Introduction to Valuation Bond Valuation

1
Introduction to ValuationBond Valuation

• Financial Management
• P.V. Viswanath
• For a First course in Finance

2
Lesson Objectives

• To look at the difference between economics and
  finance
• To introduce the notion of future dollars as
  traded goods.
• To introduce the price of future dollars
• To relate the price of money to interest rates.
• To use these rates to price Treasury securities
• To introduce the notion of arbitrage

3
Absolute and Relative Pricing

• In economics, we tend to price goods and assets
  by considering the factors affecting the supply
  and demand for them.
• The number of goods and assets are very many.
  Each of them is different in some way or another
  from the other.
• Computing the price of one good does not allow us
  to price another good, except to the extent that
  other goods are substitutes or complements for
  the first good.
• In finance, the number of assets can be
  reasonably characterized in terms of a smaller
  number of basic characteristics.
• Hence most assets can, to a first approximation
  be priced by considering them as combinations of
  more fundamental assets.

4
The Fundamentals of Economics

• One of the issues that economics analyzes is the
  determination of prices of goods.
• For example, what determines the price of eggs?
• We have a supply curve that is, a schedule of
  quantities of eggs that their current possessors
  would be willing to sell and the prices at which
  they would be willing to sell them.
• The higher the price, the more theyd be willing
  to sell.

5
The Supply Curve
Price ( per unit)
Quantity of Eggs
6
The Demand Curve

• We can also imagine the different amounts of eggs
  that people would be willing to buy and the
  prices at which they would buy those quantities.
• The lower the price, the more would be demanded.

7
The Demand Curve
Price ( per unit)
Quantity of Eggs
8
The Determination of the Price of Eggs
Price ( per unit)
Quantity of Eggs
9
Economics and Finance

• Finance, like Economics, is interested in the
  prices of goods.
• But the goods that financial analysts are
  interested in, are quite different.
• As you might imagine, financial economists are
  interested in money (or purchasing power) and in
  the price of money.
• But what does it mean to talk about the price of
  money? In what currency would you pay to acquire
  money?

10
Money and Time

• The answer is that access to resources today is
  not the same as access to resources tomorrow
  that is, money available today is not the same as
  money available tomorrow.
• You can buy something today only if you have the
  money to buy it with today. Having access to
  money, which will be available tomorrow wont
  allow you to necessarily buy things today!
• This means that we can talk of different kinds of
  money.
• And, denoting time by the subscript t, we can
  talk of the price of time 1 (tomorrow) money in
  terms of time 0 (today) money.

11
More on the price of money

• Lets assume that all prices are denominated in
  t0 dollars (todays money).
• Then, just as we might say that the price of a
  book is 10, the price of a subway token is 2
  and the price of a cup of Starbucks coffee is
  3.50, we could also say
• The price of a t1 dollar is 0.90, the price of
  a t2 dollar is 0.7831 and the price of a t3
  dollar is 0.675.

12
The price of coffee, said differently

• This might sound a little strange to you, but
  lets put it slightly differently.
• Going back to a cup of coffee, we said its price
  was 3.50, but if we know that 3.50 1, we
  could equally well say that the price of a book
  is 1.
• Then even if we were all in the US and Starbucks
  only accepted US dollars, there would be no
  problem if Starbucks had its price list
  denominated in euros.

13
More ways to price coffee

• Lets take this further.
• Suppose Starbucks required everybody to play the
  following game in order to figure out the price
  of its offering.
• Suppose they took the actual dollar price of a
  coffee multiplied it by 2 and added 3 to it and
  called it java units (J).
• A cup of coffee that normally cost 3.5 would be
  listed as costing 10J.
• Then if we saw a cappuccino listed at 13J, we
  would simply subtract 3 to get 10, then divide by
  2 to get a price of 5.
• It would be a little weird, but nothing
  substantive would change.

