investments - lecture notes valuation 2013-08

Ch 17 ‐ Bond Yields and Prices ‐ Jones

Ch 10 ‐ Bond Prices and Yields ‐ BKM

Bonds are when a corporation or government borrows money, paying interest in the form of a "coupon,"

and at the end of the prescribed life of the bond, the borrower must pay back the "par value" or "face

value" of the bond. Conveniently, "face value" can be abbreviated as FV, and this corresponds with

the FV part of our calculations below.

Much of this will be basic review, but terms you should know:

FACE VALUE (a.k.a. "PAR")

COUPON RATE

MATURITY DATE

NOTE: These and other optional features are detailed in the contract between the

issuer (company, government, etc) and the holder of the bond which is called an

INDENTURE

YIELDS YIELD TO MATURITY The 'i' on your calculator, also the "discount rate"

YIELD TO CALL The 'i' with 'N' changed to represent time to "call"

... plus you add a "call premium" to the FV

CURRENT YIELD = Coupon Rate / Current Price of the Bond

NOTE that NONE of the yields are specified in the indenture.

BOND TYPES:

ZERO

STRIPS

CALLABLE

CONVERTIBLE

PUTTABLE

FLOATING RATE

Additional terms and calculations:

What's the difference between a bond and preferred stock?

Nominal return vs. Real return ... (1+Nominal return) / (1+Inflation) ‐ 1

Flat or Clean price of a bond vs. the Invoice or "Dirty" price of a bond

17 Bonds 48

BOND PRICING (review)

Using the notation you should understand well by now, a sample bond might look like this:

A 5 year bond, paying a 10% coupon once/yr, priced at $1000 when issued, $1000 face value

The yield (i) is also 10% in this case (it may differ from the coupon, we'll get into this soon).

Remember:

The coupon rate is not the yield, the yield is not the coupon rate. They may be the

same (as this example shows), but the yield is our DISCOUNT rate. The coupon is

what determines how much the bond is paying as a % of face value.

N i PV PMT FV

5 10.0% ‐1000 100 1000

= yield or

discount rate 100.00$ 100.00$ 100.00$ 100.00$ 1,100.00$

0 1 2 3 4 5

($1,000.00)

What is different here is that the final cash flow is not equal to our previous payments because it includes

the $1000 par value of the bond that is due the bond holder at maturity. An adjustment needs to be made

in how we enter and represent the cash flow components in the problem.

All we need to do is separate CF5 into its two parts:

The face value of the bond can be treated as a lump sum, 1,000.00$

and this is a simple annuity. 100.00$ 100.00$ 100.00$ 100.00$ 100.00$

0 1 2 3 4 5

($1,000.00)

Therefore, the PV of a bond is the PV of its annuity cash flows PLUS the PV of its face value.

1

PMT * [ ( 1 + i )^N ] FV

i ( 1 + i )^N

FV * PV factor ( i, N )

PV of the lump sum

+=PVbond

PMT * Annuity factor( i, N )

PV of the annuity portion

1 ‐

17 Bonds 49

Your financial calculator and Excel make this very easy to calculate, but you must make sure you enter the

components of the calculation correctly.

A common mistake here is to enter $1100 for the FV.

N i PV PMT FV Remember that these CFs need separate discounting.

5 8.0% ? 100 1000

1,000.00$

100.00$ 100.00$ 100.00$ 100.00$ 100.00$

0 1 2 3 4 5

($1,079.85)

N i PV PMT FV Remove the FV and this calculates the PV of the annuity

5 8.0% ? 100 0

‐$

100.00$ 100.00$ 100.00$ 100.00$ 100.00$

0 1 2 3 4 5

($399.27)

N i PV PMT FV Remove the FV and this calculates the PV of the annuity

5 8.0% ? 0 1000

1,000.00$

‐$ ‐$ ‐$ ‐$ ‐$

0 1 2 3 4 5

($680.58)

100.00 x ( 1.080 ^ 5.0 )

8.0% 1.080 ^ 5.0

1

100.00 x ( 1.469 ) 1000

8.0% 1.469

100.00 x ( )

8.0%

100.00 x ( )

8.0%

100.00 x ( 3.993 ) + 680.58 NOTE: This is the "PV Factor of an Annuity"

PVbond 1,079.85 = 399.27 + 680.58

1 ‐

+1000

1

1 ‐

+

1 ‐

+

0.681

680.58

0.319

+ 680.58

17 Bonds 50

More common mistakes to avoid

First, make sure you know the difference between a regular annuity and an annuity due.

