability and asset turnover