14
Rates

• So now, lets go back to the price of money we
  said that the price of a t1 dollar was 0.90,
  and that the price of a t2 dollar was 0.7831.
• Now clearly the price of a t1 dollar, which is
  0.90 today, will rise to 1 at t1.
• Hence providing todays price of a t1 dollar is
  equivalent to providing the rate of change of the
  price over the coming period.
• I have exactly the same information in each case.
• This rate of change is also my rate of return
  over the next year if I buy a t1 dollar, today,
  and is also known as the interest rate.
• In our example, this works out to (1-0.90)/0.90
  or 11.11

15
Rates

• What about the price of a t2 dollar, which we
  said was 0.7831?
• Once again, the price of this t2 dollar would be
  1 at t2 (in t2 dollars, of course).
• We could compute the gross return on this
  investment, in the same way, as 1/0.7831 1.277
  or a return of 27.70.
• But this is a return over two periods, and we
  cannot compare it directly to the 11.11 that we
  computed earlier.
• The solution to this problem is to annualize the
  two-period return

16
Computing Annualized Rates

• We computed the return on buying a t2 dollar at
  27.70.
• Suppose the one-period return on this is r that
  is, the return from holding this t2 dollar from
  now until t1 is r. Then, every dollar invested
  in this specialized investment could be sold at
  (1r) at t1.
• Now, if we assume the return on this t2 dollar
  if held from t1 to t2 is also r, then the
  (1r) value of our outlay of one t0 dollar in
  this investment would be (1r)(1r) or (1r)2.
• But we already know from our return computation,
  that this is exactly 1.277 (that is 1 plus the
  27.7).
• Hence we equate (1r)2 to 1.277 and solve for r.

17
Annualized Rates

• This involves simply taking the square-root of
  1.277, which is 13.
• Of course, we wont get exactly 13 in each of
  the two periods.
• The 13 rate is, rather, a sort of average return
  over the two periods, that results in a 27.7
  over the two years.
• We can now take 0.675, the price of a t3 dollar
  and also convert it to a rate of return.
• In this case, we take the cube root of (1/0.675),
  which works out 14

18
Yield to maturity

• So now we have the current prices of t1, t2,
  and t3 dollars, or 0.90, 0.7831 and 0.675
  respectively.
• Alternatively, the information in these prices
  could also be presented as rates of return, which
  in our case are 11.11, 13 and 14 respectively.
• These rates are also called yields-to-maturity.
• Yields-to-maturity, in general, are the
  annualized total returns that you would get if
  you held a particular financial instrument to
  maturity.
• In this case, the returns each year are only from
  price appreciation, while in other cases, there
  may be annual cash payments received by the
  investor, as well.

19
Using the rates

• We have assumed, up to this point, that
  purchasing a t1 dollar is riskless. That is, the
  person who sold us the t1 dollar today, in
  return for the 0.90, would, in fact, pay us 1
  at time t1.
• We will continue with this assumption, for now.
• Note, as well that buying a t1 dollar is
  equivalent to lending money for one period, while
  selling a t1 dollar is equivalent to borrowing
  money for 1 period.
• A bond is precisely a promise to pay its holder
  some combination of future dollars.
• Corporations, governments and other entities who
  need funds for the continuing operations issue,
  that is sell, such bonds.

20
Treasury Bonds

• Consider now a bond issued by the Treasury
  Department, which essentially acts as banker for
  the Federal Government.
• These bonds, or promises to pay are considered
  default-free, i.e. we fully expect the Treasury
  to live up to its promises.
• We can, therefore, evaluate and price these
  Treasury or T-bonds using the risk-free yields
  that we established before.
• This need not be true of other governmental
  institutions, such as municipalities, such as the
  City of New York or Federal agencies such as PATH
  the Port Authority of New York and New Jersey.

21
Treasury Bonds

• On Feb. 29th 2008, the Treasury issued a 2 note
  with a maturity date of February 28, 2010 with a
  face value of 1000, which was sold at auction.
• The price paid by the lowest bidder was
  99.912254 of face value.
• This means that the buyer of this bond would get
  every six months 1 (half of 2) of the face
  value, which in this case works out to 10.
• In addition, on Feb. 28, 2010, the buyer would
  get 1000.

22
Terminology

• The maturity of this bond is 2 years.
• The coupon rate on this bond is 2
• The face value of this bond is 1000
• The price paid for this bond is 999.123
• The yield-to-maturity obtained by this buyer is
  2.045, i.e. the average rate of return for this
  buyer if s/he held it to maturity.