See the last page of this section. Know how to toggle between "begin" and "end" mode on

your calculator.

Many bonds will be described as having annual payments (an annual coupon), but most will have

2 or more payments per year.

You need to adjust the

PMT (derived from the Coupon Rate)

RATE or "i"

... and the NUMBER of PERIOD or "N"

It is possible that "i" equals the coupon rate when the bond is issued, but it will rarely be equal to that

value after the bond is issued.

The coupon rate is the percentage of the face value of the bond that is paid out annually.

Therefore, the coupon rate = Payment / Face Value of the bond

e.g. $100 / $1000

10% coupon

But the coupons are frequently paid semi‐annually or possibly quarterly, therefore the annual coupon

rate could be 10%, but when paid semi‐annually, the coupon becomes

$50 per PMT, paid twice per year

10 year, semiannual bond, 10.00% coupon, 9.00% YTM

Par is $1,000.00

How do you set this up and what's the price?

N i PV PMT FV

20 4.50% ? $50.00 $1,000.00

‐1065.04

10 year, semiannual bond, 7.00% coupon, $900.00 current price

Par is $1,000.00 What is the YTM?

N i PV PMT FV

20 ? ‐900.00 $35.00 $1,000.00

4.25%

8.50% = answer … semi‐annual PMTs

mean the yield calculated is only

semi‐annual, therefore, multiply *2

17 Bonds 51

7.5 year, semiannual bond, 12.50% coupon, $917.25 current price

Par is $1,000.00 What is the YTM?

N i PV PMT FV

15 ? ‐917.25 $62.50 $1,000.00

7.17%

14.34% = answer … remember "*2"

6 year, semiannual bond, 9.50% YTM $1,157.30 current price

Par is $1,000.00

N i PV PMT FV

12 0.0475 ‐1157.30 ? $1,000.00

$65.00

Annual Payment $130.00

Coupon Rate 13.00%

You hold a zero coupon bond when there is a sudden change in interest rates overnight.

5 years to maturity, YTM is 8.00% , and rates change by RISING

1.00% overnight. What happens to the value of your bond (in % and $ terms)?

Note, zero coupon bonds assume semi‐annual compounding. 2

N i PV PMT FV

10 4.00% ‐$675.56 0 1000

10 4.50% ‐$643.93 0 1000

‐$31.64 ‐4.68%

You hold an annual coupon bond for 1 year, receiving the 8.00% coupon just before selling.

When purchased it has 8 years to maturity, and the YTM is 10.50% . Over the

year, interest rates have FALLEN by ‐0.50%

What is the HPR for this investment?

N i PV PMT FV

8 10.50% ‐$869.02 80 1000

7 10.00% ‐$902.63 80 1000

Capital gain on the sale of the bond $33.61 3.87%

(is that it? are we done?)

33.61 + 80.00

= (Cap gain + One PMT) / PV0

= (P1 ‐ P0 + Income) / P0

13.07%HPR869.02

=

17 Bonds 52

YIELD CURVE and the TERM STRUCTURE OF INTEREST RATES

Expectations Hypothesis

Yields to maturity are determined by expectations of future short term rates

These also include expectations about inflation.

FORWARD RATES

If we know a multi‐year bond rate, we can extract a "forward" rate that is inferred

by the multi‐year rate.

If you "know" the 1 year rate of 8%, you can solve for the second rate if you're given

only the two year rate of 8.995% ...

2 year ret. 1.08995^2

1 year ret. 1.0800= 1.1000

the "forward"

rate for Year 2=

17 Bonds 53

Use the following zero coupon bonds and their yields to maturity to determine:

The one‐year interest rate starting one year from now.

Bond Years to Maturity Yield to Maturity

a 1 7.25%

b 2 7.75% 1.08252331 8.252%

c 3 8.25% 1.09256971 9.257%

d 4 8.50% 1.0925347 9.253%

e 5 8.75% 1.09755774 9.756%

Isolate the one year rate by:

1.0775 ^2 1.1610

1.0725 ^1 1.0725

Forward rate ONE YEAR from now 1.0825233 ‐ 1 = 0.08252331

8.2523%

LIQUIDITY PREFERENCE ‐‐ LIQUIDITY PREMIUM

= = 1.08252331

17 Bonds 54

A Regular Annuity vs. an Annuity Due:

A regular annuity has the cash flows structured at the END of the periods. An Annuity Due

has the cash flows at the BEGINNING of the periods. This significantly affects the value of

CFs.

example: N PMT i

5 $100 8%

0 1 2 3 4 5

Reg Ann. 100$ 100$ 100$ 100$ 100$

$399.27 =PV of the regular annuity

Ann. Due 100$ 100$ 100$ 100$ 100$

$431.21 =PV of the annuity due

(use the annuity due or "BEGIN" method on your calculator)

IMPORTANT:

THIS is why you must make sure your calculator doesn't get set to "BEGIN" mode by

accident. The default setting, and the one you'll use almost all the time, is to make your

Time Value of Money (TVM) calculations in "END" mode ‐‐ where the CFs are made or

arrive at the "end" of the periods.

