# Introduction to Valuation Bond Valuation

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```					Introduction to Valuation
Bond Valuation

Financial Management

P.V. Viswanath

For a First course in Finance
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

P.V. Viswanath                    2
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.

P.V. Viswanath                      3
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 they’d be willing to
sell.

P.V. Viswanath                  4
The Supply Curve
Price
(\$ per unit)                        S

Quantity of Eggs

P.V. Viswanath                      5
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.

P.V. Viswanath                   6
The Demand Curve
Price
(\$ per unit)

D

Quantity of Eggs

P.V. Viswanath                      7
The Determination of the Price of Eggs
Price
(\$ per unit)                          S

The curves intersect at
equilibrium, or market-
clearing, price. At P0 the
quantity supplied is equal
P0
to the quantity demanded
at Q0 .

D

Q0               Quantity of Eggs

P.V. Viswanath                                8
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?

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Money and Time

today is not the same as money available tomorrow.
 You can buy something today only if you have the money to
available tomorrow won’t 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.

P.V. Viswanath                     10
More on the price of money

 Let’s assume that all prices are denominated in t=0
dollars (today’s 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 t=1 dollar is \$0.90, the price of a t=2
dollar is \$0.7831 and the price of a t=3 dollar is
\$0.675.

P.V. Viswanath                  11
The price of coffee, said differently

 This might sound a little strange to you, but let’s
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.

P.V. Viswanath                 12
More ways to price coffee

 Let’s 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.

P.V. Viswanath                        13
Rates
 So now, let’s go back to the price of money: we said that the
price of a t=1 dollar was \$0.90, and that the price of a t=2
dollar was \$0.7831.
 Now clearly the price of a t=1 dollar, which is \$0.90 today,
will rise to \$1 at t=1.
 Hence providing today’s price of a t=1 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 t=1 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%

P.V. Viswanath                     14
Rates

 What about the price of a t=2 dollar, which we said was
\$0.7831?
 Once again, the price of this t=2 dollar would be \$1 at t=2
(in t=2 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

P.V. Viswanath                     15
Computing Annualized Rates

 We computed the return on buying a t=2 dollar at 27.70%.
 Suppose the one-period return on this is r%; that is, the
return from holding this t=2 dollar from now until t=1 is r%.
Then, every dollar invested in this specialized investment
could be sold at \$(1+r) at t=1.
 Now, if we assume the return on this t=2 dollar if held from
t=1 to t=2 is also r%, then the \$(1+r) value of our outlay of
one t=0 dollar in this investment would be \$(1+r)(1+r) or
(1+r)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 (1+r)2 to 1.277 and solve for r.

P.V. Viswanath                     16
Annualized Rates

 This involves simply taking the square-root of 1.277, which
is 13%.
 Of course, we won’t 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 t=3 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%

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Yield to maturity

 So now we have the current prices of t=1, t=2, and t=3
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.

P.V. Viswanath                      18
Using the rates
 We have assumed, up to this point, that purchasing a t=1
dollar is riskless. That is, the person who sold us the t=1
dollar today, in return for the \$0.90, would, in fact, pay us
\$1 at time t=1.
 We will continue with this assumption, for now.
 Note, as well that buying a t=1 dollar is equivalent to
lending money for one period, while selling a t=1 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.

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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.

P.V. Viswanath                     20
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.
\$1000.

P.V. Viswanath                21
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.

P.V. Viswanath                  22
Pricing this bond
 Let’s 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.
P.V. Viswanath                    23
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

P.V. Viswanath                                  24
Yield Curves for Feb. 1-12, 2008

Yield Curves

2/1/2008
5
2/4/2008
Yield-to-Maturity

4
2/5/2008
3                                             2/6/2008
2                                             2/7/2008

1                                             2/8/2008
2/11/2008
0
0        10       20            30   40   2/12/2008

Maturity

P.V. Viswanath                    25
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%.

P.V. Viswanath                     26
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?

P.V. Viswanath                     27
Discounting
 We know the value at t=3 of this promise would be exactly
200.
 The value at t=2of the promise would have to be such that
would yield a return of exactly 4% over the last period, i.e.
from t=2 to t=4.
 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 t=1 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 t=0 of a promise
to pay \$200 in n periods is 200/(1.04)n.

P.V. Viswanath                          28
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 t=1 would sell
today for 1/1.01052 = 0.979326.