23
Pricing this bond

• Lets assume for now, that we do not know the
  price of this bond. How can we price this bond?
• What we do know is that a holder of this bond
  would receive 10 in 6 months, another 10 in 1
  year, 10 again in 1.5 years and 1010 in 2
  years.
• We also know that a 6-month T-bill issued on Feb.
  21, 2008 sold for 98.968667.
• A T-bill is a promise to pay money 6 months in
  the future. With the given price then, a buyer
  would get a 1.042 return for those 6 months.
• This is often annualized by multiplying by 2 to
  get a bond-equivalent yield of 2.084.
• We also know that yields of bonds generally are
  higher for higher maturities.

24
Yield Curves for Feb. 1-12, 2008
Date 1mo 3mo 6mo 1yr 2yr 3yr 5yr 7yr 10yr 20yr 30yr
02/01/08 1.75 2.10 2.15 2.13 2.09 2.22 2.75 3.13 3.62 4.31 4.32
02/04/08 2.15 2.27 2.22 2.17 2.08 2.23 2.78 3.18 3.68 4.37 4.37
02/05/08 2.22 2.19 2.13 2.06 1.93 2.08 2.66 3.08 3.61 4.32 4.33
02/06/08 2.12 2.10 2.10 2.05 1.96 2.11 2.67 3.08 3.61 4.36 4.37
02/07/08 2.19 2.17 2.13 2.08 1.99 2.21 2.79 3.21 3.74 4.50 4.51
02/08/08 2.24 2.23 2.12 2.05 1.93 2.10 2.69 3.11 3.64 4.41 4.43
02/11/08 2.35 2.31 2.13 2.06 1.93 2.10 2.67 3.09 3.62 4.38 4.41
02/12/08 2.55 2.31 2.12 2.06 1.94 2.13 2.71 3.13 3.66 4.43 4.46
25
Yield Curves for Feb. 1-12, 2008
26
Pricing a Treasury bond

• Suppose we believe that the current environment
  of uncertainty will continue.
• We might believe that investors will be even more
  unwilling to invest in securities that have any
  default risk.
• In that case, they will be willing to buy
  Treasury securities at lower yields.
• We use this to estimate current bond-equivalent
  yields.
• Suppose we estimate the current bond-equivalent
  yields for 6-month money, 1 year money, 18-month
  money and 2 year money as 2.07, 2.1, 2.11 and
  2.14.

27
Discounting

• Keeping in mind that what we have are
  bond-equivalent yields, i.e. yields computed on a
  six-monthly basis and then doubling to get the
  annual yield, we will compute the current prices
  of future dollars.
• To do this, we need to employ a procedure called
  discounting.
• Suppose the required risk-free rate of return on
  future dollars is 4 per period.
• Now, if I have a certain, default-free promise of
  200 in 3 periods, what is the value of this
  promise today?

28
Discounting

• We know the value at t3 of this promise would be
  exactly 200.
• The value at t2of the promise would have to be
  such that would yield a return of exactly 4 over
  the last period, i.e. from t2 to t4.
• Suppose the required value is S. Then, we would
  need (200-S)/S 1.04.
• Solving this equation, we find S 200/1.04.
• What would the value at t1 be?
• Applying the same principle, we see that it must
  be S/1.04 or 200/1.042.
• Analogously, we can see that the value at t0 of
  a promise to pay 200 in n periods is 200/(1.04)n.

29
Pricing the 2-yr T-bond

• Coming back to our T-bond, the annualized yield
  on 6-month money is 2.07 hence the six-month
  yield is 2.07/2 or 1.035.
• Hence a promised dollar-payment at t 0.5 would
  sell today for 1/1.01035 or 0.989756 today.
• The annualized yield on 1-year money is 2.1
  hence the six-month yield is 2.10/2 1.05.
• Hence a promised dollar-payment at t1 would sell
  today for 1/1.01052 0.979326.

30
Pricing the 2-yr T-bond

• The annualized yield on 1.5 year money is 2.11
  hence the six-month yield is 2.11/2 1.055.
• Hence the price today of a promised
  dollar-payment at t1.5 is 1/1.010553 0.96901,
  using the discounting method.
• The annualized bond-equivalent yield on 2-year
  money is 2.14.
• The price today of a promised dollar-payment at
  t2 is 1/1.01074 0.95832

31
Pricing the 2-yr T-bond

• So now we know that our bond pays 10 in 6
  months, another 10 in 1 year, 10 again in 1.5
  years and 1010 in 2 years.
• We also know that one dollar promised for each of
  those dates is worth, today, 0.989756,
  0.979326, 0.96901 and 0.95832 respectively.
• Our bond, therefore, must sell for 10(0.989756)
  10(0.979326) 10(0.96901) 1010(0.95832)
  997.28.