TI BA II +: To toggle between END mode and BGN mode (and back)

2ND ‐ PMT (BGN); 2ND ‐ ENTER (SET); 2ND ‐ CPT (QUIT)

HP 12c: To enter BEG (Begin) mode "g" ‐ 7 ("BEG")

To return to END mode "g" ‐ 8 ("END")

Note that calculators usually only display when you're in "Begin" mode

(the non‐default setting)

17 Bonds 55

PV of a Growing (or "Graduated") Annuity

Starting with our PV formula for annuities, we would like to account for annuities where the CF

grows over time.

1

PVA = PMT1 * [ ( 1 + i )^N ]

i

Doing so is simple, and we need only add growth ("g") to both our discount factor and the denominator.

( 1 + g )

PVGA = PMT1 * [ ( 1 + i ) ]

( i ‐ g )

This is an annuity we valued earlier: a $100 cash flow received for 5 years with an 8% discount rate.

We can discount the individual cash flows using the PV FACTORS. We can use the PV formula in Excel

or the formula above and all return the same PV for the annuity.

Initial

example: N Cash Flow i g

5 $100 8% 3%

PV Factors 0 1 2 3 4 5

PV 8% 1.00 0.926 0.857 0.794 0.735 0.681

No Growth CFs ‐$ $100.00 $100.00 $100.00 $100.00 $100.00

The discounted PMTs ‐$ 92.59$ 85.73$ 79.38$ 73.50$ 68.06$

PVA = 399.27$ (sum of discounted CFs)

$399.27 =PV( 8%, 5, 100)

399.271 = 100 * ( 1 ‐ 1/((1+.08)^5))/.08

The annuity below is identical to the above except we are growing the CF by 3% per year. The same

valuation techniques can apply, but note that Excel and your typical financial calculators do not

allow you to plug in a "g" for the growth rate.

You will need to either individually discount the CFs by the PV factor OR use the PV formula for a

growing annuity.

PV Factors 0 1 2 3 4 5

PV 8% 1.00 0.926 0.857 0.794 0.735 0.681

Growing CFs ‐$ $100.00 $103.00 $106.09 $109.27 $112.55

The discounted PMTs ‐$ 92.59$ 88.31$ 84.22$ 80.32$ 76.60$

PVGA = 422.04$ (sum of discounted CFs)

n/a Excel does not provide a "growth" PV formula

422.04 = 100 * ( 1 ‐ (( 1 + .03 ) / ( 1 + .08 ))^5) / (.08 ‐ .03)

1 ‐

1 ‐ ( )^N

17 Bonds 56

Important:

NOTE that the growth annuity formula is the same as the regular annuity formula but it accounts

for growth. If growth (g) is zero, the annuity has the same value as the "no growth" PV annuity.

if g=0 : 399.271004 = 100 * ( 1 ‐ (( 1 + 0.00 ) / ( 1 + 0.08 ))^5) / (0.08 ‐ 0.00)

Initial

example: N Cash Flow i g

5 $100 8.0% 3.0%

( 1 + g )

PVGA = PMT1 * [ ( 1 + i ) )

( i ‐ g )

1.030

100.00 x ( 1.080 )

0.080 ‐0.030

100.00 x ( )

0.050

PVGA = 100.00 x = 100.00 x 4.2204

= 422.04

1 ‐ 0.78898

0.050

1 ‐ ( )^N

1 ‐ ( ) ^ 5.00.954

1 ‐ ( ) ^ 5.0

17 Bonds 57

Ch 18 ‐ Bonds: Analysis and Strategy ‐ Jones

Ch 11 ‐ Managing Bond Portfolios ‐ BKM

INTEREST RATE RISK

Interest Rate Sensitivity

i. Bond prices and yield are inversely related

ii. An increase in YTM results in a smaller Price than an equal decrease in YTMiii. Prices of LT bonds tend to be more sensitive to interest rate changes than ST bonds

iv. Sensitivity of bond prices to yield increases at a declining rate as maturity increases

v. Interest rate risk is inversely related to the bond’s coupon rate.

vi. Price sensitivity is inversely related to current YTM

Change in bond price as a function of change in yield to maturity

DURATION

What is duration?

