1. How to Value Bonds and Stocks

2. What is a Bond?
A bond is a legally binding agreement between a borrower and a lender

3. Bond Terminology Face value (F) or Principal Coupon rate
For a corporate bond this is generally $1,000
Coupon rate
This is a Stated Annual rate
Determines the coupon payment
Coupon payment (C )
Zero- coupon bond
Yield to maturity
Rating

4. Yield to Maturity
YTM is the return that the bond is offering if you bought it today and held it till maturity
The YTM is determined by the riskiness of the bond, which is a function of:
Time to maturity
Longer term bonds are riskier
Risk of default
Risk is measured by bond ratings

5. Pure Discount Bonds Makes no coupon payments
Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs)
Example: T-Bill
Yield to maturity comes only from the difference between the purchase price and face value
A pure discount bond cannot sell for more than par. WHY?

6. Pure Discount Bonds Present value of a pure discount bond at time 0:
Information needed for valuing pure discount bonds:
Time to maturity (T) = Maturity date - today’s date
Face value (F)
Discount rate (r)
Present value of a pure discount bond at time 0:

7. Pure Discount Bond: Example
Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.

8. Coupon Bonds
Make periodic coupon payments in addition to repaying the principal
Coupon payments are the same each period
Coupon payments are typically semi-annual.

9. Valuing a Coupon Bond
The value of a bond is simply the present value of it’s future cash flows
We value a bond is a package of two investments:
Present value of the coupon payments
Present value of the principal repayment

10. Coupon Bond Pricing Equation
An annuity plus a lump sum

11. Coupon Bond Pricing: BA II plus
N = The number of coupon payments
I/Y = The rate corresponding to the coupon frequency
PV = The price of the bond today
PMT= The amount of the coupon payment
FV = The principal that will be repaid

12. Valuing a Corporate Bond
DuPont issued a 30 year maturity bonds with a coupon rate of 7.95%.
Interest is paid semi-annually
These bonds currently have 28 years remaining to maturity and are rated AA.
The bonds have a par value of $1,000
Newly issued AA bonds with maturities greater than 10 years are currently yielding 7.73%
What is the value of DuPont bond today?

13. DuPont example (continued)
Annual interest ($) =
Semiannual coupon payment =
Semiannual discount rate =
Number of semiannual periods=
PV=

14. Level Coupon Bond: Example (Given)
Consider a U.S. government bond with a 6 3/8% coupon that expires in December 2010.
The Par Value of the bond is $1,000.
Coupon payments are made semi-annually (June 30 and December 31 for this particular bond).
Since the coupon rate is 6 3/8%, the payment is $
On January 1, 2006 the size and timing of cash flows are:
The require annual rate is 5%

15. Level Coupon Bond: Example (Given)
Coupon Rate 6 3/8%, pay semi-annually
10 Semi-Annual Payments of $
Maturity December 2010, Start Jan. 2006
The Par Value of the bond is $1,000.
The require annual rate is 5%
N = 10, I/Y = 2.5, PV=???, PMT = , FV=1,000::: PV = $1,060.17

16. Valuing a Corporate Bond (Given)
Value a bond with the following characteristics (calculator):
Face value: $1,000
Coupon rate (C ): 8%
Time to maturity: 4 years
Discount rate: 9%
Present Value: $967.02
You should know how to get any one of these numbers given the other 4.

17. YTM and Bond Prices
How are prices and YTM related?

18. Coupon Rate and YTM Coupon rate = YTM Coupon rate > YTM

19. YTM and Bond Value
When the YTM < coupon, the bond trades at a premium.
1300
1200
Bond Value
When the YTM = coupon, the bond trades at par.
1100
1000
6 3/8
800
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Discount Rate
Coupon Rate
When the YTM > coupon, the bond trades at a discount.

20. Computing Yield to Maturity
Finding the YTM requires trial and error if you do not have a financial calculator
If you have a financial calculator, enter N, PV, PMT, and FV,
Remembering the sign convention
PMT and FV need to have the same sign, PV the opposite sign

21. YTM with Semiannual Coupons
A bond has a 10% coupon rate, 20yrs to maturity, makes coupon payments semi-annually, a $1,000 face, and is selling at $1,197.93
Is the YTM more or less than 10%?
What is the semi-annual coupon payment?
How many periods are there?
What is the YTM?

