fin 819: lecture 31 valuation of common stocks and bonds how to apply the pv concept

FIN 819: lecture 3 1

Valuation of Common Stocks and Bonds

How to apply the PV concept


Today’s plan

Review what we have learned in the last lecture

Valuing stocks• Some terms about stocks

• Valuing stocks using dividends

• Valuing stocks using earnings

• Valuing stocks using free cash flows


Today’s plan (Continue)

Bond valuation and the term-structure of interest rates• Terminology about bonds

• The valuation of bonds

• The term structure of interest rates

• Use duration to measure the volatility of the bond price


What have we learned in the last lecture?

Payback rule• Shortcomings

IRR rule• Shortcomings

Free-cash flow calculation


Some specific questions in the calculation of cash flows

Include all incidental effects Do not forget working capital requirements Forget sunk costs Include opportunity costs Be careful about inflation Depreciation Financing


Free cash flows calculation

Free cash flows = cash flows from operations + cash flows from the change in working capital + cash flows from capital investment and disposal


Calculating cash flows from operations

Method 1• Cash flows from operations =revenue –cost

(cash expenses) – tax payment Methods 2

• Cash flows from operations = accounting profit + depreciation

Method 3• Cash flows from operations =(revenue –

cost)*(1-tax rate) + depreciation *tax rate


A summary example 2

Now we can apply what we have learned about how to calculate cash flows to the IM&C’s Guano Project (in the textbook), whose information is given in the following slide.


IM&C’s Guano Project

Revised projections ($1000s) reflecting inflation


IM&C’s Guano ProjectCash flow analysis ($1000s)


IM&C’s Guano Project

NPV using nominal cash flows

$3,519,000or 519,3

20.1

444,3

20.1

110,6

20.1

136,10

20.1

685,10

20.1

205,6

20.1

381,220.1

630,1600,12

76

5432

NPV


New formula

In chapter 4, it is argued that

FCF=earnings –net investment

Net investment = total investment - depreciation

Do you agree with this formula? Why?


Example

A project costs $2,000 and is expected to last 2 years, producing cash income of $1,500 and $500 respectively. The cost of the project can be depreciated at $1,000 per year. If the tax rate is 50%, what are the free cash flows?


One more question

Mr. Pool is now 40 years old and plans to invest some fraction of his current annual income of $40,000 in an account with an annual real interest rate of 5%, starting next year until he retires at the age of 70 to accumulate $500,000 in real terms. If the real growth rate of his income is 2%, what fraction of his income must be invested?


Some terms about stocks

Book Value – The value of the stocks according to the balance sheet.

Liquidation Value - Net proceeds that would be realized by selling the firm’s assets and paying off its creditors.

Market Value Balance Sheet - Financial statement that uses market value of assets and liabilities.


Some terms about stocks

Secondary Market - market in which already issued securities are traded by investors.

Dividend - Periodic cash distribution from the firm to the shareholders.

P/E Ratio - Price per share divided by earnings per share.

Dividend yield – Dividends per share over the price of per share


Example

IBM has a trading price of $70 per share. Its annual earnings per share is $5. Its annual dividend per share is $3.5. What are the P/E and the dividend yield?

P/E=70/5=14 Dividend yield=3.5/70 or 5%


Valuing Common Stocks using dividends (first approach)

Dividend Discount Model - Computation of today’s stock price which states that share value equals the present value of all expected future dividends plus the selling price of the stock.

H - Time horizon for your investment.

PDiv

r

Div

r

Div P

rH H

H01

12

21 1 1

( ) ( )

...( )


Example

George has bought one IBM share in the beginning of this year and decides to hold this share until next year. The expected dividend this year is $10 per share and the stock is expected to sell at $110 per share in the end of the year. If the cost of the capital is 10%, what is the current stock price?


Solution

P0=(110+10)/(1+0.1)=$109.1


Valuing common stocks using dividends

Example

Current forecasts are for XYZ Company to pay dividends of $3, $3.24, and $3.50 over the next three years, respectively. At the end of three years you anticipate selling your stock at a market price of $94.48. What is the price of the stock given a 12% expected return?


Solution

00.75$

)12.1(

48.9450.3

)12.1(

24.3

)12.1(

00.3

0

3210

P

P


Valuing common stocks using dividends

If we forecast no growth, and plan to hold out stock indefinitely, we will then value the stock as the PV of a PERPETUITY.

r

EPSor

r

DivPperpetuityPV 11

0)(

Assumes all earnings are paid to shareholders.