Frederick Macaulay: the "effective maturity of a bond"

weighted average of the times until each payment with weights proportional

to the PV of the payment

‐ weighted average

‐ times until each payment

‐ with weights proportional to

‐ the PV of the payment

Unofficial definition: The time it takes to ____________________

GET DA' MONEY

...in ___________________ terms.

PRESENT VALUE

It is used to measure a bond's ____________ to changes in _________________

SENSITIVITY INTEREST RATES.

18 Managing Bonds 58

To calculate duration, start with a sample bond …

N i Coupon

10 10% 10%

2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019

0 1 2 3 4 5 6 7 8 9 10 PV of CFs

First, write out the cash flows for our bond ‐‐‐ and don't forget the PMT + the FV in the last period100$ 100$ 100$ 100$ 100$ 100$ 100$ 100$ 100$ 1,100$ $1,000

Then create a table of present value (discount) factors, i.e. 1/(1+r)^(t) or 1*(1+r)^(‐1)1.00 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 0.386

Calculate the PV of the cash flows from the bond91$ 83$ 75$ 68$ 62$ 56$ 51$ 47$ 42$ 424$ 1,000$

Determine the weights of the CFs *relative to* the PV of the bond9.1% 8.3% 7.5% 6.8% 6.2% 5.6% 5.1% 4.7% 4.2% 42.4% 100.0%

w(t) = ( CF(t)/(1 + y)^t ) / Price

Multiply the weights and the time (t) it takes to receive each cash flow, sum these values up,

and you have duration

D = sum of ( "t" * w(t) ) DURATION0.09 0.17 0.23 0.27 0.31 0.34 0.36 0.37 0.38 4.24 6.76

Note, this result is confirmed using the =Duration(...) function in excel 6.76

OK, there is a simpler looking version, especially if you have a "shorter" bond ...

N i Coupon FV

4 8% 10% $1,000

Time to

Payment

Cash

Flow *

PV factor

(PV of $1) PV of CF

Weight

(PVCF/

Sum of

PVCFs) t * W

1 $100 0.926 $92.59 0.087 0.087

2 $100 0.857 $85.73 0.080 0.161

3 $100 0.794 $79.38 0.074 0.223

4 $1,100 0.735 $808.53 0.758 3.033

Sum of the PV CFs 1,066.24 1.00 3.504 3.504


* Note that the Cash Flow (CF) is not just equal to the PMT. When doing this calculation, it also

includes the FV at the end.

N i Coupon FV

5 10% 5% $1,000

Time to

Payment

Cash

Flow

PV factor

(PV of

$1)** PV of CF

Weight

(PVCF/

Sum of

PVCFs) t * W

1 $50 0.909 $45.45 0.056 0.056

2 $50 0.826 $41.32 0.051 0.102

3 $50 0.751 $37.57 0.046 0.139

4 $50 0.683 $34.15 0.042 0.169

5 $1,050 0.621 $651.97 0.804 4.022

Sum of the PV CFs 810.46 4.488 4.488

** Note that you may not need to calculate the PV factor. Just calculate the PV of the CF.

N i Coupon FV

4 12% 8% $1,000

t CF PV factor PV of CF Weight t * W

1 $80 0.893 $71.43 0.081 0.081

2 $80 0.797 $63.78 0.073 0.145

3 $80 0.712 $56.94 0.065 0.194

4 $1,080 0.636 $686.36 0.781 3.125

Sum of the PV CFs 878.51 3.546 3.546 ***

*** You can check your result (for fun) using the excel formula =DURATION(DATE(2000,1,1),DATE(2000+N ,1,1),COUPON ,YIELD ,PMTs per year)

Also note, you can check your result using the TI BA II+ Pro, but the duration

you will get is 3.1661 , but this is the MODIFIED DURATION.

3.5461


D of a Perpetuity = ( 1 + y ) / y

From this we can use an alternate duration calculation in a single formula

assuming this is an annuity , i.e. note the process above can be used for

varying cash flows, for which there are needs and uses (as you'll practice).

Start with the duration of a perpetuity and then adjust for coupon

rate (c), yield (y) and time to maturity (N).