32
Arbitrage

• What we have done is to treat our 2 year 2
  coupon bond as a portfolio of four other
  zero-coupon bonds and then priced it as the sum
  of the values of those zero-coupon bonds.
• But what will guarantee that this price equality
  will hold?
• Heres where the efficient functioning of markets
  comes into play.
• A process called arbitrage ensures that the price
  of a combination of other financial securities
  does not deviate too much from the price implied
  the prices of those other securities.

33
Arbitrage

• Suppose, for example that our two-year bond sold
  for 996.
• Then a bond trader could buy a bond at this
  price, then, himself, issue the corresponding
  four zero-coupon bonds and sell them at their
  market prices.
• He would then end up with a profit of 1.28 per
  bond.
• If the bond sold for, say, 998, he could buy the
  zero-coupon bonds and then create a synthetic
  coupon bond and sell it at the higher price and
  make a profit of 998-997.28 or 72 cents per bond.

34
Relative pricing of financial assets

• Consider first riskless financial assets, i.e,
  assets that are claims on riskless cashflows over
  time.
• Consider a fundamental asset, i, defined by a
  claim to 1 at time t i.
• There can be T such fundamental assets,
  corresponding to the t 1,..,T time units.
• Then, any arbitrary riskless financial asset that
  is a claim to ci at time i, i 1,..,T can be
  considered a portfolio of these T fundamental
  assets.
• Hence, the price, P of any such asset is related
  to the prices of these first T fundamental
  assets.
• In fact, the price of this asset would simply be

35
Relative pricing of risky financial assets

• What about risky financial assets?
• We can equivalently imagine, for every level of
  risk, a set of T fundamental risky assets. Then,
  for any arbitrary risky asset of this level of
  risk, we can equivalently write

• Of course, this is not entirely satisfactory,
  because wed have TxM fundamental assets
  corresponding to each of M levels of risk. We
  will come back to this when we talk about the
  CAPM.
• In any case, we need to examine how this pricing
  is established in the market-place.

36
Arbitrage and the Law of One Price

• Law of One Price In a competitive market, if two
  assets generate the same cash (utility) flows,
  they will be priced the same.
• How is this enforced?
• If the law is violated if asset 1 sells for
  more than asset 2, then investors can make a
  riskless profit by buying asset 2 and selling it
  as asset 1!
• In practice we need to take transactions costs
  into account.
• Also, it may be difficult to execute the two
  transactions at the same time prices might
  change in that interval this introduces some
  risk.

37
Exchange Rates and Triangular Arbitrage

• Consider the exchange rates reigning at closing
  on January 30.
• The yen/euro rate was 157.87 yen per euro
• The euro/ rate was 1.4835 per euro.
• The yen/ rate was 106.4 yen per dollar.
• If we start with a dollar, we can buy 106.4 yen
  these can then be used to buy 106.4/157.87 or
  0.674 euros, which can, in turn, be used to
  acquire 0.9998, which is very close to a dollar.

38
Triangular Currency Arbitrage

• Suppose the euro/ rate had been 1.50 per euro.
• Then, it would have been possible to start with
  one dollar, acquire 0.674 euros, as above, and
  then get (0.674)(1.5) or 1.011, or a gain of
  1.1 on the initial investment of a dollar.
• This would imply that the dollar was too cheap,
  relative to the euro and the yen.
• Many traders would attempt to perform the
  arbitrage discussed above, leading to excess
  supply of dollars and excess demand for the other
  currencies.
• The net result would be a drop a rise in the
  price of the dollar vis-à-vis the other
  currencies, so that the arbitrage trades would no
  longer be profitable.

