The Discount Cash Flow Valuation Model

1
Bonds

• The Discount Cash Flow Valuation Model
• Basics of Bonds and their Valuation

2
Bond Definitions

• Bond A bond is a security that represents a loan
  made by investors to the issuer.
• Par value (face value) The issuer promises to
  pay the face/par/maturity value of the bond when
  it matures.
• Coupon payment The issuer may promise to pay the
  investor a regular coupon payments every period
  until the bond matures.
• Coupon rate Percentage of face value.
• Maturity date Duration of the contract.
• Yield or Yield to maturity Average rate of
  return.

3
Valuing Coupon Bonds

• Value of a Level-coupon bond PV of coupon
  payment annuity PV of face value

4
Present Value of Cash Flows as Rates Change

• Bond Value PV of coupons PV of par
• Bond Value PV annuity PV of lump sum
• Remember, as interest rates increase present
  values decrease
• So, as interest rates increase, bond prices
  decrease and vice versa

5
Valuing a Par Bond with Annual Coupons

• Consider a bond with a coupon rate of 10 and
  annual coupons. The face value is 1000 and the
  bond has 5 years to maturity. The yield to
  maturity is 10. What is the value of the bond?
• Using the formula
• B PV of annuity PV of lump sum
• B 1001 1/(1.10)5 / .11 1000 / (1.10)5
• B 379.08 620.92 1000

6
Valuing a Discount Bond with Annual Coupons

• Consider a bond with a coupon rate of 10 and
  annual coupons. The par value is 1000 and the
  bond has 5 years to maturity. The yield to
  maturity is 11. What is the value of the bond?
• Using the formula
• B PV of annuity PV of lump sum
• B 1001 1/(1.11)5 / .11 1000 / (1.11)5
• B 369.59 593.45 963.04

7
Valuing a Premium Bond with Annual Coupons

• Suppose you are looking at a bond that has a 10
  annual coupon and a face value of 1000. There
  are 5 years to maturity and the yield to maturity
  is 8. What is the price of this bond?
• Using the formula
• B PV of annuity PV of lump sum
• B 1001 1/(1.08)5 / .08 1000 / (1.08)5
• B 399.27 680.27 1079.54

8
Graphical Relationship Between Price and
Yield-to-maturity
Bond Price
Yield-to-maturity
9
Coupon Bond Principles

• 1 For par bonds yield-to-maturity coupon
  rate.
• 2 for premium bonds (bond price gt face value)
  ytm lt coupon rate.
• 3 for discount bonds ( bond price lt face
  value) ytm gt coupon rate

10
Interest Rate Risk

• Price Risk
• Change in price due to changes in interest rates
• Long-term bonds have more price risk than
  short-term bonds
• Low coupon rate bonds have more price risk than
  high coupon rate bonds
• Reinvestment Rate Risk
• Uncertainty concerning rates at which cash flows
  can be reinvested
• Short-term bonds have more reinvestment rate risk
  than long-term bonds
• High coupon rate bonds have more reinvestment
  rate risk than low coupon rate bonds

11
Price Risk
12
Computing Yield-to-Maturity

• The yield-to-maturity is the discount rate that
  makes the present value of the cash flows from
  the bond equal to the current price of the bond.
  
• Finding the YTM requires trial and error if you
  do not have a financial calculator.
• Example What is the yield-to-maturity of a
  1,000 par value, 10 coupon rate bond coming due
  in 3 years that currently sells for 1076.
• 1076 100(PVAFr,T) (1,000)(1r)-3
• gt r YTM 7.10

13
Yield-to-Maturity, what does it tell?

• They allow you to compare different kinds of
  bonds those with dissimilar coupons, different
  market prices, and different maturities.
• YTM will equal your total earnings if
• You hold the bond to maturity,
• Coupons are reinvested at an interest rate equal
  to YTM.

14
Yield-to-Maturity, what does it tell?

• So, It is a promised annual rate of return. Why?
• Because CFs (coupons) may not be reinvested at
  the same rate as YTM
• Yield-to-maturity is the rate implied by the
  current bond price

15
Current Yield and YTM

• The current yield of a coupon bond is the ratio
  of its annual coupon payment to its current
  price.
• Example The current yield of the previous bond
  is
• 100 / 1076 0.09293 or 9.293
• Current yield measures the portion of an
  investors holding period return that comes in
  the form of interest income.
• Relationship between current yield and YTM
• YTM Current Yield Capital Gain(Loss)