22. YTM with Annual Coupons (Given)
Consider a bond with a 10% annual coupon rate, 15 years to maturity, and a par value of $1,000. The current price is $
Will the YTM be more or less than 10%?
MORE
What is the YTM?
N = 15
I/Y = ???? = 11%
PV =
PMT = 100
FV = 1000

23. The effect of changes in interest rates on bond prices
Known as interest rate risk
Consider two identical 8% coupon bonds except that one matures in 4 years, the other matures in 10 years
Calculate the change in the price of each bond if interest rates fall from 8% to 6%, if interest rates rise from 8% to 10%

24. Interest Rates and Time to Maturity
The longer a bond has till maturity, the greater the price impact of a change in interest rates
WHY?

25. Interest Rates and Bond Prices
Bond Prices and Interest Rates have an Inverse Relationship

26. Pricing Stocks
Remember: The value of any asset is the present value of its expected future cash flows.
Bond Cash flows are:
Stock produces cash flows from:

27. Stock Valuation Terminology
Dt or Divt –dividend expected at time t
P0 – market price of stock at time 0
Pt – expected mkt price of stock at time t
g- expected growth rate of dividends
rs or re- required rate of return on equity
D1 / P0 – expected one-year dividend yield
(P1 - P0)/ P0 – expected one year capital gain
The stocks total return = div yield + cap. gain

28. Valuing Common Stock
The price of a share is simply the present value of the expected future cash flows
An investor planning on selling his share in a year is willing to pay:
The investor buying the share next year plans on selling it a year later so he is only willing to pay:

29. Keep Going Using summation: P0 = H Dh / (1 + r)h + PH / (1 + r)H
This process can be repeated into the future
Using summation:
P0 = H Dh / (1 + r)h + PH / (1 + r)H
What happens to PH as H approaches infinity?

30. Dividend Valuation Model
As H approaches infinity PH goes to zero
Because of this we only need to be concerned with the stock’s future dividends
The price of a stock is equal to the present value of its expected future dividends

31. Constant Dividend
How do you value a stock that will pay a constant dividend?
Hint: what does the cash flow stream look similar to?

32. Constant Dividend Example
What is the value of a stock that is expected to pay a constant dividend of $2 per share?
The required rate of return is 10%

33. Growing Dividends
Now we are assuming that the firm’s dividends will grow at a constant rate, g forever
This is similar to a:
So the price of a share is:

34. Growing Dividend Example
Geneva steel just paid a dividend of $2.10. Dividend payments are expected to grow at a constant rate of 6%. The appropriate discount rate is 12%. What is the price of Geneva stock?
Div1 =
P0 =

35. Valuing Stock with Changing g
Find the PV of dividends during the period of non-constant growth, PA
Find the price of the stock at the end of the non-constant growth period, PN
Discount the price found in 2 back to the present, PB
Add the two present values (1+3) to find the intrinsic value (price) of the stock P0 = PA + PB

36. Differential Growth Rates
Dividends will grow at g1 for N years and g2 thereafter
Step 1: An N-year annuity growing at rate g1
Step 2: A growing perpetuity at rate g2
PN = DivN+1 / (R-g2)
Step 3: PB = PN / (1+R)N
Step 4: P0 = PA + PB

37. Non-Constant Growth Example (Given)
Websurfers Inc, a new internet firm is expected to do very well during its initial growth period. Investors expect its dividends to grow at 25% for the next 3 years. Obviously one cannot expect such extraordinary growth to continue forever, and it is expected that dividends will grow at 5% after year 3 in perpetuity. Its current dividend is $1/share. Required rate of return on the stock = 10%. Calculate what the current price should be.

38. Websurfer Inc, Example (Given)
1*1.253
*1.052
= 2.15
PA=[(1*1.25)/( )]*[1-{1.25/1.10}3] = 3.90
PN ={1*1.253*1.05}/( ) = 41.00
PB =41.00/(1.103) = 30.80
P0 = PA + PB = = $34.70
1*1.253*1.05 = 2.05
1*1.25 = 1.25
1*1.252 =1.56
1*1.253 = 1.95

39. A Differential Growth Example
A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity.
What is the stock worth? The discount rate is 12%.

40. Solution
A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity. R=12%
PA =
PN =
PB =
P0 = PA + PB =

41. Important Parameters
The value of a firm depends on the discount rate, the growth rate, and the initial dividend.

42. The Discount Rate The market consensus of the firm’s required rate
This is the Market Capitalization Rate
Return that an investor expects to make
This is similar to what for a bond?

