Security Valuation

DCF Approach to

Stock and Bond Valuation

DCF Valuation

• In general, the value of a security is the sum of the present values of the (expected) future cash flows that accrue to the owner of that security.– Expected cash flows since we have to be concerned about risk.– The presence of risk implies we must use a discount rate consistent

with market conditions and appropriate for the risk involved.– Different discount rates may be appropriate for different cash

flows.– Discount all future cash flows to the present and the sum of the

present values equals the securities current value.• Correct application of this approach may be used to value

all the securities you will come across. For some securities (i.e. options, a topic for later) there are easier ways.

Bond Valuation - Terminology• Face or par value (F), is the amount the bond promises to pay at its

maturity date.• Coupon interest: The bond is quoted as a coupon rate of C per year

and usually makes actual payments of C/2 every 6 months. C/F is defined as the coupon interest rate, a rate that is constant over the life of the bond.

• Call provision, call protection, call premium.• Default risk.• Discount rate, r, the market determined appropriate rate of return

of this type of security.• Yield to maturity (yield) is the discount rate that equates the

bond’s promised payments to its market price, V.• Current yield C/V

Pure Discount Bonds

• Pure discount bonds are the simplest bonds possible. They pay no interest and promise only to return the face value at maturity.

• T-bills (up to a year in maturity) and Strips.

– The value of a three year $1,000 face value strip, when the market rate is 6% is

maturity toperiods is T ;)1(

:formulaValuation Tr

FV

62.839$)06.1(

000,1$3V

Consol Bonds

• Consol bonds do not have a maturity, they make periodic interest payments, forever.

• Who but the English?

– If a consol bond you are buying from a cockney broker pays ₤50 per year and the interest rates in London are 7%, how much will you pay?

r

CV :formulaValuation

29.714£07.0

50£V

Level Coupon Bonds

• Level coupon bonds are the most common bonds. They pay a semiannual coupon, quoted as C per year, face of F, T years to maturity, and an annual discount rate of r.

• There are two components of this valuation:– The coupon payments comprise an annuity.– The lump sum payment of face value at

maturity.

Level Coupon Bonds cont…

• The two pieces:– Annuity of C/2 for 2T periods (6 months).– Lump sum of F at the end of 2T periods.

• Technical note: r is the stated annual discount rate and we are using semiannual compounding. This is the current value assuming you receive the first coupon 6 months from today.

1 2 T-1 T

C/2 C/2 C/2C/2 C/2C/2 C/2 C/2F

0

)r+(1

F

rrr

C = P 2TT0

2/)2/1)(2/(

1

2/

1

2 2

Bond Pricing Example

• Dupont issued 30-year bonds with a coupon rate of 7.95%. These bonds currently have 28 years remaining to maturity and are rated AA. Newly issued AA bonds with similar maturities are currently yielding 7.73%. The bonds have a face value of $1000. What is the value of this Dupont bond today?

Bond Pricing Example cont…

• Annual coupon payment = 0.0795*$1000=$79.50• Semiannual coupon payment = $39.75 = $79.50/2• Semiannual discount rate = 0.0773/2 = 0.03865• Number of semiannual periods = 28*2 = 56

06.1025$03865.0

1000

)03865.01)(03865.0(

1

03865.0

175.39

56

56

)(1

= V 0

Bond Prices and Interest (Discount) Rates

When The Discount Rate Is Equal To The Coupon Rate The Bond Will Sell At Par

When The Discount Rate Is Above The Coupon Rate The Bond Will Sell At A Discount To Par

When The Discount Rate Is Below The Coupon Rate The Bond Will Sell At A Premium To Par

At The Instant Before Maturity All Bonds Will Sell At Par• Why Do These Relations Hold?• What Feature Of A Bond Is The Primary Determinant

Of Its Price Sensitivity To Interest Rates?

Bond Prices and Time to Maturity

600.00

800.00

1000.00

1200.00

1400.00

1600.00

30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

Years to Maturity

Pric

e

7%10%13%

Discount RatesWhat is the coupon rate?

Bond Prices and Interest (Discount) Rates

0.00

200.00

400.00

600.00

800.00

1000.00

1200.00

1400.00

1600.00

1800.00

2000.00

5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%

Discount Rate

Pri

ce

5

10

30

What is the common coupon rate of these bonds?Why is the long maturity bond more volatile?

Years to maturity

Yield to Maturity (or Call)

• The yield to maturity is the discount rate that equates the bond’s current price with its stream of promised future cash flows.

• This is the yield that you would receive if you held the bond to maturity (and were able to reinvest the coupon payments at this rate).