16
Current Yield vs. Yield to Maturity

• Yield to maturity current yield capital gains
  yield
• Previous example 10 coupon bond, face value of
  1000, 3 years to maturity, 1076 price
• Current yield 100 / 1076 .0929 9.293
• Price in one year 1052.36 USD, assuming no
  change in YTM.
• Capital gain yield (1052.361076) / 1076
• -.02197 -2.197
• YTM 9.293 2.197 7.09,

17
Bond Values with Semiannual Compounding
18
Semi-annual bonds -Example

• Suppose that the Genesco 15 year, 15 bond paid
  interest semi-annually rather than annually.
  What would be its price upon issue if current
  rates are 15 on similar bonds?
• INT 1,000 x 0.15 150 ?INT/2 150/275
• N 15 ? 2N 30
• M 1,000
• kd 15 ? kd /2 15/2 7.5
• PV 1,000

19
Example Valuing Bonds w. Semi-Annual Payments

• Find the present value (as of January 1, 2002),
  of a 6.375 coupon T-bond with semi-annual
  payments, and a maturity date of December 2009 if
  the YTM is 5-percent.

20
Yield to Call

• Some bonds are calleable, they can be called back
  by the issuer before the maturity. Condional upon
  the market interest rates, the issuer may prefer
  to use this option. Why?
• .....
• In this case, we compute the yield to maturity of
  the bond as if you receive the call price and the
  bond is called on its earliest date.

21
Yield to Call - Example

• Suppose that AZ Inc has a 10 year 8 coupon bond
  outstanding that can be called at the end of year
  5 for a 5 premium.
• Further suppose that its current market price is
  112.42 and that it has been outstanding for 2
  years.
• If you buy this bond, what yield you would
  probably obtain out of this investment?

22
Yield to call - Example

• So, we have PV112.42, INT8, N8, M100 ? i
  YTM 6,
• Since the bond has been outstanding for 2 years,
  the bond can be called in 3years.
• Since the bond is selling for premium, it is most
  likely that the firm will call the bonds in 3
  years. Why?
• Since the call premium is 5, then its maturity
  value will be 100 x 1.05 105 if called
  (assuming a 100 face value).
• So, what is the yield for PV112.42, INT8,
  N3, M105
• The YTM between 5.5 and 5.75. Find out the
  number yourself.

23
Bond Pricing Theorems

• Bonds of similar risk (and maturity) will be
  priced to yield about the same return, regardless
  of the coupon rate
• If you know the price of one bond, you can
  estimate its YTM and use that to find the price
  of the second bond
• This is a useful concept that can be transferred
  to valuing assets other than bonds

24
Differences Between Debt and Equity

• Debt
• Not an ownership interest
• Creditors do not have voting rights
• Interest is considered a cost of doing business
  and is tax deductible
• Creditors have legal recourse if interest or
  principal payments are missed
• Excess debt can lead to financial distress and
  bankruptcy

• Equity
• Ownership interest
• Common stockholders vote for the board of
  directors and other issues
• Dividends are not considered a cost of doing
  business and are not tax deductible
• Dividends are not a liability of the firm and
  stockholders have no legal recourse if dividends
  are not paid
• An all equity firm can not go bankrupt

25
The Bond Indenture

• Contract between the company and the bondholders
  and includes
• The basic terms of the bonds
• The total amount of bonds issued
• A description of property used as security, if
  applicable
• Sinking fund provisions
• Call provisions
• Details of protective covenants

26
Bond Classifications

• By holder
• Registered vs. Bearer Forms
• Security
• Collateral secured by financial securities
• Mortgage secured by real property, normally
  land or buildings
• Debentures unsecured
• Seniority

27
Bond Characteristics and Required Returns

• The coupon rate depends on the risk
  characteristics of the bond when issued
• Which bonds will have the higher coupon, all else
  equal?
• Secured debt versus a debenture
• Subordinated debenture versus senior debt
• A bond with a sinking fund versus one without
• A callable bond versus a non-callable bond

28
Examples of Credit Ratings

• Moody's SPs Fitchs DCRs Definition
• Aaa AAA AAA AAA Prime. Maximum Safety
• Aa1 AA AA AA High Grade High Quality
• Aa2 AA AA AA
• Aa3 AA- AA- AA-
• A1 A A A Upper Medium Grade
• A2 A A A
• A3 A- A- A-
• Baa1 BBB BBB BBB Lower Medium Grade
• Baa2 BBB BBB BBB
• Baa3 BBB- BBB- BBB-
• Ba1 BB BB BB Non Investment Grade
• Ba2 BB BB BB Speculative
• Ba3 BB- BB- BB-
• B1 B B B Highly Speculative
• B2 B B B
• B3 B- B- B-
• Caa1 CCC CCC CCC Substantial Risk
• Caa2 CCC - - In Poor Standing