( 1 + y )

y

N y Coupon FV

4 12.0% 8.0% $1,000

1.120 1.120 + 4 ( .080 ‐ .120 )

0.12 0.080 ( 1.1204

‐ 1 ) + .120

1.120 + ‐0.160

0.080 x 0.574 + .120

0.960

0.166

D = 3.546

See the Duration Exercise spreadsheet under Course Documents

NOW THAT YOU CAN CALCULATE DURATION, HOW CAN YOU USE IT?

1) The main use is to calculate the change in price movements ...

P / P = -D x [ ( (1+y) / (1+y) ]

D* = D / (1+y) = "Modified" DurationThe modification is the adjustment for yield,

and it allows for an easier to use formula for Δ P

P / P = -D* x [ ( (1+y) ] = -D*y

2) We can also use it to PASSIVELY MANAGE bond portfolios

3) There is another calculation related to duration called CONVEXITY.

Convexity is generally desirable because ΔP is higher when yields fall than when

yields rise

P / P = -D* y + ½ x Convexity x (y)2

-D = [ ( 1 + y ) + N ( c - y) ]

c [ ( 1 + y )N - 1 ] + y

D = ‐

5.79=

‐9.33

9.33 ‐ 9.33 ‐


PASSIVE BOND MANAGEMENT

IMMUNIZATION: A strategy to shield assets from interest rate movements

If we match the interest rate exposure of assets and liabilities ‐‐

whether rates rise of fall, we have limited/removed

____________ and ____________________

PRICE RISK REINVESTMENT RISK

Example:

You are a pension fund manager who has obligations over the next 4 years oft CF PV of CF Weight t * W

1 5 million $4.46 0.416 0.416 Sum PVCFs

2 2 $1.59 0.149 0.297 $10.74 Million

3 3 $2.14 0.199 0.597 DURATION

4 4 $2.54 0.237 0.947 2.257

You are not able to easily match the CFs of your

assets to those of your liabilities (the obligations).

How would you immunize your portfolio if interest rates are 12%

ACTIVE BOND MANAGEMENT

Forecast interest rate movementsIf anticipate declining interest rates – increase portfolio duration, and vice versaGenerate abnormal returns only if information or insight is superior to the rest of the market

CONVEXITY

DURATION creates a linear estimate of the potential price changes.

Actual price change is curved.


The graph below ("Bond Value vs Yield") shows

this across a wide range of prices and yields. In

reality, changes in yield tend to be smaller, and

duration is relatively effective at estimating

price changes due to small changes in yield.

Mispricing error due to convexity is less

noticeable here when we graph a smaller

range for the change in yield based on the

previously used data0

500

1,000

1,500

2,000

2,500

0 0.05 0.1 0.15

Bond Value

Yield to Maturity

Bond Value vs Yield

680

700

720

740

760

780

800

0.1025 0.1075 0.1125 0.1175 0.1225 0.1275


Ch 10 ‐ Common Stock Valuation

Ch 13 ‐ Equity Valuation

We use fundamental analysis to identify stocks that are mispriced relative to some

measure of “true” value

Book value

Liquidation value

Replacement cost – Tobin’s Q (market value to replacement cost)

Intrinsic value vs. market price

Expected HPR should equal required return

Intrinsic value – [ E(D1) + E(P1) ] / ( 1 + k ) … seeking +/‐ alpha

MEASURING / ESTIMATING INTRINSIC VALUE

An investment should be valued as the present value of its expected cash flows.

This is easier to determine for fixed or steady cash flows, but we can apply similar

techniques to value the cash flows of common stock.

N

VCS = Ʃ CFN / ( 1 + i )N

t=1

Above is the same formula we applied to valuing the PV of an annuity, but the annuity

cash flows are fixed. Here we assume they can vary, but the concept is the same.

We can discount each individual cash flow to current PV dollars, or we can apply some

of the formulas below to the stocks we want to value.

What varies from what we've examined before is:

The timing of the cash flows

For example, do we sell our stock in one year vs. ten years?

Does this make a difference in the present value?

The potential growth of the cash flows

10 Equities 64

First: What is a stock worth today that you expect to hold for only a year or two?

We start with this to emphasize the typical CFs from a stock ‐‐‐

1) the dividends a stock pays

2) the price for which we expect to sell the stock at the end of

our holding period

i = 12% 0 1 2 3 4 5

PV factor 1.000 0.893 0.797 0.712 0.636 0.567

We will discount the CFs below "manually" using the PV factors above

Let's say we have a stock that we expect to be able to sell for 20.00$

in the future. It pays 2.40$ annual dividends at the end of the year.

Note that the price of the stock today has nothing to do with how long we

hold the stock ‐ assuming our dividend yield is equal to our discount rate.