39
Risk Arbitrage

• In this case, trading will continue until there
  are no more riskfree profit opportunities.
• Thus, arbitrage can ensure that the sorts of
  pricing relationships referred to above can be
  supported in the marketplace, viz

• What if there are still opportunities that will,
  on average, lead to profit, but the investors
  intending to benefit from this profit will have
  to take on some risk?
• Presumably investors will trade off the risk
  against the expected profit so that there will be
  few of these expected profit opportunities, as
  well this brings us to the notion of the
  informational efficiency of financial markets.

40
Efficient Markets Hypothesis EMH

• An assets current price reflects all available
  information this is the EMH.
• If it didnt, there would be an incentive for
  investors to act on that information.
• Suppose, for example, that investors noticed that
  good news led to stock prices rising slowly over
  two consecutive days.
• This would mean that at the end of the first day,
  the good news was not all incorporated in the
  stock price.

41
Efficient Markets Hypothesis

• In this situation, it would be optimal for
  traders to buy even more of a stock that was
  noted to be rising on a given day, since the
  stock would rise more the next day, giving the
  trader an unusually good chance of making money
  on the trade.
• But if many traders pursue this strategy, the
  stock price would rise on the first day, itself,
  and the informational inefficiency would be
  eliminated.
• Empirically, financial markets seem to be
  reasonably close to being efficient.
• This allows us to price financial assets with
  respect to fundamentals without worrying about
  deviations from these fundamental prices.

42
Stock Price Fundamentals

• What determines the price of a stock? Or, in
  other words, why would an investor hold stocks?
• The answer is that s/he expects to receive
  dividends and hopefully benefit from a price
  increase, as well.
• In other words, P0 PV(D1) PV(P1)
• However what determines P1?
• Again, using the previous logic, we must say that
  its the expectation of a dividend in period 2
  and hopefully a further price rise. Continuing,
  in this vein, we see that the stock price must be
  the sum of the present values of all future
  dividends.

43
Dividend Mechanics

• Declaration date The board of directors declares
  a paymentRecord date The declared dividends are
  distributable to shareholders of record on this
  date.Payment date The dividend checks are
  mailed to shareholders of record.
• Ex-dividend date A share of stock becomes
  ex-dividend on the date the seller is entitled to
  keep the dividend.   At this point, the stock is
  said to be trading ex-dividend. 

44
Dividend Discount Model

• What is the price of a stock on its ex-dividend
  date?
• Using the previous logic, we see that its simply

• where k is the appropriate discount rate to
  discount the dividends consistent with their
  riskiness.
• We assume that the one-period ahead discount rate
  is the same for all periods. That is, we use the
  same rate to discount D1 to time 0, as we use to
  discount D2 to time 1.

45
Gordon Growth Model

• If we assume that the dividend is growing at a
  rate of g per annum forever, this formula
  simplifies to

• We see that the price of a stock is higher, the
  higher the level of dividends, the higher the
  growth rate of dividends and the lower the
  required rate of return or the discount rate, k.

46
Two essential concepts

• Cash flows at different points in time cannot be
  compared and aggregated. All cash flows have to
  be brought to the same point in time, before
  comparisons and aggregations are made.
• The concept of a Time Line

47
Cash Flow Types and Discounting Mechanics

• There are five types of cash flows -
• simple cash flows,
• annuities,
• growing annuities
• perpetuities and
• growing perpetuities

48
I. Simple Cash Flows

• A simple cash flow is a single cash flow in a
  specified future time period.
• Cash Flow CFt
• ____________________________________________
• Time Period t
• The present value of this cash flow is-
• PV of Simple Cash Flow CFt / (1r)t
• The future value of a cash flow is -
• FV of Simple Cash Flow CF0 (1 r)t

49
Application The power of compounding - Stocks,
Bonds and Bills

• Between 1926 and 1998, Ibbotson Associates found
  that stocks on the average made about 11 a year,
  while government bonds on average made about 5 a
  year.
• If your holding period is one year,the
  difference in end-of-period values is small
• Value of 100 invested in stocks in one year
  111
• Value of 100 invested in bonds in one year
  105

50
Holding Period and Value
51
The Frequency of Compounding

• The frequency of compounding affects the future
  and present values of cash flows. The stated
  interest rate can deviate significantly from the
  true interest rate
• For instance, a 10 annual interest rate, if
  there is semiannual compounding, works out to-
• Effective Interest Rate 1.052 - 1 .10125 or
  10.25
• The general formula isEffective Annualized Rate
  (1r/m)m 1where m is the frequency of
  compounding ( times per year), andr is the
  stated interest rate (or annualized percentage
  rate (APR) per year