• The yield to call is the discount rate that equates the bond’s current price with its stream of promised cash flows until the expected call date.

• Given two bonds, equivalent in all respects except that one is callable, which bond will have a higher price?

YTM – Example

• On 9/1/95, PG&E bonds with a maturity date of 3/01/25 and a coupon rate of 7.25% were selling for 92.847% of par, or $928.47 per $1,000 of face value. What is their YTM?

• Semiannual coupon payment = 0.0725*1000/2 = $36.25.

• Number of semiannual periods to maturity = 30*2 – 1 = 59.

YTM – Example cont…

• r/2 can only be found by trial and error. However, calculators and spread sheets have algorithms to speed up the search.

• Searching reveals that r/2 = 3.939% or a stated annual rate of r = 7.877%.

• This is an effective annual rate or annual YTM of:

)2/r+(1

1000

)2/r1)(2/r(

1

2/r

125.36$ = 47.928$ 5959

%03.81)03939.1( 2

Term Structure of Interest Rates• We have been talking (and we will frequently continue

to talk) as if the interest rate were constant across all future periods.

• One look at the WSJ bond pages and you know I’m lying to you.

• Its not just because I enjoy doing so.

• We shouldn’t leave this discussion without introducing the term structure of interest rates.

• The term structure of interest rates is the structure of yields on debt instruments which differ only in their times to maturity.

Measuring the Term Structure

• We have been dealing with “spot rates,” and thinking of the same “spot rate” for each maturity.

• The formulas are the same, we just need to allow a subscript to differentiate different maturities.

• Knowing the face value and price you can calculate the spot rates or knowing face and the spot rate you can calculate the price of the zero’s with different maturities.

1)1(

1

00

T

TTT P

Frand

r

FP

Examples• Suppose that a two year zero has a face of $1,000 and a

current price of $800. What is the two-year spot rate?

• Once we have computed all the spot rates we can find the current value of any stream of cash flows.

• A government bond that matures in 4 years promises to pay an annual coupon of 6%. The spot rate for year 1 is 6%, for year 2 is 6.5%, for year 3 is 7%, and for year 4 is 7.5%. How do we find the value of this bond?

%8.111)25.1(1800$

000,1$ 2/12

1

2

r

21.952$)5.07.1(

060,1$

)07.1(

60$

)065.1(

60$

06.1

60$4320 P

Forward Rates• An idea you need to be aware of is forward rates.• Suppose that the one year spot rate is 5% and the two year spot

rate is 10%. If you want to invest for two years but buy a one year bond how can we think about your investment?

• Because there is a rate that applies for now till one year from now and a rate that applies for now till two years from now, there is implicitly defined a rate between one year from now and two years from now.

t=0 t=1 t=2

P0 P0(1+r1) P0(1+r1)(1+E(r1,2))

P0P0(1+r2)2

Forward Rates – Example

• If the one year spot rate is r1 = 5% and the two year spot rate is r2 = 10% then we define the forward rate between the end of year one till the end of year two, f1,2 by the equation:

• We won’t dwell on these rates but you need to know about them because they can be handy.

• How would you estimate what the price of a coupon bond will be one year from now?

222,11 )1()1)(1( rfr

Preferred Stock Valuation

• Preferred Stock– Preferred stock has a fixed dividend payment.– Preferred dividends can be omitted without

placing the firm in default.– Preferred stock has no maturity date.

• Does preferred stock have the same risk as the firm’s debt?

• Preferred stock looks like a perpetuity.

Preferred Stock Valuation cont…

• Preferred stock is typically valued as a perpetuity. Given the promised dividend payment, Divp, and the discount rate, rp, the value of a share of preferred stock is:

p

pp r

DivP

Preferred Stock Example

• On 8/24/01 Sears preferred stock had a dividend of $2.22 per share and was selling at $26.25 per share. What rate of return were investors requiring on Sears preferred stock?

%46.825.26$

22.2$

p

pp P

Divr

Common Stock Valuation - Terminology

• Dt =dividend per share of stock at time t.

• P0=market price of the stock at time 0 (now).

• Pt=market price of the stock at time t. (Prior to date t, this would be the expected price).

• g=expected growth rate in dividend payments.

• rs=required rate of return.

• D1/P0=dividend yield during period 1.

• [P1 - P0]/P0= capital gain rate during period 1.

Common Stock Valuation - Terminology

• Dividend yield is the annual dividend over the current price.

• PE (price earnings ratio) is the current price divided by the sum of the last 4 quarters announced earnings.

• Get to know how to read the information contained in the WSJ stock quotes, it can be surprisingly informative.