CFs PV of a share today that we hold one year (receive dividend, then sell stock)

Dividends 2.40$

Selling price 20.00$

VCS = 20.00$ 20.00$ ‐$ ‐$ ‐$ ‐$

CFs

Dividends 2.40$ 2.40$ 2.40$


VCS = 20.00$ 2.14$ 1.91$ 15.94$ ‐$ ‐$

CFs

Dividends 2.40$ 2.40$ 2.40$ 2.40$ 2.40$


VCS = 20.00$ 2.14$ 1.91$ 1.71$ 1.53$ 12.71$

Note that depending on the expected CFs from a stock and the discount rate,

the holding period may have zero impact on the PV

10 Equities 65

The Dividend Discount Model

The "DDM" uses the dividends as sole source of the cash flows on which we

will base our valuation of the company's stock.

If we assume no growth to the dividends, the stream of cash flows is similar

to a PERPETUITY, and the model we use to value them is the same.

No Growth DDM:

CF D

r i

Constant Growth DDM (often called the "Gordon model"):

CF D0(1+g) D1

r ‐ g i ‐ g i ‐ g

Note the lack of any notation used for the "Div" or dividend in the "No

Growth" model. Because the Dividends in a no growth model, by definition,

do not grow. They do not change. Div0 is equal to Div1, and it's equal to Div3

and Div15 for that matter.

In the Growth DDM, the cash flows, the dividends, are assumed to grow at

the rate equal to "g." Therefore we use "g" to calculate Div1 and this goes

into a the calculation of our value of common stock ‐‐ VCS.

Free Cash Flow Models

Larger, mature companies tend to pay dividends, but many large firms in a

modern economy choose to pay lower or no dividends to fund future growth.

Therefore, models can be used to estimate what a firm could pay out in dividends

FCFE1

k ‐ g

Free Cash Flow to Equity

FCFE = Net Income + Depreciation ‐ Debt Repayments ‐ Capital Exp.

‐ Change in Working Capital + New Debt Issues

V0 =Expected FCFE

k ‐ g=

VCS = =

VCS = = =

10 Equities 66

Two‐Stage growth

Many companies, especially younger, fast growing firms, need to be considered

as having not one growth rate, but two (or more).

For this model, we assume a company has a rapid period of growth, and it then

levels off after the company matures and becomes larger.

We refer to these two growth rates merely as:

g 1 : The initial, faster period of growth, e.g. from now to 3

or maybe even 10 years out

g 2 : The second stage of lower, constant growth

The formula for doing this will look complex, but by breaking the stream of

cash flows into two parts, it takes the shape of

1) An annuity of CFs of the growth period "N"

2) The PV of the stream of dividends after the growth

period "N"

Therefore the value of our common stock looks like this:

( 1 + g1 )

D0 (1+g1)* [ ( 1 + i ) ] +

( i ‐ g1 )

( 1 + g1 )

D0 (1+g1)* [ ( 1 + i ) ] +

( i ‐ g1 )

Three‐Stage Growth Model ‐‐

The two‐stage growth model assumes an instantaneous switch from the high

growth rate to the lower rate, e.g. growing at 20% for 5 years and then

suddenly dropping to 5% thereafter.

A three‐stage model assumes a period in the middle where the growth declines

linearly from the high‐growth period to the low‐growth period.

VCS = PVGA + PV0 ( PVGrowthPerp @ N )

1 ‐ ( )^N DN+1

( i ‐ g2 )

( 1 + i )N

( 1 + i )N

1 ‐ ( )^N D0 (1+g1)

N (1+g2)

( i ‐ g2 )

10 Equities 67

This is more realistic and is represented by the equation below:

D0 (N1 + N2)

i ‐ g2 2

You may skip this, but note there are many versions of this model. It is primarily used

to capture a more realistic decline in growth, but some models also capture changing

dividend payout rates.

The Earnings and/or PVGO Model ‐‐

Thus far, we assume a given growth rate (or rates) as given, but we know from

the DuPont deconstruction of ROE that management has significant control

over the firm and how that relates to returns.

First, a firm must invest in future growth. It does not magically appear, and

this growth is a function of two factors: the amount of investment in future

growth, and the ability of management to capitalize on that investment

effectively.