Example

Suppose that a stock is going to pay a dividend of $3 every year forever. If the discount rate is 10%, what is the stock price for the following cases:• (a) you invest and hold it forever?

• (b) you invest and hold it for two years?

• (c) you invest and hold it for 20 years?


Solution

(a) P0=3/0.1=$30 (b)P0=PV (annuity) + PV( the stock price at

year 2) = 3/1.1 + 3/1.12+(3/0.1)/1.12

= 3/0.1=$30

(c) P0=PV (annuity of 20 years) + PV (the stock price at the year of 20) =$30


Valuing Common Stocks

Gordon Growth Model: A version of the dividend growth model in which dividends grow at a constant rate (Gordon Growth Model).

Stocks can be valued as a perpetuity with a growth rate, if you want to hold this stock forever, that is

gr

DivP

1

0


Example

Suppose that a stock is going to pay a dividend of $3 next year. Dividends grow at a growth rate of 3%. If the discount rate is 10%, what is the stock price for the following cases:• (a) you buy and hold it forever?

• (b) you buy and hold it for two years?

• (c) you buy and hold it for 20 years?


Solution (a) P0=3/(0.1-0.03)=$42.86 (b)P0=PV (annuity) + PV( the stock price at

year 2) = 3/1.1 + 3*1.03/1.12+(3*1.032/(0.1-

0.03))/1.12

= 3/(0.1-0.03)=$42.86 (c) P0=PV (annuity of 20 years) + PV (the stock price at the year of 20) =$42.86


Capitalization rate

Expected Return - The percentage yield that an investor forecasts from a specific investment over a set period of time. Sometimes called the market capitalization rate.

Expected Return

rDiv P P

P1 1 0

0


Example

If Fledgling Electronics is selling for $100 per share today and is expected to sell for $110 one year from now, what is the expected return if the dividend one year from now is forecasted to be $5.00?


Solution

According to the formula,

15.100

1001105Return Expected


Capitalization rate

The formula for the capitalization rate can be broken into two parts.

Capital. Rate = Dividend Yield + Capital Appreciation

Expected Return

rDiv

P

P P

P1

0

1 0

0


Using dividends models to derive the capitalization rate

Capitalization Rate can be estimated using the perpetuity formula, given minor algebraic manipulation.

gP

Divr

gr

DivP

0

1

10


Valuing Common Stocks

Example-

If a stock is selling for $100 in the stock market, the cost of capital is 12% and the next year dividend is $3, what might the market be assuming about the growth in dividends?

$100$3.

.

.

00

12

09

g

g


Some terms about dividend growth rates

If a firm elects to pay a lower dividend, and reinvest the funds, the stock price may increase because future dividends may be higher.

Payout Ratio - Fraction of earnings paid out as dividends

Plowback Ratio - Fraction of earnings retained by the firm.


Deriving the dividend growth rate g

Growth can be derived from applying the return on equity to the percentage of earnings plowed back into operations.

g = return on equity X plowback ratio

SharePer Equity Book

EPS

Equityon Return

ROE

ROE


Example

Our company forecasts to pay a $5.00 dividend next year, which represents 100% of its earnings. This will provide investors with a 12% expected return. Instead, we decide to plow back 40% of the earnings at the firm’s current return on equity of 20%. What is the value of the stock before and after the plowback decision?


Solution

Without growth

With growth

75$08.012.0

6.0*5

08.02.0*4.0

0

P

g

67.41$12.0

50 P


Example (continued)

If the company did not plowback some earnings, the stock price would remain at $41.67. With the plowback, the price rose to $75.00.

The difference between these two numbers (75.00-41.67=33.33) is called the Present Value of Growth Opportunities (PVGO).


Valuing common stocks using earnings

We often use earnings to value stocks as

What is the relationship between this formula and the dividend growth formula?

PVGOr

EPSP 1

0


Example Firm A has a market capitalization rate of 15%. The

earnings are expected to be $8.33 per share next year. The plowback ratio is 0.4 and ROE is 25%. Every investment in year i is to yield a simple perpetuity starting in year (i+1) with each cash flow equal to total investment times ROE. All the investments have the same capitalization rate.• (a) Using the formula P=ESP1/r + PVGO to calculate the

stock price• (b) If ROE is increased, what will happen to the stock price?

Why?• (c) Use the dividend model to calculate the stock price?• (d) What have you found?• (e) Think about why you have this kind of result?