Common Stock Valuation• What would you pay for a share of stock today?• To answer this question, ask: why would you buy it?• Suppose you have a one year holding period horizon.

• D1 and P1 represent expectations.• Note also that we can rearrange this to see that return is dividend

yield plus the capital gain yield.

• Before date 1 this is the expected return, at or after date 1 this is the realized return.

sss r

PD

r

P

r

DP

111

11110

0

01

0

1

P

PP

P

Drs

Common Stock Valuation cont…

• What determines P1?

• An investor purchasing the stock at time 1 and holding it until time 2 would be willing to pay:

• Substitute this into the equation for P0 from the last slide and find:

)1(22

1sr

PDP

22

221

0 )1()1()1( sss r

P

r

D

r

DP

Common Stock Valuation cont…

• Repeat this process N times and find:

• If we continue to apply the same logic (let N get really big) we find that:

• The current market value (price) of a share of stock is the present value of all its expected future dividends.

Ns

NN

s

N

sss r

P

r

D

r

D

r

D

r

DP

)1()1(....

)1()1()1( 33

221

0

1t

ts

t0

)r+(1DP

Stock Valuation if Dividends Display Constant Growth (Forever)

• If the dividend payments on a stock are expected to grow at a constant rate, g, and the discount rate is rs, then the value of the stock at time 0 is

• g must be less than rs for this to be valid.• If g = 0 this collapses to the perpetuity formula.

– If g is negative this works for shrinking dividends.

• Labeled the Gordon growth model.• Why would prices change?

gr

DP

s 1

0

Example

• Geneva Steel just paid a dividend of $2.10. Geneva’s dividend payments are expected to grow at a constant rate of 6%. The appropriate discount rate is 12%. What is the current price of Geneva Stock?

• D0 = $2.10 D1 = $2.10(1.06) = $2.226

10.37$06.012.0

226.2$0

P

Aside: Estimating the Required Return from the Current Price

• We are focusing on valuation – determination of the price.• Suppose you observe a price that you consider reliable, and instead

wish to infer rs. Rearrange the constant growth valuation formula to obtain:

• Example: US East stock currently sells for $22. Its most recent dividend was $1.50, and dividend growth of 6% is expected.

• This method is often used in utility regulation. Leaves us wondering: what is g? (Seems to be an important question.)

gP

Drs

0

1

%23.1306.00723.006.022$

59.1$

59.1$)06.1(50.1$1

sr

D

Estimating the Growth Rate: Illustration

• A common starting point for estimating the growth rate is to assume:– The firm’s ROE is constant over time and across projects.– The proportion of the firm’s income paid out as dividends (payout

ratio) is also constant.– The firm will have no future financings.

• Then, income and dividends will both grow at the same rate as owner’s equity, and owners equity will grow only due to retained earnings.

• The growth rate will be ROE(1 – payout ratio) = SGR.• Very sensitive to the assumptions, don’t use this method if

they are not representative of actual conditions.

Common Stock Valuation Example

• In early 1996 (from a case used in another class).• ROE = 13%, payout ratio = 45%

– Implying g = 0.13(1 – 0.45) = 0.0715.– 1995 dividend was $1.64.– Thus, D1 = $1.64(1.0715) = 1.757.– Calculating rs = 0.11, we have

(The actual price was $45 per share, explaining my choice of examples more than anything.)

64.45$0715.011.0

757.1$0

P

Non-constant Growth in Dividends

• Firms often go through lifecycles.– Fast growth.

– Growth that matches the economy.

– Slower growth or decline.

• A super normal growth stock is one that is experiencing rapid growth. But, supernormal growth is, by definition, only temporary. Why?

Valuation of Non-constant Growth Stocks

• Could just derive all expected dividend payments individually and discount them. Tedious.

• Find the present value of the dividends during the period of rapid growth.

• Project the stock price at the end of the rapid growth period. This will be the discounted value of the subsequent dividends. Discount this price back to the present.

• Add these two present values to find the intrinsic value (price) of the stock.

Example

• Batesco Inc. just paid a dividend of $1. The dividends of Batesco are expected to grow by 50% next year (time 1) and 25% the year after that (year 2). Subsequently, Batesco’s dividends are expected to grow at 6% in perpetuity.

• The proper discount rate for Batesco is 13%.

• What is the fair price for a share of Batesco stock?

Example cont…

• First, determine the dividends.

• D0 = $1 g1 = 50%

• D1 = $1(1.50) = $1.50 g2 = 25%

• D2 = $1.50(1.25) = $1.875 g3 = 6%

• D3 = $1.875(1.06) = $1.9875

0 321 4

......