We therefore get our "g" from the retained earnings and our ROE, specifically

it is

g = b * ROE

growth rate = Retention ratio * Return on Equity

For example: If a firm has an ROE of 15%, we would expect it to be able to

grow at a rate of g as follows:

g = b * ROE

Growth

rate

Retention

% ROE %

0.0% 0.0% 15.0%

3.8% 25.0% 15.0%

7.5% 50.0% 15.0%

11.3% 75.0% 15.0%

15.0% 100.0% 15.0%

VCS = * ( ( 1 + g2 ) + * ( g1 ‐ g2) )

10 Equities 68

This should make it clear that if a firm pays out all it's earnings (e.g. EPS) as

dividends (D), then there are no fund to invest in growth, therefore "g" is

zero, and we can value the firm with the same formula we use for a

perpetuity (above) and our No Growth Dividend Discount Model (DDM).

No Growth DDM:

CF D EPS1

r i k

See that we can use "r", "i", and "k" as notation for the same thing

… the required rate of return, the discount rate, the interest rate,

the WACC, etc.

However, if we're going to invest in growth, we won't be paying 100% of our

earnings as dividends, therefore we separate the value of the firm into two

pieces, the value from the expected cash flows, and the value of the growth

opportunities (the PVGO) we create with our new investment using the

retained earnings.

EPS1

k

We solve for the PVGO as

NPV1

k ‐ g

Note that because the valuation models contain various assumptions or rely to varying degrees

on historical data or ratios, the INTRINSIC VALUES they produce can vary significantly.

Example: Honda Motor Co. 2008

VCS = = =

RE1 * ( ROE/k ‐ 1 )

k ‐ g=

VCS = + PVGO

PVGO =

10 Equities 69

PRICE TO EARNINGS RATIOS

‐‐ This will be covered during our discussion of RATIOS in "Company Analysis" (Ch15 ‐ Jones)

P/E ratios can contain a great deal of information about a company, but we

shouldn't draw conclusions that are too broad from this data point alone.

High P/E companies are usually associated with higher growth

Understand the difference between

P/E TTM and P/E Forward

Low P/E firms, therefore are often lower growth, BUT this does not mean

they are low risk.

A LOW P/E could be the result of a high "k" ...

P0 1 ‐ 40%

E1 12% ‐ 6%

= 10.0

Higher risk stocks should have a higher return, therefore the "k" is higher.

P/E

g

A common rule of thumb is that a fairly priced stock will have a P/E ratio close to

its growth rate. This would mean the PEG ratio should be about 1.0 for a fairly

priced stock.

Where the P/E ratio is less than the growth rate OR the PEG < 1.0, the firm may

be a bargain. (Source: Peter Lynch, One Up on Wall Street)

In reality, PEG ratios typically fluctuate between 1.0 and 1.5.

PEG Ratio =

=( 1 ‐ b )

k ‐ g=

10 Equities 70

We can see why higher P/E ratios indicate ample growth opportunities because...

... we can express the forward P/E ratio as

P0 1 PVGO

E1 k E1 / k

And we know that the price of a stock can be expressed as

D1

k‐ g

D1 equals the amount of earnings we are not retaining, or …

D1 = E1 ( 1 ‐ b )

and since growth is a function of ROE and the retained earnings

as g = ROE * b, the latter being the plowback or retention ratio

D1

k‐ g

therefore … P0 ( 1 ‐ b )

E1 k ‐ g

THIS is the P/E ratio for a firm growing a long‐run sustainable pace.

Ex: Price/Share $12 Has a forward P/E of 12.0

EPS $1

The firm's required rate of return 12%

Return on Equity is 15%

The firm pays out earnings at a 40%

Therefore the firm's sustainable P/E ratio would be 13.3

P0

E1 ( 12% ‐ 15% x 60% )

40%

3%

1 ‐ 60%

k ‐ (ROE * b )

=

( 1 ‐ b )

k ‐ (ROE x b )

=( 1 ‐ b )

=

P0

P0

==

= =E1 ( 1 ‐ b )

k ‐ (ROE x b )

= + [ ]

=

10 Equities 71

What are the implications of a firm having a sustainable P/E ratio less than

its current P/E ratio?

What about a negative value for a sustainable P/E ratio? What does this

indicate?

Pitfalls of P/E ratios

Earnings are "accounting earnings."

Analysts focus on "earnings quality" at times to account for inflation,

one time charges, etc.

Earnings can be "managed"

Other ratios ‐ Comparative Valuation

Price to Book

Price to Cash Flow

Price to Sales

10 Equities 72

Ch 15 ‐ Company Analysis ‐ Jones

Ch 14 ‐ Financial Statement Analysis & Ratios ‐ BKM

This chapter reviews a number of accounting measures and ratios we use in fundamental

analysis. I won't be spending much time reviewing the basics.