P.V. Viswanath               29
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 t=1.5 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
t=2 is 1/1.01074 = 0.95832

P.V. Viswanath               30
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.

P.V. Viswanath                31
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?
 Here’s 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.

P.V. Viswanath                     32
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.
P.V. Viswanath                   33
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.                              T
 In fact, the price of this asset would simply be           
P *  ci Pi
i 1

P.V. Viswanath                           34
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:        T
P   ci Pi
*

i 1

 Of course, this is not entirely satisfactory, because we’d 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.

P.V. Viswanath                          35
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.

P.V. Viswanath                       36
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.
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.

P.V. Viswanath               37
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.

P.V. Viswanath                         38
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:
T
P   ci Pi
*

i 1
 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.
P.V. Viswanath                           39
Efficient Markets Hypothesis – EMH

 An asset’s current price reflects all available
information– this is the EMH.
 If it didn’t, 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.

P.V. Viswanath                  40
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.

P.V. Viswanath                       41
Stock Price Fundamentals

 What determines the price of a stock? Or, in other
words, why would an investor hold stocks?
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
it’s 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.

P.V. Viswanath                  42
Dividend Mechanics

 Declaration date: The board of directors declares a payment
Record 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.

P.V. Viswanath                      43
Dividend Discount Model

 What is the price of a stock on its ex-dividend date?
 Using the previous logic, we see that it’s simply
D1        D2               Dn
P0                    ...              ...
1  k (1  k ) 2
(1  k ) n

 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.
P.V. Viswanath                  44
Gordon Growth Model

 If we assume that the dividend is growing at a rate
of g% per annum forever, this formula simplifies
to:
D1
P0 
kg
 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.

P.V. Viswanath                    45
Two essential concepts

1. 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
2. The concept of a Time Line:

P.V. Viswanath              46
Cash Flow Types and Discounting
Mechanics
 There are five types of cash flows -
   simple cash flows,
   annuities,
   growing annuities
   perpetuities and
   growing perpetuities

P.V. Viswanath   47
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 / (1+r)t
 The future value of a cash flow is -
FV of Simple Cash Flow = CF0 (1+ r)t

P.V. Viswanath                      48
Application: The power of
compounding - Stocks, Bonds and
Bills
 Between 1926 and 1998, Ibbotson Associates found
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

P.V. Viswanath                      49
Holding Period and Value

P.V. Viswanath      50
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 is
Effective Annualized Rate = (1+r/m)m – 1
where m is the frequency of compounding (# times per year), and
r is the stated interest rate (or annualized percentage rate (APR) per
year

P.V. Viswanath                                  51
The Frequency of Compounding

Effective Annual
Frequency     Rate   t          Formula          Rate
Annual        10%    1          r                10.00%

Semi-Annual   10%    2          (1+r/2)2-1       10.25%
Monthly       10%    12         (1+r/12)12-1     10.47%

Daily         10%    365        (1+r/365)365-1   10.52%
Continuous    10%               er-1             10.52%

P.V. Viswanath                           52
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

P.V. Viswanath                 53
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]

A      1 
PV of an Annuity  PV ( A, r , n)  1        n 
r  (1  r ) 

P.V. Viswanath                    54
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 -
 -
1
1 
   (1.10)5 
PV of \$1000 each year for next 5 years = \$1000                 \$3,791
 .10       
           

 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).

P.V. Viswanath                               55
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).
              
    r         
Annuity given Present Value = A(P V, r,n) = P V         1
 -
1              
   (1 + r)n   

P.V. Viswanath                             56
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                              
   0.0067      
Monthly Payment on Mortgage = \$200,000             1          \$1473.11
 -
1               
   (1.0067)360 

P.V. Viswanath                                    57
Future Value of an Annuity

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

 + r)n - 1 
(1
FV of an Annuity = FV(A,r,n) = A            
    r      

P.V. Viswanath                  58
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) -

(1.10)5 - 1 

FV of \$1,000 each year for next 5 years = \$1000             = \$6,105
 .10       

 The notation that will be used for the future value of an
annuity will be FV(A,r,n).

P.V. Viswanath                                 59
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 -
     r      
Annuity given Future Value = A(FV, r,n) = FV 
(1+ r)n - 1 


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.