1.50 1.875 1.9875 2.107

g1=50% g2=25% g3=6% g4=6%

Example cont…• Supernormal growth period:

• Constant growth period. Value at time 2:

• Discount Pc to time 0 and add to Ps:

• What if supernormal growth lasted 5 yrs at 50%?

796.2$ = )(1.13

1.875+

(1.13)1.50

= )r+(1

D+)r+(1

D = P 22s

2

s

1s

$28.393 = 0.06-0.13

1.9875 =

g-rD = P

3s

3c

035796.2 .$2 = )(1.13

28.393+ =

)r+(1P+P = P 22

s

cs0

Stocks That Pay No Dividends

• If investors value dividends, how much is a stock that pays no dividends worth?

• A stock that will literally never pay dividends in any form has a value of zero.

• In actuality, a company that has not paid dividends to date can be worth a lot, if the company had good investment projects or if it has assets that can be liquidated.– McDonald’s started in the 1950’s but paid its first dividend in

1975. The market value of McDonald’s stock was in excess of $1 billion prior to 1975.

– Anyone familiar with Intel’s history?

Valuing Operations Instead of Dividends

• Stocks can be (and often are) valued based on earnings and/or operating cash flows instead of dividends. Let OCF denote operating cash flow (after taxes and after all working capital corrections).

• Let F denote the net cash flow to the firm from financings (new debt and equity issues less any debt repaid or equity repurchased).

• Let I denote net new capital investment taken by the firm.

• Then using the cash flow identity, dividends can be stated as

Dt = OCFt + Ft – It.

• So we can value the firm by discounting future operating cash flows, financing flows, and requisite capital investments instead of dividends.

• An “NPV” approach.

Valuing Operations cont…

• Let NPVGO represent the net present value of the firm’s future investments (growth options). This is the present value of the operating cash flows those investments will create less the present value of the capital outflows that will be required to develop them.

• Let NPVF represent the net present value of the firm’s future financing transactions. This is the present value of the proceeds from financings less the present value of the resulting obligations --- interest and principal for debt, dividend dilution for equity (a good starting point is NPVF=0: why?).

• Let PVA denote the present value of the future cash flows from the firms existing assets.

• Let PVL denote the present value of the future cash flows associated with the firm’s existing liabilities.– All should be stated on a per share basis if we want the price per share.

Valuing Operations cont…• The following valuation approach is equivalent to the discounted

dividend approach:

P0 = PVA - PVL + NPVGO + NPVF • Equivalent, even though it does not directly involve dividend

projections at all!• Observations regarding RWJ’s Chapter 5 discussion:

– They assume no future financings. (More generally, NPVF = 0 is probably a very good approximation).

– They assume no existing debt, so PVL = 0. – They assume that existing assets pay a perpetuity in the amount of EPS per

period. So, PVA = EPS/rs.

• So, with their special restrictions, we have:

P0 = EPS/rs + NPVGO.

XCORP Example

• Suppose that Xcorp’s current assets produce net cash flows of $1 million per year in perpetuity. The discount rate for Xcorp is 15%.

• What is the market value of Xcorp?0 321 4

......

1 million 1 million 1 million1 million

millionmillion

r

CFPVA

s

67.6$15.0

1$1

XCORP Example cont…

• Now suppose that Xcorp has an R&D project that will require cash infusions of $1 million in each of the next three years. Subsequently, the project will generate additional cash flow of $0.75 million per year in perpetuity. Xcorp’s net cash flow including the project is:

• What is the market value of Xcorp with the project?

0 321 4......

0 million 0 million 1.75 million0 million

XCORP Example cont…

• Xcorp’s cash flow can be divided up into two pieces:

• The cash flow from current assets:

• Plus the cash flows from the new project:

0 321 4......

1 million 1 million 1 million1 million

0 321 4......

-1 million -1 million 0.75 million-1 million

XCORP Example concluded

• The NPV of the project at time 0 is:

• Xcorp’s value with the project is:

million

NPVGO

004.1$

15.0

75.0

)15.1(

1

)15.1(

1

)15.1(

1

15.1

1332

million

NPVGOPVAP

671.7$004.1$667.6$0

The Discounted Dividend and NPV Approaches Are Equivalent

• Fresno Corporation has one asset and one growth opportunity. • The existing asset is a factory which generates operating cash

flow of $100,000 per year. This will continue for 10 years only, with no salvage value.

• The growth opportunity would require an investment of $2 million at t=2, and will return $350,000 to Fresno in each year from t=3 to t=10.

• The growth opportunity will be funded by selling $2 million in zero coupon bonds at t=2. The bonds will be repaid at t=10.