52
The Frequency of Compounding

53
II. Annuities

• An annuity is a constant cash flow that occurs at
  regular intervals for a fixed period of time.
  Defining A to be the annuity,
• A A A A
• 0 1 2 3 4

54
Present Value of an Annuity

• The present value of an annuity can be calculated
  by taking each cash flow and discounting it back
  to the present, and adding up the present values.
  Alternatively, there is a short cut that can be
  used in the calculation A Annuity r
  Discount Rate n Number of years

55
Example PV of an Annuity

• The present value of an annuity of 1,000 at the
  end of each year for the next five years,
  assuming a discount rate of 10 is -
• The notation that will be used in the rest of
  these lecture notes for the present value of an
  annuity will be PV(A,r,n).

56
Annuity, given Present Value

• The reverse of this problem, is when the present
  value is known and the annuity is to be estimated
  - A(PV,r,n).

57
Computing Monthly Payment on a Mortgage

• Suppose you borrow 200,000 to buy a house on a
  30-year mortgage with monthly payments. The
  annual percentage rate on the loan is 8.
• The monthly payments on this loan, with the
  payments occurring at the end of each month, can
  be calculated using this equation
• Monthly interest rate on loan APR/12 0.08/12
  0.0067

58
Future Value of an Annuity

• The future value of an end-of-the-period annuity
  can also be calculated as follows-

59
An Example

• Thus, the future value of 1,000 at the end of
  each year for the next five years, at the end of
  the fifth year is (assuming a 10 discount rate)
  -
• The notation that will be used for the future
  value of an annuity will be FV(A,r,n).

60
Annuity, given Future Value

• If you are given the future value and you are
  looking for an annuity - A(FV,r,n) in terms of
  notation -

Note, however, that the two formulas, Annuity,
given Future Value and Present Value, given
annuity can be derived from each other, quite
easily. You may want to simply work with a
single formula.
61
Application Saving for College Tuition

• Assume that you want to send your newborn child
  to a private college (when he gets to be 18 years
  old). The tuition costs are 16000/year now and
  that these costs are expected to rise 5 a year
  for the next 18 years. Assume that you can
  invest, after taxes, at 8.
• Expected tuition cost/year 18 years from now
  16000(1.05)18 38,506
• PV of four years of tuition costs at 38,506/year
  38,506 PV(A ,8,4 years) 127,537
• If you need to set aside a lump sum now, the
  amount you would need to set aside would be -
• Amount one needs to set apart now
  127,357/(1.08)18 31,916
• If set aside as an annuity each year, starting
  one year from now -
• If set apart as an annuity 127,537
  A(FV,8,18 years) 3,405

62
Valuing a Straight Bond

• You are trying to value a straight bond with a
  fifteen year maturity and a 10.75 coupon rate.
  The current interest rate on bonds of this risk
  level is 8.5.
• PV of cash flows on bond 107.50 PV(A,8.5,15
  years) 1000/1.08515 1186.85
• If interest rates rise to 10,
• PV of cash flows on bond 107.50 PV(A,10,15
  years) 1000/1.1015 1,057.05
• Percentage change in price -10.94
• If interest rate fall to 7,
• PV of cash flows on bond 107.50 PV(A,7,15
  years) 1000/1.0715 1,341.55
• Percentage change in price 13.03

63
III. Growing Annuity

• A growing annuity is a cash flow growing at a
  constant rate for a specified period of time. If
  A is the current cash flow, and g is the expected
  growth rate, the time line for a growing annuity
  looks as follows

64
Present Value of a Growing Annuity

• The present value of a growing annuity can be
  estimated in all cases, but one - where the
  growth rate is equal to the discount rate, using
  the following model
• In that specific case, the present value is equal
  to the nominal sums of the annuities over the
  period, without the growth effect.