It would be worthwhile to review all the ratios covered in this chapter if you are not

very familiar with them. Having the formulas in your notes for the exam would also be

wise.

From the section on Company Stock Valuation, note the discussion RE: P/E ratios and

the PEG ratio. This topic tends to cross into both "chapters" of each text.

In addition to the usual ratios and P/E, we will focus on Return on Equity (ROE), how

managerial decisions can directly affect it, and how the resulting ROE directly affects firm value.

15 Company Analysis ‐ Ratios 73

The DuPont System ‐ Deconstructing Return on Equity

ROE Net Income Measure of Management Effectiveness

Equity Also: Return on Book Equity, Return on Net Worth

Is ROE the same as ROA? How does it differ?

ROE Net Income = Net Income x Assets

Equity Equity Assets

= Net Income x Assets

Assets Equity

ROA Equity Multiplier

ROE Net Income x Sales x Assets

Equity Sales Assets


Sales Assets Equity

Profit Margin Total Asset Turnover Equity Multiplier

(Sales Multiple) (Leverage Ratio)

1.60$ x 32.00$ x 16.00$

32.00$ 16.00$ 10.00$

5.0% 200% 1.60

ROE = NI / Equity 1.60$ 16.0% Debt to Equity

= ROA * Eq. Multiplier 10.00$ 60%

ROA 1.60$ 10.0%

ROA * = NI / Assets 16.00$

* Note that some authors/analysts suggest using Operating Income instead of Net Income in this calculation


Note how the various decisions available to management in running the firm lead to

changes in ROE, and this leads directly to changes in firm value.

EPS1 2.15 Div1 RE1

Payout Ratio 45.0% 0.97$ 1.18$

ROE 17.0%

Required Return 13.0%

Growth Rate (g) 9.35%

Note that if we use the

Value without Growth 16.54 Constant Growth DDM

Value of Growth Oppt's 9.97 the value is ...

Value of Stock 26.51 26.51

ROE

k

k ‐ g

ROE Value10% 12.90

11% 13.92

12% 15.12

13% 16.54

14% 18.25

15% 20.37

16% 23.04

17% 26.51

18% 31.21

19% 37.94

20% 48.38

‐ 1 )VCS =

EPS1+

k

RE1 * (

‐

10.00

20.00

30.00

40.00

50.00

60.00

10% 12% 14% 16% 18% 20%

Stock Value

Return on Equity

Stock Value as ROE Rises






ROE Net Income = Net Income x Assets

Equity Equity Assets

= Net Income x Assets

Assets Equity



Equity Sales Assets


Sales Assets Equity

Profit Margin Total Asset Turnover Equity Multiplier

(net) (Sales Multiple) (Leverage Ratio)

1.60$ x 32.00$ x 16.00$

32.00$ 16.00$ 10.00$

5.0% 200% 1.60

ROE = NI / Equity 1.60$ 16.0% Debt to Equity

= ROA * Eq. Multiplier 10.00$ 60%

ROA 1.60$ 10.0%

ROA * = NI / Assets 16.00$

* Note that some authors/analysts suggest using Operating Income instead of Net Income in this calculation

76





ROE Net Income = Net Income x Assets = Net Income x Assets

Equity Equity Assets Assets Equity


ROE Net Income x Pretax Profit x EBIT x Sales x Assets

Equity Pretax Profit EBIT Sales Assets

ROE Net Income x Pretax Profit x EBIT x Sales x Assets

Pretax Profit EBIT Sales Assets Equity

NI x ( EBIT ‐ Int. Exp. ) x EBIT x Sales x Assets

( EBIT ‐ Int. Exp. ) EBIT Sales Assets Equity

Tax Burden Interest Burden Margin Asset Turnover Leverage

(share after taxes) (share after interest pmts) (gross, not net) (Sales Multiple) (Equity Multiplier)

1.33$ x 2.30$ x 3.84$ x 32.00$ x 16.00$

2.30$ 3.84$ 32.00$ 16.00$ 10.00$

value used for ROE 0.58 x 0.60 x 0.12 x 2.00 x 1.60

is actually the amount "kept" for both values, not really the "burden"

42.2% 40% 12% 2.0 x 1.6 x

Net Tax % Net Int. % of EBIT or 200% or 160%

ROE 1.33$ 13.3% 0.133 Debt to Equity

10.00$ ROE = NI / Equity = ROA * Eq. Multiplier 60%

ROA 1.33$ 8.3%

16.00$ ROA = NI / Assets

formulas

DuPont 5‐way 77

investments - lecture notes valuation 2013-08

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