29
Issuer Government and Agencies

• Treasury Securities
• Federal government debt
• T-bills pure discount bonds with original
  maturity of one year or less
• T-notes coupon debt with original maturity
  between one and ten years
• T-bonds coupon debt with original maturity
  greater than ten years
• Municipal Securities
• Debt of state and local governments
• Varying degrees of default risk, rated similar to
  corporate debt
• Interest received is tax-exempt at the federal
  level

30
Example 7.4

• A taxable bond has a yield of 8 and a municipal
  bond has a yield of 6
• If you are in a 40 tax bracket, which bond do
  you prefer?
• 8(1 - .4) 4.8
• The after-tax return on the corporate bond is
  4.8, compared to a 6 return on the municipal
• At what tax rate would you be indifferent between
  the two bonds?
• 8(1 T) 6
• T 25

31
Zero-Coupon Bonds

• Make no periodic interest payments (coupon rate
  0)
• The entire yield-to-maturity comes from the
  difference between the purchase price and the par
  value
• Cannot sell for more than par value
• Sometimes called zeroes, or deep discount bonds.
• Treasury Bills and principal-only Treasury strips
  are good examples of zeroes

32
Pure Discount Bonds Example

• At the 6 month treasury auction the issue sells
  for a price of 97.
• What is the effective annual rate you would earn
  if you purchased this bond on the issue and held
  it until maturity?

33
Solution

• If you buy the bond for 97, you will get 100 in
  6 months, so the 6 month rate is
• To express this on an annual basis, you have to
  take into account that you can reinvest the
  money. So in the next 6 months you can earn the
  same rate again. Thus after a year at this rate
  you will have
• So the effective annual rate is

34
Turkish Treasury Bills - Example

• Example
• Here is a line from Reuter page on March 19,
  1997
• Value 19March97
• Maturity 04June97
• Average Price 87303
• Simple Yield 68.94
• Compounded Yield 90.34
• Now, let us verify how these values are computed

35
Turkish Treasury Bills (contnd)

• Number of days from 19 March 97 to 04 June 97 77
  days.
• Period rate of return over (77/365) years
• r (Face / PV) - 1 (100,000 / 87303) - 1
  0.145435
• Annual Simple Rate 0.145435 (365/77) 0.6894
• Compounded yield
• EAR ( 1 0.6894/(365/77))(365/77) 1
• EAR ( 1.145435)(365/77) 1 0.903482 or 90.34

36
Floating-Rate Bonds

• Coupon rate floats depending on some index value
• Examples adjustable rate mortgages and
  inflation-linked Treasuries
• There is less price risk with floating rate bonds
• The coupon floats, so it is less likely to differ
  substantially from the yield-to-maturity
• Coupons may have a collar the rate cannot go
  above a specified ceiling or below a specified
  floor

37
Other Bond Types

• Disaster bonds
• Income bonds
• Convertible bonds
• Put bonds
• There are many other types of provisions that can
  be added to a bond and many bonds have several
  provisions it is important to recognize how
  these provisions affect required returns

38
Bond Markets (Second Hand)

• Primarily over-the-counter transactions with
  dealers connected electronically
• Extremely large number of bond issues, but
  generally low daily volume in single issues
• Makes getting up-to-date prices difficult,
  particularly on small company or municipal issues
• Treasury securities are an exception

39
Bond Quotations

• Example for corporate bond quotation
• ATT 6s09 6.4 177 93 7/8 ¼
• Company ATT
• Coupon rate 6 coupon payment per year 60
• Bond matures in 2009
• Current yield 6.4 computed as annual coupon
  divided by current price
• Bonds traded 177
• Quoted price 93 7/8 of face value, so if face
  value is 1000, the price is 938.75. Bond prices
  are quoted as a percent of par, just as the
  coupon is quoted as a percent of par.
• Price change increase by ¼ percent, so the
  dollar change is .0025(1000) 2.50

40
Treasury Quotations

• Highlighted quote in Figure 7.4
• 8 Nov 21 12505 12511 -46 5.86
• Matures in November 2021
• Bid price is 125 and 5/32 percent of par value.
  If you want to sell 100,000 par value T-bonds,
  the dealer is willing to pay 1.2515625(100,000)
  125,156.25
• Ask price is 125 and 11/32 percent of par value.
  If you want to buy 100,000 par value T-bonds,
  the dealer is willing to sell them for
  1.2534375(100,000) 125,343.75
• The difference between the bid and ask prices is
  called the bid-ask spread and it is how the
  dealer makes money.
• The price changed by 46/32 percent from the
  previous day or 1437.50 for a 100,000 worth of
  T-bonds
• The yield is 5.86