Simple Solution

(a)

g=10%, EPS1/r=8.33/0.15=$55.56

PVGO=NPV1/(r-g)=2.22/(0.15- 0.1)=$44.44, P=$100

(b) The price will be increased

(c) P=Div1/(r-g)=5/(0.15-0.1)=$100


Valuing common stocks using FCF (free cash flows)

The value of a business or stock is usually computed as the discounted value of FCF out to a valuation horizon (H).

The horizon value is sometimes called the terminal value .

HH

HH

r

PV

r

FCF

r

FCF

r

FCFPV

)1()1(...

)1()1( 22

11


FCF and PV

HH

HH

r

PV

r

FCF

r

FCF

r

FCFPV

)1()1(...

)1()1( 22

11

PV (free cash flows) PV (horizon value)


FCF and PV

Free Cash Flows (FCF) should be the theoretical basis for all PV calculations.

FCF is a more accurate measurement of PV than either Div or EPS.

The market price does not always reflect the PV of FCF.

When valuing a business for purchase, always use FCF.


FCF and PV

ExampleGiven the cash flows for Concatenator Manufacturing Division, calculate the PV of near term cash flows, PV (horizon value), and the total value of the firm. r=10% and g= 6%

66613132020202020(%) growth .EPS

1.891.791.681.59.23-.20-1.39-1.15-.96-.80- FlowCash Free

1.891.781.681.593.042.693.462.882.402.00Investment

3.783.573.363.182.812.492.071.731.441.20Earnings

51.3173.2905.2847.2643.2374.2028.1740.1400.1200.10ValueAsset

10987654321

Year


FCF and PV

4.2206.10.

59.1

1.1

1 value)PV(horizon 6

6.3

1.1

23.

1.1

20.

1.1

39.1

1.1

15.1

1.1

96.

1.1

.80-PV(FCF) 65432

Solution


FCF and PV

$18.8

22.4-3.6

value)PV(horizonPV(FCF)s)PV(busines


How to estimate the horizon value?

It is very difficult to forecast or estimate the horizon value. There are several ideas that may be used to estimate the horizon value.• Competition

• Constant growth rate


Another example

Imagine Corporation has just paid a dividend of $0.40 per share. The dividends are expected to grow at 30% per year for the next two years and at 5% per year thereafter. If the required rate of return in the stock is 15% (APR), calculate the current stock price.


Solution

Answer: First: visualize the cash flow pattern;

• C1, C2 and P2 Then, you know what to do? P0 = [(0.4 *1.3)/1.15] + [(0.4 *

1.3^2)/(1.15^2)] + [(0.4 * 1.3^2*1.05)/((1.15^2 * (.15 -.05))] = $6.33

FIN 819: Lecture 2

Another cash flow problem!Firm Excellent has an old packaging machine that can be used for another 3 years. The remaining book value for this old machine is $15,000, which can be depreciated in the next three years by using the straight-line depreciation approach. If the old packaging machine is used, the annual total cost of operation, labor and maintenance is $10,000. In the market, the new packaging machine is available now at the price of $50,000, which will increase by 5% annually. Based on Excellent’s experience, the new packaging machine will be used forever. The capital investment cost of the new package machine will be depreciated in the next 10 years by using the straight-line depreciation approach. The annual total cost of operation, labor and maintenance will be $8,000 when the new package machine is used.

Questions:

a. What is the valuation horizon used in this problem?

b. Should Excellent invest in the new packaging machine now or

wait three years later?


Bonds

Bond – a security or a financial instrument that obligates the issuer (borrower) to make specified payments to the bondholder during some time horizon.

Coupon - The interest payments made to the bondholder.

Face Value (Par Value, Principal or Maturity Value) - Payment at the maturity of the bond.

Coupon Rate - Annual interest payment, as a percentage of face value.


Bonds

A bond also has (legal) rights attached to it:• if the borrower doesn’t make the required

payments, bondholders can force bankruptcy proceedings

• in the event of bankruptcy, bond holders get paid before equity holders


An example of a bond

A coupon bond that pays coupon of 10% annually, with a face value of $1000, has a discount rate of 8% and matures in three years.• The coupon payment is $100 annually

• The discount rate is different from the coupon rate.

• In the third year, the bondholder is supposed to get $100 coupon payment plus the face value of $1000.

• Can you visualize the cash flows pattern?


Bonds

WARNINGWARNINGThe coupon rate IS NOT the discount rate used in the Present Value calculations. The coupon rate merely tells us what cash flow the bond will produce.

Since the coupon rate is listed as a %, this misconception is quite common.