65
The Value of a Gold Mine

• Consider the example of a gold mine, where you
  have the rights to the mine for the next 20
  years, over which period you plan to extract
  5,000 ounces of gold every year. The price per
  ounce is 300 currently, but it is expected to
  increase 3 a year. The appropriate discount rate
  is 10. The present value of the gold that will
  be extracted from this mine can be estimated as
  follows

66
IV. Perpetuity

• A perpetuity is a constant cash flow at regular
  intervals forever. The present value of a
  perpetuity is-

67
Valuing a Consol Bond

• A consol bond is a bond that has no maturity and
  pays a fixed coupon. Assume that you have a 6
  coupon console bond. The value of this bond, if
  the interest rate is 9, is as follows -
• Value of Consol Bond 60 / .09 667

68
V. Growing Perpetuities

• A growing perpetuity is a cash flow that is
  expected to grow at a constant rate forever. The
  present value of a growing perpetuity is -
• where
• CF1 is the expected cash flow next year,
• g is the constant growth rate and
• r is the discount rate.

69
Valuing a Stock with Growing Dividends

• Southwestern Bell paid dividends per share of
  2.73 in 1992. Its earnings and dividends have
  grown at 6 a year between 1988 and 1992, and are
  expected to grow at the same rate in the long
  term. The rate of return required by investors on
  stocks of equivalent risk is 12.23.
• Current Dividends per share 2.73
• Expected Growth Rate in Earnings and Dividends
  6
• Discount Rate 12.23
• Value of Stock 2.73 1.06 / (.1223 -.06)
  46.45

70
What are bonds?

• A borrowing arrangement where the borrower issues
  an IOU to the investor.

Time
0
Price
Coupon Payments Coupon Rate x FV/ 2 Paid
semiannually
1
Investor
Issuer
2
.
.
Face Value (FV)
T
71
Bond Pricing

• A T-period bond with coupon payments of C per
  period and a face value of F.
• The value of this bond can be computed as the sum
  of the present value of the annuity component of
  the bond plus the present value of the FV, where
  is the present value of an
• annuity of 1 per period for T periods, with a
  discount rate of r per period.

72
Bonds with semi-annual coupons

• Normally, bonds pay semi-annual coupons
• The bond value is given by
• where the first component is, once again, the
  present value of an annuity, and y is the bonds
  yield-to-maturity.

73
Bond Pricing Example

• If F 100,000 T 8 years the coupon rate is
  10, and the bonds yield-to-maturity is 8.8,
  the bond's price is computed as
• 
  106,789.52

74
The Relation between Bond Prices and Yields

• Consider a 2 year, 10 coupon bond with a 1000
  face value. If the bond yield is 8.8, the price
  is 50 1000/(1.044)4 1021.58.
• Now suppose the market bond yield drops to 7.8.
  The market price is now given by 50
  1000/(1.039)4 1040.02.
• As the bond yield drops, the bond price rises,
  and vice-versa.

75
Bond Prices and YieldsA Graphic View
76
Bond Yield MeasurementDefinitions

• Yield to MaturityA measure of the average rate
  of return on a bond if held to maturity. To
  compute it, we define the length of a period as 6
  months, and then calculate the internal rate of
  return per period. Finally, we double the
  six-monthly IRR to get the bond equivalent yield,
  or yield to maturity. This is more commonly used
  in the marketplace.
• Effective Annual YieldTake the six-monthly IRR
  and annualize it by compounding. This measure is
  less commonly used.

77
Bond Yield Measurement Examples

• An 8 coupon, 30-year bond is selling at
  1276.76. First solve the following equation
• This equation is solved by r 0.03. (You will
  see later how to solve this equation.)
• The yield-to-maturity is given by 2 x 0.03 6
• The effective annual yield is given by (1.03)2 -
  1 6.09

78
Computing YTM by Trial and Error

• A 3 year, 8 coupon, 1000 bond, selling for
  949.22
• Period Cash flow Present Value
• 9 11 10
• 1 40 38.28 37.91 38.10
• 2 40 36.63 35.94 36.28
• 3 40 35.05 34.06 34.55
• 4 40 33.54 32.29 32.91
• 5 40 32.10 30.61 31.34
• 6 1040 798.61 754.26 776.06
• Total 974.21 925.07 949.24
• The bond is selling at a discount hence the
  yield exceeds the coupon rate. At a discount
  rate equal to the coupon rate of 8, the price
  would be 1000. Hence try a discount rate of 9.
  At 9, the PV is 974.21, which is too high. Try a
  higher discount rate of 11, with a PV of
  925.07, which is too low. Trying 10, which is
  between 9 and 11, the PV is exactly equal to
  the price. Hence the bond yield 10.