43. Where does “r” come from?
We generally estimate r from one of the dividend valuation models
Using constant dividend growth model:
In practice, estimates of r have a lot of estimation error
Rearrange and solve for R:

44. Where does “R” come from?
What is D1/P0?
What is g?

45. Classifying Stocks
Firms are often classified based on where investors expect to earn their return from
“Income/Value stocks”: have a higher dividend yield
“Growth stocks”: have a higher growth component
As long as both are equally risky, the return should be the same

46. Where does “g” come from?
From analysts' estimates
I/B/E/S, Google, Yahoo, or WSJ
From earnings re-investment
g = plowback ratio * ROE
How much does the firm reinvest, and what is the return on the investment
Look Familiar?

47. Link between stock prices and earnings
A “new valuation model” :
Consider a firm with a 100% payout ratio, so Div = EPS and earnings remain flat.

48. Present Value of all Future Growth Opportunities (PVGO)
The price is composed of the value of the firm’s current assets (100% payout firm) and the firm’s growth opportunities
Growth opportunities are opportunities to invest in positive NPV projects.
P0 =

49. Who cares about PVGO?
For what type of stock is the PVGO more important?
Growth or Value stocks

50. PVGO Example
Assume that a firm has 2 potential projects. Project A & B with NPV’s of $2m, and $3m, respectively. The firm pays out all its earnings as dividends, and paid a dividend of $1/share last year. It has 200,000 shares outstanding. Assume the discount rate is 10%.
What is the share price, if the cash flow from the firm's existing assets are expected to remain the same in perpetuity, and the firm takes on Project A, and B?  

51. Stock Value Represents:
Present value of expected future dividends,
Present value of free cash flow,
Present value of average future earnings under a no-growth policy plus the present value of growth opportunities

52. Price-Earnings Ratio
The price-earnings ratio is calculated as the current stock price divided by annual EPS.
The Wall Street Journal uses last 4 quarter’s earnings
Many analysts use this to determine how the market feels about a company

53. Price/Earnings Ratio Is selling at a high P/E good?
Why might the P/E be high:

54. Problem 1 (Given)
A firm is expected to grow at 25% for the next 3 years. Its growth is expected to decline to 15% for the following 4 years. It is then expected to grow at 5% in perpetuity. Find the current share price if the current dividend is $1 and the discount rate is 10%. 

55. Problem 1 (Given) PA=[(1*1.25)/(0.10-0.25)]*[1-{1.25/1.10}3]=3.89
A firm is expected to grow at 25% for the next 3 years. Its growth is expected to decline to 15% for the following 4 years. It is then expected to grow at 5% in perpetuity. Find the current share price if the current dividend is $1 and the discount rate is 10%.
PA=[(1*1.25)/( )]*[1-{1.25/1.10}3]=3.89
PN1 ={1*1.253*1.15}/( )]*
[1-{1.15/1.10}4]=8.76
PB =8.76/(1.103) = 6.58
PN2 ={1*1.253*1.154*1.05}/( )] = 71.80
PC =71.80/(1.107) = 36.83
P0 = PA + PB + Pc = = $47.30

56. Problem 2 (Given)
Consider a firm whose dividend growth is expected to decline gradually. For the next two years, the growth is expected to be 20%. In the following years, it is expected to grow at 18%, 13% and 10%. From year 6 onwards, dividends are expected to grow at 5% for perpetuity. Assume the current dividend is $1 and the required rate of return is 10%. What is the current price?

57. Problem 2: Phase 1 – Years 1-5 (Given)
DIV1 = 1.00 * 1.20 = 1.20
DIV2 = 1.20 * 1.20 = 1.44
DIV3 = 1.44 * 1.18 = 1.70
DIV4 = 1.70 * 1.13 = 1.92
DIV5 = 1.92 * 1.10 = 2.11
PA = 1.2/ / / / /1.15 = 6.18

58. Problem 2 Phase 2 – Years 6- (Given)
DIV6 = 2.11 * 1.05 = 2.22
PN = DIV6 / (r-g) = 2.22/( ) = 44.4
PB = 44.4 * 1/1.15 = 27.57
Current Price = = $33.75

59. Quick Quiz
How do you find the value of a bond, and why do bond prices change?
What is a bond indenture, and what are some of the important features?
What determines the price of a share of stock?
What determines g and R in the DGM?
Decompose a stock’s price into constant growth and NPVGO values.
Discuss the importance of the PE ratio.

60. Why We Care Basic real world application of the time value of money
Foundation of Investment/ Financial Analysis