P.V. Viswanath                         60
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

P.V. Viswanath                                   61
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%

P.V. Viswanath                          62
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 –

P.V. Viswanath                 63
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:
     (1+ g) 
n
 -
1              
     (1+ r) 
n
PV of an Annuity = P V(A,r,g,n) = A(1 + g)               
   (r - g) 

              


 In that specific case, the present value is equal to the
nominal sums of the annuities over the period, without the
growth effect.

P.V. Viswanath                         64
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 –
   (1.03)20 
1 -
   (1.10)20 
PV of extracted gold = \$300* 5000 * (1.03)                   \$16,145,980
 .10 - .03 

            


P.V. Viswanath                                 65
IV. Perpetuity

 A perpetuity is a constant cash flow at regular
intervals forever. The present value of a perpetuity
is-
A
PV of Perpetuity =
r

P.V. Viswanath                   66
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

P.V. Viswanath                67
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 -
CF1
PV of Growing P erpetuity =
(r - g)

where
    CF1 is the expected cash flow next year,
    g is the constant growth rate and
    r is the discount rate.

P.V. Viswanath                68
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

P.V. Viswanath                       69
What are bonds?
A borrowing arrangement where the borrower issues
an IOU to the investor.
Time
Price                   0

Coupon Payments       1
= Coupon Rate x FV/ 2 2     Investor
Issuer               Paid semiannually    .
.

Face Value (FV)         T

P.V. Viswanath                 70
Bond Pricing

A T-period bond with coupon payments of \$C per period and a
face value of F.
0    1      2   3    4    5                      T
0     C     C     C     C    C                      C + FV
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 ArT is the present value of an
annuity of \$1 per period for T periods, with a discount rate of r%
per period.
F
CA    T

(1  r ) T
r

P.V. Viswanath                      71
Bonds with semi-annual coupons

Normally, bonds pay semi-annual coupons:
0    0.5   1     1.5   2        2.5                      T

0    C/2   C/2   C/2   C/2      C/2                      C/2+F

The bond value is given by: C A 2 T                F
(1  y / 2 ) 2 T
y /2
2

where the first component is, once again, the present
value of an annuity, and y is the bond’s yield-to-
maturity.

P.V. Viswanath                        72
Bond Pricing Example

If F = \$100,000; T = 8 years; the coupon rate is
10%, and the bond’s yield-to-maturity is 8.8%, the
bond's price is computed as:

(01)100000
.            1   1 16          100000
x      1          
2       .044      1044  
.         (1044)16
             
= \$106,789.52
.

P.V. Viswanath                   73
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 A.044 + 1000/(1.044)4 = 1021.58.
4

 Now suppose the market bond yield drops to
4
7.8%. The market price is now given by 50  A.039 +
1000/(1.039)4 = 1040.02.

 As the bond yield drops, the bond price rises, and
vice-versa.

P.V. Viswanath                   74
Bond Prices and Yields
A Graphic View

B o n d P rice

B o n d Y ield

P.V. Viswanath                    75
Bond Yield Measurement
Definitions
 Yield to Maturity
A 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 Yield
Take the six-monthly IRR and annualize it by
compounding. This measure is less commonly used.

P.V. Viswanath                  76
Bond Yield Measurement: Examples

 An 8% coupon, 30-year bond is selling at \$1276.76. First
solve the following equation:
60
40          1000
1276.76                 
t 1 (1  r ) t
(1  r ) 60
 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%

P.V. Viswanath                      77
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%.
P.V. Viswanath                             78
Computing YTM by Trial and Error:
A Graphic View

Bond Price

974.21

949.25

925.07

9%   10% 11%
Bond Yield
P.V. Viswanath                79
Coupons and Yields
 A bond that sells for more than its face value is
   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.

P.V. Viswanath                      80
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 today’s
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%

P.V. Viswanath                      81
Non-flat Term Structures
 If we now have a 10% coupon FV=1000 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.

P.V. Viswanath                       82
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

P.V. Viswanath                                  83
Yield Curves for Feb. 1-12, 2008

Yield Curves

2/1/2008
5
2/4/2008
Yield-to-Maturity

4
2/5/2008
3                                             2/6/2008
2                                             2/7/2008

1                                             2/8/2008
2/11/2008
0
0        10       20            30   40   2/12/2008

Maturity

P.V. Viswanath                    84
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.

P.V. Viswanath                 85
Time Pattern of Bond Prices: Graphic View

Assuming yields are constant
and coupons are paid continuously
Par Bonds

Discount Bonds

Maturity date   Time
P.V. Viswanath                          86
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

P.V. Viswanath                       87

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