• Fresno currently has 100,000 equity shares outstanding.• For simplicity, we will assume that interest/discount rates are

zero.

Valuing Fresno’s Operations

• P0 = PVA - PVL + NPVGO + NPVF– PVL = 0 (No existing debt).

– NPVF = 0 (Fair terms on the future financing)

– PVA = $100,000x10 = $1,000,000 (no discounting)

– NPVGO = $350,000x8 - $2,000,000 = $800,000

• P0 = $1,000,000 + $800,000 = $1,800,000 in total, or $18 per share.

Valuing Fresno’s Dividends• Fresno pays $100,000 per year in regular dividends. At t = 10 Fresno

will pay off its debt and pay out all remaining cash as a liquidating dividend.

• We need to determine the size of the liquidating dividend, and obtain the present value (no discounting) of the dividend stream.Year BofY Cash OCF I F D EofYCash

1 0 100 0 0 100 02 0 100 2000 2000 100 03 0 450 0 0 100 3504 350 450 0 0 100 7005 700 450 0 0 100 10506 1050 450 0 0 100 14007 1400 450 0 0 100 17508 1750 450 0 0 100 21009 2100 450 0 0 100 245010 2450 450 0 -2000 900 0

Valuing Fresno’s Dividends cont…

• Fresno will pay 9 dividends of $100,000 each, plus a liquidating dividend of $900,000.

• With a zero discount rate the present value of the dividend stream is $1,800,000, or $18 per share.

• Does the fact that the discounted dividend approach gives the same answer as the present value of operations approach rely on the zero discount rate assumption?

• So, use whichever method is easier to implement!

Price – Earnings (P/E) Ratios

• The investment community relies heavily on P/E ratios.– P/E’s are one of the items reported for every NYSE and

NASDAQ stock in daily newspapers.– Some analysts do simple valuation by obtaining the

product of earnings per share and the “P/E multiplier”

• Using the RWJ framework,

P0 = EPS/rs +NPVGO,so the ratio of price to earnings per share is:

P0/EPS = 1/rs + NPVGO/EPS

P/E Ratios – Example• If NPVGO = 0 and r = .15, then P/E = 1/.15 = 6.67

• If NPVGO = 0 and r = .08, then P/E = 1/.08 = 12.50.

• If the Net Present Value of Future Investments is five times as large as current EPS and r = .08, then P/E = 1/.08 + 5 = 17.50

• So, high P/E’s require either low discount rates or lots of good future investments (relative to current earnings), i.e. earnings growth. Note, though, that earnings growth obtained through negative NPV investment won’t help. P/E ratios should not help anyone “pick stocks.”

• Is the current P/E ratio in U.S. markets high or low by historical standards?

Residual Income Valuation and “EVA”

• A long overlooked, but recently rediscovered, approach to valuation is the “residual income” approach.– Let BVt denote the book value of equity (per share) at time t.

– Let NIt denote the net income per share for the interval ending at time t.

– Let DIVt denote the dividend per share at time t.

• The “clean surplus” accounting relation is:

BVt = BVt-1 + NIt – DIVt, or

DIVt = NIt – BVt .

• Does this always hold?

Residual Income and EVA cont…

• Substitute the “clean surplus” relation into the “share price is the present value of all future dividends” expression and rearrange it:

– Where ROE is return on equity, Nit/Bt-1.– Where rs is the discount rate for equity, (Bt – Bt-1)/Bt-1.

• Why would they call this a “residual income” approach?

1

100 )1(

)(

tt

s

tst

r

BrROEBP

Residual Income – Example

• Jensen company currently has a book value per share of $20. It earns a return on equity of 15%, and its cost of equity capital (discount rate) is 10%. For simplicity, assume that book value and ROE will remain constant forever. (This is solely so we can compute present values based on level perpetuities.)

– The key to pushing share price above book value is…

30$10.0

120

10.0

20)10.015.0(200

P

EVA Analysis

• The residual income approach underlies “economic value added” techniques that are currently popular.

• EVA = Income – Opportunity Cost of Invested Capital.• Which is just “residual income.”• In the previous example, EVA is:

.15(20) - .10(20) = 3 – 2 = $1(per share, per period)

Valuation Techniques: Summary

• Financial assets (and some real assets) can be valued by discounted cash flow techniques: compute the present value of all expected future cash flows to be given off by the asset.

• For bonds, this is mainly a matter of time value mechanics and the selection of the appropriate discount rate.

• For stocks, DCF techniques can be implemented either by discounting the forecasted dividend stream, or by discounting future flows to equity. The important issues are the firm’s inherent ability to generate cash flows and the riskiness of these cash flows.