41
Clean vs. Dirty Prices

• Clean price quoted price
• Dirty price price actually paid quoted price
  plus accrued interest
• Example 8 Nov 21 13223 13224 -12 5.14
• Assume today is July 15, 2005 (last coupon
  payment May 15, 05)
• Number of days since last coupon 61
• Number of days in the coupon period 184
• Accrued interest (61/184)(.04100,000)
  1326.09
• Prices (based on ask)
• Clean price 132,750
• Dirty price 132,750 1,326.09 134,076.09
• So, you would actually pay 134,076.09 for the
  bond

42
Inflation and Interest Rates

• Real rate of interest change in purchasing
  power
• Nominal rate of interest quoted rate of
  interest, change in purchasing power and
  inflation
• The ex-ante nominal rate of interest includes our
  desired real rate of return plus an adjustment
  for expected inflation

43
The Fisher Effect

• The Fisher Effect defines the relationship
  between real rates, nominal rates and inflation
• (1 R) (1 r)(1 h), where
• R nominal rate
• r real rate
• h expected inflation rate
• Approximation
• R r h

44
Example 7.6

• If we require a 10 real return and we expect
  inflation to be 8, what is the nominal rate?
• R (1.1)(1.08) 1 .188 18.8
• Approximation R 10 8 18
• Because the real return and expected inflation
  are relatively high, there is significant
  difference between the actual Fisher Effect and
  the approximation.

45
Term Structure of Interest Rates

• Term structure is the relationship between time
  to maturity and yields, all else equal
• For instance, each and every bond the Treasury
  issues has a second hand market (some issues are
  more liquid then others). Thus every issue has a
  price, and so every issue has a yield.
• Yield curve graphical representation of the
  term structure
• The collection of all these yields is plotted on
  a curve with the maturity of the issue on the x
  axis and the yield on the y axis --- this curve
  is called the yield curve.

46
Example of Yield Curve

• Below is the Treasury yield curve for 2/11/03

47
Shape of the Yield Curve

• Normal upward-sloping, long-term yields are
  higher than short-term yields
• Inverted downward-sloping, long-term yields are
  lower than short-term yields

48
The shape of the Treasuries yield curve

• The yields on Treasuries depend upon
• Real rate of interest opportunity cost of
  deferred consumption in real terms.
• Expected inflation investors must be
  compensated for anticipated loses in purchasing
  power.
• Maturity risk premium investors demand
  compensation for their interest rate risk
  exposure.

49
Figure 7.6 Upward-Sloping Yield Curve
50
Decomposition of Yields to Maturity

• We can decompose the YTM of Government bonds into
  the following components
• ytm real rate of interest inflation premium
  maturity risk premium
• Corporate bonds face additional risks credit
  risk liquidity risk
• Credit risk The risk that coupons and the
  principal may not be paid off.
• A bonds credit risk is often captured by its
  bond rating.
• Bonds issuers pay credit rating firms to rate
  their debt.
• Liquidty risk The more thinly traded a bond, the
  wider the bid/ask spread. Thus the most costly
  it is to trade that bond.

51
Bond Prices with a Spreadsheet

• There is a specific formula for finding bond
  prices on a spreadsheet
• PRICE(Settlement,Maturity,Rate,Yld,Redemption,
  Frequency,Basis)
• YIELD(Settlement,Maturity,Rate,Pr,Redemption,
  Frequency,Basis)
• Settlement and maturity need to be actual dates
• The redemption and Pr need to given as of par
  value
• Click on the Excel icon for an example

52
Example - About T-Papers

• On 15/08/1995, a bond with 9 month maturity
  (15/05/1996) is bought at a price that would
  yield 84.70 annual. Six months later, on
  13/02/1996 the same T-Bond is sold at a price
  that would yield 71 annual. What would be your
  return over the investment period should you sell
  the bond on 13/02/1996?
• Purchasing price on 15/08/1995
• Number of days to maturity 274
• Price 100,000 / (1 0.8470274/365) 61,131
  TL
• Selling price on 13/02/1996
• Number of days to maturity 92
• Selling Price 100,000 / (1 0.7192/365)
  84,820 TL

53
About T-Papers (contnd)

• Periodic rate of return (over 274 92 182
  days)
• (84,820 61,131) / 61,131 23,689 / 61,131
  0.3875 or 38.75
• Annual rate of return (simple)
• 0.3875365/182 0.7771 or 77.71
• Annual rate of return (compounded)
• Investment duration is 182 days and the period
  rate of return is 0.3875.
• EAR (1.3875)365/182 1 0.9274 or 92.74.