Bond Valuation

The price of a bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return.

Nr

cpn

r

cpn

r

cpnPV

)1(

000,1...

)1()1( 21


Zero coupon bonds

Zero coupon bonds are the simplest type of bond (also called stripped bonds, discount bonds)

You buy a zero coupon bond today (cash outflow) and you get paid back the bond’s face value at some point in the future (called the bond’s maturity )

How much is a 10-yr zero coupon bond worth today if the face value is $1,000 and the effective annual rate is 8% ?

PV

Facevalue

Time=tTime=0


Zero coupon bonds (continue)

P0=1000/1.0810=$463.2 So for the zero-coupon bond, the price is

just the present value of the face value paid at the maturity of the bond

Do you know why it is also called a discount bond?


Coupon bond

The price of a coupon bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return.

)()()1()1(

11

)1(

)(....

)1()1( 21

parPVannuityPVr

par

rrrcpn

r

parcpn

r

cpn

r

cpnPV

tt

t


Bond Pricing

Example

What is the price of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 5.6%.


Bond Pricing

Example

What is the price of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 5.6%.

77.010,1$

)056.1(

060,1

)056.1(

60

)056.1(

60321

PV

PV


Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 6 %?

000,1$

)06.1(

060,1

)06.1(

60

)06.1(

60321

PV

PV


Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 15 %?

51.794$

)15.1(

060,1

)15.1(

60

)15.1(

60321

PV

PV


Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually?


Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually?

91.010,1$

)028.1(

030,1

)028.1(

30...

)028.1(

30

)028.1(

306521

PV

PV


Bond Pricing

Example (continued)

Q: How did the calculation change, given semi-annual coupons versus annual coupon payments?


Bond Yields

Current Yield - Annual coupon payments divided by bond price.

Yield To Maturity (YTM)- Interest rate for which the present value of the bond’s payments equal the market price of the bond.

ty

parcpn

y

cpn

y

cpnP

)1(

)(....

)1()1( 21


An example of a bond

A coupon bond that pays coupon of 10% annually, with a face value of $1000, has a discount rate of 8% and matures in three years. It is assumed that the market price of the bond is the present value of the bond at the discount rate of 8%. • What is the current yield?

• What is the yield to maturity.


My solution

First, calculate the bond price P=100/1.08+100/1.082+1100/1.083

=$1,051.54 Current yield=100/1051.54=9.5% YTM=8%


Bond Yields

Calculating Yield to Maturity (YTM=r)

If you are given the market price of a bond (P) and the coupon rate, the yield to maturity can be found by solving for r.

tr

parcpn

r

cpn

r

cpnP

)1(

)(....

)1()1( 21


Bond Yields

Example

What is the YTM of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? The market price of the bond is $1,010.77

77.010,1$

)1(

060,1

)1(

60

)1(

60321

PV

rrrPV


Bond Yields

In general, there is no simple formula that can be used to calculate YTM unless for zero coupon bonds

Calculating YTM by hand can be very tedious. We don’t have this kind of problems in the quiz or exam

You may use the trial by errors approach get it.


Bond Yields (3)

Can you guess which one is the solution?

(a) 6.6%

(b) 7.1%

(c) 6.0%

(d) 5.6% My solution is (d).


The rate of return on a bond

price bondor investmentchange price+incomeCoupon

=return of Rate

Example: An 8 percent coupon bond has a price of $110 dollars with maturity of 5 years

and a face value of $100. Next year, the expected bond price will be $105. If you

hold this bond this year, what is the rate of return?

investment ofcost

profit=return of Rate


My solution

The expected rate of return for holing the bond this year is (8-5)/110=2.73%• Price change =105-110=-$5

• Coupon payment=100*8%=$8

• Profit=8-5=$3

• The investment cost or the initial price=$110


Some new terms

So far, we consider one discount rate for all the cash flows

In fact, the discount rate for one period cash flows can be different from the discount rate for two-period cash flows.

Spot interest rate: the actual interest rate available today (t=0)

Future interest rate: the spot rate in the future (t>0)


Example

Spot rates (r) Investment Horizon r 1 6% 2 6.5% 3 7% 4 7.2%


The Yield Curve

Term Structure of Interest Rates: is the relationship between the spot rates and their maturity dates

Yield Curve - Graph of the term structure.


The term structure of interest rates (Yield curve)


Value the bond (revisit)

If we are given the term structure of interest rates, we know the discount rates for cash flows in different time periods.