79
Computing YTM by Trial and Error A Graphic View
80
Coupons and Yields

• A bond that sells for more than its face value is
  called a premium bond.
• The coupon on such a bond will be greater than
  its yield-to-maturity.
• A bond that sells for less than its face value is
  called a discount bond.
• The coupon rate on such a bond will be less than
  its yield-to-maturity.
• A bond that sells for exactly its face value is
  called a par bond.
• The coupon rate on such a bond is equal to its
  yield.

81
Non-flat Term Structures

• There is an implicit assumption made in the
  previous slide that the annualized discount rate
  is independent of when the cashflows occur.
• That is, if 100 to paid in year 1 are worth
  94.787 today, resulting in an implicit discount
  rate of (100/94.787 -1) 5.5, then 100 to be
  paid in year 2 are worth (in todays dollars),
  100/(1.055)2 89.845. However, this need not
  be so.
• Demand and supply for year 1 dollars need not be
  subject to the same forces as demand and supply
  for year 2 dollars. Hence we might have the 1
  year discount rate be 5.5, the year 2 discount
  rate 6 and the year 3 discount rate 6.5

82
Non-flat Term Structures

• If we now have a 10 coupon FV1000 three year
  bond, which will have cash flows of 100 in year
  1, 100 in year 2 and 1100 in year 3, its price
  will be computed as the sum of 100/(1.055)
  94.787, 100/(1.06)2 89.00 and 100/(1.065)3
  910.634 for a total of 1094.421.
• We could, at this point, compute the
  yield-to-maturity of this bond using the formula
  given above. If we do this, we will find that
  the yield-to-maturity is 6.439 per annum.
• This is not the discount rate for the first or
  the second or the third cashflow. Rather, the
  yield-to-maturity must, in general, be
  interpreted as a (harmonic) average of the actual
  discount rates for the different cashflows on the
  bond, with more weight being given to the
  discount rates for the larger cashflows.

83
Yield Curves for Feb. 1-12, 2008
Date 1mo 3mo 6mo 1yr 2yr 3yr 5yr 7yr 10yr 20yr 30yr
02/01/08 1.75 2.10 2.15 2.13 2.09 2.22 2.75 3.13 3.62 4.31 4.32
02/04/08 2.15 2.27 2.22 2.17 2.08 2.23 2.78 3.18 3.68 4.37 4.37
02/05/08 2.22 2.19 2.13 2.06 1.93 2.08 2.66 3.08 3.61 4.32 4.33
02/06/08 2.12 2.10 2.10 2.05 1.96 2.11 2.67 3.08 3.61 4.36 4.37
02/07/08 2.19 2.17 2.13 2.08 1.99 2.21 2.79 3.21 3.74 4.50 4.51
02/08/08 2.24 2.23 2.12 2.05 1.93 2.10 2.69 3.11 3.64 4.41 4.43
02/11/08 2.35 2.31 2.13 2.06 1.93 2.10 2.67 3.09 3.62 4.38 4.41
02/12/08 2.55 2.31 2.12 2.06 1.94 2.13 2.71 3.13 3.66 4.43 4.46
84
Yield Curves for Feb. 1-12, 2008
85
Time Pattern of Bond Prices

• Bonds, like any other asset, represent an
  investment by the bondholder.
• As such, the bondholder expects a certain total
  return by way of capital appreciation and coupon
  yield.
• This implies a particular pattern of bond price
  movement over time.

86
Time Pattern of Bond Prices Graphic View
Assuming yields are constant and coupons are
paid continuously
87
Time Pattern of Bond Prices in Practice

• Coupons are paid semi-annually. Hence the bond
  price would increase at the required rate of
  return between coupon dates.
• On the coupon payment date, the bond price would
  drop by an amount equal to the coupon payment.
• To prevent changes in the quoted price in the
  absence of yield changes, the price quoted
  excludes the amount of the accrued coupon.
• Example An 8 coupon bond quoted at 96 5/32 on
  March 31, 2008, paying its next coupon on June
  30, 2008 would actually require payment of
  961.5625 0.5(80/2) 981.5625