Then

Here r1, r2, …, rt are spot rates for period 1, 2, …, t, respectively.

ttr

parcpn

r

cpn

r

cpnPV

)1(

)(....

)1()1( 22

11


Question

Which kind of the yield curve can make you use a single discount rate for the bond valuation?

For what kind of bonds, YTM is the same as spot rates?


Example

Please use the following information to value a 10%, four years coupon bond, if the spot rates are:

Year Spot rate

1 5% 2 5.4%

3 5.7%

4 5.9%


Solution

The interest payment is $100 every year.

5.144,1$

)059.1(

)000,1100(

)057.1(

100

)054.1(

100

)05.1(

1004321

PV


A problem

A 6 percent six-year bond yields 12% and a 10 percent six year bond yields 8 percent. Please calculate six-year spot rate.


Forward rate

Forward Rate - The interest rate, fixed today, on a loan made in the future at a fixed time according to the term structure of the interest rates.

The forward rate is implied by the term structure of interest rates and doesn’t exist in a financial market


Forward rate

Another way to look at the forward rate is that bonds can be priced in the following way:

ttr

parcpn

r

cpn

r

cpnPV

)1(

)(....

)1()1( 22

11

)1()1(

)(....

)1)(1()1( 11211

1 tt

t fr

parcpnfr

cpn

r

cpnPV


Forward rate calculation

One period forward rate can be calculated by using the spot rates as follows:

Where fn is the forward rate from period (n-1) to period n, rn is the n-period spot rate and rn-1 is the spot rate for the (n-1)-period spot rate.

11)1(

)1(1

nn

nn

nr

rf


Example

Using the information for spot rates given in the previous example, what is the forward rate(f2) from year 1 to year 2?


Solution

Here n=2,

%8.5

058.105.1

054.1

)1(

)1(1

2

2

11

22

2

f

r

rf


What moves the interest rate?

Nominal interest rate• What is it?

Real interest rate• What is it?

Inflation• What is it?

Can the nominal interest rate be less than zero?

Can the real interest rate be less zero?


What moves the interest rate?

Irving Fisher’s theory• Nominal r = (1+Real r)(1+ expected inflation)-1

• Real r is theoretically somewhat stable

• Change in inflation drives the change in the interest rate

This theory doesn’t work bad in the past 50 years.

Why do we care about the movement of the interest rate?


Bond price volatility

When the interest rate changes, what will happen to the bond price?

Then what decides the sensitivity of the bond price to the interest rate change?

ty

parcpn

y

cpn

y

cpnP

)1(

)(....

)1()1( 21


Duration

To capture the sensitivity of the bond price to the interest rate change, financial economists have defined a measure called Duration (or Macaulay Duration)

Duration is a weighted average of the maturity for the cash flows of the bond


Example

A two-year 5% coupon bond with YTM of 10%.

Maturity Cash flows

1 50

2 50+1000


Duration

Mathematically

Volatility (percent) =Duration/(1+YTM) Change in bond price = volatility*change in

interest rates. If YTM is small, Change in bond price =

Duration*change in interest rates.

1)(

1

1

t

ii

ii

t

ii

wandP

CPVw

iwDuration


Example

Calculate the Duration, volatility and the change of the bond price for 1% change in the interest rate for a bond that is 6 7/8%, 5-year coupon bond with 4.9% YTM.


Solution

i Ci PV(Ci) wi __ _i* wi _______

1 68.75 65.54 .060 0.060

2 68.75 62.48 .058 0.115

3 68.75 59.56 .055 0.165

4 68.75 56.78 .052 0.209

5 68.75 841.39 .775 3.875 1085.74 1.00 4.424


Solution

Volatility=4.424/(1.049)=4.22 Price change=4.22*1%=4.22%


Example (practice question 8)

Please use the following information to answer the questions in the next slide.

Year Spot rate

1 5% 2 5.4%

3 5.7%

4 5.9%

5 6.0%


Questions

(a) What are the discount factors for each date?

(b) What is the forward rates for each period?

(c) Calculate the PV of 5 percent and 10 percent five-year note?

(d) Which note is going to have a higher YTM?


Solution

(a)df1=1/1.05=0.95

df2=1/1.0542=0.90

(b) f2=1.0542/1.05-1=5.8%

f3=1.0573/1.0542-1=6.3%

(c) PV(5%)=$959.34

PV(10%)=$1171.43

(d) 5% note has a higher YTM.

fin 819: lecture 31 valuation of common stocks and bonds how to apply the pv concept

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